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University of Southern California Dissertations and Theses
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Demonstration of error suppression and algorithmic quantum speedup on noisy-intermediate scale quantum computers
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Demonstration of error suppression and algorithmic quantum speedup on noisy-intermediate scale quantum computers
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Demonstration of error suppression and algorithmic quan tum sp eedup on noisy-in termediate scale quan tum co mputers b y Bib ek P okharel A Dissertation Presen ted to the F A CUL TY OF THE USC GRADUA TE SCHOOL UNIVERSITY OF SOUTHERN CALIF ORNIA In P artial F ulfillmen t of the Requiremen ts for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) Ma y 2023 A c kno wledgmen ts I cannot compile an exhaustiv e list of p eople and reasons that I am grateful for. Ev en if I could, it w ould not fit in to a few pages. Instead, here is a necessary but insufficien t list. I am grateful to m y paren ts, who allo w ed me to pic k m y in terests and w ere supp ortiv e ev en though I am the first in m y family to pursue suc h a long and obscure academic trac k in a strange foreign coun try . They treat m y ac hiev emen ts as their o wn and ha v e supp orted me from thousands of miles a w a y . I am most thankful to m y advisor, Daniel Lidar. His passion for researc h and for attempt- ing to answ er meaningful and impactful questions is infectious. He taugh t me to do researc h seriously , with as m uc h care as p ossible, and to b e proud of the final pro duct. He alw a ys made the time to read ev erything I wrote carefully and found mistak es and inconsistencies that w ould ha v e gone unnoticed. He pushed me to not b e satisfied with a shallo w under- standing, ev en under pressure. He allo w ed me to w ork at m y o wn pace and ga v e me the time and structure to gro w as a researc her, and for that, I will forev er b e in his debt. I w as first in tro duced to quan tum computation b y Arjendu P attana y ak, m y undergraduate researc h advisor. He to ok the time to explain and re-explain quan tum mec hanics, ga v e me problems that w ere m uc h harder than I could handle, and encouraged me to go to conferences where I understo o d next to nothing. I am grateful that he pushed me out of m y comfort zone and taugh t me the difference b et w een researc h and studying. I w as Eleanor Rieffel’s in tern for only three mon ths, but she con tin ued to guide me ev en when it w as no longer her official resp onsibilit y . She alw a ys emphasized the need for clarit y and precision in comm unication - this has b een a gen uine pro ductivit y m ultiplier. When I w ork hard to explain m y though ts clearly , it is often easy for others to tell me exactly wh y ii I w as wrong. P ainful as it ma y b e, I ha v e found this to b e the b est w a y to mak e progress. My first course in quan tum information w as with Iv an Deutsc h. I am still amazed at ho w thorough his lectures w ere. Iv an w ork ed with us, his studen ts, and painfully w en t through all the n uances of whatev er problem w e w ere solving. He is a though tful and p erceptiv e teac her, and I am thankful to ha v e had the c hance to learn from him. One of the reasons I applied to USC w as b ecause I had read T o dd Brun’s w ork on quan tum c haos and the arro w of time. I w as impressed with the clarit y of this written w ork and disco v ered that he is ev en more brillian t in p erson. He has alw a ys tak en the time to answ er m y questions ab out ph ysics, no matter ho w trivial, and the answ ers w ere consisten tly insigh tful. His sharpness is matc hed only b y his kindness. I w as fortunate to share m y time at USC with A dam, Am y , An urag, Hum b erto, Huo, Jenia, Josh, Ka W a, Matthew, Milad, Mostafa, Nic, P at, Razieh, Ric hard, Siddharth, T ameem, Victor, Vina y , and Zihan. They are kind, patien t, and forev er ready to share their exp ertise. Some of the most memorable con v ersations of m y Ph.D. career w ere with Namit. Ev en when the time w as in short supply , he alw a ys made time to discuss things i n excruciating detail. Time, after all, is the rarest of resources, and I am grateful that he is willing to share it with me. I am fortunate to ha v e Haimeng’s un w a v ering willingness to help me no matter the request. She is alw a ys a call a w a y . She alw a ys listened to m y ram blings with utmost atten tion and and nev er hesitated to tell me ab out the inconsistencies in m y though t pro cess. I w ould also lik e to thank m y furry companions, Bato o and Momo, for sitting b esides me while I w ork ed and trying to lo ok in terested. Last but definitely not the least, I am grateful to m y wife, Hiy an thi. Hiy an thi has b een there for me b efore I w as in terested in Ph ysics and w as adaman t ab out b eing an economist. Ev en though she has nev er fully understo o d m y obsession with P h ysics and m y unin vited ran ts ab out the elegance of mathematics, she has alw a ys b een supp ortiv e. She listened to m y frustrations and tried her b est to relate to them. She ev en read m y incomprehensible iii man uscripts and ga v e helpful feedbac k. She supp orted me financially and emotionally while I pursued m y dream, and I w ould not ha v e b een able to do this without her. iv T able of Con ten ts A c kno wledgmen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of T ables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Chapter One: In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Success metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Chapter T w o: Fidelit y impro v emen t on sup erconducting qubits . . . . . . . . . . 14 2.1 Metho dology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Single-qubit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Dephasing vs. sp on taneous emission . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Dep endence on the pulse in terv al . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Discussion and Concl usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Supplemen tary Informatio n . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Chapter Three: Better-than-classical Gro v er searc h . . . . . . . . . . . . . . . . . 37 3.1 Gro v er’s Algorithm: bac kground and implemen tation . . . . . . . . . . . . . 39 3.2 Op en system mo de l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Decouple then comput e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 T w o-qubit enco ded Gro v er algorithm protected b y quan tum error detection . 45 3.5 3-qubit to 5-qubit G ro v er protected b y dynamical decoupling . . . . . . . . . 57 v 3.6 Discussion and Concl usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.7 Supplemen tary Informatio n . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Chapter F our: Demonstration of algorithmic quan tum sp eedup . . . . . . . . . . . 83 4.1 Quan tum sp eedup quan tified . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Exp erimen tal Imple men tation . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Discussion and Concl usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 Supplemen tary Informatio n . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 vi List of T ables 2.2.1 Fit parameters when Eq. ( 2.1 ) is used to fit the mean fidelities in Fig. 2.2.2 . The first deca y constan t, λ , is significan tly increased under DD. The sec- ond deca y constan t, α , is effectiv ely infinite for all ev olutions other than IBMQX5’s free ev olution. The mo dulation frequencyγ v anishes for IBMQX5 under DD and is near zero for A corn under free ev olution, consisten t with purely exp onen tial fidelit y deca y , i.e., Mar k o vian ev olution. . . . . . . . . . . 19 2.6.1 Ph ysical P arameters-IBMQX5 - A ccessed 06/19/2018. The minim um, a v er- age, and maxim um CNOT gate fidelit y are 0.8417, 0.9330, and 0.9513 resp ec- tiv ely . The gate (readout) fidelit y is 1 min us gate (readout) error. . . . . . . 24 2.6.2 Ph ysical parameters for Rigetti A corn - A ccessed 4/13/18. The minim um, a v erage, and maxim um Con trolled-Z gate fidelit y are 0.72, 0.865, and 0.917 resp ectiv ely . The gate (readout) fidelit y is 1 min us gate (readout) error. . . . 25 2.6.3 Single qubit pulse times, n um b er of qubits, and shots p er exp erimen t for the IBMQX5 and 19Q-A corn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.4 Ph ysical P arameters for IBMQX4 - A ccessed 06/21/2018. The minim um, a v erage, and maxim um CNOT gate fidelit y are 0.8738, 0.9441, and 0.9774 resp ectiv ely . The gate (readout) fidelit y is 1 min us gate (readout) error. . . . 26 2.6.5 Fit parameters for A corn (considering only the15 activ e qubits) and IBMQX4 when Eq.( 2.1 ) is used to fit the mean fidelities i n Fig. 2.6.5 . . . . . . . . . . 29 vii 2.6.6 P erformance summary of the differen t DD pulses w e implemen ted on the IBMQX5. The XX and Y Y sequences are sp ecific to pure dephasing errors. The (ZI) N sequence is suppresses pure SE errors. All three of these sequences underp erform the XY4 sequences, but p erformance is b etter after suppression of pure dephasing errors. The GA sequences are discussed in App endix 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.1 Error outcome table for the [[4,2,2]] co de. The table describ es the effect of single-qubit P auli errors E on eac h of the enco ded computati onal basis el- emen ts jb 1 b 2 i . The X , Y , and Z -t yp e errors map eac h jb 1 b 2 i to a distinct subspace after deco ding: C ⊥ X = span(fj0100i,j0110i,j0011i,j0001ig) , C ⊥ Y = span(fj1100i,j1110i,j1011i,j1001ig) , andC ⊥ Z = span(fj1000i,j1010i,j1111i,j1101ig) , resp ectiv ely , suc h that C ⊥ = C ⊥ X C ⊥ Y C ⊥ Z . Th us, eac h subspace uniquely determines the error t yp e, whic h w e use to p erform algorithmic error tomog- raph y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.7.1 The b est-p erforming DD sequence at eac h problem size for b oth QPUs. These sequences w ere determined b y implemen tingn+1 oracles of the form 0 k 1 n−k for the n -qubit Gro v er problem. . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.7.2 Device sp ecifications for Jakarta and Nairobi on July 14, 2022. 1QG, 2QG, and R O denote 1-qubit gate, t w o-qubit gat e, and readout, resp ectiv ely [ 1 ]. . 70 4.5.1 Device sp ecifications for Mon treal and Cairo on Marc h 12, 2022. 1QG and 2QG denote 1-qubit gate and 2-qubit g ate, resp ectiv ely . R O denotes readout. 106 viii List of Figures 2.2.1 Mean fidelit y o v er 16 qubits of IBMQX5 and 15 qubits of A corn, for initial states jψi = i[cos(θ/2)j0i + sin(θ/2)j1i] . Results sho wn are under DD using XY4 and under free ev olution. IBMQX5: after N = 40 pulses, i.e., 10 rep etitions of the base XY4 sequence. DD impro v es the fidelit y only for states with θ≳π/3 . A corn: after N = 192 pulses, i.e., 48 rep etitions of the base XY4 sequence. DD impro v es the fidelit y only for states with θ≳ π/4 . Throughout w e rep ort 2σ error bars (95% confidence in terv als) calculated using the b o otstrap metho d (for more details see Section 2.6 , Sec. C). . . . 17 2.2.2 Mean fidelit y , after a v eraging o v er all qubits, and all 36 initial conditions in t yp e 2 preparation, as a function of the n um b er of pulses for IBMQX5 (b ottom axis) and A corn (top axis). The pulse in terv al is the shortest p ossible: 90 ns for IBMQX5, 50 ns for A corn. Solid lines are fits to Eq. ( 2.1 ), with fit parameters as p er T able 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 DD p erformance as a function of pulse in terv al τ for IBMQX5 (in units of 90 ns). The in tersection time t in t of the fidelit y curv es under free ev olution and DD, and the deca y-time exp onen t λ , as a function of the pulse spacing, τ . Linear fits yields t int =3.5(τ/90 ns)+108 and λ=4.3(τ/90 ns)+88.0 . 20 ix 2.4.2 Probabilities of differen t computational basis states after DD for initially pre- pared Bell states jΦ + i = 1 √ 2 (j00i+j11i) and jΨ + i = 1 √ 2 (j01i+j10i) , as a function of the n um b er of pulses, for b oth IBMQX5 and A corn. T op ro w: free ev olution. Bottom ro w: ev olution under DD. The j00i state is fa v ored under free ev olution. The solid horizon tal line indicatesp=0.25 , the limit of a fully mixed state. F or jΨ + i there is no noticeable difference in the p erfor- mance with or without DD. F orjΦ + i on IBMQX5, after N 20 , p 11 0.25 , suggesting that at this p oin t all information has essen tially b een scram bled. On A corn, complete scram bling ofjΦ + i o ccurs after N 30 . Ov erall, DD is more effectiv e at slo wing do wn the deca y to the fully mixed state for A corn than for IBMQX5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.1 Connectivit y b et w een the qubits in IBMQX5. The qubit at the tail (tip) of an arro w is the con trol (target) i n a con trolled-U gate. . . . . . . . . . . . . 26 2.6.2 Connectivit y b et w een the qubits in IBMQX4. The qubit at the tail (tip) of an arro w is the con trol (target) i n a con trolled-U gate. . . . . . . . . . . . . 27 2.6.3 IBMQX4 results for the fidelities under DD compared to free ev olution, after a v eraging o v er 36 initial conditions (t yp e 2) and a ll 5 qubits. . . . . . . . . 27 2.6.4 Connectivit y b et w een the qubits within A corn. Qubit 3 is disconnected. Qubits 2,12, 15,18 w ere not used as their p erformance v aried substan tially o v er time. Based on [ 2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.5 A corn c hip results for the mean fidelit y a v erages o v er 36 initial conditions (t yp e 2) and all 15 activ e qubits for differen t dates. . . . . . . . . . . . . . . 29 x 2.6.6 Bo otstrapping example: The left plot represen ts frequency coun ts (y-axis) v ersus fidelit y (x-axis) for a data set from the IBMQX5, tak en from 36 dif- feren t initial conditions (t yp e 1) and 16 qubits. The righ t plot represen ts the samples after b o otstrapping the original data set. The mean and confidence in terv als w ere then calculated based on the b o otstrapp ed distribution (righ t plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6.7 Correlation plot of the fidelities under DD compared to free ev olution, for IBMQX5 and A corn. Initial conditions are color co ded as a function of θ . T op panel: θ 2 [0, π 3 ) ; middle panel: θ 2 [ π 3 , 2π 3 ) ; b ottom panel: θ 2 [ 2π 3 ,π] . In (a) the initial state is close to the ground state j0i and DD is w orse than free ev olution. In (b) the initial state is a close to an equal sup erp osition, th us susceptible to dephasing, and DD is o v erall b etter than free ev olution, esp ecially for A corn. In (c) the initial state is close to the excited state j1i and DD is again b etter than free ev olution at in termediate N for IBMQX5, and at all N for A corn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6.8 Infidelit y scaling a s a function of τ for di fferen t n um b ers of pulses N . . . . . 33 2.6.9 The slop ea (10 ) and in terceptblogN deriv ed from plottinglog( p 1F) as a function of log(τ) for IBMQX5, at differen t n um b ers of pulses N . The solid blac k line is log(N)+c 0 and the dotted blac k line is log(N)/2+c 0 , where c 0 is the in tercept at N =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.10 Fidelit y deca y results for the GA-based DD sequences compared to XY4 and free ev olution, on IBMQX5. Left: GA 8a ; middle: GA 16a ; righ t: GA 32a . A small impro v emen t o v er XY4 is seen for t he GA 16a sequence. . . . . . . . . . 34 xi 3.1.1 Circuit description for Gro v er’s algorithm. The relativ e amplitudes of all the states at eac h stage of the algorithm are sho wn. Starting with an equal sup erp osition state, the oracle assigns a relativ e phase difference of π to the mark ed state. The amplitude amplification step the n p erforms an in v ersion ab out the mean, allo wingjmi to ha v e a larger probabilit y amplitude than all other states. This round of querying and amplifying is rep eatedq times. The optimal n um b er of rounds for the n -qubit Gro v er problem is q opt =b π 4 2 n/2 c . The only m ulti-qubit op eration required to implemen t b oth the oracle and the amplitude amplification step is C n−1 Z (v ertical line in the Oracle and Amplitude Amplification b o xes). . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 Gro v er and DD. The timeline for one oracle query for 4-qubit Gro v er with the mark ed state j1111i is sho wn. Qubits q 4 and q 6 are sp ectators in this example. Recall that eac h oracle query for 4-qubit Gro v er requires t w o C 3 Z gates. C 3 Z requires 14 CNOT s, and the en tire circuit uses 28 CNOT s; see Section 3.7 for circuit compilation details. The pre-DD circuit elemen ts are gra y ed out, and the colored lines represen t the DD pulses. The DD sequence exemplified here uses four pulses for illustration purp oses; in realit y , w e used longer sequences. The sc heme demonstrated highligh ts four primary features of our implemen tation: (1) all idle in terv als, including the ones on inactiv e qubits, are filled, (2) only one rep etition of eac h sequence is p erformed, and the pulse in terv al is adjusted accordingly , (3) eac h pulse in the sequence can b e unique, (4) a single qubit can exp erience m ultiple DD rep etitions if there are m ultiple idle in terv als. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 xii 3.4.1 Enco ded t w o-qubit Gro v er. The t w o-qubit single-query (q = 1 ) Gro v er cir- cuit enco ded using the [[4,2,2]] co de for the mark ed statej01i is sho wn. The enco ding step (left dashed b o x) prepares the enco ded initial state j00i = 1 √ 2 (j0000i+j1111i) (the 4-qubit GHZ state) from the ph ysical initial state j0000i . The enco ded Gro v er circuit is implemen ted b y con v erting eac h ph ys- ical gate of the n = 2 case of Fig. 3.1.1 in to its logical coun terpart, whic h is then con v erted in to a ph ysical 4 -qubit implemen tation (middle left and righ t dashed b o xes); see Section 3.7 for details. P = diag(1,i) is the phase gate. W e p ost-select the measured results b y deco ding (righ t dashed b o x) and dis- carding an y measuremen t outcome for whic h the co de detects errors, i.e., do es not result in one of the four dec o ded statesfj0000i,j0010i,j0111i,j0101ig . . 45 3.4.2 Algorithmic error tomograph y on Jakarta. The bitstring observ ed after U dec either corresp onds to a mark ed en try or an error tabulated in T able 3.4.1 . T op: exp erimen tal results. The n um b ers in eac h b o x are the empirical p ercen tage probabilities for detectedX ,Y , orZ -t yp e errors, with2σ standard deviation. Logical error p ercen tage probabilities are sho wn in the first column of eac h table. Eac h ro w corresp onds to a differen t mark ed state. The probabilities in eac h ro w do not sum to unit y since w e do not displa y the probabilit y of obtaining the correct mark ed state. Left: without DD protection. Righ t: with DD protection. Bottom: the same for the sim ulated mo del. . . . . . . . 49 3.4.3 Algorithmic error tomograph y on Nairobi. Data en tries are as in Fig. 3.4.2 , except that only data without DD is sho wn. Go o d agreemen t is observ ed b et w een the results of our error mo del and the exp erimen t. In particular, compared to the error tomograph y table for Jakarta (Fig. 3.4.2 ), w e do not observ e an asymmetry in Z errors across mark ed states. . . . . . . . . . . . 50 xiii 3.4.4 T w o-qubit Gro v er results. T w o-qubit single-query Gro v er failure probabil- it y results without (Unenc) and with (Enc) p ostselection using the [[4,2,2]] co de on Jakarta and Nairobi are sho wn. The transparen t b o xes represen t the theoretically exp ected failure probabilities from the mo del describ ed in Section 3.2 , whic h do es not include DD; their cen ters corresp ond to the a v er- age o v er mark ed states, and their b oundaries corresp ond to 95% confidence in terv als after b o otstrapping. The colored bars represen t the exp erimen tal results (see the legend), and the exp erimen tal error bars (blac k for Jakarta with DD and Nairobi, or pink for Jakarta without DD) corresp ond to 95% confidence in terv als after b o otstrapping. Dark green app ears where the pink and ligh t green colors (i.e., Jakarta with and without DD) o v erlap. In the Unenc case, w e run t w o iden tical copies of the t w o-qubit Gro v er problem to equalize resources with the Enc case and c ho ose the cop y with the highest success probabilit y . Also sho wn are the results with MEM using iterativ e Ba y esian unfolding (see Section 3.7 for details). F ailure probabilities with and without DD protection are sho wn for Jakarta but not for Nairobi, where the sim ulated and observ ed error tomograph y and failure probabilities are in agreemen t (see Section 3 ). The presence of DD do es not affect the success probabilit y in the enco ded implemen tation, and as a result, the pink bars are mostly hidden b ehind the green bars. Ho w ev er, the nature of detected errors, ev en in the enco ded case, is affected b y DD (see Fig. 3.4.2 ). All data for differen t runs on the same QPU w ere collected on the same da y; data from differen t QPUs w ere collected on di fferen t da ys. . . . . . . . . . . . . . . . . 51 xiv 3.4.5 T w o-qubit Gro v er results for Jakarta (top) and Nairobi (b ottom). The left and righ t panels sho w the output distribution for all p ossible oracles for t w o setups: unenco ded and enco ded with MEM. As in Fig. 3.4.4 , Unenc cor- resp onds to t w o copies of unenco ded t w o-qubit Gro v er, of whic h the b est result is rep orted. Enc corresp onds to the results enco ded using the [[4,2,2]] co de. The Enc results are rep orted after p ostselection. Let p s (m,b,e) b e the observ ed success probabilit y for mark ed state m , detected state b , and exp erimen t t yp ee2f Enc,Unenc+MEM,Enc+MEMg . The success probabil- it y c hanges from orange to green when p s (m,b,e) > max m,b p s (m,b, Unenc) . Error bars corresp ond to 95% confidence i n terv als. . . . . . . . . . . . . . . 52 3.4.6 P erformance of DD sequences. A v erage success probabilit y for 5-qubit Gro v er with t w o oracle queries on Nairobi. The DD sequences are rank ed in order of decreasing success probabilit y . The t w o dotted lines represen t success proba- bilities corresp onding to a random and classical strategy , resp ectiv ely . R GA8a and R GA8c are tied as the b est-p erforming sequences. F ree denotes the result of an unprotected implemen tation. Error bars corresp ond to 99% confidence in terv als. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 xv 3.4.7 5-qubit Gro v er results on Nairobi. Left: a v erage success probabilit y with and without DD or MEM for 5-qubit Gro v er implemen ted on Nairobi. The b o xes corresp ond to the theoretically exp ected success probabilities. The quan tum oracle is queried t wice; in the ideal case, the success probabilit y is 0.602. The unprotected (F ree) ev olution is on par with a random guess, significan tly w orse than the optimal classical strategy (dashed v ertical line), and just adding MEM do es not c hange the result. In con trast, the DD-assisted implemen tation crosses the classical threshold, and the results impro v e ev en more with MEM, up to a success probabilit y of0.15 . Error bars corresp ond to 99% confidence in terv als. Middle and righ t: the complete input-output maps for all2 5 mark ed states, without and with DD + MEM, are sho wn. States are sorted b y increasing Hamming w eigh t; in the F ree case, lo w Hamming w eigh t states ha v e a higher success probabilit y (more green on the left). This is lik ely to b e a consequence of amplitude damping (sp on taneous emission), whic h fa v ors the j0i state of eac h qubit. In the unprotected case (F ree, middle), there is no discernible correlation b et w een the input mark ed state and the output detected state. In the protected case (DD + MEM righ t), blac k-to- purple signifies b etter-than-classical success probabilit y , and this threshold is crossed for all 32 mark ed states. The DD sequence used here is R GA8a [ 3 ], whic h w as the top-p erforming sequence in our DD surv ey (see Fig. 3.4.6 ). . . 57 xvi 3.4.8 Success probabilities v ersus problem size. Nairobi (green) and Jakarta (or- ange) success probabilities for n2f3,4,5g are sho wn for DD-protected and unprotected implemen tations. The translucen t bands indicate the theoreti- cally estimated success probabilities using the mo del describ ed in Section 3.2 . W e p erformed q = 2 queries to the quan tum oracle in all cases. T he ideal success probabilitie s are 0.945 , 0.908 , and 0.602 for n = 3 , 4 , and 5 , re- sp ectiv ely . The white lines corresp ond to the success probabilities for the classical strategy and random sampling from the unsorted list (q =0 ). Error bars corresp ond to 99% confidence in terv als. . . . . . . . . . . . . . . . . . . 58 3.7.1 P erformance of DD sequences, expanding on the results sho wn in Fig. 3.4.6 . A v erage success probabilit y forn=3,4,5 with t w o oracle queries on Jakarta (top) and Nairobi (b ottom). The DD sequences are rank ed in order of decreas- ing success probabilit y . The t w o dotted lines represen t success probabilities corresp onding to a random and classical strategy , resp ectiv ely . F or n > 3 , the unprotected ev olution (F ree) is marginally b etter than c ho osing an ele- men t randomly and do es not cross the classical threshold. DD protection is necessary to cross the classical threshold, and the R GA and UR sequences with few er than 12 pulses are the b est p erformers. Error bars corresp ond to 99% confidence in terv als. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.7.2 P erformance under differen t oracle query n um b ers. Success probabilities are sho wn as a function of t he n um b er of oracle queries for Jakarta (top) and Nairobi (b ottom). All results included MEM, and error bars represen t 99% confidence in terv als. Dashed red lines corresp ond to the optimal classical success probabilit y . Except for n = 3 , the classical threshold is crosse d only with DD. In our exp erimen ts, w e set q = 2 , whic h is the optimal n um b er of rep etitions for all instances other than n = 5 on Jakarta. Error bars corresp ond to 99% confidence in terv als. . . . . . . . . . . . . . . . . . . . . 66 xvii 3.7.3 3-qubit, 4-qubit, and 5-qubit Gro v er on Jakarta after t w o oracle queries, com- plemen ting Fig. 3.4.7 , whic h only sho ws Nairobi for n = 5 . Eac h ro w rep- resen ts a problem size in ascending order. In a ro w, the horizon tal bar plot on the left sho ws the success probabilit y under no error suppression and mit- igation (F ree), with measuremen t error mitigation (F ree + MEM), with DD protection (DD), and with DD protection and measuremen t error mitigation (DD + MEM ). The dashed horizon tal line and the b o xes represen t the clas- sical and the theoretically exp ected success probabilit y , resp ectiv ely . The second and third columns sho w the input-output map for F ree and DD + MEM, highligh ting the impro v emen t offered b y these strategies. The states are sorted b y increasing Hamming w eigh t. The transition from green to blac k o ccurs at the classical success probabilit y threshold. With DD protection, the classical threshold is crossed in a ll cases. . . . . . . . . . . . . . . . . . . . . 68 3.7.4 Same as Fig. 3.7.3 , but for Nairobi. . . . . . . . . . . . . . . . . . . . . . . 69 3.7.5 Device connectivit y . Jakarta and Nairobi devices are built using the IBM Quan tum F alcon r5.11H pro cessors and ha v e sev en qubits. . . . . . . . . . . 71 3.7.6 Nairobi and MEM for unenco ded t w o-qubit Gro v er. The horizon tal bar plot on the left sho ws the failure probabilit y for the unenco ded t w o-qubit Gro v er algorithm for the unmitigated data and under t w o MEM tec hniques: iterativ e Ba y esian unfolding (IBU) and resp onse matrix in v ersion (In v). The three heat maps sho w the output distribution under no mitigation, IBU, and In v, going from left to righ t. The input mark ed states are on the v ertical axis, and the prin ted n um b ers on the diagonal represen tp s for the resp ectiv e mark ed states. The off-diagonal elemen ts in the output distribution are written explicitly to emphasize the presence of negativ e probabilities under In v in the righ t-most figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.7.7 Same as Fig. 3.7.6 but for enco ded t w o-qubit Gro v er. . . . . . . . . . . . . 71 xviii 3.7.8 Scaling calibration metrics. The ℓ 1 -norm based distance D b et w een the ob- serv ed and sim ulated outputs for t w o-qubit Gro v er is sho wn as a function of the scaling parameters λ i . The mo del parameters are scaled b y setting T 1 !λ −1 1 T 1 , T 2 !λ −1 2 T 2 and p D !λ g p D . The left and righ t columns repre- sen t unenco ded and enco ded implemen tations, resp ectiv ely . The dotted blac k line represen ts the default setting λ i =1 . The grey band in the Jakarta case (top ro w) is D for the DD-protected circuit v ersions. F or Nairobi (b ottom ro w), ev en without error suppression, the fit b et w een theory and exp erimen t is already quite go o d, and our optimization aims to impro v e the fit further. Th us, in the b ottom ro w (Nairobi), the grey band corresp onds to the distance b et w een sim ulated and observ ed at λ i =1 . . . . . . . . . . . . . . . . . . . 78 3.7.9 Algorithmic error tomograph y on Jakarta for optimized parameters. The plot sho ws the results of AET on the enco ded t w o-qubit Gro v er algorithm after setting (λ 1 ,λ 2 ,λ g ) = (1,1,6) . Eac h ro w of the error tomograph y table corresp onds to a mark ed state, and eac h column represen ts logical errors and X,Y,Z t yp e errors. Ev en after rescaling the mo del parameters, compared to the error tomograph y table for Jakarta (Fig. 3.4.2 ), w e do not see an asymmetry in Z errors across mark ed states. . . . . . . . . . . . . . . . . . 79 xix 3.7.10 P ostselection for 4-qubit and 5-qubit Gro v er: Success probabilities are sho wn with and without p ostselection for Nairobi (green) and Jakarta (orange). Here w e only consider the DD-protected circuits. 3- qubit Gro v er do es not undergo p ostselection. F or 4-qubit and 5-qubit Gro v er, w e p ostselect to coun t only the exp erimen ts for whic h the ancilla qubit (q1 in Fig. 3.7.5 ) is in j0i . The white lines corresp ond to the success probabilities for the classical strategy and random sampling from the unsorted list (q = 0 ). With p ostselection, for all problem sizes, the DD-protected quan tum q = 2 strategy outp erforms the classical strategy for q 3 . Without p ostsel ection, the b etter-than- classical requiremen t is met b y all implemen tations other than 5-qubit Gro v er on Jakarta, where w e ac hiev e a break ev en. Error bars corresp ond to 99% confidence in terv als. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.0.1 Circuit for the BV algorithm, including DD pulses. The oracle sho wn enco des the unkno wn bitstringb=111000 for the ssBV-6 problem. A con trolled-NOT (CNOT, or CX) or iden tit y gate is p erformed from qubiti to the ancilla qubit if b i = 1 or 0 , resp e ctiv ely . Note that the quan tum and classical oracles are iden tical in the ssBV-n problem, and so b oth tak e timet r /jbj to run, where jbj is the Hamming w eigh t of b . Eac h BV-n circuit requires n+1 qubits. A Hadamard gate (H ) is applied to eac h qubit b efore and after the oracle, and eac h qubit is measured in the computational basis, for a total circuit depth jbj+3 (with equalit y only for fully connected arc hitectures). DD pulses (P i ) are turned on during idle times. . . . . . . . . . . . . . . . . . . . . . . 85 xx 4.2.1 F ull output distribution for BV-6 from Cairo. Oracles f b are n um b ered from 0 to 63 , corresp onding to b 2 f0 6 ,...,1 6 g , sorted b y increasing Hamming w eigh t. Ideally , the output state for oraclef b (v ertical axis) isb , but in realit y , other bitstrings (horizon tal axis) are observ ed as w ell. Green dots on the diagonal corresp ond top s >1/2 , wherep s is the empirical frequency (success probabilit y) with whic h b w as output for oracle f b . Success probabilities are rep orted with 5σ confidence in terv als. . . . . . . . . . . . . . . . . . . . . . 87 xxi 4.3.1 Time-to-solution (TTS) as a function of problem size or n um b er of data qubits n . W e rep ort TTS(n) = 1 2 n P b TTS(n,b) , where TTS(n,b) is giv en b y Eq. ( 1.26 ), withp d =0.99 andt r (n) replaced b yt r (n,b) , since eac h oracle (lab eled b y its secret stringb2f0,1g n ) tak es a differen t time to run. Results for Mon treal and Cairo are sho wn b y the orange and blue sym b ols, resp ec- tiv ely , and filled (empt y) sym b ols represen t results with (without) DD; dotted lines are guides to the ey e. The asymptotic classical scaling TTS C (n) 2 n is sho wn as white grid lines, and the h yp othetical, ideal quan tum scaling TTS Q (n) / n of eac h QC is indicated b y the dashed lines (for QC-sp ecific parameter v alues see SI). The w orst-case scaling fit for eac h curv e is sho wn b y the solid lines, whose slop es λ are rep orted in the b ottom legend, with uncertain ties represen ting 95% confidence in terv als. Without DD, the TTS curv es terminate atn ′ max =16 (n ′ max =20 ) for Mon treal (Cairo), since w e find p s = 0 for n>n ′ max . Moreo v er, λ> 1 without DD, indicating a w orse-than- classical scaling. With DD protection, on Cairo, thep s >0 range is exte nded to n = 23 , and λ is just b elo w the break ev en p oin t of 1 , but the uncertain t y is to o large to conclude that quan tum sp eedup has o ccurred. In con trast, the Mon treal scaling with DD do es exhibit quan tum sp eedup, as explained in the text. Since t w o-qubit op erations and readout durations are shorter for Cairo, it exhibits a consisten tly lo w er absolute TTS than Mon treal. W e rep ort 5σ confidence in terv als from b o otstrapping for eac h data p oin t; error bars are mostly co v ered b y the sym b ols. . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.2 Results forλ hmax , the maxim um lo cal slop e of eac h of the curv es in Fig. 4.3.1 fornh max , i.e., the w orst-case-scaling when Fig. 4.3.1 is restricted toh max + 1 qubits. Only Mon treal with DD exhibits an unam biguous quan tum sp eedup, with λ hmax w ell b elo w 1 for all nh max . Error bars represen t 2σ confidence in terv als. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 xxii 4.5.1 Equiv alen t circuits used in our reduction from the circuit for b = 1 k 0 n−k to the circuits forb=1 k 0 m−k withm2[k,n1] . Illustrated on the left is BV-2 with the b = 11 oracle (m = 2 and k = 2 ), and on the righ t BV-4 with the b=1100 oracle (n=4 andk =2 ). The left circuit is obtained from the righ t circuit b y tracing o v er the last nk =2 data qubits. . . . . . . . . . . . . 97 4.5.2 Comparison of the effect of crosstalk without and with DD (the UR 14 se- quence). TTS(n) is sho wn for ibmq_jakarta. While TTS standard TTS reduced without DD protection, the TTS results are statistically indistinguishable in the presence of DD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5.3 Sc hematic of the lattice connectivit y for 27-qubit devices with the hea vy-hex la y out [ 4 ]. The dashed lines connect qubits that are m ultiplexed together for readout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5.4 The h yp othetical, ideal TTS for Mon treal and Cairo as a function of n . A deviation from linear scaling is se en for small n . . . . . . . . . . . . . . . . 103 4.5.5 Results from Cairo with a differen t logical to ph ysical qubit mapping than used in Fig. 4.3.1 , using all 27 qubits. These runs w ere p erformed on Jan uary 2nd, 2022. Due to a significan t readout error in Q19, there is an abrupt jump in the TTS atn=15 . Consequen tly , qubits Q19, Q20, and Q22 w ere left out of all subsequen t exp erimen ts. . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5.6 Example of an ssBV implemen tation using ancilla sw apping. Here w e con- sider the circuit for implemen ting the oracle with b = 11 for ssBV-2 on a linear arc hitecture with qubit connectivit y Q2-Q1-QA (left). The standard sw apping tec hnique (second from left) requires 5 CNOT s (second from righ t), but c ho osing to sw ap the ancilla allo ws the circuit to b e implemen ted with just 3 CNOT s (righ t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 xxiii 4.5.7 The n um b er of CNOT s used to implemen t the b = 1 n oracle is sho wn as a function of n . This n um b er w ould scale as 2n for a c hain, but for the hea vy- hex arc hitecture, it scales as1.76n . As b oth Mon treal and Cairo use the same arc hitecture, this n um b er is the sam e for b oth. . . . . . . . . . . . . . . . . 105 4.5.8 Binary tree represen ting the oracle computation for the construction used in our exp erimen ts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5.9 Binary tree represen ting the oracle computation for the construction using divide-and-rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5.10log(F QEB ) is plotted as a function of problem size. W e ha v e extracted the exp onen t λ ps suc h that F QEB = c2 (1−λps )n . Note that for classical algorithm solving the BV problem, λ ps = 1 . Here, ev en without DD, w e are observing b etter-than-classical p erformance. See Fig. 4.5.10 to understand wh y the TTS has a w orse scaling than p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.5.11 TTS i fori2f exp, a vg,ming with their resp ectiv e scaling factorsλ i are sho wn for Mon treal and Cairo, with and without DD. Note thatλ exp >λ a vg >λ min = λ ps , where λ ps is the same as in Fig. 4.5.11 . . . . . . . . . . . . . . . . . . . 114 xxiv Chapter One: In tro duction 1.1 Dynamical decoupling Building and studying noisy programmable quan tum computers [ 5 ] has a dual purp ose: (1) to in v estigate the promise of quan tum computing b y implemen ting simple computational tasks and (2) to help understand ho w error correction, suppression, and mitigation strategies can help us reac h fault-tolerance. All suc h strategies incur some resource o v erhead: ancillas, increased run time, more op erations, or m ultiple rep etitions of the exp erimen t. Dynamical Decoupling [ 6 – 9 ] is the least resource-in tensiv e error suppression strategy . While DD is fully compatible with quan tum error correction [ 10 , 11 ], its m ost economical form requires only a sequence of pulses that are strategically inserted in to the quan tum circuit. In other w ords, DD requires no enco ding, ancillas, extra measuremen ts, or p ost-pro cessing. In this thesis, w e explore the p oten tial of DD in impro ving the p erformance of v arious programmable sup er- conducting devices. Often w e use DD in conjunction with other error mitigation strategies lik e measuremen t error mitigation [ 12 , 13 ] and quan tum error detection [ 14 ]. W e start b y demonstrating the effectiv eness of DD in impro ving qubit fidelities and conclude b y sho wing substan tial algorithmic p erformance impro v emen t due to DD protection. Error suppression via DD has a long and w ell-studied history [ 14 ] b efore it w as consid- ered seriously as a w a y to impro v e the p erformance of programmable quan tum computers. The foundational demonstration of DD (ev en though the term DD w as coined later) w as p erformed b y Erwin Hahn in 1950 [ 15 ]. The Hahn ec ho exp erimen t considers an equal su- p erp osition state undergoing decoherence. By applying a π -pulse ab out the y -axis in the middle and end of the ev olution, the quan tum state can b e refo cused and its broadening narro w ed. Up to a rotation of the Blo c h sphere, the Hahn ec ho is equiv alen t to anXX pulse sequence, whereX pulses are applied in the middle and end of the ev olution. This metho d of puncturing the system ev olution with a sequence of clev erly placed pulses w as later general- ized [ 6 , 7 , 9 , 16 ] as a metho d to decouple a system from its en vironmen t dynamically , hence the name DD. As w e will discuss in the next c hapter, an XYXY sequence, i.e., alternating X and Y pulses at equal in terv als b et w een the start and end of the ev olution is ev en more effectiv e than the XX pulse. In particular, the XYXY sequence is univ ersal as it cancels all single qubit system-en vironmen t Hamiltonian terms up to the first order in the ev olution time τ . Ov er the last t w o decades, n umerous generalizations of these pulse sequences ha v e b een considered – in terms of the exact unitaries applied, pulse placemen t, and consideration of v arious pulse shap es [ 17 ]. V arious c hoices of these design parameters allo w for DD sequences that remo v e sp ecific undesired system-en vironmen t terms while lea ving others unaffected. Often, but not alw a ys, the goal is to cancel the system-bath in teraction altogether. DD sequences can also b e made robust [ 3 , 18 – 21 ] to pulse imp erfections suc h as flip angle errors, pulse width errors, rotation axis missp ecification, and pulse shap e. DD sequences ha v e b een used to suppress decoherence in trapp ed ions [ 22 , 23 ], photonic qubits [ 24 ], electron para- magnetic resonance [ 25 ], n uclear magnetic resonance (NMR) [ 26 – 28 ], trapp ed atoms [ 29 ] and nitrogen v acancies in diamond [ 30 ]. Ref. [ 17 ] reviews v arious DD sequences and ex- p erimen tal implemen tation of these sequences across v arious quan tum platforms. In this w ork, our fo cus is not on designing or ev en systematically comparing v arious DD sequences. W e refer in terested readers to Ref. [ 31 ] for a systematic comparison of more than 60 dif- feren t pulse sequences on QCs similar to the ones used in this w ork. Nor do w e consider sequences with non-uniform pulse in terv als, i.e., otherwise promising sequences suc h as Uhrig DD (UDD [ 32 ]), quadratic DD (QDD [ 33 ]) and nested UDD (NUDD [ 34 ]) are ignored. In- stead, w e will pro vide pro of of principle demonstrations of p erformance impro v emen ts in 2 v arious settings emplo ying state-of-the-art DD sequences on programmable sup erconducting QCs [ 1 , 35 ]. 1 Idealized DD sequences DD is p erformed b y applying a carefully c hosen sequence of pulses to the quan tum device, suc h that the time-a v erage of the system-en vironmen t in teraction is canceled [ 36 ]. T o b e more sp ecific, giv en a time-indep enden t Hamiltonian H , w e can decomp ose it in to the follo wing comp onen ts H =H 0 S +H 1 S +H B +H SB =H 0 S +H B +H err (1.1) Here H S and H B act only on the system and the bath, resp ectiv ely; the terms that act on b oth the system and the bath are in the system-bath in teraction term H SB . H 0 S is the system comp onen t w e wish to k eep, and w e are indifferen t to H B , whic h do es not affect the system. The free ev olution of the system is f τ U(τ)=exp(iτH). (1.2) DD sequences are added via a con trol termH c (t) suc h that the effect of H err =H 1 S +H SB can b e suppressed. Let us assume thatH c (t) comprises ideal and instan taneous pul ses generated b y Ha miltoniansfH P k g n k=1 applied at in terv alsfτ k g n k=1 , i.e., H c (t)= X k Ω(tτ k )H P k where ∆/2 Z ∆/2 Ω(t)dt=Ω 0 ∆=π/2, (1.3) where∆ is the pulse width andΩ(t) describ es the pulse shap e. F or delta-function lik e pulses, lim ∆→0 Ω(t)=Ω 0 δ(t) , the instan taneous unitaries corresp onding to the pulses will b e ˆ P k =e −i π 2 H P k , (1.4) 3 and the ev olution under DD error suppression is ˜ U(T)=f τn ˆ P n ...f τ 2 ˆ P 2 f τ 1 ˆ P 1 (1.5) DD sequences with uniform pulse gaps, where τ n = τ , are often describ ed only in terms of the pulse, i.e., ˆ P 1 ˆ P 2 ˆ P n or simply ˆ P 1 ˆ P 2 ... ˆ P n sequence. ˜ U(T) can b e decomp osed in to desirable and undesirable terms ˜ U(T)=G +B. Ab o v e, G = e −iTH 0 S B ′ (T) where B ′ (T) is an a rbitrary unitary acting only on the bath. Ov erall, the goal of error suppression via DD is to minimize the distance b et w een ˜ U(T) and G , i.e., to minimize η DD =kBk. (1.6) Herekk signifies the sup op erator norm (the largest singular v alue of A ): kAk sup {|v⟩} kAjvik kjvik = sup {|v⟩ s.t. ∥|v⟩∥=1} kAjvik. (1.7) Theoretical in v estigations of DD sequences pro vide b ounds on η DD with resp ect to the rel- ev an t energy scales β kH B k,J kH err k , ϵ = β +J . Moreo v er, attempts at designing sophisticated DD sequences fo cus on minimizing η DD with resp ect to the pulse gap τ , pulse width ∆ , and the flip angle error ϵ r , all of whic h w e discuss b elo w. 2 Basic single qubit sequences F or the ma jorit y of our implemen tations w e fo cus on single-qubit DD sequences. The Hahn ec ho sequence [ 15 ], whic h is also called the XX sequence, is the simplest p ossible DD se- quence. Consider the effect of this sequence on arbitrary single qubit system-en vironmen t coupling, H SB = P α=x,y,z σ α B α . Here, Xe −iτH S B Xe −iτH S B = e −i2τσx⊗B x +O(τ 2 ) . In 4 w ords, theXX sequence do es not cancel all the single-qubit system bath in teraction terms, as eac h pulse in XX comm utes with the P auli matrix X . W e can concatenate the XX sequence with the ZZ sequence to get ZXf τ Xf τ ZXf τ Xf τ = Yf τ Xf τ Yf τ Xf τ , i.e., the XYXY sequence (also called XY4) [ 36 ]. The XY4 sequence is univ ersal in that it cancels all single-qubit system bath in teraction terms to the first order in τ , i.e., ˜ U(T)=I B ′ +O(τ 2 ) . T o b e more precise [ 11 ], η XY4 =(4Jτ) 1 2 (4ετ)+ 2 9 (4ετ) 2 +O τ 3 (1.8) Ev en in this idealized analysis, where w e are not accoun ting for pulse imp erfections, XY4 only pro vides first order protection against arbitrary single qubit errors. Higher order pro- tection can b e ac hiev ed with concatenated dynamical decoupling (CDD) sequences but at the cost of adding exp onen tially more pulses. The n -th order CDD n sequence is defined recursiv ely using a base sequence CDD 0 . Using XY4 as the base sequence, w e get CDD n XY4(CDD n−1 )=(CDD n−1 )f τ Xf τ CDD n−1 f τ Yf τ (CDD n−1 )f τ Xf τ (CDD n−1 )f τ Y. (1.9) This impro v es the error measure suc h that η CDDn =4 n(n+3)/2 (cετ) n (Jτ)+O τ n+2 , (1.10) where the constan t c = O(1) . While the higher order protection pro vided b y the CDD is desirable, CDD n sequences ha v e 4 n terms. Later w e sho w that w e w ere rarely able to imple- men t CDD n for n> 3 b ecause with finite pulses CDD 4 w as m uc h longer than the timescale of concern. It is w orth noting that it is p ossible to obtain higher order decoupling without incurring an exp onen tial cost, particularly b y using the Uhrig DD (UDD) [ 37 ] sequence and the family of sequences deriv ed from it, whic h incurs a lo w-degree p olynomial scaling in the order n . These sequences ha v e non-uniform pulse in terv als and require precise con trol 5 o v er the timing and shap e of the pulses that implemen t the DD sequences. In this thesis w e did not attempt to implemen t non-uniform pulse sequences (see later c hapters for the pragmatic reasons for this c hoice). Ho w ev er, UDD and other non-uniform sequences w ere tested on similar devices in Ref. [ 31 ], whic h found that non-uniform sequences – while quite effectiv e – w ere not alw a ys b etter than uniformly spaced DD sequences that are robust to pulse imp erfections. 3 Sequences with finite pulse width In the previous section (see Eq. ( 1.3 )), w e considered lim ∆→0 Ω(t) = Ω 0 δ(t) . Relaxing this idealization means that the pulses will not b e instan taneous and therefore the undesired Hamiltonian term H err will affect the system ev en during pulse application. Therefore, a more realistic pulse unitary is P k exp ( i Z τ k +∆/2 τ k −∆/2 dt[Ω(tτ k )H P k +H err +H B ] ) . (1.11) Under this pulse application, b oth η XY4 and η CDDn acquire anO(∆) dep endence: η ∆ XY4 =4J∆+η XY4 , (1.12) η ∆ CDDn >16J∆+η CDDn . (1.13) In fact, ev en the errors for non-uniform UDD sequences ha v e a linear dep endence on ∆ . F or P auli matrix based sequences, the only w a y to get rid of theO(∆) dep endence is to use the Eulerian DD (EDD) sequence EDDXYXYYXYX. (1.14) 6 EDD is a palindromic v ersion of the XY4 sequence and has η ∆ EDD =(8Jτ) 1 2 (8ετ)+ 2 9 (8ετ) 2 +O τ 3 ,∆ 2 . (1.15) In this w ork, w e refer to EDD also as R GA8c. Later c hapters, in particular Chapter 3 sho ws that EDD is one of most effectiv e sequences when suppressing errors during a quan tum computation. 4 Flip angle errors and robust sequences Another source of error is mis-application of the pulse sequences. In particular, the pulses migh t ha v e a systematic o v er or under rotation due to miscalibration. F or single qubit pulses, these flip angle errors can b e mo deled as P r j =exp n i π 2 (1+ε r )σ α o , (1.16) where the error is quan tified using ε r . Belo w w e describ e t w o sequence families - robust genetic algorithm based (R GA) and univ ersally robust (UR) sequences. Both are kno wn to mitigate these systematic flip-angle t yp e errors. A crucial comp onen t of R GA sequences are conjugate pulses ¯ P k defined suc h that ¯ P k exp ( i Z τ k +∆/2 τ k −∆/2 dt[Ω(tτ k )H P k +H err +H B ] ) . (1.17) Notice the additional min us sign in fron t of Ω compared to the definition of P k in Eq. ( 1.11). Ref. [ 3 ] utilizes the fact that giv en a set of pulses that comprise the sequence and the n um b er of pulses that are applied after a uniform in terv al, the goal of finding an optimal DD sequence is a com binatorial optimization problem. They solv e this optimization problem n umerically using genetic algorithm-based approac hes while accoun ting for finite pulse width and flip angle errors. 7 The first few genetic algorithm (GA) based sequences are [ 3 ]: GA 4 :=P 1 f τ P 2 f τ P 1 f τ P 2 f τ (1.18a) GA 8a :=If τ P 1 f τ P 2 f τ P 1 f τ If τ P 1 f τ P 2 f τ P 1 f τ , (1.18b) GA 16a :=P 3 ( GA 8a )P 3 ( GA 8a ), (1.18c) GA 32a := GA 4 [ GA 8a ]. (1.18d) where P 1 ,P 2 are single-qubit P auli op erators suc h that P 1 6= P 2 2fX,Y,Zg . W e set P 1 = X,P 2 = Y for GA 4 and P 1 = X,P 2 = Z for GA 8a . With this c hoice GA 4 = XY 4 . While comp etitiv e against higher-order sequences lik e CDD, these sequences are not robust against finite width or against flip angle errors. Robust genetic algorithm (R GA) sequences w ere generated to address flip angle errors. The R GA sequences listed b elo w are either first-order sequences withη =O(τ 2 ) or concatenations of these first-order sequences. W e do not detail the con tribution of∆ andε r toη here as they are presen ted in detail in Ref. [ 3 ]. Nonetheless, it is w orth noting that sequences lab eled with subscript ‘a ’ w ere found to b e effectiv e against flip angle errors, and those with subscript ‘ c ’ are effectiv e against finite pulse width errors. RGA 4 ¯ Y X ¯ Y X (1.19a) RGA 4p ¯ Y ¯ X ¯ Y ¯ X (1.19b) RGA 8c XY XY Y XY X (1.19c) RGA 8a X ¯ Y X ¯ Y Y ¯ XY ¯ X (1.19d) RGA 16b RGA 4p ([RGA 4p ]) (1.19e) RGA 32a RGA 4 ([RGA 8a ]) (1.19f ) RGA 32c RGA 8c ([RGA 4 ]) (1.19g) This is a partial list of all the sequences presen ted in Ref. [ 3 ], as here w e only fo cused on 8 sequences with up to 32 pulses. As detailed in later c hapters, DD sequences are applied when qubits are idle during computation, and so far, w e did not consider algorithms where there w as enough idle time b et w een op erations to fit sequences with more than 32 pulses. The UR family of sequences w as in tro duced in Ref. [ 21 ], where a system Hamiltonian with semiclassical dephasing noise w as considered. The sequences w ere deriv ed while enforcing that this dephasing noise is canceled while placing no restrictions on the pulse shap e, the magnitudes of flip-angle errors, or the missp ecification of the axis of rotation. The UR n pulses are defined recursiv ely using the follo wing form ula UR n =(π) ϕ 1 (π) ϕ 2 ...(π) ϕn (1.20a) ϕ k = (k1)(k2) 2 Φ (n) +(k1)ϕ 2 (1.20b) Φ (4m) = π m ,Φ (4m+2) = 2mπ 2m+1 . (1.20c) Here (π) ϕ is rotation ab out the axis at an angleϕ from the +x -axis. W e c ho oseϕ 1 =0 , and ϕ 2 = Φ(n) so that all UR n sequences are palindromic. In Ref. [ 21 ], the pulse application errors are quan tified using α,β,γ,δ,p suc h an imp erfect X -pulse w ould b e implemen ted as U pulse (α,β,p)= 2 6 4 p 1pe iα p pe iβ p pe −iβ p 1pe −iα 3 7 5 . (1.21) p , α,β corresp ond to unkno wn but systematic errors in pulse application. The errors accu- m ulated during free ev olution are mo deled as unkno wnH err (t)=∆σ z , i.e., dephasing errors. So, δ ∆τ , whic h includes the free ev olution duration τ , accoun ts for the errors accum u- lated during free ev olution. The fidelit y b et w een the DD protected ev olution ˜ U(T) and the desired ev olutionG is computed t o b e F = 1 2 Tr ˜ U † G =12(1p) n/2 sin 2 h n 2 (α+δπ/2ϕ 2 /2) i (1.22) Note that a linear increase in the n um b er of pulses n , whic h is alw a ys ev en, leads to an 9 increase in the order of suppression of flip angle errors ϵ r = 1p . Ho w ev er, as these se- quences are implemen ted using noisy gates, whic h are not guaran teed to ha v e only systematic and unc hanging error parameters, exp erimen tal implemen tation will ev en tually exp erience a decrease in p erformance with an increasing n um b er of pulses. In our in v estigations, the optimal UR n w as for n opt 20 . Notably , as this sequence w as deriv ed under a semiclassical mo del, a clear understanding of η URn is lac king. In fact, a theoretical analysis of URDD in the con text of an arbitrary quan tum bath is an in teresting op en question. URDD w as one of the most effectiv e sequences w e tested, whic h further motiv ates a rigorous study of these sequences. In Ref. [ 31 ] w e considered the exp erimen tal effectiv eness of CDD, R GA, and UR sequence families alongside non-uniform pulse in terv al sequences lik e UDD, in extending single qubit lifetimes. Ref. [ 31 ] also in v estigates the p erformance of sequences with more than 32 pulses and compares the effect of a v arying n um b er of pulses and c hanging the pulse gap τ . W e refer in terested readers to Ref. [ 31 ] more a comprehensiv e surv ey of DD sequences not considered in this thesis. 1.2 Success metrics The cen tral goal of this w ork is to in v estigate ho w error suppression can b e used to mean- ingfully impro v e the p erformance of programmable quan tum computers for sp ecific com- putational tasks. The metrics for success then dep end on the problem of concern. W e consider t w o metrics and their a v erages and generalizations: fidelit y and time-to-solution (TTS). These metrics are redefined in the corresp onding c hapters as appropriate, but here w e pro vide a brief o v erview and motiv ation for c ho osing these metrics. Giv en a desired quan tum state jψi and an observ ed quan tum state jϕi , the Uhlmann fidelit y [ 38 ] isF = T r h p p σρ p σ i 2 . When the quan tum states are pure this form ula simpli- fies to F = jhϕjψij 2 . F requen tly , w e encoun ter situations where the desired quan tum state 10 σ =jψihψj is pure but the obs erv ed quan tum state ρ is mixed. In these cases, F =jhψjρjψij 2 . (1.23) Moreo v er, if jψi =Ujbi where jψi is a computational basis elemen t with b2f0,1g n and U is an unitary , then F = hbjU † ρUjbi 2 . (1.24) In other w ords, if w e apply the gate U † after preparing ρ but b efore the measuremen t then the fidelit y F is simply the probabilit y of observing the bitstring b . F or b oth the algorithms considered in this thesis, the desired jψi = jbi , i.e., the desired answ er at the end of the computation is a classical bitstring b . In those cases, F can b e in terpreted as the success probabilit y p s . While equiv alen t to fidelit y , w e rep ort success probabilities in these circumstances b ecause the classical coun terpart of the algorithms also ha v e a rep orted success probabilit y . Comparison of classical and quan tum success probabilities at fixed problem sizes can b e misleading as other resources, most notably the time tak en to arriv e at the solution, are ignored when computingp s . So the second metric w e consider i s the time-to-solution (TTS). TTS is a w ell-established metric used to quan tify device p erformance [ 39 ]. TTS is defined in the con text where a calculation is p erformed m ultiple times with the success probabilit y p s , with p s remaining constan t o v er iterations. If eac h iteration is indep enden t of the last, then the n um b er of times the exp erimen t m ust b e rep eated so that the exp erimen t succeeds at least once with desired probabilit y p d is giv en b y (1p s ) R =1p d . (1.25) Generally , w e set p d =0.99 and if eac h iteration tak es time t r then TTS is defined as TTS=t r R =t r log(1p d ) log(1p s ) . (1.26) 11 TTS is a function of p s , but it also accoun ts for the tradeoff b et w een higher success proba- bilities and longer computation times. Unfortunately , a simple comparison of classical and quan tum TTS can also b e misleading as t r , the time p er iteration, can v ary substan tially b et w een differen t classical and quan tum computers. As w e explain in detail in Chapter 4 , instead w e fo cus on the v alue of the ratio of TTS at the large p ossible problem sizen max , i.e., S max = lim n→n max S(n) for S(n) = TTS C (n) TTS Q (n) . S max allo ws for a t r and n indep enden t measure of sp eedup b et w een classical and quan tum computers. Ho w ev er, computingS max is often not p ossible for noisy devices, and in those scenarios, w e restrict ourselv es to the fidelit y-based measures discussed ab o v e. 1.3 Summary of results W e start with the simplest p ossible test of error suppression due to DD. In Chapter 2 , w e con- sider the impact of DD on single-qubit and t w o-qubit fidelities. Using v arious sup erconducting- qubit based platforms, w e demonstrate substan tial fidelit y gains due to DD relativ e to unpro- tected ev olution. W e also pro vide evidence that single-qubit based DD sequence are capable of protecting en tangled t w o-qubit states. The eff ect of error suppression and error detection on a b ona-fide quan tum algorithm, in particular the Gro v er searc h algorithm, is in v estigated in Chapter 3 . W e consider Gro v er searc h algorithm up to fiv e qubits on v arious sup erconducting-qubit-based programmable quan tum computers. Aided b y the[[4,2,2]] error-detection co de, w e demonstrate 99.4(0.2)% success probabilit y for t w o-qubit Gro v er. F or 3-qubit to 5-qubit Gro v er, w e emplo y a surv ey of the p erformance of three families of robust DD sequences – CDD, R GA and UR. In the end, w e rep ort b etter-than-classical success probabilities at all problem sizes. In Chapter 4 , w e consider the Bernstein-V azirani algorithm. The problem of concern is to guess then bits of a secret bitstring b y querying an oracle that enco des the bitstring. After one call to the oracle, the pla y er is allo w ed one guess. If the guess is correct, the pla y er wins; if the guess is wrong, the game con tin ues with a new oracle enco ding a new bitstring. In the 12 classical case, the probabilit y of guessing the correct string is exp onen tially small (inn ), so it tak es the pla y er an exp onen tial n um b er of rounds to win. In con trast, in the quan tum case, the success probabilit y after eac h call is 1. If the pla y er has access to a p erfect, noiseless quan tum computer, the pla y er wins after j ust a single round. Using robust DD sequences, w e rep ort a quan tum sp eedup in the scaling of the TTS with resp ect to problem size. The sp eedup w e rep ort is not exp onen tial but p olynomial, giv en that the QC w e used that is far from noiseless. Nonetheless, our result establishes that the curren t generation of rela- tiv ely noisy , in termediate-scale quan tum (NISQ) computers is already capable of deliv ering an unam biguous quan tum sp eedup and that dynamical decoupling pla ys a crucial role in ac hieving this sp eedup. 13 Chapter T w o: Fidelit y impro v emen t on sup erconducting qubits T ext for this c hapter is adapted from [ 40 ]. T w o decades after the first detailed quan tum computing prop osals [ 41 – 44 ], rudimen tary gate-mo del quan tum computers (QCs) based on sup erconducting transmon qubits with co- herence times in the microseconds range are finally a v ailable and remotely accessible via public cloud-based services. In terest in these platforms, made publicly a v ailable so far b y IBM, Rigetti, and Alibaba, has b een high, and n umerous exp erimen ts ha v e b een rep orted demonstrating a v ariet y of quan tum proto cols [ 45 – 48 ] and algorithms [ 49 – 51 ]. Giv en their presen t in termediate scale of 10 -20 fairly noisy qubits, gates, and measuremen ts [ 5 ], the cur- ren t QCs are particularly v ery w ell suited to tests of simple quan tum error correction and suppression proto cols. Indeed, a v ariet y of quan tum error correction (QEC) exp erimen ts on cloud based platforms ha v e b een rep orted [ 52 – 58 ]. Ho w ev er, so far this b o dy of w ork has not offered a demonstration that QEC can result in impro v emen ts for general decoherence while applying standard initialization, gates, and readout op erations (w e review these stud- ies in Section 2.6 , Sec. A). The main reason app ears to b e that the o v erhead in tro duced b y QEC results in error rates that are to o high to b e comp ensated b y the sc hemes that ha v e b een tried so far, and claims of impro v emen t ha v e had to resort to clev erly a v oiding the execution of actual initialization and k ey gate op erations [ 58 ]. Here, rather than attempting to demonstrate error correction, w e fo cus on error suppres- sion. Sp ecifically , w e seek to mitigate the effects of decoherence using dynamical decoupling (DD) [ 6 , 7 , 9 , 16 ], one of the simplest strategies a v ailable in the to olkit of quan tum error mitigation [ 14 ]. W e demonstrate that DD is capable of extending the lifetimes of single-qubit states as w ell as en tangled t w o-qubit states. T o the b est of our kno wledge, this amoun ts to the first unequiv o cal demonstration of successful decoherence mitigation in cloud-based sup erconducting qubit platforms. Moreo v er, as a test of the robustness of our results w e p erformed DD exp erimen ts on three of the cloud-based systems, the 16 -qubit IBMQX5, 5 - qubit IBMQX4, and the 19 -qubit Rigetti A corn c hips. Giv en their similarities they pro vide suitable platforms for indep enden t tests of the p erformance of DD, and w e exp ect that the lessons dra wn will ha v e wide applicabilit y . 2.1 Metho dology The nativ e single gates on the IBM and Rigetti platforms are rotationsR α (ϕ)=exp[i(ϕ/2)σ α ] , with α 2 fy,zg (see Section 2.6 , Sec. B, for more details ab out these platforms). Ar- bitrary single-qubit unitaries can b e applied b y sp ecifying Euler angles θ,ϕ,λ suc h that U(θ,ϕ,λ) = iR z (ϕ)R y (θ)R z (λ) . Since DD is exp ected to pro vide quan tum error suppres- sion for arbitrary initial states, w e tested the p erformance of DD on a v ariet y of initial states b y rep eatedly preparing single-qubit states of the form jψi = U(θ,ϕ,λ)j0i , where j0i and j1i are computational basis states (eigenstates of σ z ). It should b e noted that in transmon qubits thej0i andj1i states are, resp ectiv ely , the ground and first excited states; this has imp ortan t implications as discussed b elo w. DD pulses w ere applied as the gates X = iexp[i(π/2)σ x ] = U(π,0,π) and Y = iexp[i(π/2)σ y ] = U(π,2π,0) . On the IB- MQX5 (A corn) c hip eac h single-qubit pulse lasted 80 ns (40 ns), with a 10 ns buffer of free ev olution b et w een pulses, and eac h suc h run w as rep eated8192 (1000 ) times. Iden tit y pulses w ere implemen ted as free ev olutions lasting90 ns (50 ns) on the IBMQX5 (A corn) c hip. Since measuremen ts are only p ossible in theZ basis, w e appliedU † (θ,ϕ,λ) at the end of eac h run and measured the final state of eac h qubit in the Z basis. Our k ey p erformance metric is the fidelit y b et w een the input and the output states, de- fined as the total n um b er of j0i states empirically observ ed divided b y the total n um b er of rep etitions. W e considered t w o t yp es of initial conditions. In “t yp e 1”, θ w as v aried in 16 15 equidistan t steps in the range [0,π] , with ϕ = λ = 0 . This corresp onds to a sequence of states (sup erp ositions for 0 < θ < π ) of the form jψi = cos(θ/2)j0i+sin(θ/2)j1i (up to a global phase). In “t yp e 2”, w e considered a set of 30 random initial conditions sampled uniformly from the Blo c h sphere along with the 6 eigenstates of the P auli matrices, i.e., j0i,j1i,ji = 1 √ 2 (j0ij1i),jii= 1 √ 2 (j0iij1i) . 2.2 Single-qubit results W e first tested the dep endence on the initial state using t yp e 1 preparation. as sho wn in Fig. 2.2.1 . Under free ev olution, the fidelit y is relativ ely high for θ 0 (corresp onding to the ground state j0i ) on b oth devices, and approac hes a clear minim um for θ 5π 8 , i.e., a sup erp osition state sligh tly biased to w ards j1i . On b oth devices the free ev olution fidelit y rises to w ards the excited state j1i , but remains w ell b elo w that of the ground state. Th us coheren t sup erp osition states undergo significan t dephasing and the excited state j1i undergo es sp on taneous emission (SE) and relaxes to the ground state. The situation is dramatically differen t under DD. When compared at the pulse n um b er for whic h DD exhibits the highest error suppression [N =40 (192) for IBM (Rigetti)], on b oth devices the θ -dep endence is essen tially eliminated, as sho wn in Fig. 2.2.1 . It is clear that the o v erall fidelit y (a v eraged o v er θ ) increases significan tly , while DD reduces the fidelit y of states close to the ground state. This is consisten t with the XY4 sequence suppressing all single-qubit error t yp es equally . F or more details see Section 2.6 , Sec. D. Figure 2.2.2 sho ws the results under t yp e 2 preparation. F or IBMQX5, DD significan tly reduces the fidelit y deca y up to N 110 pulses. The free ev olution fidelit y deca ys rapidly but has a shallo w minim um at N 60 , then surpasses the fidelit y under DD for N > 110 , whic h con tin ues to deca y exp onen tially . This exp onen tial deca y is consisten t with Mark o vian dynamics. The situation is rather differen t for A corn. First, w e note that the initial fidelit y (deter- 16 — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◼ ◻ ◆ ◇ 0.4 0.5 0.6 0.7 0.8 0.9 1. 1/8 1/4 3/8 1/2 5/8 3/4 7/8 Figure 2.2.1: Mean fidelit y o v er 16 qubits of IBMQX5 and 15 qubits of A corn, for initial states jψi = i[cos(θ/2)j0i + sin(θ/2)j1i] . Results sho wn are under DD using XY4 and under fre e ev olution. IBMQX5: after N = 40 pulses, i.e., 10 rep etitions of the base XY4 sequence. DD im- pro v e s t he fidelit y only for states with θ≳ π/3 . A corn: after N = 192 pulses, i.e., 48 rep etitions of the ba se XY4 sequence. DD impro v es the fidelit y only for states with θ≳π/4 . Throughout w e rep ort 2σ error bars (95% confidence in terv als) calculated using the b o otstrap metho d (for more details se e Section 2.6 , Sec. C). mined b y the initialization and readout errors) is lo w er for A corn than for IBMQX5: 0.91 and 0.96 , resp ectiv ely . Second, the fidelit y under DD is consisten tly greater than under free ev olution, and the roles are rev ersed: free ev olution is v ery nearly Mark o vian (exp o- nen tial deca y) while under DD it exhibits a recurrence. These fidelit y differences suggest that the en vironmen ts are differen t for the t w o QCs, with the nativ e IBMQX5 en vironmen t b eing non-Mark o vian, while that of A corn is more Mark o vian. Con v ersely , DD remo v es the non-Mark o vian comp onen t for IBMQX5, while it in tro duces a non-Mark o vian comp onen t for A corn. W e ma y sp eculate that the non-Mark o vianit y is due to residual lo w-frequency noise (e.g., 1/f ) in the IBMQX5 case, and that the DD pulses themselv es in tro duce lo w frequency noise in the A corn case. T o quan tify the fidelit y deca y with and without DD w e fit the data to a mo dulated 17 — — — — — — — —— —— — — — — —— ——— —— — —— —— ——— — — — —— ——— — — — — — — — —— —— — — — — —— ——— —— — —— —— ——— — — ——— ——— — — — — — — — — — — — — — — —— ————— — —— —————————————— — — — — — — — — — — — — — — —— ————— — —— —————————————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 100 200 300 400 500 600 0.5 0.6 0.7 0.8 0.9 1.0 — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0.5 0.6 0.7 0.8 0.9 1.0 0 33 66 100 133 166 200 Figure 2.2.2: Mean fidelit y , after a v eraging o v er all qubits, and all 36 initial conditions in t yp e 2 preparation, as a function of the n um b er of pulses for IBMQX5 (b ottom axis) and A corn (top axis). The pulse in terv al is the shortest p ossible: 90 ns for IBMQX5, 50 ns for A corn. Solid lines are fits to Eq. ( 2.1 ), with fit parameters as p er T able 2.2.1 . exp onen tial deca y with three free parameters λ,α (dimensionless deca y times) and γ (di- mensionless mo dulation frequency): F(N)=cf(N)+c 0 , f(N)=e −N/λ cos(Nγ)+e −N/α . c= F Nmax F 0 f(N max )1 , c 0 =F 0 c. (2.1) Here F 0 is the initial fidelit y , F Nmax is the fidelit y at N = 592 (192) for IBMQX5 (A corn). The deviation of F Nmax from 1 accoun ts for the initialization errors, readout errors, and decoherence that w ere not cancelled b y DD, as w ell as the errors accum ulated during the application of the imp erfect DD pulses, arising from imp erfect con trol o v er the pulse shap e, duration, and in terv al. T able 2.2.1 summarizes the v alues of the fit parameters. While λ quan tifies the sharp deca y during the b eginning of the ev olution, ev olution at longer timescales is quan tified b y α . The most significan t finding for IBMQX5 is that the initial deca y time c haracterized b y λ is more than tripled in the presence of DD. While 18 Mac hine Ev olution F 0 10 −2 F N max 10 −2 λ α γ IBMQX5 F ree 96.50.1 55.60.7 28.91.2 9105 0.730.12 IBMQX5 DD 96.50.1 53.10.1 88.40.3 1 0 A corn F ree 90.80.4 59.80.6 68.11.3 1 0.140.11 A corn DD 90.80.4 77.10.4 74.90.9 1 0.50.03 T able 2.2.1: Fit parameters when Eq. ( 2.1 ) is used to fit the mean fidelities in Fig. 2.2.2 . The first deca y constan t, λ , is significan tly increased under DD. The second deca y c onstan t, α , is effectiv ely infinite for all ev olutions other than IBMQX5’s free ev olution. The mo dulation fre- quency γ v anishes for IBMQX5 under DD and is near zero fo r A corn under free ev olution, consis- ten t with purely exp onen tial fidelit y deca y , i.e., Mark o vian ev olution. the impro v emen t in deca y time is m uc h more mo dest for A corn, the result is a sense ev en b etter than for IBMQX5, in that DD impro v es its fidelit y for all N w e w ere able to test. W e also tested DD on the 5 -qubit IBMQX4, with similar results (see Sec tion 2.6 , Sec. B.2, for details). 2.3 Dephasing vs. sp on taneous emission Figure 2.2.1 sho ws that b oth dephasing and SE pla y imp ortan t roles. This is studied in more detail in Section 2.6 , Sec. D, where w e sho w that for initial states close to the ground state j0i , DD is w orse than free ev olution, but for sup erp osition states susceptible to dephasing and states close to the excited state j1i susceptible to SE, there is a clear b enefit in using the XY4 sequence. In ligh t of this, it is in teresting to try to address one of these error sources at a time. A DD sequence that suppresses only dephasing (σ z ) errors is (XI) N or (YI) N (N rep etitions of XI or YI ), since X and Y an ticomm ute with σ z . Lik ewise, SE is suppressed b y (ZI) N , since Z an ticomm utes with σ − . W e rep ort on results for these sequences in Section 2.6 , Sec. E; they underp erform XY4, as exp ected, but b oth lead to a substan tial slo wing do wn of fidelit y deca y , with dephasing suppression b eing the dominan t effect, accoun ting for nearly 90% of the v alue of λ under XY4. This can b e view ed as an example of using DD as a diagnostic to ol, to iden tify the relativ e dominance of differen t decoherence c hannels [ 69 , 70 ]. 19 — — — — — — — — — — — — — — — — — — — — — — — — ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1 2 3 4 5 6 40 60 80 100 120 Figure 2.4.1: DD p erformance as a function of pulse in terv al τ for IBMQX5 (in units of 90 ns). The in tersection time t in t of the fidelit y curv es under free ev olution and DD, and the deca y-time exp onen t λ , as a function of the pulse spacing, τ . Linear fits yields t int =3.5(τ/90 ns)+108 and λ =4.3(τ/90 ns)+88.0 . 2.4 Dep endence on the pulse in terv al —It is w ell kno wn from DD theory that p erformance dep ends strongly on the pulse in terv al τ [ 14 ]. W e th us consider the in tersection time of the fidelit y curv es under free ev olution and DD for IBMQX5, denoted t in t , whic h represen ts the duration o v er whic h DD impro v es the fidelit y o v er free ev olution. The dep endence on τ is sho wn in Fig. 2.4.1 . W e observ e that, as exp ected, t in t deca ys to first order in τ , implying that as the pulse in terv al increases, DD b ecomes less effectiv e. Also sho wn in Fig. 2.4.1 is the deca y-time exp onen t λ as a function of τ , whic h b eha v es similarly: λ deca ys from an initial v alue of 88 to 60 , still t wice as large as that of free ev olution ( 29 ). Somewhat surprisingly , b oth t in t and λ deca y non- monotonically withτ , a finding that is not captured b y standard DD theory and presen ts an in teresting op en theoretical problem. A dditional analysis is presen ted in Section 2.6 , Sec. F. Protection of t w o-qubit en tangled states.—T o ev aluate the p erformance of DD in preserv- ing en tangled states, w e initialized qubit pairs in Bell states of the formjΦ + i= 1 √ 2 (j00i+j11i) 20 — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ 0.1 0.2 0.3 0.4 0.5 — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ 0 20 40 60 80 0.1 0.2 0.3 0.4 0.5 — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ 0 20 40 60 80 — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ 0 20 40 60 80 — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — ● ● ● ● ● ● ● — — — — — — — — — — — — — — □ □ □ □ □ □ □ ○ 00 ◇ 11 ● 01 □ 10 0 20 40 60 80 Figure 2.4.2: Probabilities of differen t computational basis states after DD for initially prepared Bell statesjΦ + i = 1 √ 2 (j00i +j11i) andjΨ + i = 1 √ 2 (j01i +j10i) , as a function of the n um b er of pulses, for b oth IBMQX5 and A corn. T op ro w: free ev olution. Bottom ro w: ev olution under DD. Thej00i state is fa v ored under free ev olution. The solid horizon tal line indicates p = 0.25 , the limit of a fully mixed state. F orjΨ + i there i s no noticeable difference in the p erformance with or without DD. F orjΦ + i on IBMQX5, after N 20 , p 11 0.25 , suggesting t hat at this p oin t all information has essen tially b een scram bled. On A corn, complete scram bling ofjΦ + i o ccurs after N 30 . Ov erall, DD is more effectiv e at slo wing do wn the deca y to the fully mixed state for A corn than for IBMQX5. andjΨ + i= 1 √ 2 (j01i+j10i) , follo w ed b y an XY4 DD sequence (higher order DD sequences for en tanglemen t protection are kno wn as w ell [ 71 ]). Ideally , one w ould p erform the measure- men ts in t he Bell bas is and re p ort the c orresp onding fidelities. Ho w ev er, w e found that due to the relativ ely large errors in tro duced b y CNOT gates and the high readout errors, Bell ba- sis measuremen ts yielded v ery noisy data whic h w as difficult to dra w meaningful conclusions from. Therefore w e instead p erformed a measuremen t of b oth qubits in the computational basisfj00i,j01i,j10i,j11ig . Letp ij b e the probabilit y of measuring the computational basis statejiji , withi,j 2f0,1g . Our results are plotted in Fig. 2.4.2 , whic h sho ws the probabilities p ij that w ere measured after initializing the system in a Bell state and letting it ev olv e either freely or under DD. Under ideal conditions one w ould exp ect to ha v ep 00 =p 11 =0.5 forjΦ + i andp 01 =p 10 =0.5 for jΨ + i . Instead, for b oth QCs, Fig. 2.4.2 (top ro w) sho ws a strong bias for j00i o v er j11i up on initialization (N =0 ) for thejΦ + i case, with some con tamination b y thej01i andj10i states. F or the jΨ + i case, Fig. 2.4.2 (b ottom ro w) sho ws con tamination b y j00i and j11i 21 up on initialization (stronger for IBMQX5 than for A corn), and a curious bias to w ards j01i o v er j10i for A corn. W e attribute these effects to the single-digit p ercen tage readout errors (see Section 2.6 , Sec. B) and the CNOT gate errors. Clearly , the preparation of the Bell states is itself prone to substan tial errors on b oth QCs. As men tioned earlier, SE pla ys a k ey role and consequen tly the main effect under free ev olution is a sharp increase in p 00 with N on b oth devices. Under DD, the main b eneficial effect is that this dominance of the ground state j00i is suppressed. Ho w ev er, on IBMQX5 for b othjΦ + i andjΨ + i a nearly uniform distribution o v er all four comput ational basis states is reac hed after 100 pulses. The trend is similar for A corn, but the deca y to the fully mixed state is slo w ed d o wn more b y DD than for IBMQX5, and DD manages to k eep the original ratio of p 00 p 11 up to 50 pulses. Ov erall, it is clear that en tanglemen t is rapidly lost, but is slo w ed do wn somewhat b y DD. 2.5 Discussion and Conclusion Our results demonstrate the undeniable usefulness of DD on protot yp e QCs for the suppres- sion of inheren t decoherence, a feature whic h has y et to b e demonstrated unconditionally using QEC [ 52 – 58 ] (see Section 2.6 , Sec. A). It is remarkable that p erformance impro v e- men t w as ac hiev able despite significan t pulse implemen tation imp erfections. Therefore, w e conclude that, giv en a quan tum circuit, it is already adv an tageous to p erform dynamically decoupled ev olution rather than free ev olution b et w een computational gates [ 11 ]. In the future, as the error rates of measuremen t and m ulti-qubit gates are reduced, it should b ecome p ossible to more accurately assess the effectiv eness of DD. W e an ticipate that reduction in m ulti-qubit errors will alleviate the restrictions placed b y connectivit y of the qubits as it will b e p ossible to p erform more SW AP gates without corrupting the states. In suc h scenarios, h ybrid QEC-DD [ 10 , 11 , 72 ] metho ds could b e exp erimen tally assessed and w ould constitute an attractiv e near-term target for higher p erformance gains than is enabled b y either sc heme alone. 22 Another attractiv e prosp ect for future exp erimen ts is the implemen tation of higher-order DD sequences. Indeed, w e ha v e already tested higher-order sequences based on genetic algorithms [ 3 ], and found a small impro v emen t o v er XY4 (see Section 2.6 , Sec. G). The success of suc h sequences in pro viding b etter fidelit y impro v emen ts than the XY4 sequence will dep end on impro v ed pulse con trol (suc h as the abilit y to fine-tune pulse in terv als, needed to implemen t UDD [ 32 ] and QDD [ 33 ]), reduction of the pulse in terv al and duration, etc. Implemen tation of robust DD sequences [ 3 , 18 – 21 ] is another particularly promising v en ue. 2.6 Supplemen tary Information 1 Mac hine Sp ecifications The IBMQX5 and Rigetti A corn c hips used in our exp erimen ts ha v e similar figures of merit, including T 1 and T 2 times, single-qubit gate errors, readout errors, etc. Details and ad- ditional information regarding the ph ysical parameters of the systems are pro vided in T a- bles 2.6.1 and 2.6.2 . Since these parameters fluctuate on a daily basis a date of access is also included. Y et more information is summarized in T able 2.6.3 . IBMQX5 W e ran exp erimen ts on this 16 -qubit c hip with circuits written in Op en Quan tum Assem bly Language [ 60 ]. The qubit connectivit y graph is illustrated in Fig. 2.6.1 , with the corresp ond- ing cross-talk and relaxation times rep orted in [ 62 ]. The c hip comprises sup erconducting transmon qubits, coupled via coplanar w a v eguides. Eac h single-qubit pulse to ok 90 ns to im- plemen t (including the buffer time of 10 ns), with the states ev olving freely b et w een the pulses, and eac h exp erimen t w as p erformed 8192 times. The single qubit gate error and readout errors are on the order of10 −3 . Dephasing times (T 2 ) for the qubits w ere b et w een30100µ s with a set of times giv en in T able 2.6.1 . The results rep orted here w ere from exp erimen ts done 23 Qubit T 1 [µ s] T 2 [µ s] Gate error [10 −3 ] Readout Error [10 −2 ] Gate Fidelit y Readout Fidelit y 0 47.0 29.4 1.95 4.78 0.9980 0.9522 1 35.1 54.7 3.82 5.00 0.9962 0.9500 2 35.7 43.3 3.72 4.45 0.9963 0.9554 3 54.3 80.3 2.25 8.95 0.9977 0.9105 4 39.3 44.7 2.13 7.76 0.9979 0.9224 5 43.3 57.2 1.68 5.83 0.9983 0.9417 6 55.2 91.7 2.37 4.10 0.9976 0.9589 7 28.9 27.8 3.17 4.01 0.9968 0.9599 8 59.5 101.6 1.13 5.86 0.9989 0.9413 9 48.6 82.9 1.10 11.37 0.9989 0.8862 10 27.5 40.9 4.41 11.76 0.9956 0.8824 11 57.3 102 1.81 5.03 0.9982 0.9497 12 47.5 55 1.39 13.24 0.9986 0.8676 13 51.8 97.1 1.63 4.25 0.9984 0.9574 14 40.6 72.3 2.11 6.51 0.9979 0.9349 15 37.3 72.8 3.90 10.53 0.9961 0.8946 Mean 44.3 70.0 2.41 7.09 0.9976 0.9291 SD 7.4 19.1 1.06 3.10 0.0011 0.0310 T able 2.6.1: Ph ysical P arameters-IBMQX5 - A ccessed 06/19/2018. The minim um, a v erage, and maxim um CNOT gate fidelit y are 0.8417, 0.9330, and 0.9513 resp ectiv ely . The gate (readout) fidelit y is 1 min us gate (readout) e rror. 24 Qubit T 1 [µ s] T ∗ 2 [µ s] Gate error [10 −3 ] Readout Error [10 −2 ] Gate Fideli t y Readout Fidelit y 0 15.2 7.2 0.1 0.01 0.9999 0.9501 1 17.6 7.7 0.1 0.01 0.9999 0.9514 2 18.2 10.8 4.3 0.43 0.9957 0.7446 3 31.0 16.8 9.2 0.92 0.9908 0.886 4 23.0 5.2 16.0 1.6 0.984 0.8465 5 22.2 11.1 14.1 1.41 0.9859 0.8443 6 26.8 26.8 25.1 2.51 0.9749 0.8217 7 29.4 13.0 14.1 1.41 0.9859 0.9292 8 24.5 13.8 15.6 1.56 0.9844 0.9526 9 20.8 13.8 42.3 4.23 0.9577 0.887 10 17.1 10.6 15.0 1.5 0.985 0.9482 11 16.9 4.9 27.0 2.7 0.973 0.973 12 8.2 10.9 13.0 1.3 0.987 0.9625 13 18.7 12.7 28.0 2.8 0.972 0.9635 14 13.9 9.4 16.0 1.6 0.984 0.9557 15 20.8 7.3 18.0 1.8 0.982 0.8032 16 16.7 7.5 30.0 3.0 0.97 0.9413 17 24.0 8.4 21.4 2.14 0.9786 0.9598 18 16.9 12.9 33.7 3.37 0.9598 0.9305 19 24.7 9.8 13.7 1.37 0.9863 0.9488 Mean 20.33 11.03 17.83 1.78 0.9822 0.9065 SD 5.50 4.83 10.82 1.08 0.0108 0.0632 T able 2.6.2: Ph ysical parameters for Rigetti A corn - A ccessed 4/13/18. The minim um, a v erage, and maxim um Con trolled-Z gate fidelit y are 0.72, 0.865, and 0.917 resp ectiv ely . The gate (read- out) fidelit y is 1 min us gate (readout) e rror. P arameter IBMQX5 (IBM) 19Q-A corn (Rigetti) Single qubit pulse time (in ns) 90 100 Num b er of qubits 16 19 (15 used) Shots p er exp erimen t 8192 1000 In terface QASM p yQuil T able 2.6.3: Single qubit pulse times, n um b er of qubits, and shots p er exp erimen t for the IB- MQX5 and 19Q-A corn. 25 Figure 2.6.1: Connectivit y b et w een the qubits in IBMQX5. The qubit at the tail (tip) of an ar- ro w is the con trol (target) in a con trolled-U gate. Qubit T 1 [µ s] T 2 [µ s] Gate error [10 −3 ] Readout Error [10 −2 ] Gate Fidelit y Readout Fidelit y 0 50.8 14.7 0.86 4.80 0.9991 0.9520 1 50.0 64.6 1.46 5.30 0.9985 0.9470 2 47.9 45.0 1.29 9.80 0.9987 0.9020 3 37.4 15.1 3.44 5.70 0.9966 0.9430 4 56.0 30.5 0.94 7.00 0.9991 0.9300 Mean 48.4 34.0 1.60 6.52 0.9984 0.9348 SD 6.8 21.2 1.06 2.01 0.0010 0.0201 T able 2.6.4: Ph ysical P arameters for IBMQX4 - A ccessed 06/21/2018. The minim um, a v erage, and maxim um CNOT gate fidelit y are 0.8738, 0.9441, and 0.9774 resp ectiv ely . The gate (read- out) fidelit y is 1 min us gate (readout) e rror. during the p erio d 6/19/18-6/22/18 unless stated otherwise. F or the Bell-state exp erimen ts the follo wing qubits w ere paired together: (0,1),(2,15),(3,14),(5,12),(6,11),(7,10),(8,9) and the exp erimen ts w ere p erformed on 3/14/18. Substan tial daily v ariations are common, but our r esults are qualitativ ely robust to suc h v ariations. IBMQX4 W e also ran exp erimen ts on the5 -qubit IBMQX4 c hip, whose connectivit y graph is sho wn in Fig. 2.6.2 . Eac h single-qubit pulse to ok60 ns to implemen t (including the buffer time of 10 ns), with the states ev olving freely b et w een the pulses, and eac h exp erimen t w as p erformed 8192 times. The single qubit gate error and readout errors are on the order of 10 −3 . Dephasing times (T 2 ) for the qubits w ere b et w een 30100µ s with a set of times giv en in T able 2.6.4 . 26 Figure 2.6.2: Connectivit y b et w een the qubits in IBMQX4. The qubit at the tail (tip) of an ar- ro w is the con trol (target) in a con trolled-U gate. — — — — — — —— —— — — — ——— — —— — — — — — — — — — — — ——— — — ——— — — — — — — —— —— — — — ———— —— — — — — —— — — —— — ———— — — —— — — — — — — — — — — — — — — — — — — — — — —— —— — ————————— ——— — — — — — — — — — — — — — — — — — — — — — ———— — ——————— ————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 100 200 300 400 500 600 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2.6.3: IBMQX4 results for the fidelities under DD compared to free ev olution, after a v er- aging o v er 36 initial conditions (t yp e 2) and all 5 qubits. The results rep orted here w ere from exp erimen ts done during the p erio d 6/19/18-6/21/18. W e tested the p erformance of the XY4 sequence on the IBMQX4. As sho wn in T able 2.6.5 and Fig. 2.6.3 , w e found impro v emen ts that are qualitativ ely consisten t with the IBMQX5 results. W e again see b oth a dominance of Mark o vian exp onen tial deca y along with a nearly 3 -fold increase in λ , from 44.7 2.8 under free ev olution to 128.0 0.8 under DD. The in tersection b et w een free and DD ev olution o ccurs att int =21616 , whic h is t wice the gate depth for whic h DD impro v es the a v erage fidelit y on IBMQX5. 27 Figure 2.6.4: Connectivit y b et w een the qubits within A corn. Qubit 3 is disconnected. Qubits 2,12, 15,18 w ere not used as their p erformance v aried substan tially o v er time. Based on [ 2 ]. Rigetti A corn The 19 -qubit A corn c hip w as accessed via Rigetti’s F orest, using circuits written in Quil [ 35 ] with remote access pro vided through the p yQuil in terface [ 2 ]. A corn comprises a com bination of fixed frequency (qubits 0-4 and 10-14) and tunable (qubits 5-9 and 15-19) transmon qubits, capacitiv e coupled. Within the curren t la y out, qubit 3 w as disabled due to p erformance issues, lea ving 19 qubits functional for programming. The single qubit gate time w as 50 ns (for all qubits other than 2, 18 whic h ha v e gate times 100 ns) with dephasing times (T ∗ 2 ) v arying b et w een1040µ s as summarized in T able 2.6.2 . The connectivit y graph is illustrated in Fig. 2.6.4 . Readout error w as on the lev el of the IBM c hip; while the relaxation times presen ted for the IBMQX5 w ere longer than those for A corn, b oth c hips exhibited relaxation times within the microsecond range with single-qubit fidelities greater than 0.98 . W e accessed the A corn c hip m ultiple times b et w een 3/1/18 and 5/9/18. The results rep orted here are from 4/3/18 unless stated otherwise. F or the Bell-state exp erimen ts, the follo wing qubits w ere paired together: (0,5),(1,6),(4,9),(11,16),(12,17),(14,19) and the exp erimen ts w ere p erfor med on 3/21/18. A corn’s cry ogenic system housing suffered a failure (a brok en scroll pump) and the c hip w as out of service from 4/17/18 on w ards [ 63 ]. During the repair, the c hip w as brough t bac k to ro om temp erature and this thermal cycle 28 — — — — — — — — — — — — — — — — — — — — — — — — — — ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ — — — — — — — — — — — — — — — — — — — — — — — — — — ● ● ● ● ● ● ● ● ● ● ● ● ● — — — — — — — — — — — — — — — — — — — — — — — — — — □ □ □ □ □ □ □ □ □ □ □ □ □ — — — — — — — — — — — — — — — — — — — — — — — — — — ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ — — — — — — — — — — — — — — — — — — — — — — — — — — ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ — — — — — — — — — — — — — — — — — — — — — — — — — — ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ○ ● □ ■ ◇ ◆ 0 50 100 150 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2.6.5: A corn c hip results for the mean fidelit y a v erages o v er 36 initial conditions (t yp e 2) and all 15 activ e qubits for differen t da tes. Mac hine A ccessed Ev olution F 0 10 −2 F Nmax 10 −2 λ α γ A corn 03/11/18 F ree 91.80.3 60.00.6 69.71.3 1 0.160.09 A corn 04/03/18 F ree 90.80.4 59.80.6 68.11.3 1 0.140.11 A corn 05/03/18 F ree 92.70.2 60.90.6 57.21.2 1 0 A corn 03/11/18 DD 91.90.3 78.40.4 71.30.9 1 0.490.03 A corn 04/03/18 DD 90.80.4 77.10.4 74.90.9 1 0.500.03 A corn 05/03/18 DD 92.70.2 68.40.3 72.61.1 1 0.360.02 IBMQX4 06/21/18 F ree 95.70.2 55.10.3 44.72.8 1452 0.370.13 IBMQX4 06/21/18 DD 95.70.2 52.90.3 128.0.8 1 0.050.02 T able 2.6.5: Fit parameters for A corn (considering only the 15 activ e qubits) and IBMQX4 when Eq.( 2.1 ) is used to fit the mean fidelities in Fig. 2.6.5 . and other p ossible con tamination significan tly affected p erformance, making qubits 2,3,15,18 un usable and also affecting m ultiple t w o-qubit gates (qubit 12 w as v ery noisy and w e did not use it). Owing to these factors, w e only used 15 total qubits in our analyses. In Fig. 2.6.5 and T able 2.6.5 w e sho w ho w p erformance w as affected b efore and after this repair. The main effect w as a decrease in the final fidelit y , due to a smaller recurrence, as captured b y the smaller γ v alue at the later date. 29 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0 20 40 60 80 100 120 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0 200 400 600 800 Figure 2.6.6: Bo otstrapping example: The left plot represen ts frequency coun ts (y-axis) v ersus fidelit y (x-axis) for a data set from the IBMQX5, tak en from 36 differen t initial conditions (t yp e 1) and 16 qubits. The righ t plot represen ts the samples after b o otstrapping the original data set. The mean and confidence in terv als w ere then calculated based on the b o otstrapp ed distribution (righ t plot). 2 Statistical metho ds T o sp eed up data collection our DD exp erimen ts w ere p erformed in parallel on all qubits (or qubit pairs for the en tanglemen t exp erimen ts) in the resp ectiv e QCs. The mean v al- ues and error bars rep orted w ere computed after a v eraging o v er all qubits and subsequen t b o otstrapping. The general b o otstrapping tec hnique implemen ted w as based on Ref. [ 64 ]. Bo otstrapping a data set w as p erformed b y taking the mean of N resamples of x p oin ts from the data set (with replacemen t). This generated a second represen tativ e set of data (N large) from whic h the mean, standard deviation, and confidence in terv als w ere then calculated. An example is giv en in Fig. 2.6.6 . F or the b o otstrapp ed data presen ted here, the data w as resampled 5000 times (with eac h sample b eing the same size as the original data) with 576 (36 initial conditions16 qubits) samples d ra wn from the IBMQX5 data. In Fig. 2.4.1 for the in tersection time, t in t , the fidelit y curv es, F(N) , are parametrized as a function of F 0 , F Nmax , λ,α , and γ . Since the errors in λ,α and γ ha v e a Gaussian distribution, w e can generate m ultiple curv es (with differen t v alues of λ,α and γ sampled from the resp ectiv e Gaussians). These newly generated curv es ha v e their o wn in tersection 30 DD τ/90 ns λ α γ t int F ree 1 28.91.2 9105 0.730.12 0 XY4 1 88.40.3 1 0 1084 XY4 2 73.50.7 1 0 957 XY4 3 72.30.5 1 0 995 XY4 4 74.80.4 1 0 1004 XY4 5 67.00.6 1 0 916 XY4 6 61.60.7 1 0 856 (XI) N 2 79.20.7 1 0.220.02 895 (YI) N 2 79.70.7 1 0.210.03 895 (ZI) N 2 63.91.0 1 0.290.03 8710 GA8a 1 78.20.3 1 0 953 GA16a 1 95.50.6 197.40.6 0.190.01 1153 GA32a 1 88.00.7 1 0 1048 T able 2.6.6: P erformance summary of the differen t DD pulses w e implemen ted on the IBMQX5. The XX and Y Y sequences are sp ecific to pure dephasing errors. The (ZI) N sequence is suppresses pure SE errors. All three of these sequences underp erform the XY4 sequences, but p erformance is b etter after suppression of pure dephasing errors. The GA sequences are discussed in App endix 6 . times, t in t . W e then rep ort the mean and 2σ error bars for the in tersection times generated this w a y . 3 DD vs free ev olution correlation plots, as a function of initial state As seen in Fig. 2.2.2 , the ev olution of initial states of the form cos(θ/2)j0i + sin(θ/2)j1i v aries dep ending on θ . Ov erall, w e find t w o qualitativ ely differen t b eha viors dep ending on whether θ 2 [0, π 3 ) or 2 [ π 3 ,π] . W e mak e this explicit b y plotting the fidelit y for eac h state under DD vs under free ev olution. Eac h data p oin t in Fig. 2.6.7 corresp onds to a single initial condition on a single qubit of the resp ectiv e mac hines. Data p oin ts corresp onding to eac h initial condition ha v e b een color-co ded using θ with θ =0 set to blue and θ =π set to 31 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.6.7: Correlation plot of the fidelities under DD compared to free ev olution, for IBMQX5 and A corn. Initial conditions are color co ded as a function of θ . T op panel: θ 2 [0, π 3 ) ; middle panel: θ 2 [ π 3 , 2π 3 ) ; b ott om panel: θ 2 [ 2π 3 ,π] . In (a) the initial state is close to the ground state j0i and DD is w orse than free ev olution. In (b) the initial state is a close to an equal sup erp osi- tion, th us susceptible to dephasing, and DD is o v erall b etter than free ev olution, esp ecially for A corn. In (c) the initial state is close to the excited statej1i and DD is again b etter than free ev olution at in termediate N for IBMQX5, and at all N for A corn. red. Differen t data p oin ts of the same color refer to fidelities for the same initial condition acquired from differen t qubits. P oin ts ab o v e the diagonal indicate an adv an tage for DD o v er free ev olution. 32 F or initial conditions that are closer to the state j0i , w e find that most of the p oin ts remain b elo w the diagonal, indicating that p erforming DD in fact reduces their fidelit y o v er time. F or states that are farther from j0i , most of the p oin ts are ab o v e the diagonal. F or IBMQX5 the impro v emen ts tend to disapp ear as time increases, while they are retained for A corn. Ov erall, free ev olution preserv es states close to the ground state b etter than DD, but b oth sup erp osition states (susceptible to dephasing) and states close the excited state j1i (susceptible to SE) b enefit from DD relativ e to free ev olution. Ov erall, fidelities impro v e more substan tially on A corn under DD than on IBMQX5, consisten t with Fig. 2.4.1 . — — — — — — — — — — — — — — — — — — — — — — — — ◼ ◼ ◼ ◼ ◼ ◼ — — — — — — — — — — — — ▲ ▲ ▲ ▲ ▲ ▲ — — — — — — — — — — — — □ □ □ □ □ □ — — — — — — — — — — — — △ △ △ △ △ △ ◼ ▲ □ △ 0.0 0.2 0.4 0.6 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 Figure 2.6.8: In fidelit y scaling as a function of τ for differen t n um b ers of pulses N . 4 DD tailored for dephasing and sp on taneous emission As w e ha v e noted, b oth dephasing and SE pla y imp ortan t roles in reducing the fidelit y of free ev olution. W e th us tested sequences tailored to eac h of these noise sources. T o suppress pure dephasing, whic h results from a system-bath in teraction term of the form σ z B (whereB is a bath op er ator), it suffices to apply the the (XI) N or (YI) N sequence, 33 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 100 200 300 400 500 600 -1.5 -1.0 -0.5 0.0 0.5 1.0 Figure 2.6.9: The slop e a (10 ) and in tercept blogN deriv ed from plotting log( p 1F) as a function of log(τ) for IBMQX5, at differen t n um b ers of pulses N . The solid blac k line is log(N)+ c 0 and the dotted blac k line is log(N)/2+c 0 , where c 0 is the in tercept at N = 0 . — — — — — — — —— —— — — — — —— ——— —— — ———— ——— — — ——— ——— — — — — — — — —— —— — — — — —— ——— —— — —— —— ——— — ———— ——— — — — — — — — — — — — — — — —— —————— —— —————————————— — — — — — — — — — — — — — — —— —————— —— —————————————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ — — — — — — — — — — — —— — — — ——— ——— — ————— —————————— — — — — — — — — — — — —— — — — ——— ——— — ————— ————— ————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 100 200 300 400 500 600 0.5 0.6 0.7 0.8 0.9 1.0 — — — — — — — —— —— — — — — —— ——— —— — ———— ——— — — ——— ——— — — — — — — — —— —— — — — — —— ——— —— — —— —— ——— — ———— ——— — — — — — — — — — — — — — — —— —————— —— —————————————— — — — — — — — — — — — — — — —— —————— —— —————————————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ — — — — — — — — — — — — —— — ————— ————— ———— ————————— — — — — — — — — — — — — —— — ————— ————— ———— ————————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 100 200 300 400 500 600 0.5 0.6 0.7 0.8 0.9 1.0 — — — — — —— —— —— — — — — —— ——— —— — ———— ——— — ——————— — — — — — —— —— —— — — — — —— ——— —— — —— —— ——— — ———— ——— — — — — — — — — — — — — — — —— —————— ———————————————— — — — — — — — — — — — — — — —— —————— ———————————————— ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 100 200 300 400 500 600 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2.6.10: Fidelit y deca y results for the GA-based DD sequences compared to XY4 and free ev olution, on IBMQX5. Left: GA 8a ; middle: GA 16a ; righ t: GA 32a . A small impro v emen t o v er XY4 i s seen for the GA 16a sequence. since the error σ z an ticomm utes with b oth X and Y . T o suppress SE, whic h results from a system-bath in teraction term of the form σ − B (a consequence of the inevitable coupling of an y quan tum system to the v acuum electromagnetic field [ 65 ]), it suffices to apply the the (ZI) N sequence, since the error σ − =j0ih1j = (σ x +iσ y )/2 an ticomm utes with Z . In b oth cases, this results in an effectiv e ev olution wherein the errors are time-rev ersed under the DD s equence. The results are giv en in T able 2.6.6 . Unsurprisingly , the p erformance of the sp ecialized (XI) N , (YI) N , and (ZI) N sequences is w orse than that of the univ ersal XY4 sequence. Ho w ev er, they accoun t for a substan tial increase in the initial deca y time λ , with the λ v alue for the (XI) N , (YI) N sequences accoun ting for nearly 90% of the λ v alue of the XY4 sequence. T able 2.6.6 sho ws that p erformance is b etter after suppression of pure dephasing 34 errors, whic h indicates that these errors dominate o v er SE errors. 5 Infidelit y as a function of pulse spacing F or ideal DD pulses the distance b et w een the free and dynamically decoupled states is b ounded as [ 66 , 67 ]: D[ρ S (T),ρ 0 S (T)] 1 2 e 2cT 2 /N 1 2c T 2 N , (2.2) whereD(ρ,σ)= 1 2 jjρσjj 1 is the trace-norm distance b et w een quan tum statesρ andσ ,T is the total ev olution time,ρ S (T) (ρ 0 S (T) ) is the state of the system under DD (free ev olution), N is the total n um b er of (non-iden tit y) DD pulses, and c is a constan t directly related to the op erator norms of the bath and system-bath Hamiltonian. Since T =τN , w e ha v e: 1 q F[ρ S (T),ρ 0 S (T)]D[ρ S (T),ρ 0 S (T)]2cτ 2 N, (2.3) where F is the Uhlmann fidelit y [ 38 ]: F(ρ 1 ,ρ 2 ) = (k p ρ 1 p ρ 2 k 1 ) 2 , with ρ 1 (T) and ρ 2 (T) b eing the state of the system under DD and free ev olution, after total ev olution timeT =Nτ . Figure 2.6.8 sho ws the scaling of the infidelit y . An extraction of the slop es and offsets of the straigh t line fits sho wn (with additional results for other τ andN v alues not sho wn here) leads to Fig. 2.4.1 . Next, w e rewrite Eq. ( 2.3 ) as log(1 p F)alog(τ)+blog(N)+c ′ . (2.4) where for ideal pulses a = 2 and b = 1 . The b ound ( 2.4 ) is c hec k ed in Fig. 2.6.9 . The slop ea , capturing the pulse in terv al dep endence, is significan tly smaller than the theoretical upp er b ound of 2 giv en in Eq. ( 2.4 ), but consisten t with it. The offset blogN , capturing the pulse n um b er dep endence, is also consisten t with the theoretical upp er b ound with b = 1 35 (in fact, it more closely matc hes b = 1/2 ). It is not surprising that the b ound is not tigh t in the realistic case of finite-width pulses, but it is in teresting that the slop e a b ecom es negativ e for sufficien tly large N (≳ 200 ). The in terpretation of the b ound ( 2.4 ) for a> 0 is that increasing N or τ while k eeping the other v ariable fixed increases the infidelit y b ound, i.e., is exp ected to reduce DD’s p erformance. Ho w ev er, a negativ e slop e in fact implies the opp osite: a decreasing infidelit y with increasing pulse in terv al τ , at fixed N . This to o is a nonstandard finding and presen ts another in teresting op en theoretical problem. 6 Higher order sequences based on genetic algorithms The results for all GA-based sequences are sho wn in Fig. 2.6.10 , along with XY4 and free ev olution. Fit results are rep orted in T able 2.6.6 . W e find that GA 32a p erformance is similar to GA 4 , while GA 8a do es sligh tly w orse. Ho w ev er, GA 16a is the b est sequence w e ha v e found so far, sligh tly outp erforming XY4. W e direct in terested readers to Ref. [ 31 ], where the single qubit fidelit y impro v emen t due to R GA (an extension of GA) and v arious other DD sequence families is compared in detail. 36 Chapter Three: Better-than-classical Gro v er searc h T ext for this c hapter is adapted from [ 73 ]. The b est p ossible classical strategy for finding a particular “mark ed” elemen t in an un- sorted list of length N requires querying half of the elemen ts in the list on a v erage; a quan tum computer (QC) can do this in quadratically few er queries using Gro v er’s searc h al- gorithm [ 74 ]. This algorithm is optimal and pro v ably b etter than all classical strategies [ 75 ]. As one of the first algorithms with a pro v able quan tum sp eedup, Gro v er searc h is often used as a subroutine for other quan tum algorithms [ 76 , 77 ]. Ov er the last t w o decades, Gro v er searc h has b een implemen ted on v arious quan tum computing platforms [ 78 – 81 ], alb eit for relativ ely small N . Enco ding a list of length N requires n =dlog 2 (N)e qubits. The list can b e queried clas- sically or using quan tum queries; in b oth cases, one finds the mark ed elemen t with some probabilit y , whic h w e refer to as the classical or quan tum success probabilit y . The largest implemen tation of Gro v er’s algorithm to date is forn=8 qubits, but without demonstrating a b etter-than-classical quan tum success probabilit y [ 78 ]. Suc h b etter-than-classical p erfor- mance has b een ac hiev ed for n = 3 [ 79 , 80 ] and n = 4 [ 81 ] qubits. Here, emplo ying t w o sev en-qubit IBM Quan tum Exp erience (IBMQE) transmon qubit platforms ibm_nairobi (Nairobi) and ibmq_jakarta (Jakarta), w e demonstrate higher success probabilities than all previous implemen tations, for n5 . Key to our demonstrations is the use of error suppression and mitigation strategies. In particular, w e use the [[4,2,2]] quan tum error-detecting co de [ 82 , 83 ], whic h enco des k = 2 logical qubits in to n = 4 ph ysical qubits and detects arbitrary single-qubit errors, to demonstrate a significan t success probabilit y enhancemen t relativ e to using t w o copies of n = 2 ph ysical qubits. These success probabilities are further impro v ed b y com bining error detection with measuremen t error mitigation [ 13 , 84 ]. W e use the quan tum error detection results to p erform what w e call algorithmic error tomograph y: for eac h algorithm execution w e compute the probabilit y of an output X , Y , or Z error (corresp onding to the three P auli matrices) on one of the four ph ysical qubits, or a logical error. This allo ws us to compute a detailed map of the errors that arise after executing the en tire algorithm. In this sense, algorithmic error tomograph y pro vides a holistic and complemen tary p ersp ectiv e to tec hniques suc h as gate se t tomograph y [ 85 , 86 ], whic h instead fo cus on individual gates. W e demonstrate b etter-than-classical p erformance for three or more ph ysical qubits b y em- plo ying error suppression via dynamical decoupling (DD) [ 6 – 8 , 16 ]. T o w ard this end, w e con- sider three robust DD families: univ ersally robust (UR) [ 21 ], concatenated DD (CDD) [ 87 ], and robust genetic algorithm (R GA) [ 3 ] sequences. W e find that robust sequences with few pulses are vital in ac hieving b etter-than-classical algorithmic p erformance. W e compare the exp erimen tally obtained results for Gro v er’s algorithm with an error mo del based on the concatenation of amplitude damping, phase damping , and dep olarization maps. Eac h map is parameterized b y the calibration metrics pro vided b y the IBM Quan tum Exp erience (IBMQE) bac k end [ 1 ]. W e test this mo del using the observ ed success probabilities and the algorithmic error tomograph y results; the latter pro vides a m uc h more stringen t test. W e find go o d agreemen t with the mo del, but only after using DD. W e in terpret this in terms of the suppression of crosstalk b y DD [ 88 , 89 ], whic h is unaccoun ted for b y the error mo del. In summary , w e demonstrate a b etter-than-classical Gro v er searc h for up to 5 qubits, enabled b y quan tum error detection and dynamical decoupling. That is, w e demonstrate algorithmic p erformance that is enhanced b ey ond the break-ev en p oin t – where protected op erations outp erform their unprotected coun terparts – and the capabilities of the b est p ossible classical algorithm executing the same task. Along the w a y , w e in tro duce algorithmic 38 Figure 3.1.1: Circuit description for Gro v er’s algorithm. The relativ e amplitudes of all the states at eac h stage of the algorithm are sho wn. Starting with an equal sup erp osition state, the oracle assigns a relativ e phase difference of π to the mark ed state. The amplitude amplification st ep then p erforms an in v ersion ab out the mean, allo wingjmi to ha v e a larger probabilit y amplitude than all other states. This round of querying and amplifying is rep eated q times. The optimal n u m b e r of rounds for the n -qubit Gro v er problem is q opt = b π 4 2 n/2 c . T he only m ulti-qubit op- eration required to implemen t b oth the oracle and the amplitude amplification step is C n−1 Z (v ertical line in the Oracle and Amplitude Amplification b o xes). error tomograph y – a c haracterization of errors afflicting an en tire quan tum algorithm based on the syndromes of a quan tum error detecting co de. The structure of this pap er is as follo ws. In Section 3.1 , w e summarize Gro v er’s algo- rithm’s salien t asp ects and discuss its implemen tation. In Section 3.2 , w e describ e the op en system mo del w e use to compute the theoretically exp ected algorithmic p erformance. De- tails ab o ut our dynamical decoupling implemen tation are in Section 3.3 . Section 3.4 fo cuses on the p erformance of Gro v er’s algorithm onn=2 qubits with and without error detection. Algorithmic error tomograph y is in tro duced in Section 3.4 as w ell. The results for2<n5 , where DD pla ys a crucial role in ac hieving b etter-than-classical p erformance, are giv en in Section 3.5 . W e conclude with observ ations and the implications of our results in Section 3.6 . 3.1 Gro v er’s Algorithm: bac kground and implemen tation 1 Problem Description Informally , the Gro v er problem is to searc h an unsorted list with N = 2 n elemen ts for a mark ed elemen t. F ormally , the goal is to find the mark ed n -bit bitstringm using the smallest 39 n um b er of queries of an oracle that implemen ts a function f m :f0,1g n 7!f0,1g defined as f m (x)=δ x,m . Classically , afterq queries, the probabilit y of correctly iden tifying the mark ed elemen t, whic h hereafter w e refer to as the success probabilit y , is p C s (q,N)=(q+1)/N (see Section 3.7 ). Consequen tly , the classical algorithm requires O(N) queries. Gro v er’s algorithm pro vides a quadratic quan tum sp eedup, requiring onlyO( p N) queries [ 74 ]. This scaling remains v alid with more than one mark ed elemen t [ 90 ], or ev en for an arbitrary initial amplitude distribution o v er the list elemen ts [ 91 ]. In the original setting of a single mark ed elemen t, the state after q queries to the oracle is jψ q i=sin[(2q+1)θ]jmi+cos[(2q+1)θ]jm ⊥ i, (3.1) wherejm ⊥ i= 1 √ N−1 P x̸=m jxi andθ =arcsin 1 √ N . Th us, the quan tum success probabilit y is p Q s (q,N) = sin 2 [(2q+1)θ] , and the theoretically optimal n um b er of queries is q opt = b π 4 p Nc . Note that p C s (q,N) max m,b p s (m,b, Unenc) . Error bars corresp ond to 95% confidence in terv als. higher than predicted; see the leftmost column of Fig. 3.4.4 . F ortunately , in the enco ded case, Jakarta’s failure probabilit y o v erlaps with the prediction bands (Fig. 3.4.4 , third column 52 from the left). Ho w ev er, a closer lo ok at the detected errors via AET rev eals a differen t discrepancy . The sim ulated results for Jakarta (b ottom-left table in Fig. 3.4.2 ) do not matc h the empirical error profile (top-left table of Fig. 3.4.2 ), whic h has significan tly stronger Z errors and also a state-dep enden t asymmetry in these errors. In other w ords, Jakarta do es not matc h the sim ulations for unenco ded or enco ded circuits without DD. In con trast, for Nairobi, Fig. 3.4.3 sho ws that the AET sim ulation results agree with the empirically observ ed ones. This also holds for the sim ulated failure probabilities (Fig. 3.4.4 ). T o in v estigate Jakarta’s observ ed discrepancy , w e first attempt to systematically amplify p D ,T 1 , and T 2 b y m ultiplying eac h quan tit y b y a phenomenologically determined v ariable λ i (see Section 2 ). This leads to a b etter o v erlap b et w een predicted and observ ed success probabilities but do es not repro duce the AET asymmetry seen in Fig. 3.4.2 . This sho ws the limitations of the phenomenological mo del of Section 3.2 and highligh ts the lev el of detail pro vided b y AET. Ho w ev er, the Jakarta discrepancy is effectiv ely remo v ed after the application of DD. Our t w o-qubit Gro v er implemen tation uses four qubits, lea ving three inactiv e qubits in the 7-qubit QPUs used in our exp erimen ts. As there are no idle in terv als in the unenco ded t w o-qubit Gro v er circuit, w e applied the XY4 sequence on the inactiv e qubits – q 2 ,q 4 , and q 6 (see Section 3.7 ). W e applied the XY4 sequence to b oth the activ e and inactiv e qubits for the enco ded case. Due to the relativ e sparsit y of idle in terv als in the t w o-qubit Gro v er circuits, w e did not attempt to implemen t robust sequences, whic h require more pulses than XY4. Fig. 3.4.4 sho ws ho w the failure probabilit y and the rates of v arious detected errors on Jakarta are affected b y the presence of DD. F or the unenco ded case (the first t w o columns from left of Fig. 3.4.4 ), DD impro v es the p erformance sligh tly , and the discrepancy b et w een the predicted and observ ed failure probabilities is remo v ed. The impro v emen t b y DD in the unenco ded t w o-qubit Gro v er case is in concurrence with Refs. [ 88 , 89 ], whic h sho w ed the efficacy of the XY4 sequence in suppressing static ZZ crosstalk in sup erconducting qubits. In other w ords, these results confirm that ZZ crosstalk – whic h is w ell-do cumen ted for 53 sup erconducting QCs [ 101 ] – lik ely con tributes to the observ ed p erformance b eing sligh tly w orse than exp ected from the mo del. A dding the XY4 sequence remo v es most of the empirical-theoretical discrepancies in b oth the magnitude and the asymmetry of the errors exhibited b y the AET profiles, as seen b y comparing the top and b ottom righ t of Fig. 3.4.2 . With DD, the enco ded circuits ha v e a w eak er state-wise asymmetry inZ -errors than seen in the left column of Fig. 3.4.2 . Moreo v er, the DD-protected circuits more closely repro duce the distribution of detected errors predicted b y the mo del of Section 3.2 than the same circuits without DD. This observ ation – that the agreemen t b et w een our theoretical mo del and the exp erimen tal results impro v es under DD – is further v alidated b elo w. The close agreemen t w e found for Nairobi b et w een our (crosstalk-free) mo del and the exp erimen tal results without DD or MEM (Fig. 3.4.3 and the first and third from left columns of Fig. 3.4.4 ) suggests that crosstalk do es not pla y a significan t role in this QPU. Fig. 3.4.4 do es exhibit a significan t discrepancy b et w een the mo del and the exp erimen tal Nairobi results when MEM is included (second and last columns of Fig. 3.4.4 ). As w e sho w in Section 3.7 , this discrepancy arises from the c hoice to mitigate readout errors using iterativ e Ba y esian unfolding (IBU) [ 13 ]. Finally , Fig. 3.4.5 complemen ts the first and last columns of Fig. 3.4.4 , as w ell as the AET results, and sho ws the output distributions for Jakarta and Nairobi for the t w o-qubit Gro v er case, with and without enco ding and MEM. The main observ ation is that for the unenco ded case, the maxim um success probabilit y is obtained for the mark ed state j00i , whic h is also the QPU’s ground state; this is unsurprising giv en the dominance of amplitude damping errors. With enco ding plus error mitigation, the o v erall p erformance increases and b ecomes indep enden t of the mark ed state. 54 4 Success probabilit y: b ey ond break-ev en impro v emen t W e no w fo cus on the effect of error detection on t w o-qubit Gro v er p erformance as seen in Fig. 3.4.4 . Due to the shallo w circuit depth, ev en without an y error detection, p e s (1,4) 93.0% - already m uc h higher than the classical success probabilit y p C s (1,4) = 1 2 . A dding error detection impro v es the success probabilit y to 96.0% . The effect of MEM is similar to that of error detection: the success probabilit y increases to 97.0% . Com bining error detection with MEM results in additional impro v emen t: w e obtain success probabilities of 98.5% on Nairobi and 99.5% on Jakarta. Due to error detection and MEM, Jakarta’s success probabilities increase b y an order of magnitude. This impro v emen t o v er the unenco ded case is non-trivial, considering that the [[4,2,2]] co de can only detect w eigh t-1 errors, and the enco ded circuit requires six t w o-qubit gates. In con trast, the unenco ded v ersion requires only t w o. The relativ ely high success probabilities w e observ e in the enco ded case suggest that most errors, ev en those due to the t w o-qubit gates, manifest as w eigh t-1 errors. This sho ws, alb eit for a relativ ely small problem size, that error detection can more than offset the extra errors in tro duced due to increased circuit depth and complexit y . W e ha v e demonstrated an algorithmic b ey ond break-ev en impro v emen t using error detec- tion in the sense that the protected algorithm clearly outp erforms its unprotected coun ter- part. Previous break-ev en impro v emen ts w ere at the individual gate lev el [ 102 , 103 ]. Here w e ha v e demonstrated suc h an impro v emen t at the lev el of the execution of an en tire algorithm, alb eit of a fixed size. The holy grail is to demonstrate the implemen tation of an algorithm for a family of problem sizes at the logical lev el with higher fidelit y than the same algorithm executed at the ph ysical lev el. A c hieving this in our setting w ould require increasing the problem and co de sizes. The family of [[2k + 2,k,2]] subsystem quan tum error detecting co des is an attractiv e option in this regard since all their logical op erators can b e c hosen to b e 2-lo cal [ 104 ], whic h simplifies the circuit design. An exp erimen tal implemen tation of suc h 55 0.00 0.05 0.10 0.15 0.20 Success Probability Free CPMG RGA2x UR24 UR18 RGA32 CDD2 RGA32a RGA16b UR32 UR6 RGA4 XY4 UR12 RGA8a RGA8c Random Classical Figure 3.4.6: P erformance of DD sequences. A v erage success probabilit y for 5-qubit Gro v er with t w o oracle queries on Nairobi. The DD sequences are rank ed in order of decreasing success prob- abilit y . The t w o dotted lines represen t success probabilities corresp onding to a random and clas- sical strategy , resp ectiv ely . R GA8a and R GA8c are tied as the b est-p erforming sequences. F ree denotes the result of an unprotected implemen tation. Error bars corresp ond to 99% confidence in terv als. larger co des and problem sizes remains a co v eted goal. 56 0.00 0.05 0.10 0.15 0.20 Success Probability DD + MEM DD Free + MEM Free Classical 0 8 16 24 Detected State 0 8 16 24 Marked State Free 0 8 16 24 Detected State 0 8 16 24 Marked State DD + MEM 0.00 0.05 0.10 0.15 0.20 Success Probability Figure 3.4.7: 5-qubit Gro v er results on Nairobi. Left: a v erage success probabilit y with and with- out DD or MEM for 5-qubit Gro v er implemen ted on Nairobi. The b o xes corresp ond to the the- oretically exp ected success probabilities. The quan tum oracle is queried t wice; in the ideal case, the success probabilit y is 0.602. The unprotected (F ree) ev olution is on par with a random guess, significan tly w orse than the optimal classical strategy (dashed v ertical line), and just adding MEM do es not c hange the result. In con trast, the DD-assisted implemen tation crosses the classi- cal threshold, and the results impro v e ev en more with MEM, up to a success probabilit y of 0.15 . Error bars corresp ond to 99% confidence in terv als. Middle and righ t: the complete input-output maps for all 2 5 mark ed states, without and with DD + MEM, are sho wn. States are sorted b y increasing Hamming w eigh t; in the F ree case, lo w Hamming w eigh t states ha v e a higher success probabilit y (more green on the left). This is lik ely to b e a consequence of amplitude damping (sp on taneous emission), whic h fa v ors thej0i state of eac h qubit. In the unprotected case (F ree, middle), there is no discernible correlation b et w een the input mark ed state and the output de- tected state. In the protected case (DD + MEM righ t), blac k-to-purple signifies b etter-than- classical success probabilit y , and this threshold is crossed for all 32 mark ed states. The DD se- quence used here is R GA8a [ 3 ], whic h w as the top-p erforming sequence in our DD sur v ey (see Fig. 3.4.6 ). 3.5 3-qubit to 5-qubit Gro v er protected b y dynamical decoupling Crossing the classical threshold in Gro v er’s searc h for an increasingly larger n um b er of qubits is a meaningful goal, not only b ecause the quadratic sp eedup offered b y Gro v er’s algorithm leads to a more dramatic impro v emen t as the problem size increases but also b ecause it b ecomes more c hallenging to realize the sp eedup exp erimen tally as the con trolled phase gate C n−1 Z is ann -qubit en tangling op eration. In the implemen tation of Ref. [ 78 ], 5-qubit Gro v er required nearly a thousand t w o-qubit gates, and for 8-qubit Gro v er, nearly 15000 gates w ere used. Notably , this exp onen tial increase in the n um b er of t w o-qubit gates with problem size is b ecause Ref. [ 78 ] did not use ancilla qubits to mak e the circuits shallo w er. It is p ossible to implemen t C n−1 Z with circuits where t w o-qubit gates scale linearly withn (see Section 3.7 ). 57 3 4 5 Problem Size 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Success Probability Random Query=1 Query=2 Query=3 Jakarta Theory Nairobi Theory Jakarta Free Jakarta DD Nairobi Free Nairobi DD Figure 3.4.8: Success probabilities v ersus problem size. Nairobi (green) and Jakarta (orange) suc- cess probabilities for n 2f3,4,5g are sho wn for DD-protected and unprotected implemen tations. The translucen t bands indicate the theoretically estimated success probabilities using the mo del describ ed in Section 3.2 . W e p erformed q = 2 queries to the quan tum oracle in all cases. The ideal success probabilities are 0.945 , 0.908 , and 0.602 for n = 3 , 4 , and 5 , resp ectiv ely . The white lines c orresp ond to the success probabilities for the classical strategy and random sampling from the u nsorted list (q = 0 ). Error bars corresp ond to 9 9% confidence in terv als. 58 Ref. [ 81 ] emplo y ed shallo w er circuits for C n−1 Z and solv ed 5-qubit Gro v er with sligh tly b etter-than-random success probabilities but without b etter-than-classical p erformance. W e use an efficien t, ancilla-assisted implemen tation of generalized T offoli-t yp e gates [ 81 , 105 ] to implemen t C n−1 Z (see Section 3.7 ). Our implemen tation, whic h builds up on the circuits from Ref. [ 81 ], uses 8 , 14 , and 22 CNOT s for a single C n−1 Z gate for n = 3 , 4 , and 5 , resp ectiv ely . The deep est circuit w e implemen t is t w o oracle queries for 5-qubit Gro v er, totaling 88 CNOT s. Despite b eing far shallo w er than Ref. [ 78 ]’s implemen tation, this is still a relativ ely deep circuit; e.g., the quan tum supremacy demonstration of Ref. [ 106 ] and the algorithmic quan tum sp eedup demonstration of Ref. [ 97 ] in v olv ed circuits of depth up to 40 and 44 , resp ectiv ely . As w e detail in this section, o wing to error suppression via DD, w e crossed the classical probabilit y threshold for all problem sizes, including for 5-qubit Gro v er. 1 Implemen tation with dynamical decoupling In con trast to t w o-qubit Gro v er, where w e restricted DD implemen tation to XY4, n 3 has ample idle in terv als. Therefore w e can implemen t robust sequences (R GA, CDD, UR) requiring more than four pulses. Eac h of these families has m ultiple mem b ers that are parameterized b y the n um b er of pulses in the sequence. W e restrict our implemen tation to DD sequences with few er than 32 pulses and, for eac h circuit, only consider sequences that w e can fit in the idle in terv als a v ailable in the quan tum circuit. A t eac h problem size n , there are 2 n p ossible oracles, eac h corresp onding to one mark ed statejbi , whereb2f0,1g n . W e pro ceed as follo ws to a v oid implemen ting this exp onen tially large set of oracles. Giv en 0k n , there are n k distinct bitstrings that are iden tical to 0 k 1 n−k up to qubit p erm utation. Recall that mark ed states differ only b y whether X or I gates surround the C n−1 Z gate. Th us, w e only consider then+1 oracles with mark ed states j0 k 1 n−k i , k2f0,...,ng . W e then estimate the a v erage success probabilit y b y computing hp(n)i= 1 2 n n X k=0 n k p(j0 k 1 n−k i). (3.12) 59 W e use hp(n)i as the metric for selecting the optimal DD sequences among those w e tested and to iden tify the exp erimen tally optimal n um b er of queries q e opt . Once the optimal DD sequence and q e opt are iden tified for eac h n , w e run the unprotected and the DD-protected Gro v er’s algorithm again at q e opt , but this time for all 2 n oracles. 2 Results 3 Optimal DD sequence and n um b er of queries Our first goal is to iden tify the b est DD sequence and q opt . The determinations made in this step inform our c hoices for the next step. F or conciseness, in this section, w e fo cus on the results of the largest problem size w e implemen ted, i.e.,n=5 . Section 3.7 sho ws the results for 3 n 5 on b oth Nairobi and Jakarta. The p erformance of v arious DD sequences for Nairobi for 5-qubit Gro v er are compared in Fig. 3.4.6 b y computinghp(5)i . The unprotected ev olution (F ree) is marginally b etter than c ho osing an elemen t randomly and do es not cross the classical threshold. DD protection is necessary to cross this threshold, but the t w o- pulse sequences R GA2x and CPMG still result in w orse-than-classical p erformance. The R GA and UR sequences p erform w ell, particularly those with few er than 12 pulses. R GA8a and R GA8c are tied as the b est-p erforming sequences; w e c ho ose R GA8a for the next step, where w e implemen t all of the 2 5 oracles. The p erformance impro v emen t seen due to robust sequences is consisten t across problem sizes and devices, as detailed in the Section 3.7 . W e also use hp(n)i to iden tify the exp erimen tally optimal n um b er of oracle queries q e opt . The theoretically optimal n um b er of rep etitions for n = 3,4,5 is q opt = 2,3,4 , resp ectiv ely . Ho w ev er, Section 3.7 sho ws that in realit y , the theoretically exp ectedq opt often leads to w orse p erformance than q e opt . F or the DD-protected implemen tation, q = 2 maximizes the success probabilit y p s in all cases other than 5-qubit Gro v er on Jakarta, where the p erformance at q =2 is comparable toq =1 . As DD protection is necessary to cross the classical threshold, for simplicit y of analysis and to maximize p s w e set q =2 from here on. 60 4 Better-than-classical p erformance Fig. 3.4.7 sho ws our results for the 5-qubit Gro v er problem on Nairobi with and without DD. Ev en in our relativ ely shallo w-depth implemen tation, b efore error suppression via DD (whether F ree or F ree + MEM), the final results are indistinguishable from randomly guessing the mark ed state. The results c hange significan tly when w e implemen t DD. With DD, the classical threshold is crossed b y all mark ed states. A dding MEM impro v es the results sligh tly , but only when accompanied b y DD. This dramatic impro v emen t due to DD holds for other problem sizes as w ell. Fig. 3.4.8 sho ws the success probabilities after t w o oracle queries on b oth devices for 3 n 5 (see Section 2 for the role of p ostselection in these results). A t the t w o smaller problem sizes (n = 3,4 ), the unprotected implemen tations are b etter than random sampling, but the success probabilit y is relativ ely lo w. F or n = 4 , the unprotected quan tum Gro v er circuit do es not exceed the classical single-query threshold. It is effectiv ely on par with random sampling for n = 5 . In con trast, for all problem sizes, the DD-protected quan tum strategy at q = 2 outp erforms the classical strategy for q 3 . DD-protected Gro v er p erformance at n = 3,4,5 is equiv alen t to classical q = 4,5,3 for Jakarta and q = 4,5,4 for Nairobi, resp ec tiv ely . Th us, DD is essen tial in attaining a b etter-than-classical p erformance. The translucen t bands in Fig. 3.4.8 and the b o xes in Fig. 3.4.7 (left) sho w the theoretically exp ected results computed from the op en system mo del with the IBMQE-supplied param- eters. The success probabilit y in this unprotected Gro v er case (dashed lines in Fig. 3.4.8 ) is considerably lo w er than the theoretical exp ectation. This discrepancy is lik ely due to crosstalk and non-Mark o vianit y , whic h are w ell-do cumen ted for IBMQE’s sup erconducting qubit-based QPUs. Once w e use DD, the observ ed fidelities impro v e and are close to the theoretical predictions. This impro v emen t is exp ected giv en DD’s abilit y to reduce the effect of crosstalk [ 88 , 89 ] a nd non-Mark o vian effects. With DD, the algorithmic p erformance approac hes the exp ectations based on our error mo del. Ho w ev er, w e emphasize that this mo del do es not predict the QPU’s p erformance 61 under DD; it simply tells us what the p erformance w ould b e if the rep orted calibration met- rics corresp onded to observ ed dynamics. The o v erlap b et w een the theoretically predicted (translucen t) and the DD-protected (solid) p erformance implies that DD successfully miti- gates the errors that our simple mo del do es not accoun t for. Ho w ev er, the mo del do es not pro vide an upp er b ound on the p ossible p erformance impro v emen t due to error suppres- sion. F or instance, b etter-optimized sequences could suppress idle-time errors further, and dynamically corrected gates can suppress errors during op erations [ 87 , 107 ]. W e note that the restriction to n 5 arose not b ecause of circuit width but depth. In particular, w e used t w o oracle queries, though theoretically q opt = 4 at n = 5 . The gap b et w een the theoretically and exp erimen tally optimal n um b er of queries is exp ected to gro w with problem size. As is true for an y quan tum algorithm, optimizing circuit compilation and increasing metrics suc h as T 1 , T 2 , and gate fidelities are all vital for scalabilit y . 3.6 Discussion and Conclusion W e implemen ted Gro v er’s algorithm of v arious sizes on m ultiple sup erconducting qubit de- vices. T o our kno wledge, this is the largest successful demonstration of Gro v er’s algorithm for whic h the quan tum strategy outp erforms its classical coun terpart. F or t w o-qubit Gro v er, w e fo cused on error detection via the [[4,2,2]] co de and sho w ed that it allo w ed us to ac hiev e near-optimal p erformance. Along the w a y , w e in tro duced the metho d of algorithmic error tomograph y . W e sho w ed that it pro vides a w ealth of information complemen tary to previous proto cols, suc h as gate set tomograph y or just measuring the success probabilit y of an algo- rithm. W e sho w ed that error suppression via DD is essen tial in attaining b etter-than-classical p e rformance for larger problem sizes. Gro v er’s algorithm is a demanding algorithm [ 78 ] as it requires m ultiple implemen tations of C n−1 Z – a fully en tangling op eration. The sup erconducting trimon device [ 79 ], whic h prior to our results ac hiev ed the highest success probabilit y for 3-qubit Gro v er, is an example of algorithm-tailored hardw are where C 2 Z is a nativ e gate. Constructing hardw are that can 62 nativ ely p erform suc h en tangling op erations ma y b e one path to realizing the full p oten tial of Gro v er’s algorithm. Still, it is desirable to ac hiev e this goal with more general-purp ose quan tum hardw are, as w e ha v e striv ed to do here. T o da y’s quan tum exp erimen talists ha v e v arious error mitigation to ols at their disp osal. Measuremen t error mitigation [ 12 , 13 ], dynamical decoupling, zero noise extrap olation [ 84 ], and quan tum error detection [ 14 ] are parallel strategies that address differen t kinds of errors. Whether and whic h error mitigation metho d to emplo y m ust b e decided based on the problem and a v ailable resources. In this w ork, w e c om bined MEM with DD and quan tum error detection. As exp ected, these strategies complemen t eac h other. Ho w ev er, w e found that often, MEM only b ecame useful after DD w as emplo y ed. Dynamical decoupling, whic h arguably has the lo w est resource o v erhead and requires no p ostpro cessing, w as the single most effectiv e strategy in impro ving the p erformance of our implemen tation of Gro v er’s algorithm. Our w ork adds to the gro wing literature [ 4 , 31 , 40 , 88 , 96 , 97 ] on the effectiv eness of error suppression through DD. While w e demonstrated a crossing of the classical threshold at ev ery problem size w e tested, b etter-than-classical success probabilities are not enough to claim a pro v able quan tum sp eedup [ 39 ]. Suc h a claim w ould require computing the scaling of the time-to-solution metric as a function of problem size and extending it to the largest p ossible problem size that can b e em b edded on the device. Here w e could not go to the largest p ossible problem size as ev en atn=5 , our circuit is quite deep – t w o queries required 88 t w o-qubit gates, and for a larger n um b er of queries or qubits, w e no longer observ ed a quan tum adv an tage. A c hieving quan tum sp eedup for Gro v er searc h will require devices that can implemen t circuits m uc h deep er than those used here without a catastrophic drop in fidelit y . Recen t results [ 108 , 109 ] suggest that without significan t impro v emen ts in the surface co de implemen tation, the latter will not necessarily pro vide an adv an tage o v er the t yp e of error suppression and mitigation metho ds w e ha v e explored here. Th us, our results are lik ely to b e necessary (but not sufficien t) stepping stones to w ard a quan tum sp eedup for Gro v er’s algorithm. 63 3.7 Supplemen tary Information Classical success probabilit y Let p C s (q,N) b e the classical success probabilit y after q oracle queries for an unsorted list with N elemen ts. If the oracle is nev er consulted, then w e are simply pic king an elemen t at random, and the probabilit y of finding the mark ed elemen t is p C s (0,N) = 1/N . A t the other extreme, afterN1 queries with negativ e replies from the oracle, w e are guaran teed to iden tify the mark ed elemen t b y selecting the last remaining elemen t so thatp C s (N1,N)=1 . Clearly , the probabilit y gro ws in prop ortion to q , so w e ma y conclude that p C s (q,N)= q+1 N . A complete argumen t is the follo wing. Supp ose there are N elemen ts. With zero queries, w e pic k the mark ed elemen t with probabilit y p C s (0,N) = 1/N and stop. With one query , either w e already pic k ed the correct elemen t with probabilit y p C s (0,N) = 1/N and are so informed b y the oracle, or w e are told this w as the wrong elemen t, so pic k again from the remaining set ofN1 elemen ts. The probabilit y that the mark ed elemen t w as in the set of N 1 is (N 1)/N , and the probabilit y of no w pic king the correct elemen t is 1/(N 1) : p C s (1,N)=p C s (0,N)+(N1)/N1/(N1)=2/N . With another query , w e’re no w told whether our second pic k w as correct or wrong; if the latter, w e pic k again from the remaining set of N 2 elemen ts. The probabilit y that the mark ed elemen t w as in the set of N 2 is (N2)/N , and the probabilit y of no w pic king the correct elemen t is 1/(N2) : p C s (2,N)= p C s (1,N)+(N2)/N1/(N2)=3/N . Eac h time w e increase the probabilit y of success b y 1/N . Th us, con tin uing, w e ha v ep C s (q,N)=p C s (q1,N)+(Nq)/N1/(Nq)=(q+1)/N . Surv ey of dynamical decoupling sequences In addition to the w ell-kno wn XY4 and CPMG sequences, w e consider three families of robust dynamical decoupling sequences - R GA, UR and CDD. Our results from testing these three robust families of DD sequences are sho wn in Fig. 3.7.1 . F or n > 3 , almost all DD 64 0.0 0.2 0.4 0.6 Success Probability Free RGA8c RGA8a CPMG RGA2x XY4 RGA4 UR6 Random Classical 0.0 0.1 0.2 0.3 0.4 0.5 Success Probability RGA32a UR32 Free RGA32 UR12 UR24 CDD2 RGA16b UR18 RGA2x CPMG RGA4 XY4 UR6 RGA8c RGA8a Random Classical 0.00 0.05 0.10 0.15 0.20 Success Probability Free UR32 RGA32a UR24 RGA32 CPMG RGA2x UR18 CDD2 RGA16b UR6 UR12 XY4 RGA4 RGA8c RGA8a Random Classical 0.0 0.2 0.4 0.6 Success Probability UR6 XY4 RGA4 CPMG RGA2x Free RGA8a RGA8c Random Classical 0.0 0.1 0.2 0.3 0.4 0.5 Success Probability RGA32 Free CDD2 RGA16b UR12 RGA32a UR24 UR32 UR18 RGA8a RGA8c XY4 RGA4 UR6 RGA2x CPMG Random Classical 0.00 0.05 0.10 0.15 0.20 Success Probability Free CPMG RGA2x UR24 UR18 RGA32 CDD2 RGA32a RGA16b UR32 UR6 RGA4 XY4 UR12 RGA8a RGA8c Random Classical Figure 3.7.1: P erformance of DD sequences, expanding on the results sho wn in Fig. 3.4.6 . A v- erage success probabilit y for n = 3,4,5 with t w o oracle queries on Ja karta (top) and Nairobi (b ottom). The DD sequences are rank ed in order of decreasing success probabilit y . The t w o dot- ted lines represen t success probabilities corresp onding to a random and classical strategy , re- sp ectiv ely . F or n > 3 , the unprotected ev olution (F ree) is marginally b etter than c ho osing an elemen t randomly and do es not cross the classical threshold. DD protection is necessary to cross the classical threshold, and the R GA and UR sequences with few er than 12 pulses are the b est p erformers. Error bars corresp ond to 99% confidence in terv als. 65 1 2 1 2 3 1 2 Number of Oracle Queries 0.0 0.2 0.4 0.6 Success Probability 3-qubit 4-qubit 5-qubit Classical Without DD With DD 1 2 1 2 3 1 2 3 Number of Oracle Queries 0.0 0.2 0.4 0.6 Success Probability 3-qubit 4-qubit 5-qubit Classical Without DD With DD Figure 3.7.2: P erformance under differen t oracle query n um b ers. Success probabilities are sho wn as a function of the n um b er of oracle queries for Jakarta (top) and Nairobi (b ottom). All results included MEM, and error bars represen t 99% confidence in terv als. Dashed red lines corresp ond to the optimal classical success probabilit y . Except for n = 3 , the classical threshold is crossed only with DD. In our exp erimen ts, w e set q = 2 , whic h is the optimal n um b er of rep etitions for all instances other than n = 5 on Jakarta. Error bars corresp ond t o 99% confidence in terv als. 66 sequences impro v ed the success probabilit y , but ev en among the sequences tried, there w as a considerable v ariation. Robust sequences with few er than 12 pulses p er DD cycle w ere the b est p erformers. The ev en tual decrease in the p erformance of sequences with an increasing n um b er of pulses is to b e exp ected as they are implemen ted using noisy gates, and there is a trade-off b et w een the protection pro vided b y DD and the accum ulation of gate errors. The R GA8c and R GA8a sequences p erformed consisten tly w ell and are the only sequences to cross the classical threshold on Jakarta for n = 5 . R GA8c is also commonly kno wn as the Eulerian DD (EDD) sequence, and R GA8a is a sligh tly mo dified v ersion of EDD. These palindromic sequences are kno wn to b e robust against flip-angle and finite-width errors. The b est sequence at eac h problem size is sho wn in T able 3.7.1 . Problem Size Jakarta Nairobi n=3 UR6 R GA8a n=4 R GA8a UR6 n=5 R GA8c R GA8a T able 3.7.1: The b est-p erforming DD sequence at eac h problem size for b oth QPUs. These se- quences w ere determined b y implemen ting n+1 oracles of the form 0 k 1 n−k for the n -qubit Gro v er problem. Fig. 3.7.2 sho ws the exp erimen tal success probabilities for the unprotected and DD- protected Gro v er circuits for all queriesq . Here, only the b est DD sequence from the surv ey ab o v e (listed in T able 3.7.1 ) is used in eac h case. Theoretically , for n = 3,4,5 , q opt = 2,3,4 resp ectiv ely . Unfortunately , for 5-qubit Gro v er, soft w are restrictions prev en ted us from go- ing b ey ond q = 2 and 3 on Jakarta and Nairobi, resp ectiv ely . Ho w ev er, it is already clear that the exp erimen tally optimal v alue w as reac hed in b oth cases. Recall that w e restricted our results to q = 2 oracle queries. F or n = 3 , this is b oth exp erimen tally and theoretically optimal. F or Nairobi, t w o queries ha v e the highest exp erimen tal success probabilit y for all problem sizes. F or Jakarta and n = 5 , a single query has a sligh tly higher success probabil- it y , but the difference b et w een q = 1 and q = 2 is not substan tial. Ov erall, for simplicit y of 67 0.0 0.2 0.4 0.6 Success Probability DD + MEM DD Free + MEM Free Classical 0 2 4 6 Detected State 0 2 4 6 Marked State Free 0 2 4 6 Detected State 0 2 4 6 Marked State DD + MEM 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Success Probability 0.0 0.2 0.4 0.6 Success Probability DD + MEM DD Free + MEM Free Classical 0 2 4 6 Detected State 0 2 4 6 Marked State Free 0 2 4 6 Detected State 0 2 4 6 Marked State DD + MEM 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Success Probability 0.0 0.1 0.2 0.3 0.4 Success Probability DD + MEM DD Free + MEM Free Classical 0 4 8 12 Detected State 0 4 8 12 Marked State Free 0 4 8 12 Detected State 0 4 8 12 Marked State DD + MEM 0.0 0.1 0.2 0.3 0.4 Success Probability Figure 3.7.3: 3-qubit, 4-qubit, and 5-qubit Gro v er on Jakarta after t w o oracle queries, comple- men ting Fig. 3.4.7 , whic h only sho ws Nairobi for n = 5 . Eac h ro w represen ts a problem size in ascending order. In a ro w, the horizon tal bar plot on the left sho ws the success probabilit y under no error suppression and mitigation (F ree), with measuremen t error mitigation (F ree + MEM), with DD protection (DD), and with DD protection and measuremen t error mitigation (DD + MEM). The dashed horizon tal line and the b o xes represen t the classical and the theoretically ex- p ected success probabilit y , resp ectiv ely . The second and third columns sho w the input-output map for F ree and DD + MEM, highligh ting the impro v emen t offered b y these strategies. The states are sorted b y increasing Hamming w eigh t. The transition from green to blac k o ccurs at the classical success probabilit y threshold. With DD protection, the classical threshold is crossed in all cases. 68 0.0 0.1 0.2 0.3 0.4 Success Probability DD + MEM DD Free + MEM Free Classical 0 4 8 12 Detected State 0 4 8 12 Marked State Free 0 4 8 12 Detected State 0 4 8 12 Marked State DD + MEM 0.0 0.1 0.2 0.3 0.4 Success Probability 0.00 0.05 0.10 0.15 Success Probability DD + MEM DD Free + MEM Free Classical 0 8 16 24 Detected State 0 8 16 24 Marked State Free 0 8 16 24 Detected State 0 8 16 24 Marked State DD + MEM 0.000 0.025 0.050 0.075 0.100 0.125 0.150 Success Probability 0.00 0.05 0.10 0.15 0.20 Success Probability DD + MEM DD Free + MEM Free Classical 0 8 16 24 Detected State 0 8 16 24 Marked State Free 0 8 16 24 Detected State 0 8 16 24 Marked State DD + MEM 0.00 0.05 0.10 0.15 0.20 Success Probability Figure 3.7.4: Same as Fig. 3.7.3 , but for Nairobi. 69 analysis, w e fo cused only on results for q =2 . Finally , Figs. 3.7.3 and 3.7.4 sho ws the results for all oracles at t w o queries using the DD sequence found from the surv ey ab o v e. The results are qualitativ ely iden tical on b oth devices. W e ha v e already clarified that DD is necessary to cross the classical threshold. One migh t susp ect that ma jorit y v oting ma y suffice to declare a detected state as the mark ed state if it is the mo de of its corresp onding probabilit y distribution. Ho w ev er, ev en under this criterion, for 5-qubit Gro v er, there is no w a y to detect the mark ed state without DD. Device Sp ecifications Jakarta and Nairobi are 7-qubit QPUs comprising the IBMQE F alcon r5.11 pro cessors [ 110 ]. On these transmon qubit-based devices, single qubit gates are p erformed b y driving a DRA G pulse, and t w o-qubit gates are implemen ted using ec ho ed cross-resonance gates. They ha v e quan tum v olumes of 16 and 32, resp ectiv ely [ 1 ]. The qubit connectivit y for these QPUs is sho wn in Fig. 3.7.5 . T able 3.7.2 sho ws the gate errors, readout errors and the T 1 and T 2 times for b oth devices. The qubits used for eac h n are listed b elo w in increasing order: q 0 , q 1 , q 3 , and q 5 for our enco ded t w o-qubit Gro v er implemen tation; q 0 , q 1 , and q 2 for 3-qubit Gro v er; q 0 , q 2 , q 3 , q 5 as the main qubits and q 1 as the ancilla for 4-qubit Gro v er; q 0 , q 2 , q 3 , q 5 , q 6 as the main qubi ts and q 1 as the ancilla for 5-qubit Gro v er. Jakarta Nairobi Min Mean Max Min Mean Max T 1 (µ s) 74.54 140.36 179.61 98.73 135.52 179.93 T 2 (µ s) 23.02 48.74 106.67 40.52 74.58 136.91 1QG Error (%) 0.02 0.03 0.03 0.03 0.04 0.07 2QG Error (%) 0.6 0.79 1.08 0.52 1.0 1.61 1QG Duration (µ s) 0.04 0.04 0.04 0.04 0.04 0.04 2QG Duration (µ s) 0.23 0.34 0.54 0.24 0.31 0.43 R O Error (%) 1.73 3.44 5.12 2.27 3.43 4.68 R O Duration (µ s) 5.35 5.35 5.35 5.35 5.35 5.35 T able 3.7.2: Device sp ecifications for Jakarta and Nairobi on July 14, 2022. 1QG, 2QG, and R O denote 1-qubit gate, t w o-qubit gate, and readout, resp ectiv ely [ 1 ]. 70 0 1 2 3 4 5 6 Figure 3.7.5: Device connectivit y . Jakarta and Nairobi devices are built using the IBM Quan tum F alcon r5.11H pro cessors and ha v e sev en qubits. Measuremen t Error Mitigation 0.00 0.05 0.10 Unenc + Inv Unenc + IBU Unenc 00 01 10 11 Detected State 00 01 10 11 Marked State 0.941 0.022 0.033 0.004 0.042 0.926 0.006 0.026 0.080 0.008 0.893 0.019 0.008 0.077 0.038 0.877 00 01 10 11 Detected State 00 01 10 11 Marked State 0.967 0.005 0.025 0.003 0.001 0.976 0.005 0.018 0.045 0.007 0.945 0.004 0.003 0.046 0.001 0.951 00 01 10 11 Detected State 00 01 10 11 Marked State 0.984 0.014 0.000 0.003 0.008 0.995 0.006 -0.008 0.024 0.007 0.957 0.011 0.005 0.024 0.006 0.966 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3.7.6: Nairobi and MEM for unenco ded t w o-qubit Gro v er. The horizon tal bar plot on the left sho ws the failure probabilit y for the unenco ded t w o-qubit Gro v er algorithm for the unmit- igated data and under t w o MEM tec hniques: iterativ e Ba y esian unfolding (IBU) and resp onse matrix in v ersion (In v). The three heat maps sho w the output distribution under no mitigation, IBU, and In v, going from left to righ t. The input mark ed states are on the v ertical axis, and the prin ted n um b ers on the diagonal represen t p s for the resp ectiv e mark ed states. The off-diagonal elemen ts in the output distribution are written explicitly to emphasize the presence of negativ e probabilities under In v in the righ t-most figure. 0.00 0.02 0.04 Enc + Inv Enc + IBU Enc 00 01 10 11 Detected State 00 01 10 11 Marked State 0.979 0.020 0.000 0.001 0.048 0.951 0.000 0.000 0.001 0.002 0.946 0.052 0.002 0.000 0.028 0.970 00 01 10 11 Detected State 00 01 10 11 Marked State 0.989 0.011 0.000 0.000 0.015 0.985 0.000 0.000 0.000 0.000 0.980 0.020 0.000 0.000 0.020 0.980 00 01 10 11 Detected State 00 01 10 11 Marked State 0.999 0.001 0.000 0.001 -0.015 1.015 0.000 0.000 0.000 0.000 1.010 -0.011 0.000 0.000 0.009 0.990 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3.7.7: Same as Fig. 3.7.6 but for enco ded t w o-qubit Gro v er. 71 Measuremen t error mitigation relies on data colle cted from quan tum detector tomogra- ph y [ 12 ] and classical p ost-pro cessing to address systematic measuremen t errors. The cali- bration exp erimen ts in v olv e the preparation of computational basis statesjji and acquiring the resp onse matrix M with elemen ts m kj = probabilit y( preparejjij measure bitstring k) . M is used to extract the “true” mitigated probabilit y v ector ⃗ t=f(⃗ p,M) from the observ ed probabilit y v ector ⃗ p . Differen t error mitigation metho ds differ in ho w they define f . The most commonly used MEM metho d, resp onse matrix in v ersion (In v), sets ⃗ t=M −1 ⃗ p . This is a frequen tly used MEM strategy [ 4 , 111 – 114 ]. Ho w ev er, In v is inheren tly fla w ed in the sense that M −1 need not b e sto c hastic, and as a result ⃗ t can ha v e negativ e elemen ts. Ref. [ 13 ] notes that iterativ e Ba y esian unfolding (IBU) – a w ell-established metho d to correct detector defects in high-energy ph ysics exp erimen ts – can address readout noise without compromis- ing on sto c hasticit y . IBU is a simplified form of the exp ectation-maximization metho d from mac hine learning that maximizes the lik eliho o d function. In particular, starting with a prior “truth sp ectrum” ⃗ t 0 , the error mitigated distribution ⃗ t n is obtained b y rep eatedly applying Ba y es’ rule to get t n i = X j M ji t n−1 i P k M jk t n−1 k p j . (3.13) W e rely on IBU, in particular, the p yIBU pac kage [ 100 ] to p erform MEM to a v oid dealing with negativ e probabilities. In Ref. [ 100 ], the optimal n at whic h the iteration stops is determined b y placing a lo w er limit on the ℓ 1 -distance b et w een ⃗ t n and ⃗ t n−1 . In Fig. 3.4.4 , there is a sligh t discrepancy in the observ ed and predicted fidelities in the Unenc + MEM and Enc + MEM columns for Nairobi: the success probabilit y is lo w er than exp ected. The bar plots in Figs. 3.7.6 and 3.7.7 sho w that this discrepancy is reduced if the mitigated distribution is computed using In v instead of IBU. Ho w ev er, In v leads to unph ysical results, as seen in the righ tmost output distribution map in Fig. 3.7.7 where p s > 1 for the mark ed statesj01i andj10i . In particular, when using In v, the terms in ⃗ t sum to 1 but the elemen ts t i are not guaran teed to b e in [0,1] . The off-diagonal terms under Enc + In v in 72 Fig. 3.7.7 are indeed negativ e. So while Enc + In v has the b est rep orted a v erage success probabilit y , whic h is also higher than our mo del predicts, the underlying output distribution is unph ysical. IBU, on the other hand, alw a ys returns a v alid probabilit y distribution but do es not fully in v ert the resp onse matrix M . The discrepancy in sim ulated and observ ed v alues of Nairobi’s measuremen t error mitigated failure probabilities reflects this. Despite its limitations, w e err on the side of caution and use IBU as the default MEM metho d. A more systematic and recen t critique of MEM is pro vided in Ref. [ 115 ]. Extracting the dep olarizing parameter from gate errors IBMQE devices are calibrated daily , and for eac h gate, the gate errore g and gate timeτ g are rep orted. The asso ciatedT 1 andT 2 times are also rep orted for eac h qubit. Ase g is extracted in presence of thermal relaxation errorsR=ΦA ,p D =f(e g ,τ g ,T 1 ,T 2 ) . In order to extract p D from e g , w e assume that e g =1F(DR). (3.14) Using F(DR)=(1p D )F(R)+p D F(DR), (3.15) where D is the comple tely dep olarizing c hannel, w e get 1e g =F(DR) (3.16a) =(1p D )F(R)+p D F(DR) (3.16b) =(1p D )F(R)+p D F(D) (3.16c) =(1p D )F(R)+ p D d . (3.16d) Here, d=4 n for n qubits. Consequen tly , p D =d F(R)1+e g dF(R)1 . (3.17) 73 If w e assume that there are no relaxation errors and only dep olarizing noise affects the gate error e g , then p D =(d/d1)e g . Circuit Construction Recall that implemen ting the n -qubit Gro v er’s algorithm requires the C n−1 Z gate, the only m ulti-qubit gate necessary for b oth the oracle and the amplitude amplification step. W e pro vide circuit diagrams for ho w eac hn -qubit con trolled phase gate, C n−1 Z , w as transpiled. W e rely on previously kno wn circuit designs for our circuit construction, particularly the circuits used b y Ref. [ 81 ]. 1 T w o-qubit Gro v er circuits F or t w o-qubit Gro v er, CZ do es not require transpilation as CZ =HCXH . Ho w ev er, a few n uances m ust b e considered when constructing the enco ded t w o-qubit G ro v er circuits. There are three comp onen ts to a Gro v er circuit with mark ed elemen tm=b 1 b 2 : initialization in to the statejψi , oracle query O m , and amplitude amplification A . More precisely , jψi= 1 4 X b i ,b j ={0,1} jb j i jb k i=H Hj00i (3.18a) O m =(X 1−b 1 X 1−b 2 ) CZ(X 1−b 1 X 1−b 2 ) (3.18b) A=H HO 00 H H. (3.18c) T o con v ert these circuits in to their logical coun terparts, w e note that X 1 = XIXI , X 2 = XXII , Z 1 = ZZII , Z 2 = ZIZI , H ⊗4 = SW AP 12 (H 1 H 2 ) and P ⊗4 = (Z 1 Z 2 ) CZ . It is also helpful to notice that [ SW AP 12 ,U U] = 0 for an y unitary U and [ SW AP 12 , CZ] = 0 . Moreo v er, SW AP 12 jb 1 b 1 i = jb 1 b 1 i and [ SW AP 12 ,O 00 ] = 0 as O 00 only has op erators of the 74 form U U and CZ . Consequen tly , jψi=H Hj00i (3.19a) =H H SW AP 12 j00i (3.19b) = SW AP 12 H Hj00i (3.19c) =H ⊗4 U enc j00i. (3.19d) Implemen ting the enco ded Gro v er oracle is straigh tforw ard and do es not in v olv e an y t w o- qubit op erations: O m =X 1−b 1 X 1−b 2 IZZIP ⊗4 X 1−b 1 X 1−b 2 . (3.20) Lastly , A=H HO 00 H H (3.21a) =H HO 00 H H SW AP 12 SW AP 12 (3.21b) = SW AP 12 H HO 00 SW AP 12 H H (3.21c) =H ⊗4 O 00 H ⊗4 (3.21d) =H ⊗4 IXXIIZZIP ⊗4 IXXIH ⊗4 . (3.21e) The corresp onding circuits for the mark ed statej01i are sho wn in Fig. 3.4.1 . 2 3-qubit to 5-qubit Gro v er circuits The problem of transpilation increases in complexit y with problem size. C n−1 Z can b e ac hiev ed b y finding a circuit decomp osition for the n -qubit T offoli gate C n−1 X . It is kno wn that the three-qubit T offoli gate, C 2 X , can b e implemen ted using six CNOT s [ 116 ]. Ho w ev er, this requires a fully connected arc hitecture. As no fully connected group of three qubits can 75 b e found in the QPUs w e used, w e rely on the 8 -CNOT decomp osition [ 81 ] of C 2 Z sho wn in Eq. ( 3.22 ). T † = T † T T † T † T T (3.22) Here T = Z 1/4 . F or C 3 Z and C 4 Z w e use relativ e-phase T offoli gates [ 105 ]. Breaking do wn C k Z using C a Z and C b Y suc h that a+b = k+c allo ws for C k Z to b e implemen ted with few er CNOT s as long as w e usec ancillas. In our construction, w e only use one ancilla for C 3 Z and C 4 Z . C 2 Y is sho wn in Eq. ( 3.23 ), = Y G G G † G † (3.23) where G=R y (π/4) , and C 3 Y is sho wn in Eq. ( 3.24 ): = Y H T T † H T T † T T † H T T † H iZ (3.24) Finally , using the relativ e-phase T offoli gates, C 3 Z can b e written as in Eq. ( 3.25 ): = j0i Y Y † (3.25) 76 and lik ewise, C 4 Z can b e constructed as in Eq. ( 3.26 ): = j0i Y Y † (3.26) This sc heme – where relativ e phase T offoli gates [ 105 ] are sewn together to generate a circuit forC n−1 Z (n>k+2 ) – can b e generalized. In particular, C n−1 Z can b e implemen ted using C 2 Y , C 2 Y † and C n−2 Z , whic h in turn uses C n−3 Z . As a result of this recursion, the n um b e r of CNOTS for a C n−1 Z circuit is #(C n−1 Z)=2(n3) #(C 2 Y)+#(C 2 Z). (3.27) Th us, the n um b er of CNOT s required to implemen t a single query of n -qubit Gro v er scales as O(n) . A t the same time, the n um b er of necessary ancillas is n 2 , i.e., it also scales linearly with n . As w e did b y using Eq. ( 3.26 ), this linear scaling of ancillas could b e a v oided b y considering C k Y with k > 2 while increasing the n um b er of CNOT s. Whether en tangling few er qubits b y allo wing for deep er circuits is w orth while will dep end on the QPU arc hitecture under consideration. Note that the theoretical optimal n um b er of queries q opt = O(2 n/2 ) so at q opt , the n um b er of CNOT s scales as O(2 n/2 n) where the exp onen tial comp onen t will dominate. Ho w ev er, as w e noted b efore, the exp erimen tally allo w ed n um b er of queries b efore decoherence tak es o v er migh t b e less than q opt . Op en system mo del optimization The mo del presen ted in Section 3.2 only accoun ts for amplitude damping, dephasing, and dep olarization, whic h are a subset of the errors in a sup erconducting device. Notably , this 77 0 5 10 15 20 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 D Unencoded 2-qubit Grover on Jakarta 0 5 10 15 20 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Encoded 2-qubit Grover on Jakarta 1 2 g 0.0 2.5 5.0 7.5 10.0 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Encoded 2-qubit Grover on Nairobi 1 2 g 0.0 2.5 5.0 7.5 10.0 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 D Unencoded 2-qubit Grover on Nairobi Figure 3.7.8: Scaling calibration metrics. The ℓ 1 -norm based distance D b et w een the observ ed and sim ulated outputs for t w o-qubit Gro v er is sho wn as a function of the scaling parameters λ i . The mo del parameters are scaled b y setting T 1 ! λ −1 1 T 1 , T 2 ! λ −1 2 T 2 and p D ! λ g p D . The left and righ t columns represen t unenco ded and enco ded implemen tations, resp ectiv ely . The dotted blac k line represen ts the default setting λ i = 1 . The grey band in the Jakarta case (top ro w) is D for the DD-protected circuit v ersions. F or Nairobi (b otto m ro w), ev en without error suppres- sion, the fit b et w een theory and exp erimen t is already quite go o d, and our optimization aims to impro v e the fit further. Th us, in the b ottom ro w (Nairobi), the grey band corresp onds to the dis- tance b et w een sim ulated and observ ed at λ i = 1 . mo del do es not include crosstalk, leakage to higher energy lev els, and non-Mark o vian system- en vironmen t in teraction. As w e sa w in previous sections , in tro ducing error suppression 78 Logical 0100 0110 0011 0001 1100 1110 1011 1001 1000 1010 1111 1101 00 01 10 11 Marked State 2.1 ± 0.0 3.8 ± 0.11 1.75 ± 0.02 0.48 ± 0.0 1.96 ± 0.17 1.88 ± 0.1 1.64 ± 0.35 0.3 ± 0.02 0.4 ± 0.07 7.78 ± 0.13 1.39 ± 0.14 0.08 ± 0.05 0.1 ± 0.07 3.36 ± 0.08 1.64 ± 0.13 3.74 ± 0.1 1.72 ± 0.03 0.26 ± 0.07 1.98 ± 0.01 1.63 ± 0.08 0.49 ± 0.07 0.38 ± 0.1 1.57 ± 0.08 7.86 ± 0.31 0.04 ± 0.01 0.12 ± 0.03 4.12 ± 0.27 0.55 ± 0.07 4.84 ± 0.05 4.24 ± 0.23 1.59 ± 0.01 0.55 ± 0.16 0.64 ± 0.06 1.92 ± 0.11 1.36 ± 0.37 0.14 ± 0.01 0.15 ± 0.1 7.04 ± 0.56 1.46 ± 0.57 2.19 ± 0.14 4.38 ± 0.13 0.34 ± 0.01 1.39 ± 0.08 4.92 ± 0.07 0.63 ± 0.12 0.37 ± 0.0 1.34 ± 0.17 1.72 ± 0.16 0.12 ± 0.0 0.06 ± 0.04 1.2 ± 0.15 6.98 ± 0.11 X errors Y errors Z errors 0.00 0.05 0.10 0.15 Figure 3.7.9: Algorithmic error tomograph y on Jakarta for optimized parameters. The plot sho ws the results of AET on the enco ded t w o-qubit Gro v er algorithm after setting (λ 1 ,λ 2 ,λ g ) = (1,1,6) . Eac h ro w of the error tomograph y table corresp onds to a mark ed state , and eac h column represen ts logical errors and X,Y,Z t yp e errors. Ev en after rescaling the mo del parameters, com - pared to the error tomograph y table for Jakarta (Fig. 3.4.2 ), w e do not see an asymmetry in Z errors across mark ed states. impro v es the agreemen t b et w een our mo del and the observ ations. Ho w ev er, in the absence of DD, the mo del results often pro vide an upp er b ound on the p erformance, and the device can p erform w orse than the mo del’s predictions. It is plausible to ask if rescaling the calibration metrics pro vided b y IBMQE results in an impro v ed phenomenological mo del and b etter predictabilit y of the device p erformance. T o do this, w e quan tify mo del predictabilit y b y computing theℓ 1 -norm based distance b et w een the exp erimen tally observ ed probabilit y distributionp m and the sim ulated distributionq m for mark ed statesjmi . W e define D = 1 2 n 2 n −1 X m=0 D m , D m = 1 2 2 n −1 X i=0 jp m i q m i j, (3.28) suc h that D is the distance b et w een the probabilit y distributions a v eraged o v er all the mark ed statesjmi . D = 0 implies a p erfect matc h b et w een the sim ulated and the observ ed distributions, and D = 1 means completely distinguishable distributions. W e then consider D as a function of rescaling parameters λ i suc h that T 1 ! λ −1 1 T 1 , T 2 ! λ −1 2 T 2 and p D ! 79 λ g p D . T o ev aluate the feasibilit y of this approac h, w e fo cus on the smallest problem size, n=2 . The most significan t error con tribution comes from the dep olarizing c hannel, as D is most resp onsiv e toλ g . With this in mind, w e start from the default settingλ i =1 , then optimize λ g ,λ 1 ,λ 2 in that order, using the optimal v alue from the previous scan as the input for the next parameter. Fig. 3.7.8 sho ws the distance D as a function of λ i . Recall that in the absence of DD, w e rep orted discrepancies in b oth the AET of the enco ded circuit (Fig. 3.4.2 ) and the failure probabilities of the unenco ded circuit (Fig. 3.4.4 ) for Jakarta. T o directly compare the result of optimizing λ to the effectiv eness of error suppression, Fig. 3.7.8 (top ro w) also sho ws the distance D b et w een t heory and exp erimen t after DD. F or the unenco ded exp erimen t, increasingp D b y a factor ofλ g =10 suffices for the mo del to matc h the exp erimen t. D is minimized at λ 1 ,λ 2 = 1 , i.e., there is no c hange from the default setting forT 1 andT 2 . Ho w ev er, ev en with the optimized v alues (i.e., settingλ g =10 , λ 1 ,λ 2 = 1 ), the new mo del barely impro v es up on activ ating DD, corresp onding to the grey band. Similarly , for the enco ded t w o-qubit Gro v er on Jakarta, only optimizingp D leads to a noticeable decrease inD , with the b est fit found at λ g =6 . The optimized mo del do es matc h the success probabilit y acquired b y activ ating DD (grey band, top righ t). Ho w ev er, AET at λ g = 6 , sho wn in Fig. 3.7.9 , do es not exhibit the state-wise asymmetry seen in Fig. 3.4.2 . A similar optimization on Nairobi confirms our earlier observ ation: the mo del that uses the pro vided calibration metrics fits with the observ ed results, and Nairobi do es not b enefit from parameter rescaling. T o summarize, rescaling the calibration metrics to impro v e predictabilit y is mildly effec- tiv e: the agreemen t with the exp erimen tal p s results impro v es b y increasing the dep olariza- tion. Ho w ev er, the optimized mo del do es not matc h the AET results for Jakarta. A more sophisticated mo del that includes qubit crosstalk and leakage to/from higher energy lev els app ears necessary for a b etter agreemen t. Ho w ev er, as highligh ted b efore, i n tro ducing er- 80 ror suppression via DD pro vides a reliable w a y to sim ultaneously impro v e p erformance and agreemen t with a mo del that accoun ts only for Mark o vian amplitude damping, dephasing, and dep olarization. P ostselection on 4-qubit and 5-qubit Gro v er The circuits forC 3 Z andC 4 Z require 5 and 6 qubits, resp ectiv ely [see Eqs. ( 3.25 ) and ( 3.26 )], with one ancilla qubit used to link the T offoli and relativ e-phase T offoli gates. The ancilla qubit is initialized in j0i and should b e in that state at the end of the algorithm. Conse- quen tly , when measured in the Z -basis, observing the ancilla qubit in j1i implies that an error o ccurred in the implemen tation of theC 3 Z orC 4 Z gate. W e p ostselect the 4-qubit and 5-qubit Gro v er exp erimen ts b y only considering exp erimen ts for whic h the ancilla qubit (q1 in Fig. 3.7.5 ) is measured to b e in statej0i . Fig. 3.7.10 sho ws the effect of p ostselection on success probabilities. The b etter-than-classical p erformance for Nairobi holds ev en if p ost- selection is a v oided. Ho w ev er, without p ostselection, w e see a tie with the classical result for 5-qubit Gro v er on Jakarta at q = 2 . Ov erall, there is a non-trivial increase in success probabilities due to p ostselection. Using ancilla qubits for p ostselection is con v enien t, as these ancilla qubits also allo w for m uc h shallo w er circuits for implemen ting C n−1 Z (recall Section 3.7 ). All the results w e rep ort in other sections are after p ostselection, including the surv ey of DD sequences. 81 3 4 5 Problem Size 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Success Probability Random Query=1 Query=2 Query=3 Jakarta w/ PS Jakarta w/o PS Nairobi w/ PS Nairobi w/o PS Figure 3.7.10: P ostselection for 4-qubit and 5-qubit Gro v er: Success probabilities are sho wn with and without p ostselection for Nairobi (green) and Jakarta (orange). Here w e only consider the DD-protected circuits. 3-qubit Gro v er do es not undergo p ostselection. F or 4-qubit and 5-qubit Gro v er, w e p ostselect to coun t only the exp erimen ts for whic h the ancilla qubit (q1 in Fig. 3.7.5 ) is inj0i . The white lines corresp ond to the success probabilities for the classical strategy and random sampling from the unsorted list (q = 0 ). With p ostselection, for all pr oblem sizes, the DD-protected quan tum q = 2 strategy outp erforms the classical strategy for q 3 . Without p ostselection, the b etter-than-classical requiremen t is met b y all implemen tations other than 5- qubit Gro v er on Jakarta, where w e ac hiev e a break ev en. Error bars corresp ond to 99% confidence in terv als. 82 Chapter F our: Demonstration of algorithmic quan tum sp eedup T ext for this c hapter is adapted from [ 97 ]. The quest to demonstrate a quan tum sp eedup using ph ysical hardw are for a computa- tional problem o v er a range of increasing problem sizes – an algorithmic quan tum sp eedup – has motiv ated the field of quan tum computing from its inception [ 117 ]. Man y quan tum algorithms are no w kno wn that theoretically outp erform classical algorithms in solving prob- lems of increasing size [ 74 , 118 – 129 ]. Ho w ev er, computational errors m ust b e suppressed to realize this p oten tial, esp ecially in the curren t noisy in termediate-scale quan tum (NISQ) era [ 5 ]. Better-than-classical algorithmic p erformance has b een demonstrated a n um b er of times, e.g., on ion-trap [ 80 , 130 ], sup erconducting [ 79 , 106 , 126 , 131 – 134 ], photonic [ 135 – 139 ], and R ydb erg atom [ 140 ] quan tum pro cessors. In most cases, this w as done b y ex- ceeding the corresp onding classical algorithmic success probabilit y at a fixed or small set of problem sizes [ 79 , 80 , 130 , 136 , 139 ], b y outp erforming a limited set of classical algo- rithms [ 131 , 132 , 140 ], or under complexit y-theoretic assumptions [ 106 , 133 , 135 , 137 ]. See Section 4.5 for a surv ey of existing exp erimen tal demonstrations of b etter-than-classical al- gorithmic results. T o qualify as a pro v able, unqualified algorithmic quan tum sp eedup, w e stipulate that the sp eedup – quan tified in terms of the scaling with problem size of the time-to-solution met- ric [ 39 ] – is (i) relativ e to the b est p ossible classical algorithm (“pro v able”) and (ii) free of an y assumptions or conjectures (“unqualified”). Moreo v er, to a v oid detectable finite-size effects w e stipulate that (iii) the sp eedup is exhibited up to the largest exp erimen tally accessible problem size. T o satisfy this stringen t and hitherto undemonstrated notion of a pro v able, unqualified algorithmic quan tum sp eedup, w e revisit the Bernstein-V azirani (BV) algorithm, whic h w as one of the v ery first theoretical examples of a quan tum/classical complexit y class separation [ 119 ]. In the original BV problem, an oracle outputsf b (x)=bx( mo d 2)2f0,1g , where x and b are b oth length-n bitstrings. Here x is a guess pro vided b y the user and b is a secret bitstring the user is trying to learn in as few oracle queries as p ossible. The b est classical algorithm requires n queries, since eac h query can only pro vide one new bit of information ab out b . By solving the problem with certain t y in a single query , the BV algorithm pro vides a linear sp eedup o v er the b est-classical algorithm. Here, w e consider a mo dified, single-shot v ersion of BV, denoted ssBV- n , where the hidden bitstringb c hanges after ev ery query . W e collo quially refer to this as the “BV guessing game”: after one query of the single-shot oracle, the pla y er is allo w ed one guess of the bitstring b . If the v erifier confirms that the guess is correct, the pla y er wins; if the guess is wrong, the game con tin ues with a new bitstring. In this setting, the optimal classical algorithm is to query the oracle with x = 0...01 i 0...0 (i is arbitrary), whic h rev eals b i , and then guess the remaining n 1 bits. This yields classical success probabilit y p s = 2 1−n , only t wice b e tter than a random guess (this cannot b e impro v ed [ 141 ]). In stark con trast, a pla y er with access to a quan tum computer (QC) running the original BV algorithm has success probabilit yp s =1 after eac h query , whic h b ecomes an exp onen tial adv an tage in the sp eedup ratio (defined b elo w) o v er the classical setting. 4.1 Quan tum sp eedup quan tified In a head-to-head comparison of success probabilities, p s > 2 1−n 0 , for a fixed problem size n 0 , implies a b etter-than-classical result. This is the con text in whic h b etter-than-classical results ha v e b een ac hiev ed for the Gro v er and BV algorithms [ 79 , 80 , 130 ]. Ho w ev er, as discussed in Section 1.2 , the success probabilit y at fixed problem size is not a reliable measure of quan tum sp eedup. Detecting an algorithmic sp eedup requires computing the scaling 84 Figure 4.0.1: Circuit for the BV algorithm, including DD pulses. The oracle sho wn enco des the unkno wn bitstring b = 111000 for the ssBV-6 problem. A con trolled-NOT (CNOT, or CX) or iden tit y gate is p erformed from qubit i to t he ancilla qubit if b i = 1 or 0 , resp ectiv ely . Note that the quan tum and classical oracles are iden tical in the ssBV-n problem, and so b oth tak e time t r /jbj to run, wherejbj is the Hamming w eigh t of b . Eac h BV-n circuit requires n+1 qubits. A Hadamard gate (H ) is applied to eac h qubit b efore and after the oracle, and eac h qubit is mea- sured in the computational basis, for a total circuit depth jbj + 3 (with equalit y only for fully connected arc hitectures). DD pulses (P i ) are turned on during idle times. with the problem size n of the sp eedup ratio of the classical and quan tum total run times: S(n) = TTS C (n) TTS Q (n) . Recall from Eq. ( 1.26 ) that TTS dep ends on b oth the success probabilit y p s and the p er iteration run time t r (n) . W e c ho ose to measure t r (n) in terms of the circuit execution time and the readout duration, and ignore precompilation and p ostpro cessing o v erheads, as the latter are inheren tly classical. It follo ws from the BV circuit structure (Fig. 4.0.1 ) that t r (n) = cτ 2q n +τ 0 , where 1 c 2 dep ends on the qubit connectivit y graph, with the t w o limits corresp onding to all-to-all connectivit y (c = 1 ) and a c hain 85 (c=2 ). F or our IBMQ implemen tation, w e foundc1.76 (see Section 4.5 ). The t w o-qubit gate time,τ 2q , and the sum of the single qubit and readout times,τ 0 , dep end on the sp ecific QC and can v ary b y orders of magnitude across platforms. When accoun ting for gate and measuremen t imp erfections, w e exp ect TTS Q (n) in the ssBV-n case to scale as n2 λn (with λ > 0 ), instead of as t r n , as w ould b e the case for a noiseless QC. Here λ is a (naiv e) effectiv e noise parameter that captures the effect of m ultiplying gate and measuremen t fidelities in a circuit of depth O(n) . When ssBV-n is solv ed classically , computingf b (x)=bx ( mo d 2) also tak es time/n (the cost of addingn bits), so w e obtain TTS C (n)/n/log 2 (12 1−n )n2 n−1 . W e th us exp ect S(n)2 (1−λ)n , n2[n min ,n max ], (4.1) where n max denotes the largest n um b er of data qubits a v ailable to the quan tum algorithm andn min is iden tified empirically b y excluding small-size effects. W e will declare a quan tum sp eedup if the sp eedup exp onen t λ < 1 . It is imp ortan t to emphasize that the sp eedup exp onen t m ust b e extracted using n reac hing up to and including n max , since otherwise one cannot hop e to dra w conclusions that reflect asymptotic scaling b eha vior. Using this criterion, w e demonstrate b elo w that a statistically significan t quan tum sp eedup is ac hiev ed for DD-protected ssBV-n quan tum circuits, but no sp eedup is obtained for “bare” quan tum circuits implemen ted without DD. 4.2 Exp erimen tal Implemen tation W e implemen ted ssBV-n on t w o differen t 27 -qubit QCs: Mon treal and Cairo. While similar in their connectivit y , they ha v e differen t quan tum v olumes, qubit generations, and gate fidelities(see Section 4.5 ). Si nce the BV algorithm uses one ancilla, w e ha v e n max = 26 . Giv en the unkno wn string b , the BV oracle is implemen ted b y p erforming CNOT s from a subset of the first n qubits to the ancilla qubit (n um b eredn+1 ), and the n um b er of CNOT s 86 0 16 32 48 64 Detected State 0 16 32 48 64 Oracle Number w/o DD = 72.3 ± 0.3% 0 16 32 48 64 Detected State 0 16 32 48 64 w/ DD = 80.3 ± 0.3% Classical 0.1 0.3 0.5 0.7 0.9 Success Probability (\%) Figure 4.2.1: F ull output distribution for BV-6 from Cairo. Oracles f b are n um b ered from 0 to 63 , corresp onding to b 2 f0 6 ,...,1 6 g , sorted b y increasing Hamming w eigh t. Ideally , the output state for oracle f b (v ertical axis) is b , but in realit y , other bitstrings (horizon tal axis) are observ ed as w ell. Green dots on the diagonal corresp ond to p s > 1/2 , where p s is the empirical frequency (success probabilit y) with whic h b w as output for oracle f b . Success probabilities are rep orted with 5σ confidence in terv als. is the Hamming w eigh t k = jbj (see Fig. 4.0.1 ). There are 2 n differen t oracles, and in the ssBV-n problem withn fixed, one is selected at random in eac h round. Ho w ev er, to increase our confidence in the results w e exploited the fact that giv en n andk , the circuits for all n k distinct bitstrings are iden tical up to qubit p erm utation; w e use d this symmetry and tested the n+1 p erm utationally-inequiv alen t strings b=1 k 0 n−k with 0kn for eac h n . Once again, w e emplo y ed the decouple then compute strategy describ ed in Section 3.3 . W e implemen ted one rep etition of UR 14 and UR 18 p er idle in terv al on Mon treal and Cairo, re- sp ectiv ely; see Fig. 4.0.1 . W e to ok 100K (32K) shots using Cairo (Mon treal) for eac h unique circuit. W e then sampled the corresp onding results for all BV-n oracles using b o otstrap- ping [ 64 ] and rep ort the mean TTS for BV-n along with error bars corresp onding to 5σ for the b o otstrapp ed distribution. See Section 4.5 for more exp erimen tal implemen tation details. 87 4.3 Results The Cairo results for BV-6 , b oth with and without DD, are sho wn in Fig. 4.2.1 . The oracles and outputs bitstrings are sorted b y increasing Hamming w eigh t. It is clear from these results that a higher Hamming w eigh t results in a decreasing success probabilit y without DD; this is consisten t with our exp ectation that deep er circuits ha v e a lo w er o v erall fidelit y . With DD, this problem is significan tly mitigated, whic h already suggests that error suppression through DD will b e cen tral to our quan tum sp eedup demonstration. In fact, with DD the single-shot output success probabilit y exceeds 1/2 for all oracles, whic h allo ws reac hing the b ounded-error quan tum p olynomial (BQP) threshold of2/3 for all p ossible inputs b y classical ma jorit y v ote on m ultiple rep etitions [ 119 , 130 ]. Without DD, the single-shot output success probabilit y is b elo w 1/2 for 7/64 of the inputs, so for these inputs, the BQP threshold cannot b e reac hed. With (without) DD, the a v erage single-shot success probabilit y is 80.3% (73.2% ). While this is m uc h higher than the classical single-shot probabilit y of 2 −5 3% , it do es not suffice for claiming a quan tum sp eedup, as this requires that w e demonstrate a scaling adv an tage as a function of the problem size n . Our main result is presen ted in Fig. 4.3.1 , whic h sho ws the TTS as a function of problem size n for b oth Mon treal and Cairo. White grid lines sho w the classical TTS (scaling as n2 n−1 ), and the ideal quan tum TTS (equal to t r n ) is sho wn for reference b y the t w o dashed lines – one eac h for Mon treal and Cairo. As is apparen t visually , the scaling without DD (empt y sym b ols) for b oth devices is w orse than the classical scaling at large problem sizes. W e attribute this, b ey ond the aforemen tioned exp onen tial fidelit y loss with circuit depth, to the fact that transmon-based devices suffer from sp on taneous emission errors, as a result of whic h they preferen tially generate bitstrings with lo w Hamming w eigh t, whic h is w orse than a uniformly random guess. This is also consisten t with the result sho wn in Fig. 4.2.1 (left). With DD, this problem is mitigated, so that p s > 0 is extended for Cairo (blue) to 88 □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ Montreal w/o DD Montreal w/ DD ◇ Cairo w/o DD ▲ Cairo w/ DD □ 1.13 ± 0.11 0.6 ± 0.03 ◇ 1.28 ± 0.08 ▲ 0.98 ± 0.02 2 6 10 14 18 22 26 1 μs 10 μs 100 μs 1 ms 10 ms 100 ms 1 s TTS Figure 4.3.1: Time-to-solution (TTS) as a function of problem size or n um b er of data qubits n . W e rep ort TTS(n) = 1 2 n P b TTS(n,b) , where TTS(n,b) is giv en b y Eq. ( 1.26 ), with p d = 0.99 and t r (n) replaced b y t r (n,b) , since eac h oracle (lab eled b y its secret string b 2 f0,1g n ) tak es a differen t time to run. Results for Mon treal and Cairo are sho wn b y the orange and blue sym- b ols, resp ectiv ely , and filled (empt y) sym b ols represen t results with (without) DD; dotted lines are guides to the ey e. The asymptotic classical scaling TTS C (n) 2 n is sho wn as white grid lines, and the h yp othetical, ideal quan tum scaling TTS Q (n) / n of e ac h QC is indicated b y the dashed lines (for QC-sp ecific parameter v alues see SI). The w orst-case scaling fit for eac h curv e is sho wn b y the solid lines, whose slop es λ are rep orted in the b ottom legend, with uncertain ties represen ting 95% confidence in terv als. Without DD, the TTS curv es terminate at n ′ max = 16 (n ′ max = 20 ) for Mon treal (Cairo), since w e find p s = 0 for n > n ′ max . Moreo v er, λ > 1 without DD, indicating a w orse-than-classical scaling. With DD protection, on Cairo, the p s > 0 range is extended to n = 23 , and λ is just b elo w the break ev en p oin t of 1 , but the uncertain t y is to o large to conclude that quan tum sp eedup has o ccurred. In con trast, the Mon treal scaling with DD do es exhibit quan tum sp eedup, as explained in the text. Since t w o-qubit op erations and readout durations are shorter for Cairo, it exhibits a consisten tly lo w er absolute TTS than Mon treal. W e rep ort 5σ confidence in terv als from b o otstrapping for eac h data p oin t; error bars are mostly co v- ered b y the sym b ols. n=n max =23 (excessiv e readout noise required us to treat Cairo as a device withn tot =24 , see Section 4.5 ). Most notably , it is clear that with DD the Mon treal scaling (orange) is 89 b etter than classical and extends to n=n max =26 , sugges ting a quan tum sp eedup. T o quan tify this and extract the sp eedup exp onen t λ as conserv ativ ely as p ossible, w e compute the w orst-case scaling from our data (see Section 4.5 for more details). The results are sho wn as the straigh t blue and orange lines in Fig. 4.3.1 , along with the n umerical v alues of λ in the legend. Without DD, w e obtain λ = 1.13 0.11 and 1.28 0.08 for Mon treal and Cairo, resp ectiv ely , meaning a quan tum slo wdo wn. F or Cairo, the scaling with DD is λ = 0.98 0.02 , not a statistically significan t difference from the classical scaling. Ho w ev er, the fit confirms that Mon treal with DD exhibits an algorithmic quan tum sp eedup: λ = 0.600.03 . All the rep orted uncertain ties represen t 2σ symmetric confidence in terv als (see Section 4.5 ). The difference b et w een Cairo and Mon treal agrees with the rep orted larger quan tum v olume (128 vs 64 ) of Mon treal [ 142 ], and suggests that the latter is a relev an t p erformance metric also in the presen t con text of algorithmic sp eedups. All the slop es v ary with n in Fig. 4.3.1 . One migh t th us ask what the scaling w ould app ear to b e for a h yp othetical QC with few er qubits (h max ) than the actual n max =26 ; w e address this in Fig. 4.3.2 . This figure sho ws the maxim um lo cal slop e of eac h of the curv es in Fig. 4.3.1 fornh max (see Section 4.5 ). The results clearly sho w the gro wth of the sp eedup exp onen tsλ hmax for Cairo with and without DD, and for Mon treal without DD, t o the p oin t λ > 1 or b ey ond, where no quan tum sp eedup surviv es. In con trast, the sp eedup exp onen t for Mon treal with DD is w ell within the quan tum sp eedup region of λ < 1 for all v alues of h max . 4.4 Discussion and Conclusion The ssBV-n problem has a pro v able, conjecture-free exp onen tial sp eedup o v er the b est p os- sible classical algorithm in the setting of a game in v olving an oracle and a v erifier. The main w eakness of this setting is its oracular nature: w e are forced to hide the in ternal structure of the circuit from the pla y ers since the BV circuit can b e efficien tly sim ulated classically b y 90 □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ Montreal w/o DD Montreal w/ DD ◇ Cairo w/o DD ▲ Cairo w/ DD 2 6 10 14 18 22 26 0.4 0.6 0.8 1.0 1.2 Figure 4.3.2: Results for λ hmax , the maxim um lo c al slop e of eac h of the curv es in Fig. 4.3.1 for n h max , i.e., the w orst-case-scaling when Fig. 4.3.1 is restricted to h max + 1 qubits. Only Mon treal with DD exhibits an unam biguous quan tum sp eedup, with λ hmax w ell b elo w 1 for all nh max . Error bars represen t 2σ confidence in terv als. virtue of the fact that it uses only Clifford gates [ 143 ]. In con trast, quan tum supremacy is not sub ject to oracular restrictions and is in this sense a more in teresting t yp e of quan tum adv an tage. Ho w ev er, this adv an tage only holds under certain conjectures. Another in terest- ing class of non-oracular sp eedups is quan tum constan t depth circuits vs classical logarithmic depth circuits [ 124 , 125 ] and quan tum limited-space adv an tage [ 126 ]. Here, the assumption is a classical resource constrain t. Some sort of tradeoff b et w een computational complexit y assumptions, constrain ts, and oracularit y th us app ears to b e inevitable. T o test for a quan tum sp eedup, w e compared the asymptotic scaling of the TTS metric with problem size for b oth classical and quan tum algorithms. W e demonstrated a statistically 91 significan t algorithmic quan tum sp eedup on Mon treal using this met ric. A crucial feature in our implemen tation w as error suppression through DD, without whic h the sp eedup w as not exhibited. It is natural to question whether this sp eedup can b e exp ected to con tin ue indefinitely . Giv en the flatness of λ hmax seen in Fig. 4.3.2 , extrap olating the quan tum sp eedup result for Mon treal to n sligh tly larger than n max app ears reasonable. Ho w ev er, for n n max the DD-enabled sp eedup cannot surviv e, since in the absence of a mec hanism for en trop y remo v al, suc h as fault-toleran t quan tum error correction [ 144 ], decoherence alw a ys ev en tually dominates. Th us, one should exp ect the DD-enabled quan tum sp eedup to disapp ear at some finite upp er limit on n . The fact that this upp er limit is not observ ed in our exp erimen ts satisfies a k ey goal of implemen ting a quan tum algorithm on a NISQ device, namely to c hec k whether a quan tum adv an tage is already accessible ev en b efore the adv en t of fault-tolerance, up to the largest problem sizes supp orted b y the device. W e ha v e sho wn here that, with the help of error suppression via DD, this is indeed the case. Another natural question is to what exten t the sp eedup rep orted here can b e further im- pro v ed. W e certainly exp ect that metho ds suc h as error mitigation (MEM) [ 145 ] and further DD sequence optimization [ 96 , 146 – 148 ] will ha v e suc h an effect, though TTS Q should then accoun t for the additional classical computation time they incur. F or MEM, this cost o v er- whelms the quan tum sp eedup w e ha v e observ ed [ 100 ]. Device-tailored optimization of DD sequences with adv anced lo w-lev el pulse con trol is an exciting fron tier that remains largely unexplored and app ears particularly promising. While w e fo cused on sup erconducting-qubit devices, DD protection can b e b eneficial across platforms, as all NISQ devices are affected b y computational errors suc h as decoherence and crosstalk. An ideal quan tum computer w ould reduce the exp onen tial gro wth of the classical TTS for the ssBV-n problem to a linear one. Our results are comparativ ely less impressiv e: w e demonstrated what amoun ts to a p olynomial quan tum sp eedup, b y reducing the exp onen t of the TTS scaling b elo w its classical minim um. Our w ork pro vides a path to testing suc h 92 sp eedups across platforms and algorithms in the NISQ era. 4.5 Supplemen tary Information Prior b etter-than-classical results There are six main categories where prior b etter-than-classical results ha v e b een demon- strated on NISQ devices [ 5 ]: quan tum supremacy , sim ulation, optimization, mac hine learn- ing, oracular algorithms, and computations with limited space. W e briefly review eac h in turn. But b efore w e do so, w e reiterate the con text for our claim of an algorithmic quan tum sp eedup. The commonly accepted criterion for an algorithmic quan tum sp eedup is to use a quan- tum computer to demonstrate a sp eedup o v er existing classical computers for a w ell-defined computational task. This criterion can b e made stricter [ 39 ] b y requiring either a pro v able quan tum sp eedup (whic h is not met ev en b y Shor’s algorithm) or a useful quan tum sp eedup, where the computational task also has a real-w orld application. Ref. [ 39 ] furthermore defined strong and limited quan tum sp eedups based on whether the comparison is against the b est p ossible or b est a v ailable classical algorithm. While w e do not exp ect the ssBV-n problem to result in a useful quan tum sp eedup, our w ork presen ts a strong and pro v able quan tum sp eedup for a w ell-defined computational task with em b edded information asymmetries b e- t w een the oracle and the pla y er. 1 Quan tum supremacy Quan tum supremacy [ 149 ] has b een demonstrated for efficien tly sampling pseudo-random quan tum circuits [ 106 , 133 ] and b oson sampling [ 135 ]. Assuming that it tak es exp onen tial time to solv e an NP-complete problem, a quan tum adv an tage in an in teractiv e pro v er-v erifier setting w as demonstrated in Ref. [ 137 ]. These results all rely on v arious complexit y-theoretic conjectures [ 150 – 152 ] to p erform computational tasks that are b ey ond the capabilit y of an y 93 classical computer. They are also susceptible to the b oundaries defined b y the capabilities of ev er-impro ving classical sup ercomputers and algorithms [ 153 ]. Indeed, classical sim ulation times far lo w er than the original estimates of Ref. [ 106 ] ha v e b een rep orted (e.g., Ref. [ 154 ]). Doubts ha v e b een cast on whether an asymptotic sp eedup is p ossible in the setting of noisy random circuit sampling [ 155 – 157 ]. 2 Quan tum optimization Quan tum annealing hardw are based on sup erconducting flux qubits w as used to demonstrate a sp eedup against sim ulated annealing for certain sp ecially crafted spin-glass problems [ 131 ]. Ho w ev er, the sp eedup did not hold against other classical algorithms (suc h as quan tum Mon te Carlo). Another recen t exp erimen t used a v ariational quan tum adiabatic algorithm, implemen ted using R ydb erg atom arra ys, to demonstrate a solution of the Maxim um In- dep enden t Set problem with a quan tum adv an tage as a function of problem hardness (not problem size) against sim ulated annealing [ 140 ]. The latter w as constrained to the same effectiv e circuit depth as the quan tum implemen tation. The sp eedup w as not tested against state-of-the-art classical algorithms, suc h as parallel temp ering with iso-energetic cluster mo v es [ 158 ]. 3 Quan tum sim ulation Quan tum sim ulation is the original setting of a quan tum adv an tage, as en visioned b y F eyn- man [ 117 ]. While it is unlik ely that the e xp onen tial scaling of classical calculations for general time ev olution in quan tum man y-b o dy systems will b e o v ercome, there is no pro of to the con trary , and the bar for o v ercoming classical approac hes is high, esp ecially in the con text of ground state preparation in quan tum c hemistry [ 159 ]. Heuristic argumen ts sup- p orting that “practical quan tum adv an tage” has already b een demonstrated ha v e b een made in Ref. [129 ]. Quan tum annealing has b een used to demonstrate a quan tum adv an tage in sim ulation against state-of-the-art classical sim ulation metho ds [ 132 , 160 ], but this still falls 94 short of a pro v able quan tum sp eedup. 4 Quan tum mac hine learning A quan tum linear optics de vice solv ed the quan tum coup on collection problem in few er attempts than a classical strategy [ 139 ]. Lik ewise, using a nanophotonic device, a quan tum sp eedup in learning time w as demonstrated in a reinforcemen t learning setup [ 136 ]. Ho w ev er, these demonstrations do not scale: in b oth cases the authors sho w that it is optimal to rev ert to the classical strategy ev en for the cases they tested, when the problem sizes or learning times increase b ey ond a certain threshold. Recen t w ork used h ybrid quan tum-classical learning with quan tum-enhanced exp erimen ts to demonstrate an exp erimen tal quan tum adv an tage [ 134 ]. Under the standard complexit y- theoretic conjecture that NP-complete problems cannot b e solv ed in randomized p olynomial time, this w ork pro v es an exp onen tial quan tum-classical separation in terms of the n um- b er of exp erimen ts required to ac hiev e a giv en accuracy in the mac hine learning task and demonstrates this separation exp erimen tally . In a similar v ein, a quan tum adv an tage w as demonstrated in a setting of sup ervised learning assisted b y an en tangled sensor net w ork against classical supp ort v ector mac hines (an en tanglemen t-enabled reduction in the error probabilit y for classification of m ultidimensional radio-frequency signals) [ 138 ]. Bey ond the setting of a complexit y-theoretic conjecture, an imp ortan t difference b et w een our w ork and these results is that in b oth cases this in v olv es quan tum data (i.e., learning on quan tum states), whereas in our case the problem b eing solv ed is classical: winning a guessing game in v olving purely classical data (secret bitstrings). Another imp ortan t difference b et w een these results and ours is that the former are h ybrid quan tum-classical approac hes, whereas w e are considering the scaling of a purely quan tum algorithm. The difference is significan t in that in the former case the cost of the classical pro cessing (suc h as v ariational optimization) is not fully accoun ted for. W e do not exp ect this cost to c hange an exp onen tial separation, but this p oin t mak es a direct comparison with our result sligh tly more c hallenging. 95 5 Oracular algorithms Some of our results are directly comparable to v arious NISQ implemen tations of oracular quan tum algorithms. Ref. [ 130 ] implemen ted the BV and Hidden Shift (HS) algorithms for n = 10 on an 11-qubit trapp ed-ion device, whic h w as the largest and most successful implemen tation of these algorithms at the time (later sup erseded b y BV-11 and BV-20 on sup erconducting and trapp ed-ion devices resp ectiv ely [ 78 ]). Moreo v er, they demonstrated b etter-than-classical p erformance on b oth algorithms, b y crossing the BQP threshold for BV and b y finding higher-than-classical success probabilities for HS, at the fixed problem size of n=10 . In con trast, here, w e reac hed BV-26 on a sup erconducting device (ibmq_mon treal) b y in v oking error suppression via dynamical decoupling. None of the previous oracular algorithm demonstrations explicitly used error suppression to impro v e coherence and fidelit y . BV-26 corresp onds to a maximal circuit depth of 44 CNOT s (46 when accoun ting for the Hadamard gate s at the b eginning and end), sligh tly higher than the largest depth (40 ) reac hed in quan tum supremacy exp erimen ts [ 106 , 133 ]. Refs [ 80 ] and [ 79 ] resp ectiv ely used 3-qubit trapp ed-ion and trimon devices to implemen t oracular algorithms, and 4-qubit and 5-qubit v ersions of Gro v er’s algorithm w ere imple- men ted on v arious IBMQ devices [ 81 ]. In all these cases claims of b etter-than-classical results fo cus on crossing a classical success probabilit y threshold at a fixed problem size. Ho w ev er, this argumen t is insufficien t for establishing a quan tum sp eedup, whic h m ust b e based on the scaling withn of a time-based metric suc h as the TTS [ 39 ] or related metrics [ 161 – 163 ]. 6 Computations with limited space The final category concerns the resource-limited setting, sp ecifically the setting where one compares quan tum and classical computations with limited space, that use only one com- putational classical bit or qubit with a read-only memory as input. In this setting a quan- tum adv an tage w as pro v en for the computation of n -bit symmetric Bo olean functions: they can b e implemen ted exactly on a quan tum computer, but classically only with probabilit y 96 j0i H H j0i H H j0i j0i j1i H H j0i H H j0i H H j0i H H j0i H H j1i H H Figure 4.5.1: Equiv alen t circuits used in our reduction from the circuit for b = 1 k 0 n−k to the circuits for b = 1 k 0 m−k with m2 [k,n1] . Illustrated on the left is BV-2 with the b = 11 oracle (m = 2 and k = 2 ), and on the righ t BV-4 with the b = 1100 oracle (n = 4 and k = 2 ). The left circuit is obtained from the righ t circuit b y tracing o v er the last nk = 2 data qubits. 1/2+O(n/2 n/2 ) [ 126 ]. Th us, this is a rather differen t setting from all the other categories review ed ab o v e, and it suggests that quan tum scrap space offers an adv an tage o v er analo- gous classical space, and more generally an exploration of space-time trade-offs in quan tum circuits. The algorithm w as exp erimen tally tested using the 27 -qubit ibmq_b erlin device for three-bit to six-bit symmetric Bo olean functions, demonstrating an algorithmic success probabilit y that exceeds the classical limit [ 126 ]. Reduction from ssBV-n to ssBV-m for CPTP maps Since the oracle acts trivially on thei -th qubit ifb i =0 , the Hadamard gate pairs on the last nk qubits cancel. Therefore, the only difference b et w een b = 1 k 0 n−k and b = 1 k 0 m−k is that the BV circuit forjbj=n orjbj=m applies cancelling Hadamard gates to the lastnk or mk qubits, resp ectiv ely . No w let m2 [k,n1] ; then all the circuits for b = 1 k 0 m−k , in theory , ha v e the iden tical output as the circuit forb=1 k 0 n−k , as illustrated in Fig. 4.5.1 . In tuitiv ely , w e ma y th us extract the ssBV-m results from the ssBV-n results b y running only the BV-n circuits and tracing o v er the last nm data qubits, a practice w e implemen ted in our exp erimen ts and subsequen t analysis. Let us no w pro v e the equiv alence of our pro cedure to actually running the ssBV-m circuits, 97 as long as the completely p ositiv e, trace preserving (CPTP) map go v erning the circuit in the op en system case factors in to a pro duct o v er the “mark ed” and “unmark ed” qubits, i.e., those corresp onding to a1 (mark ed) or0 (unmark ed) in the bitstringb that defines the giv en oracle. W e first consider the ideal ssBV-n algorithm without an y op en-system effects. The initial state isjψ 0 i=j0i ⊗n j1i . Applying the initial Hadamard la y er yields jψ 1 i=(H ⊗n+1 )jψ 0 i=j+i ⊗n ji. (4.2) W e then apply the oracle corresp onding to the hidden bitstringb2f0,1g n , i.e.,O f b jxijyi= jxijyf b (x)i , so that the state b ecomes jψ 2 i=O f b jψ 1 i= (4.3a) =O f b 2 4 1 p 2 n+1 X x∈{0,1} n jxi(j0ij1i) 3 5 (4.3b) = 1 p 2 n+1 X x∈{0,1} n jxi(jf b (x)ij1f b (x)i) (4.3c) = 1 p 2 n X x (1) f b (x) jxiji. (4.3d) In the last line ab o v e, w e use the fact that for an y x , f b (x) is either 0 or 1, i.e., jxi(jf b (x)j1f b (x)i)= 8 > < > : jxi(j0jj1i) f b (x)=0 jxi(j1ij0i) f b (x)=1 (4.4a) =(1) f b (x) jxi(j0ij1i). (4.4b) 98 The final Hadamard la y er is applied next, and, using f b (x)=bx (mo d 2): jψ 3 i=H ⊗n+1 jψ 2 i (4.5a) = H ⊗n 1 p 2 n X x (1) b·x jxi ! Hji =jbij1i. (4.5b) In preparation of our more general discussion b elo w, let us equiv alen tly represen t the action of ssBV-n with hidden bitstringb on some initial stateρ of then data qubits and the ancilla qubit as BV n (b)[ρ]= T r n+1 H ⊗n+1 O f b H ⊗n+1 [ρ] , (4.6) where O f b represen ts the BV oracle, and T r n+1 means that the state of the ancilla qubit (n um b ere dn+1 ) is discarded at the end, so that BV n (b)[ρ] is the state of then data qubits at the end of one run of the algorithm. Th us, if w e writeΠ b =jbihbj andjψ 0 ihψ 0 j=Π 0 n Π 1 , then it follo ws from Eq. ( 4.5 ) that BV n (b)[jψ 0 ihψ 0 j] = Π b . Previously , w e argued that if b n =1 k 0 n−k , then BV m (b m )[Π 0 m Π 1 ]= T r {m+1,m+2,...,n} [ BV n (b n )[Π 0 n Π 1 ]], (4.7) where the trace means that the states of qubitsfm+1,m+2,...,ng are discarded from the result of running BV n (b n ) . T o pro v e this claim in the absence of an y noise, note that: T r {m+1,m+2...n} [ BV n (b n )[jψ 0 ihψ 0 j]] (4.8a) = T r {m+1,m+2...n} [Π bn ] (4.8b) = T r {m+1,m+2...n} [Π 1 k 0 n−k] (4.8c) = T r {m+1,m+2...n} [Π 1 k 0 m−k Π 0 n−m] (4.8d) =Π 1 k 0 m−k (4.8e) = BV m (b m )[Π 0 m Π 1 ], (4.8f ) 99 as claimed. No w consider the case where eac h gate is represen ted not b y a unitary but a CPTP map. F or con v enience, let us ren um b er the ancilla to b e the 0 ’th qubit instead of the n + 1 ’th qubit. W e can rewrite the initial state as jψ 0 ihψ 0 j=Π 1 Π 0 n =ρ A ρ B , (4.9) where ρ A = (Π 1 Π 0 m) and ρ B = Π 0 n−m . The BV oracle do es not in tro duce an y t w o-qubit gates b et w een qubit sectors A =f0,1,..mg and B =fm+1,m+2,..ng , so it is reasonable to assume that under the coupling to the en vironmen t they remain uncoupled (as long as there is no unin tended crosstalk b et w een the t w o sectors). Therefore, the CPTP map for the noisy ssBV-n algorithm will b e BV n (b n )[ρ A ρ B ]= T r 0 [(H A H B )(O A O B ) (H A H B )[ρ A ρ B ]], (4.10) where H A,B and O A,B represen t the CPTP maps corresp onding to the exp erimen tal imple- men tation of the unitariesH (m ulti-qubit Hadamard) andO (oracle) acting on qubit sectors A,B . Recall that for arbitrary CPTP mapsU andV acting on a te nsor-pro duct space U V [ρ σ]=U[ρ] V[σ]. (4.11) 100 □ □ □ □ □ □ ◦ ◦ ◦ ◦ ◦ ◦ ▲ ▲ ▲ ▲ ▲ ▲ □ w/o DD standard ◦ w/o DD reduced w/ DD standard ▲ w/ DD reduced 1 2 3 4 5 6 10 μs 20 μs 30 μs 40 μs 50 μs TTS Figure 4.5.2: Comparison of the effect of crosstalk without and with DD (the UR 14 sequence). TTS(n) is sho wn for ibmq_jakarta . Wh ile TTS standard TTS reduced without DD protection, the TTS results are statistically indistinguishable in the presence of DD. Therefore, T r {B} [BV n (b n )[jψ 0 ihψ 0 j]] (4.12a) = T r {0,B} [(H A H B )(O A O B ) (H A H B )[ρ A ρ B ]] (4.12b) = T r {0,B} [(H A O A H A )[ρ A ] (H B O B H B )[ρ B ]] (4.12c) = T r {0} [(H A O A H A )[ρ A ]] (4.12d) =BV m (b m )[ρ A ] (4.12e) =BV m (b m )[Π 1 Π 0 m]. (4.12f ) 101 R6 R1 R2 R3 R5 R4 26 25 24 23 21 18 17 15 22 20 19 16 14 13 12 10 7 6 4 0 1 2 3 5 8 1 1 9 Figure 4.5.3: Sc hematic of the lattice connectivit y for 27-qubit devices with the hea vy-hex la y- out [ 4 ]. The dashed lines connect qubits that are m ultiplexed toge ther for readout. 102 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ Montreal, slope = 0.40 μs ◦ Cairo, slope = 0.27 μs 0 5 10 15 20 25 0 5 10 15 Figure 4.5.4: The h yp othetical, ideal TTS for Mon treal and Cairo as a function of n . A deviation from linear scaling is seen for small n . This is the CPTP map generalization of the closed-system result Eq. ( 4.7 ), and it sho ws that that the reduction from ssBV-n to ssBV-m holds rigorously also in the op en system setting, as long as as the CPTP map factors according to the qubit sectors A and B . Ref. [ 88 ] considered in detail ho w the state of the sp ectator qubits can c hange the effect of crosstalk on neigh b oring qubits. In pa rticular, it sho w ed that while crosstalk can b e exacerbated if the sp ectator qubits are in a sup erp osition state (as is the case here), this effect can b e coun teracted b y using DD. The impro v emen t w e ha v e rep orted under DD further confirms this observ ation in the con text of ssBV- n circuits. 103 ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ◇ Cairo w/o DD ▲ Cairo w/ DD ◇ 1.44 ± 0.16 ▲ 1.22 ± 0.16 2 6 10 14 18 22 26 100 ns 1 μs 10 μs 100 μs 1 ms 10 ms 100 ms 1 s TTS Figure 4.5.5: Results from Cairo with a differen t logical to ph ysical qubit mapping than used in Fig. 4.3.1 , using all 27 qubits. These runs w ere p erformed on Jan uary 2nd, 2022. Due to a signif- ican t readout error in Q19, there is an abrupt jump in the TTS at n = 15 . Consequen tly , qubit s Q19, Q20, and Q22 w ere left out of all subsequen t exp erimen ts. T o in v estigate this more closely , w e implemen ted ssBV-n for n = 1 to 6 on the 7-qubit ibmq_jakarta pro cessor, and considered b oth the standard and the reduced setup (i.e., b oth sides of Fig. 4.5.1 . The TTS results are sho wn in Fig. 4.5.2 . Without DD, w e exp ect cross- talk to in tro duce un w an ted coupling b et w een the mark ed and unmark ed qubits, whic h lo w ers the success probabilit y and hence increases the TTS. This is what w e already observ ed on Mon treal and Cairo, and is also seen on Jakarta. F urthermore, w e exp ect the pro of of the reduction from ss BV-n to ssBV-m to break do wn in the presence of cross-talk, and indeed, w e observ e in Fig. 4.5.2 that TTS standard 6= TTS reduced without DD-protection. Ho w ev er, with DD the TTS is statistically iden tical for b oth setups. This further v alidates our use of 104 Q1 • Q1 Q2 • Q2 ⇒ QA QA Q2 • Q2 Q1 • × QA = QA × Q1 Q2 • Q2 Q1 • • • QA = QA • Q1 Q2 • Q2 Q1 • QA QA • Q1 Figure 4.5.6: Example of an ssBV implemen tation using ancilla sw apping. Here w e consider the circuit for implemen ting the oracle with b = 11 for ssBV-2 on a linear arc hitecture with qubit connectivit y Q2-Q1-QA (left). The standard sw apping tec hnique (second from left) requires 5 CNOT s (second from righ t), but c ho osing to sw ap the ancilla allo ws the circuit to b e imple- men ted with just 3 CNOT s (righ t). △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ slope=1.76 0 5 10 15 20 25 0 10 20 30 40 CNOTs Figure 4.5.7: The n um b er of CNOT s used to implemen t the b = 1 n oracle is sho wn as a function of n . This n um b er w ould scale as 2n for a c hain, but for the hea vy-hex arc hitecture, it scales as 1.76n . As b oth Mon treal and Cairo use the same arc hitecture, this n um b er is the same for b oth. the reduced setup. 105 Mon treal Cairo Min Mean Max Min Mean Max T 1 (µ s) 57.57 113.2 187.1 39.93 102.19 198.93 T 2 (µ s) 22.58 99.72 198.71 18.58 114.19 290.06 1QG Error (%) 0.02 0.04 0.15 0.01 0.03 0.07 2QG Error (%) 0.57 1.35 6.09 0.52 4.64 100.0 1QG Duration (µ s) 0.04 0.04 0.04 0.02 0.02 0.02 2QG Duration (µ s) 0.27 0.43 0.63 0.16 0.31 0.71 R O Error (%) 0.79 2.59 10.66 0.47 1.39 4.89 R O Duration (µ s) 5.2 5.2 5.2 0.73 0.73 0.73 T able 4.5.1: Device sp ecifications for Mon treal and Cairo on Marc h 12, 2022. 1QG and 2QG de- note 1-qubit gate and 2-qubit gate, resp ectiv ely . R O denotes readout. Device Sp ecifications Both Mon treal and Cairo are 27-qubit devices with a connectivit y arc hitecture as sho wn in Fig. 4.5.3 . These devices feature fixed-frequency transmon qubits with micro w a v e pulses for implemen ting con trol and readout. They ha v e quan tum v olumes of 128 and 64 and are built using the IBM Quan tum F alcon r4 and r5.11 pro cessors, resp ectiv ely [ 142 ]. T 1 , T 2 times, and gate, as w ell as measuremen t errors and durations, are detailed in T able 4.5.1 . Recall that w e mo deled the h yp othetical, ideal TTS Q (n) as cτ 2q n+τ 0 . This linear mo del is actually a sligh t o v ersimplification; the ideal TTS (assuming no decoherence) for Mon treal and Cairo as a function of n is sho wn in Fig. 4.5.4 , and only b ecomes linear for n > 2 (Mon treal) or n> 5 (Cairo). The dashed lines sho wn in Fig. 4.3.1 are iden tical to the data sho wn in Fig. 4.5.4 (not the linear fits). As b oth readout and 2-qubit gate durations are higher for Mon treal, TTS Mon treal > TTS Cairo . The TTS gro wth rates (for n> 5 ) are cτ 2q = 0.40µ s and cτ 2q = 0.27µs , and are sligh tly lo w er than the a v erage 2-qubit gate durations of 0.43µ s and 0.31µ s for Mon treal and Cairo, resp ectiv ely . The in tercepts are at 5.28µ s and 0.77µ s for Mon treal and Cairo, resp ectiv ely . W e used the en tire c hip for Mon treal, but Cairo had three noisy qubits that w ere left out from the exp erimen ts rep orted in Fig. 4.3.1 . T o explain this, Fig. 4.5.5 sho ws our ssBV-n 106 results on the en tire Cairo c hip using a differen t logical-to-ph ysical implemen tation than Fig. 4.3.1 . Here, the largest problem sizes solv ed with and without DD are n = 19 and 25 resp ectiv ely . There is an abrupt jump in the TTS at n = 15 due to a large readout error in Q19. In particular, the readout error for Q19 is 12.9% , whic h is an order of magnitude higher than the a v erage readout error of 1.76% . Consequen tly , w e used a differen t logical-to- ph ysical em b edding where the three fault y qubits, Q19 and its neigh b ors Q20 and Q22 (see Fig. 4.5.3 ), w ere left out of the exp erimen t. While this reduced the largest problem size w e could solv e in the DD setting (from n = 26 to n = 23 ), it allo w ed us to extract λ without b e ing affected b y the anomalous readout error in Q19. Ov erall, w e effectiv ely treated Cairo as a 24 -qubit device in all our subsequen t exp erimen ts. Reduction of circuit depth: CNOT-efficien t SW APs for ssBV- n A crucial ingredien t in our implemen tation is a simple metho d for circuit depth reduction, needed to o v ercome the limited connectivit y of the Mon treal and Cairo c hip arc hitectures. A fully connected arc hitecture w ould allo w BV-n to b e implemen ted withn t w o-qubit gates. Both the Cairo and Mon treal connectivit y graphs are of degree three, and most qubits are connected to just t w o others; see Fig. 4.5.3 . Generally , circuit transpilers deal with the sparseness b y sw apping qubits as necessary . The BV algorithms requires us to en tangle the “mark ed” qubits (those corresp onding to a 1 in the bitstring b that defines the giv en oracle) wit h the ancilla, and so w e m ust sw ap the unmark ed qubits when the ancilla and the mark ed qubit are not directly coupled. This is equiv alen t to sw apping the ancilla instead of the mark ed qubit. While implemen ting a SW AP requires three CNOT s, b y using the circuit iden tit y CNOT 12 SW AP 12 = CNOT 21 CNOT 12 , w e can implemen t a CNOT follo w ed b y a SW AP with just t w o CNOT s. Fig. 4.5.6 illustrates this for the ssBV-2 case. Ov erall, sw apping the ancilla reduces the CNOT scaling from 4n to 2n for a linear arc hitecture. On the hea vy-hex la y out, the n um b er of CNOT s required for implemen ting ssBV-n is found to scale as 1.76n , as sho wn in Fig. 4.5.7 . F or ssBV-26 , our longest circuit, w e used 44 CNOT s. 107 x 1 +x 2 +x 3 +x 4 +x 5 +x 6 +x 7 +x 8 x 1 +x 2 +x 3 +x 4 +x 5 +x 6 +x 7 x 1 +x 2 +x 3 +x 4 +x 5 +x 6 x 1 +x 2 +x 3 +x 4 +x 5 x 1 +x 2 +x 3 +x 4 x 1 +x 2 +x 3 x 1 +x 2 x 1 x 2 x 3 x 3 x 5 x 6 x 7 x 8 Figure 4.5.8: Binary tree represen ting the oracle computation for the construction used in our exp erimen ts. This is b ecause on top of the 26 CNOT s b et w een the mark ed and the ancilla qubit, 18 CNOT s w ere needed to p erform the SW AP . If eac h of the 18 SW APs required 3 extra CNOT s instead, then ssBV-26 w ould require 80 CNOT s. W e remark that it is p ossible to reduce the depth of our circuits using a binary tree-based divide-and-conquer setup. Ho w ev er, this comes at the cost of using more qubits, whic h reduces the problem size w e can implemen t. Let us consider this in detail. Recall that the oracle implemen ts f(x) = bx mo d 2 = b 1 x 1 ++b n x n mo d 2. The circuits and trees b elo w apply for n = 8 , but the resulting form ulas can b e easily generalized to arbitrary n . Under the curre n t construction w e used, the binary tree represen ting the oracle computation w ould b e as sho wn in Fig. 4.5.8 . The corresp onding circuits w ould b e: 108 x 1 +x 2 +x 3 +x 4 +x 5 +x 6 +x 7 +x 8 x 1 +x 2 +x 3 +x 4 x 1 +x 2 x 1 x 2 x 3 +x 4 x 3 x 4 x 5 +x 6 +x 7 +x 8 x 5 +x 6 x 5 x 6 x 7 +x 8 x 7 x 8 Figure 4.5.9: Binary tree represen ting the oracle computation for the construction using divide- and-rule. x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 a Here there are three definitions of “depth”: the n um b er of op erations n op , the time tak en to p erform the computationt , and the depthd of the binary tree giv en ab o v e. F or the quan tum circuit, n op is the n um b er of CNOT s, and the time tak en to p erform the computation is prop ortional to the n um b er of CNOT s la y ers C ∥ . In our implemen tation, t Q / n op = n = C ∥ =d+1=O(n) and t C =n1=O(n) . The binary tree depth d , the parallel t w o-qubit depth C ∥ , and the time needed to imple- men t the circuit t q can all b e reduced. Ho w ev er, this divide-and-conquer strategy requires more ancillas and more CNOT s to implemen t. Consider the divide-and-conquer strategy for adding the bits sho wn in Fig. 4.5.9 . Using this strategy , p erforming bx (mo d 2) tak es t c /d = log 2 (n) steps. Crucially , ev en here, the n um b er of op erationsn op isO(n) , but due to the parallelization in v olv ed in summing the literals, t c =O(log 2 (n)) . The quan tum circuit b elo w implemen ts divide-and-conquer: 109 x 1 x 2 a 1 x 3 x 4 a 2 x 5 x 6 a 3 x 7 x 8 a 4 Note that for the quan tum implemen tation, the data qubits x i are not used to store the information ab outx j forj6=i , and therefore, ancillas are required to p erform the summation. In this v ersion, the n um b er of op erations n op =8+2=n+(log 2 n1)=O(n) , time tak en t q / C ∥ = 4 = 1+log 2 n = O(log 2 (n)) , and the binary tree depth d = log 2 (n) . Note that n op =n+(d1) and C ∥ = 1+log 2 n , where the resp ectiv e n and 1 exist b ecause w e need an extra la y er of CNOT s, whic h are implemen ted in parallel, to transfer information from data qubits x i to the ancillas. Compared to our implemen tation where t q =O(n) , this is a reduction in the time tak en the implemen t the circuit. Ho w ev er, it comes at the cost of using n/2 ancillas. Ov erall, this strategy reduces b oth the classical and quan tum computation times, but it also requires more bits and qubits and more CNOT s. More imp ortan tly , it reduces the largest problem size w e can implemen t. It is w orth noting that if the b ottlenec k w ere circuit depth, in particular, if giv en some n -qubit device, w e w ere failing to exceed the problem size 2n/3 , then this strategy could reduce the circuit depth while using the n/3 qubits as ancillas. Bo otstrapping In all our TTS figures, w e rep ort TTS a vg mean i ( TTS i ), where TTS i is the time-to-solution for the i ’th oracle. Eac h oracular exp erimen t w as rep eated for 32,000 (100,000) shots on Mon treal (Cairo), and w e coun ted ho w man y of these exp erimen ts returned the correct 110 answ er. W e b o otstrapp ed o v er the observ ed coun ts, using the metho d describ ed in [ 64 ], to get 100 sampled v ersions of eac h oracular exp erimen t. The rep orted TTS v alues and the error bars corresp ond, resp ectiv ely , to the exp ected v alue and 5σ for TTS a vg computed from these b o otstrapp ed samples. It is p ossible for the success probabilit y for a giv en oracle to b e non-zero and y et for some of the b o otstrapp ed samples to ha v e no successful coun ts. In suc h cases, w e discarded those b o otstrapp ed samples as this w ould lead to infinite TTS. An actual infinite TTS means that the solution w as not observ ed during an y rep etition of the exp erimen t, i.e., p s =0 . T o obtain the confidence in terv als for the sp eedup exp onen ts λ , w e first use Mathematica’s LinearMo delFit function to compute the w orst-case fit on eac h b o otstrapp ed sample. W e then rep ort the exp ected v alue with 2σ confidence in terv als obtained from the ensem ble of λ s. Dynamical decoupling details V arious theoretical strategies for ho w to com bine DD with quan tum computation ha v e b een describ ed [ 11 , 107 , 164 , 165 ], including a v ariet y of DD sequences. T o go b ey ond the necessary theoretical simplifications, w e exp erimen tally tested a large set of kno wn DD pulse sequences on the BV-10 oracle b=1 10 and then considered the p erformance of the top four sequences w e iden tified in this manner on the en tire ssBV- n exp erimen t (not sho wn). All suc h DD sequences outp erformed the unprotected implemen tation on b oth Mon treal and Cairo, and the sp ecific c hoice of the DD sequence had little impact on the sp eedup exp onen ts λ . Our exp erimen ts utilize the univ ersally robust (UR p ) sequences [ 21 ], whic h are kno wn to p erform w ell on sup erconducting devices [ 31 , 166 , 167 ]. W e implemen ted one rep etition of the DD sequence in all a v ailable idle time gaps and did not attempt to optimize the pulse shap e or pulse placemen t. Suc h an optimization w ould undoubtedly further impro v e p erformance, and presen ts a fruitful future researc h direction; see, e.g., Refs. [ 88 , 146 , 147 ]. 111 Measuremen t error mitigation W e did not implemen t measuremen t error mitigation (MEM) in this w ork [ 12 , 114 , 168 ]. Ho w ev er, this w as done in Ref. [ 100 ], whic h implemen ted MEM on their single-shot BV results. MEM did impro v e the success probabilities at all problem sizes. Ho w ev er, the classical run time to p erform MEM, t MEM , w as m ultiple orders of magnitudes higher than the QC’s run time (sec vs. µ sec). Moreo v er, t MEM increases with problem size. W e opted not to include MEM in our results as it w ould n ullify an y scaling adv an tage when t MEM is considered. Computation of the W orst-Case TTS Scaling Let n max denote the largest n um b er of data qubits a v ailable to the quan tum algorithm; e.g., in the BV algorithm case, whic h requires one ancilla qubit, n max = n tot 1 , where n tot is the total n um b er of ph ysical qubits a v ailable on the QC. An y scaling of S(n) that is extracted from a QC with a relativ ely smalln tot , as is in v ariably the case in the curren t NISQ era [ 5 ], is sub ject to finite-size effects. Th us, the b est one can hop e for is that extrap olations to n > n max are meaningful, and an y conclusions based on suc h extrap olations m ust b e revisited when devices with larger n tot b ecome a v ailable. With this in mind, w e estimate the scaling of the sp eedup ratio S(n) b y computing the most conserv ativ e estimate allo w ed b y the exp erimen tal data, as explained b elo w. This extrap olated scaling can b e used to compare differen t QCs. T o compute the scaling exp onen ts rep orted in Fig. 4.3.1 , w e first fit TTS / 2 λ l,u n to the data for n 2 [l,u] with l 2 [1,u 2] and u 2 [3,n ′ max ] , where n ′ max is the largest n for whic h w e ha v e TTS data, i.e.,n ′ max 2f16,20,23,26g forf Mon treal w/o DD, Cairo w/o DD, Cairo w/DD, Mon treal w/DDg , resp ectiv ely . In other w ords,λ l,n ′ max is the slop e obtained b y fitting to the data b et w een l and n ′ max . W e then obtain the asymptotic sp eedup exp onen t via λ = max l λ l,n ′ max . By taking the max , w e ensure that the sp eedup exp onen t th us 112 □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ Montreal w/o DD Montreal w/ DD ◇ Cairo w/o DD ▲ Cairo w/ DD □ ◇ ▲ 0 5 10 15 20 25 0 5 10 15 Figure 4.5.10: log(F QEB ) is plotted as a function of problem size. W e ha v e extracted the exp o- nen t λ ps suc h that F QEB = c2 (1−λps )n . Note that for classical algo rithm solving the BV prob- lem, λ ps = 1 . Here, ev en without DD, w e are observing b etter-than-classical p erformance. See Fig. 4.5.10 to understand wh y the TTS has a w orse scaling than p s . estimated is conserv ativ e, i.e., represen ts the w orst-case scaling compatible with the data. Similarly , Fig. 4.3.2 sho ws the sp eedup exp onen t λ hmax = max l λ l,hmax , i.e., the maxim um lo cal deriv ativ e of eac h of the curv es in Fig. 4.3.1 for n h max . All λ v alues are rep orted with a 2σ confidence in terv al obtained after b o otstrapping. 113 △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ 100 ns 1 μs 10 μs 100 μs 1 ms 10 ms 100 ms 1 s TTS △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ 2 6 10 14 18 22 26 100 ns 1 μs 10 μs 100 μs 1 ms 10 ms 100 ms 1 s TTS △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ◇ 2 6 10 14 18 22 26 Figure 4.5.11: TTS i for i 2 f exp, a vg,ming with their resp ectiv e scaling factors λ i are sho wn for Mon treal and Cairo, with and without DD. Note that λ exp >λ a vg >λ min =λ ps , wher e λ ps is the same as in Fig. 4.5.11 . Cross-En trop y Benc hmarking and success probabilit y scaling The linear cross-en trop y b enc hmark fidelit y for an ideal unitary circuit U is defined as [ 169 ]: F QEB =2 n h X x q U (x)p U (x)i1=2 n hhp U (x)ii1, (4.13) whereq U :f0,1g n 7![0,1] is the probabilit y densit y function of the distribution obtained b y measuring Uj0 n i , p U (x) is the probabilit y of bitstring x 2 f0,1g n computed classically for the ideal quan tum circuitU , and the outer a v eragehi is o v er random circuitsU . In realit y , 114 ideal unitary circuits are replaced b y noisy v ersions, generally trace-preserving completely- p ositiv e mapsE U . Th us the sampling is from the noisy probabilit y distribution of outcomes ˜ q U (x)=hxjE U (j0ih0j)jxi , whic h replaces q U (x) in Eq. ( 4.13 ). F QEB w as used b y Ref. [ 106 ] to exp erimen tally demonstrate quan tum supremacy for the first time, in the con text of random circuit sampling. While w e are not dealing with random circuits, w e can glean some useful information from F QEB . First, w e can drop the a v erage o v er random circuits. Second, in our case, p U (x)=δ x,b , where b is the secret string enco ded b y the BV circuit. Th us, w e obtain F QEB =2 n p s 1, (4.14) where the success probabilit y p s = ˜ q U (b) = hbjE U (j0ih0j)jbi . Th us, for our purp oses, the cross-en trop y b enc hmark fidelit y and the success probabilit y are essen tially iden tical p erfor- mance metrics. Note that when the bitstrings are extracted from a uniform probabilit y distribution,p s = 2 −n and soF QEB =0 . This is not sp ecific to BV. Th us, at an y problem size, F QEB measures ho w far a w a y the observ ed distribution is from a random distribution, whic h w e can also in terpret as the result of randomly guessing the secret string with no oracle queries. On the other hand, a fter one classical oracle query , w e ha v eF QEB =2 n p C s 1=2 n 2 1−n 1=1 for a classical solution of the ssBV-n problem. Th us, the scaling of F QEB with n in the quan tum case is informativ e ab out whether the re is a quan tum adv an tage. In particular, a p ositiv e or negativ e slop e for F QEB (n) , i.e., λ ps < 1 or > 1 , signifies b etter- or w orse-than-classical p erformanc e, resp ectiv ely . In Fig. 4.5.11 , w e plot F QEB as a function of problem size n . W e extract λ ps using the same strategy describ ed in previous sections for computing λ TTS . Note that λ ps < 1 for all cases, i.e., the QC output is b etter-than-classical. If w e w ere to consider ho w success probabilit y scales with problem size, λ ps < 1 w ould b e enough to claim a b etter-than- 115 classical p erformance for BV. Ho w ev er, recall that λ TTS < 1 only with DD protection and only on Mon treal. W e next explain ho w the run time accoun ts for the difference b et w een λ TTS and λ ps . W e rep ort TTS(n)= 1 2 n P b TTS(n,b) , TTS exp (n)= 1 2 n X b TTS(n,b), (4.15) whic h is the TTS a v eraged o v er all secret stringsb of a giv en sizen . Here TTS(n,b) is giv en b y E q. ( 1.26 ), i.e., TTS(n,b)=t r (n,b) log(10.99) log(1p s (t r (n,b))) , (4.16) where the run timet r (n,b) dep ends not only onn but also onb since eac h oracle (lab eled b y its secret string b2f0,1g n ) tak es a differen t time to run. If the run time and success probabilit y only dep ended on the problem size and w ere inde- p enden t of the sp ecific oracle, then w e w ould ha v e TTS a vg (n)=t r, a vg (n) log(10.99) log(1p s, a vg (n)) . (4.17) Here t r, a vg (n) and p s, a vg (n) are extracted from the exp erimen tal data b y a v eraging o v er all the oracles of size n . Alternativ ely , assume that the scaling of the TTS is purely due top s and thatt r do es not con tribute to the scaling. In this case, setting t r =t 0 and pic king the minim um run time as t 0 (i.e., t 0 =t r (1) ), w e then ha v e: TTS min =t 0 log(10.99) log(1p s, a vg (n)) c p s, a vg (n) . (4.18) Note that λ TTS min =λ ps as the run time is constan t across problem sizes and oracles in this case. Fig. 4.5.10 sho ws TTS exp , TTS a vg and TTS min with their resp ectiv e scaling factors . As 116 exp ected, λ min =λ ps . A ccoun ting for the run times dep endence on the problem size already leads toλ a vg >λ min . When w e accoun t for the actual dep endence oft r on the oracle and the problem size, w e obtain λ exp >λ a vg >λ min , λ exp >1 without DD and λ exp <1 for Mon treal with DD protection. Ov erall, computing the scaling of the TTS metric as a function of problem size leads to the most reliable (and more stringen t) condition for ac hieving a quan tum sp eedup o v er classical p erfo rmance. The con trast with Fig. 4.5.11 illustrates the pitfalls of using just the cross- en trop y b enc hmarking fidelit y , or just the success probabilit y . TTS exp (n) is a w orst-case metric that fully accoun ts for all the costs, and while asymptotically t r (n,b) cannot c hange the scaling asso ciated with the success probabilit y , it do es do so at the scale of the problem sizes accessible to curren t exp erimen ts using the IBMQ devices. 117 List of Publications The w ork presen ted in this thesis con tains material from the follo wing publications and preprin ts: 1. Bib ek P okharel, Namit Anand, Benjamin F ortman, and Daniel A. Lidar. Demon- stration of Fidelit y Impro v emen t Using Dynamical Decoupling with Sup erconducting Qubits. Ph ys. Rev. Lett., 121, 220502 (2018). doi:10.1103/Ph ysRevLett.121.220502 . 2. Bib ek P okharel, and Daniel A. Lidar. Better-than-classical Gro v er searc h via error detection and error suppression. arXiv:2211.04543 [quan t-ph], (2022). doi:10.48550/arXiv.2211.04543 3. Bib ek P okharel, and Daniel A. Lidar. Demonstration of algorithmic quan tum sp eedup. arXiv:2207.07647 [quan t-ph], (2022). doi:10.48550/arXiv.2207.07647 118 Other publications and preprin ts completed during the duration of the PhD but not included in this thesis are: 1. Bib ek P okharel 1 , Moses ZR Misplon 1 , W alter Lynn, P eter Duggins, Kevin Hallman, Dustin Anderson, Arie Kapulkin, and Arjendu K. P attana y ak. Chaos and dynamical complexit y in the quan tum to classical transition. Scien tific rep orts 8, no. 1: 1-10 (2018). doi:/10.1038/s41598-018-20507-w 2. Siddarth Sriniv asan, Sandesh A dhikary , Jacob Miller, Bib ek P okharel, Guillaume Rabusseau, and Byron Bo ots. T o w ards a T race-Preserving T ensor Net w ork Repre- sen tation of Quan tum Channels First W orkshop on Quan tum T ensor Net w orks in Mac hine Le arning In conjunction with NeurIPS. (2021). Link to t he article 3. Ilaria Siloi, Virginia Carnev ali, Bib ek P okharel, Marco F ornari, and Rosa Di F elice. In v estigating the Chinese p ostman problem on a quan tum annealer. Quan tum Mac hine In telligence 3, no. 1.: 1-10 (2021). doi:10.1007/s42484-020-00031-9 4. Andrew D. Maris 1 , Bib ek P okharel 1 , Sharan Ganjam Seshac hallam, Moses ZR Mis- plon, and Arjendu K. P attana y ak. Chaos in the quan tum Duffing oscillator in the semiclassical regime under parametrized dissipation. Ph ysical Review E 104, no. 2: 024206. (2021). doi:10.1103/Ph ysRevE.104.024206 5. Kabuki T akada, Shigetoshi Sota, Seiji Y unoki, Bib ek P okharel, Hidetoshi Nishimori, and Daniel A. Lidar. Phase transitions in the frustrated Ising ladder with sto quastic and nonsto quastic catalysts. Ph ysical Review Researc h 3, no. 4: 043013. (2021). doi:10.1103/Ph ysRevResearc h.3.043013 6. Bib ek P okharel, Zo e Gonzalez Izquierdo, P Aaron Lott, Elena Strbac, Krzysztof Osiew alski, Emman uel P apathanasiou, Alexei K ondrat y ev, Da vide V en turelli, and Eleanor Rieffel. In ter-generational comparison of quan tum annealers in solving hard sc heduling problems arXiv:2112.00727 [quan t-ph] (2022). A ccepted to Quan tum Infor- mation P ro cessing. doi:10.48550/arXiv.2112.00727 7. Haimeng Zhang, Bib ek P okharel, E. M. Lev enson-F alk, and Daniel A. Lidar. Predict- ing Non-Mark o vian Sup erconducting-Qubit Dynamics from T omographic Reconstruc- tion .Ph y sical Review Applied 17, no. 5: 054018. (2022). doi:10.1103/Ph ysRevApplied.17.054018 8. Nic Ezzell, Bib ek P okharel, Lina T ew ala, Gregory Quiroz, and Daniel A. Lidar. Dy- namical decoupling for sup erconducting qubits: a p erformance surv ey arXiv:2207.03670 [quan t-ph] ( 2022). doi:10.48550/arXiv.2207.03670 1 co-first author 119 9. Siddarth Sriniv asan 1 , Bib ek P okharel 1 , Gregory Quiroz, and Byron Bo ots. Scalable Measuremen t Error Mitigation via Iterativ e Ba y esian Unfolding. arXiv:2210.12284 [quan t-ph] (20 22). doi:10.48550/arXiv.2210.12284 120 References [1] IBM Quan tum, 2022. URL: https://quantum- computing.ibm.com/ . [2] Rigetti. The rigetti qpu, 2018. URL: http://pyquil.readthedocs.io/en/latest/qpu.html . [3] Gregory Quiroz and Daniel A. Lidar. Optimized Dynamical Decoupling via Genetic Algorithms. Ph ysical Review A, 88(5), No v em b er 2013. doi:10.1103/PhysRevA.88. 052306 . [4] P etar Jurcevic, Ali Ja v adi-Abhari, Lev S. Bishop, Isaac Lauer, Daniela F. Bogorin, Markus Brink, Lauren Cap elluto, Okta y Günlük, T oshinari Itok o, Naoki Kanaza w a, Abhina v Kandala, George A. Keefe, Kevin Krsulic h, William Landers, Eric P . 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Abstract (if available)
Abstract
Quantum computers must be able to function in the presence of decoherence. The simplest strategy for decoherence reduction is dynamical decoupling (DD), which requires no encoding overhead and works by converting quantum gates into decoupling pulses. Here, we demonstrate that the DD method is suitable for implementation in today’s relatively noisy and small-scale quantum computers. We start by demonstrating increases in qubit fidelity due to error suppression and then show that this also translates to algorithmic improvement. We report better-than-classical success probabilities for a complete Grover search algorithm on the largest scale demonstrated to date, of up to five qubits, using two different IBM superconducting transmon qubit platforms. This is enabled, on the four and five-qubit scale, by error suppression via robust dynamical decoupling pulse sequences, without which we do not observe better-than-classical results. Then we consider the single-shot Bernstein-Vazirani algorithm, which solves the problem of identifying a hidden bitstring that changes after every oracle query. Here, we unequivocally demonstrate an unequivocal quantum speedup over the best possible classical algorithm, quantified in terms of the scaling with the problem size of the time-to-solution metric.
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Pokharel, Bibek B.
(author)
Core Title
Demonstration of error suppression and algorithmic quantum speedup on noisy-intermediate scale quantum computers
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College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Degree Conferral Date
2023-05
Publication Date
01/26/2023
Defense Date
11/28/2022
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University of Southern California
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error suppression,OAI-PMH Harvest,quantum algorithm,quantum computing
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Lidar, Daniel (
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), Brun, Todd (
committee member
), Haas, Stephan (
committee member
), Hen, Itay (
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), Levenson-Falk, Eli (
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bbk.pokharel@gmail.com,pokharel@usc.edu
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Tags
error suppression
quantum algorithm
quantum computing