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Quantum feedback control for measurement and error correction
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Quantum feedback control for measurement and error correction
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Quantum Feedback Control for Measurement and Error Correction by Jan Florjanczyk A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements of the Degree DOCTOR OF PHILOSOPHY (Electrical Engineering) August, 2016 Copyright 2016 Jan Florjanczyk Dedication To my parents. ii Acknowledgements I would rst and foremost like to thank my advisor Prof. Todd A. Brun who welcomed me into his research group by urging me to pursue what motivated me most and promising to follow down whatever path that led me. He has not only shaped my approach to research but also my view of what a great teacher should be. I have learned from him that quality uniquely endowed upon engineers working in theoretical research: a sneaking suspicion that you can always break something mysterious down into simpler and more beautiful parts. I am grateful to Professors Daniel Lidar and Edmond Jonckheere who graciously provided me with feedback and support in my qualifying exam, in this dissertation, and at many MURI Retreats on Quantum Control. I would also like to thank Professors Ben Reichardt and Paolo Zanardi for their critique and adjudication of my qualifying exam. My thanks also go out to Mark M Wilde for inspiring me to come to USC. I extend my thanks to Jos e Ra ul Gonz alez Alonso, Kung-Chuan Hsu, Chris Cantwell, Yicong Zheng, Ching-Yi Lai, Scout Kingery, Siddharth Muthukrishnan, Milad Marvian, and Lin Jing for their encouragement and support throughout the journey that is graduate school. I am grateful for being made to feel at home at the Communication Sciences Institute by Gerrielyn Ramos, Corine Wong, and Susan Wiedem. A special thanks to Jos e Ra ul Gonz alez Alonso, Patrick Davison, and Alex Leavitt for [insert joke about board games]. Most of all I would like to thank my parents, Mirek and Anna, and my sister Ursula because they have some sort of magical wormhole that defeats Euclidean distance and keeps my family close when I need them the most. iii Table of Contents Dedication ii Acknowledgements iii List of Tables vi List of Figures vii Abstract ix 1 Introduction 1 1.1 Quantum control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Continuous measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Quantum error correcting codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Quantum channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Continuous measurements with probe feedback 11 2.1 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Constraints on weak unitary pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Interleaving unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.1 A qubit-to-qubit example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Continuous measurements with Hamiltonian feedback 26 3.1 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Two-qubit example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Non-orthogonal states example . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.3 Revealing algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 In-situ noise estimation for error correction 40 4.1 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Single-parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1 Fixed dephasing angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drifting dephasing angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Multi-parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3.1 Non-degenerate grids for Bayesian inference . . . . . . . . . . . . . . . . . . 48 iv 4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 References 55 Appendices 60 A Calculation of the O( 2 ) terms of the reversibilty condition 60 B Miscellaneous lemmas 62 v List of Tables 3.1 We list all rank-n representations of Jordan algebras that can be embedded into a span of Hermitian matrices. The third representation corresponds to the 2- dimensional embedding ofC intoR. The fourth and fth representations correspond to 2- and 4-dimensional embeddings of H into C and R. The notation diag (R n ) refers to the space of n-dimensional diagonal matrices. . . . . . . . . . . . . . . . . 31 4.1 Average gains made to the code lifetime from numerical simulations of 1000 samples at each error rate. The average Frobenius error is calculated over the entire life of the estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 vi List of Figures 1.1 The random walk construction begins at x = 0 with a weak measurement given by the step operators M (0). Each successive step of the walk contributes to the total walk operator, Eq. (1.4), via one of the two step operators and updates x accordingly. The walk terminates at either endpointX where the total walk operator is designed to match the desired instantaneous measurement. . . . . . . 5 2.1 (Model A) The circuit above performs one step of the stochastic process for a pointer valuex. Time ows from left to right. The value of the pointer variable is classical and thus symbolized by a double line. All single lines are quantum states. The probe statej(x)i P is a qubit prepared according to the pointer variable. The probe and systemj i then interact for a short time via the Hamiltonian H PS . The probe is destroyed in the projective measurementfh (x)jg. The measurement result from the detector is used to update the pointer variable from x to x and the procedure is repeated with the new value. . . . . . . . . . . . 12 2.2 (a) At the beginning of the process the system qubit is in thej+i state, indicated by the green vector. The continuous measurement procedure causes the state to walk along the blue curve on the surface of the sphere, sometimes reversing direction and doubling back along it. At the end of the process, the system qubit reaches the state M 2 j+i=p 2 . (b) The random walk undertaken by the pointer variable x, illustrated by the blue line, ends when the value of x reaches either of the boundaries illustrated by the red lines. (c) The amplitudes of the state evolve towards their post-measurement values. . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 (Model B) At each timestep, we perform a weak measurement by preparing the probeji and tuning the interaction Hamiltonian H PS (x) based on a pointer variable x. The system and probe interact for a short time and the probe is measured in an orthogonal detector basish j. . . . . . . . . . . . . . . . . . . . 26 3.2 The resulting control sets as the search for closed algebras progressively removes elements of S. Each node is an attempt to solve Eq. (3.7) and the labels on the edges are the elements removed from S as a result. . . . . . . . . . . . . . . . . . 36 3.3 In the diagrams above we've indicated with black blocks the non-zero elements of all H i following the simultaneous block diagonalization procedure. (a) All matrices in the Jordan algebra of the non-orthogonal states example 3.3.2 can be written in this form in the appropriate basis. (b), (c), (d) Block diagonal forms of the three Jordan subalgebras S 1 , S 2 , and S 3 found in section 3.3.1. . . . . . . 39 vii 4.1 Our model of feedback control for optimal use of the asymmetric code uses an on- line estimate of the parametrized noise channel to modify the codespace and stabilizers. (A) We rotate all qubits in the codespace by ^ U t . (B) Stabilizer error correction using the modied stabilizers of Eq. (4.1). (C) The estimator updates the form of the stabilizer measurements. Following the syndrome extraction step, the estimator also updates the parameter estimate, symbolized by the two direc- tions of the ow of information. Quantum information and classical information are dierentiated with single and double lines respectively. . . . . . . . . . . . . . 41 4.2 (a) Performance of the adaptive stabilizers for the [[15; 1; 7=3]] shortened Reed- Muller code in the presence of xed dephasing noise. (A) Mean lifetime of 5000 samples (100 samples for p = 10 5 ) measured in terms of error correction cycles. (B) Mean lifetime of 200 samples in the presence of drifting dephasing noise ( 2 = 0:01) measured in terms of error correction cycles. (C) Mode of the lifetime of the [[15; 1; 7=3]] code without adaptive stabilizers. (D) Optimal lifetime of the [[15; 1; 7=3]] code given perfect a priori knowledge of . (E) Expected lifetime of the [[23; 1; 7]] code. (b) One run of our Bayesian estimator for drifting according to Brownian motion 2 = 0:03 and an error rate p = 0:003.) . . . . . . . . . . . . 45 4.3 Performance of the adaptive stabilizers for the [[15; 1; 7=3]] shortened Reed-Muller code and the [[31; 6; 7=5]] code with 2500 and 30; 000 particles in the presence of a xed unital channel of eccentricities (0:7; 0:2; 0:1) and an unknown orientation. 52 4.4 Eect of particle number on lifetime factor of the [[15; 1; 7=3]] and [[31; 6; 7=5]] codes and on the average Frobenius norm distance to the nearest grid element for a random channel of eccentricity (0:7; 0:2; 0:1). The dotted lines represent the standard error around the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 viii Abstract In this thesis I detail two uses of feedback quantum control that improve our ability to manip- ulate quantum systems and store quantum information. In the rst part, I demonstrate how to decompose the standard \instantaneous" quantum measurement into a continuous process. Specically, I consider continuous measurements that are eected on a target system via interaction with a stream of probes. First, I consider feedback on the stream of probes with an interaction Hamiltonian that is xed and prove that the only decompositions possible are of measurement operators with two singular values. In the second case, I consider feedback on the interaction Hamiltonian itself and prove that a much larger class of measurements is possible. I show that a linearly controlled Hamiltonian allows for decompositions of measurement operators with a number of singular values given by the rank of a Jordan algebra formed from the linear control terms. I demonstrate how to identify such algebras from existing sets of linear controls. In the second part of this thesis I study a feedback control model for stabilizer error correction that can improve the lifetimes of error correcting codes by adaptively realigning the codespace to minimize the uncorrectable error rate. The scheme matches asymmetric error correcting codes to asymmetric noise channels by estimating the channel parameters from the weights of errors observed. I show that for dephasing noise at an unknown angle, the channel estimator can align the [[15, 1, 3]] shortened Reed-Muller code so that it fails only from weight 4 bit- ip errors. I also consider dephasing noise with a time-varying dephasing angle and nd a constant increase to the code lifetime. Finally, I demonstrate how to use a randomly-spaced parametrization of a subset of unital channels to increase the average lifetime of two asymmetric codes given no prior knowledge of the channel. ix Chapter 1 Introduction The past decade of research in quantum computing has been driven by the unique promise that a large-scale computer will allow scientists to compute the solutions to previously impossible or intractable problems. Many search [32] and optimization [24, 30] tasks inherit polynomial or even exponential speedups [63] via known quantum algorithms. A future quantum computer's ability to directly simulate the dynamics of atoms in large molecules also has promising applications for ab initio computational chemistry [57]. An integral part of building such a computer then, will be the ability to prepare and control quantum systems with a high degree of precision. 1.1 Quantum control Closed-loop control has long been successfully applied to various classical systems such as ight stabilization, electronic motors, and economic models. Closed-loop quantum control, however, must be developed with extra consideration given to the coupling between the controller and the target system. In [34] the authors describe two types of controllers coupled to a target system. In the rst case, a completely quantum controller has a joint dynamical description with the target, resembling the interaction between a system and a bath. In this case, any eort to decrease the set of possible states on the target system necessarily increases the entropy of the controller state. The completely quantum coupling also implies that any joint evolution of the target and the controller are completely reversible. In the second case, a classical controller interacts with the target system by making measurements on the target and subsequently applying coherent (or unitary) pulses according to a given control strategy. 1 In this thesis, I will explore two applications of this second control strategy. I will rst consider the case of continuous feedback as it applies to implementing continuous quantum measurements. The typical \instantaneous" quantum measurement stands out in quantum theory as the only discontinuous and irreversible process. Continuous quantum measurements, however, are just as informative as the former but can be realized over a nite interval of time. Many quantum mechanical systems have naturally slow measurement times and are well described by continuous measurements. Homodyne and heterodyne measurements are widely used in optics, and produce a continuous output current [67, 62]. Magnetic resonance force microscopy [59] can also perform single-spin measurements using continuous measurements. There is a secondary purpose to the study of continuous quantum measurement, namely, a desire to formulate a continuous-time theory of non-coherent quantum operations. Whereas the continuous-time theory of coherent processes inherits the very elegant structure of Lie algebras, the theory of non-coherent quantum operations does not exhibit an obvious analog. In 1932, Pascual Jordan began work to establish a new theory of quantum mechanics. His goal was to develop an algebra that did not rely on the existence of an a priori associative algebra of matrices and one which could be described by quantum observables alone [46]. Although the development of Jordan algebras ultimately fell short of giving quantum mechanics this new home, it gave birth to a rich area of mathematics. It may be heartening, then, that this research will show an application (although certainly far removed from his original goal) of Jordan's classication of nite-dimensional Jordan algebras to the continuous measurement problem. In the second part of this thesis, I will apply feedback control as a way to extend the lifetime of asymmetric quantum error correcting codes in the presence of asymmetric noise. Again, a classical controller will perform measurements on a target system and apply unitary pulses for optimal control. The measurements will be specialized stabilizer measurements for a chosen code and the unitary pulses will be single-qubit rotations aimed at lowering the eective probability of uncorrectable errors. 2 1.2 Continuous measurements Continuous measurements can be built up from the continuous-time limit of diusive weak mea- surements (see Def. 3 below) and are often used in experiment this way. Superconducting qubits and similar solid-state devices can be measured by a weak dispersive coupling to a microwave cavity, which can be measured in turn by homodyne measurement [17, 12, 21]. The experiments of Haroche and Raimond [16, 61] use a stream of Rydberg atoms to repeatedly probe the state of a microwave mode in a superconducting cavity. In addition, latency is suciently low in these modern experiments that it is possible to do continuous feedback in real time, as has already been demonstrated in the microwave cavity/Rydberg atom system [20, 8, 9]. Continuous feedback can also be used to perform generalized measurements on quantum systems [53, 66, 26, 27], perform continuous-time quantum error correction [52, 1, 36], and reverse wavefunction collapse [38]. A continuous-time decomposition of a generalized measurement (Denition 2) is a process that accomplishes the same transition over a nite interval of time and varies the quantum state smoothly throughout. Consider that when we construct a continuous-time implementation of a coherent (i.e. unitary) evolution U, we decompose U into N time-steps of innitesimal length U = (IiH) N : where H is the Hermitian operator associated with U. Thus, at each time-step, the state is perturbed no more than by a factor linear in (for bounded eigenvalues of H). The operator IiHe iH can be termed a \weak unitary" in this sense. As unitary operators are decomposed by weak unitary operators, so too are generalized mea- surements decomposed by weak measurement operators. Specically, the innitesimal time-steps of the continuous-time decomposition are diusive weak measurements. Much like their weak unitary counterparts, these vary the target state in a controlled way via their strength param- eter . Although diusive weak measurements allow for multiple outcomes, they do not have a straightforward composition for a desired n-outcome measurement. Consider that a sequence of N diusive k-outcome weak measurements would have k N possible realizations. Furthermore, as !1 in the continuous limit, the number of possible outcomes would also approach innity, rather than converge to some desired n. 3 Rather than solve this problem directly forn-outcome generalized measurements, we will focus rst on the 2-outcome case as described in [53]. Once a scheme is devised in the 2-outcome case, it can be easily used to decompose anyn-outcome measurement by chaining together 2-outcome de- compositions. Specically, if we wish to measure the operatorsfM i g n i=1 , then we can rst measure fM 1 ; q IM y 1 M 1 g, then, if we get the second outcome,fM 2 (IM y 1 M 1 ) 1=2 ; q IM y 2 M 2 (I M y 1 M 1 ) 1=2 g, and so on. It should be noted that a decomposition scheme also exists which directly decomposes n-outcome measurements into one unbroken continuous stochastic process [66]. Denition 1 (Continuous decomposition). We call a one-parameter family of weak measurements fM (x)g x a continuous decomposition offM 1 ;M 2 g iffM (x)g x satisfy the reversibility condition (Def. 4) and the endpoints, as given by Eq. (1.5), match M 1 and M 2 . We callfM (x)g x the step operators of this decomposition. In the work that follows I will relate diusive weak measurements to generalized measurements. Both of these occur instantaneously but the latter perturbs the state in a smoothly controlled way. Note that the outcome of each measurement is a random function of the state and thus a sequence of weak measurements forms a stochastic process. Denition 2 (Generalized quantum measurements). An-outcome generalized quantum measure- ment is an instantaneous and random transformation of a quantum statej i of the form j i! M k j i q h jM y k M k j i where n X k M y k M k =1; (1.1) and where the transition k happens with probability p k =h jM y k M k j i. Denition 3 (Diusive weak measurement [14]). A diusive weak measurement is parametrized by a \strength" parameter and has operatorsfM k g n k=1 of the form M k /1 +^ " k ; (1.2) where ^ " k is an operator of bounded norm. The authors of [53] accomplish a continuous decomposition of a 2-outcome measurement by casting the weak measurement steps as corresponding to a 1-dimensional random walk indexed 4 Figure 1.1: The random walk construction begins atx = 0 with a weak measurement given by the step operators M (0). Each successive step of the walk contributes to the total walk operator, Eq. (1.4), via one of the two step operators and updates x accordingly. The walk terminates at either endpointX where the total walk operator is designed to match the desired instantaneous measurement. by the pointer variable x. In this case the result of the process does not depend on total time but, instead, on the drift of the pointer x which is, in turn, dependent on the state. Each weak measurement step updatesx tox depending on the result. A critical detail of this construction is that any dependence on the duration of the process is accounted for by constructing step operators M (x) that cancel when applied in opposite directions. More precisely, the scheme requires the steps to be reversible as dened below. Denition 4 (Reversibility condition). A one-parameter family of weak measurement operators fM (x)g x satises the reversibility condition if M (x)M (x)/1 (1.3) for all x in a given interval. The step operators M (x) in Eq. (1.3) are chosen such that the rst operator updates x to x and the second returns it to (x) = x. If the product of the two is proportional to the identity, then the operators have no net eect on the quantum state up to a normalization constant. As one can see, there is one random walk performed by the pointer x and another random walk performed by the evolution of the system state under each weak measurement. The path of both walks is uniquely parametrized by x. It is simply easier in the analysis that follows to track the walk on the operators rather than a full description ofj i. To help clarify this construction, we provide a graphical representation in Figure 1.1. 5 The total walk operator M(x) describing the the random walk above is given by the product of step operators from the initial state at x = 0 to the current value of the pointer variable x, M(x)/ 8 > > > > > > > < > > > > > > > : bjxj=c Y j=0 M + (j) x> 0 bjxj=c Y j=0 M (j) x< 0 (1.4) The particular instantaneous 2-outcome measurement to which this decomposition corresponds is given by the endpoints of the random walk in the continuous limit, M 1 = lim !0 M(X) and M 2 = lim !0 M(X) (1.5) We've motivated our construction by drawing comparisons to weak unitary operators and this may cause our construction to seem somewhat narrow. However, the random walk construction is actually far more general when framed in the context of the following two requirements: (i). That the stochastic process must faithfully reproduce the results of the instantaneous mea- surement over multiple instances, and, (ii). That the stochastic process must depend only on the state being measured and not on the total time of the evolution. Indeed, if the above two requirements are to be met, then any stochastic evolution of the quan- tum must be path-dependent and time-independent. This means that although x(t) has many realizations, M(x(t)) has only two. 1.3 Quantum error correcting codes The last two decades of research in quantum computing have yielded remarkable advances in quantum error correction and fault-tolerant quantum computing. Error correction is a necessary component for any quantum device exposed to an environment which, by necessity of interact- ing with the quantum system, will introduce noise and unwanted dynamics. Error correction is 6 considered successful when the action of the environment is minimized in some way, either by encoding information in a subspace (as is the case with decoherence-free subspace methods [44]), or by actively countering it via recovery operations as is the case with stabilizer quantum error correction [31]. LetP n be the group of all Pauli operatorsX,Y , andZ onn qubits (possibly with overall factors of1 ori).Then, a [[n;k;d]] quantum stabilizer code protectsk logical qubits by encoding them into n physical qubits. The code itself is dened by a subgroup SP n such that S is generated by a set of stabilizer operatorsfg s g. We say a quantum statej i is in the codespace of the code if g s j i =j i for all g s . A quantum state can accumulate an error Ej i by interacting with the environment. By measuring all of the stabilizer operatorsg s we not only projectE into an element ofP n but also recover the result of all 2 nk measurements which collectively form the syndrome of E. Many dierent errors can result in the same syndrome and we denote by d the distance of the code if it can unambiguously correct errors of weightb(d 1)=2c where the weight of the error is the number of non-identity elements in E following its projection intoP n . Generally, the stabilizer code construction is designed only in terms of the weights of its correctable errors, and not in terms of the specic error model of a particular quantum device. It is natural then to consider error correction schemes that are adapted specically to certain types of errors. Many methods for doing this have already been considered including designing new error correction procedure through direct optimization [65], concatenating repetition codes with Calderbank-Shor-Steane (CSS) codes in the presence of dominant dephasing noise [5], and using asymmetric quantum codes to combat asymmetric noise [23, 37, 60, 7, 43]. Denition 5 (Asymmetric quantum stabilizer code [37]). The asymmetric quantum stabilizer code denoted by [[n;k;d X =d Z ]] encodesk logical qubits inton physical qubits and can correct errors with up to t x =b(d x 1)=2c X operators and t z =b(d z 1)=2c Z operators. In Chapter 4 we will design a system that exploits this asymmetry in the presence of noise biased towards one type of error over the other. We will primarily use the [[15; 1; 7=3]] shortened Reed-Muller code [10] which is the smallest asymmetric CSS code and has the added benet of allowing transveral non-Cliord operations [15]. Another CSS construction from [6] using Bose- Chaudhuri-Hocquenghem (BCH) codes details a [[31; 6; 7=5]] asymmetric code. It should be noted that our method is compatible with all asymmetric stabilizer codes and the advantage that our 7 feedback control scheme yields is only a function of the noise channel and the code distances. As such, the analysis contained in this thesis will apply to all codes that express asymmetric X and Z distances. 1.4 Quantum channels In general, a quantum operation is any map on the set of bounded positive operators on a Hilbert space that is completely positive I.e.: I n is positive for identity maps of any dimension n. Every such operation can be written in terms of Kraus operators A i () = X i A y i A i : (1.6) Additionally, if the operation preserves the trace, that is Tr [ ()] = Tr [] for any state then it is known as a quantum channel. Denition 6 (Unital quantum channel). A unital quantum channel is a completely positive trace-preserving map on quantum states for which the maximally mixed state is a xed point, that is (I=d) =I=d (1.7) when acts on states in a Hilbert space of dimension d. A unital channel can equivalently be expressed as the convex combination of unitary chan- nels [55] and for this reason they often appear as the result of a partial trace after a joint unitary evolution over a quantum state and an environment. A common type of unital channel is the Pauli channel. Denition 7 (Pauli channel). A Pauli channel is a channel over n qubits that can be written in the the form E () = (1p) + X i p i A i A i (1.8) for P i p i =p and A i 2fI;X;Y;Zg n . 8 In this work we will considern identical single-qubit channels acting onn distinct qubits. The state of each qubit can be represented as a vector ~ r2R 3 contained inside the Bloch sphere [49]. We write the qubit state as = I +~ r~ 2 (1.9) where~ = [X;Y;Z]. Unital channels that act on single qubits can be described by their action on the vector~ r () = I + (M~ r)~ 2 (1.10) where the Bloch matrix M is any real 33 matrix that respects the Fujiwara-Algoet conditions [28, 11, 13]. Any such matrixM can always be decomposed using the polar decompositionM =Q V P where Q V 2 SO(3) and P is positive semi-denite. When the action of the channel can be described entirely by P (i.e.: Q V =I), we have the result of Lemma 6 found in Appendix B, M = (1 2p)I + 2pQ T U 2 6 6 6 4 k 1 0 0 0 k 2 0 0 0 k 3 3 7 7 7 5 Q U : (1.11) where Q U 2 SO(3). Note that when Q U = I and p < 1=2, this reduces to a single-qubit Pauli channel. When Q U 6=I, we refer to this as an oriented Pauli channel. Denition 8 (Oriented Pauli channel). An oriented Pauli channel M is the result of a unitary channel U , followed by a Pauli channel D , and the inverse unitary channel U y. That is, M = U y D U (1.12) From the polar decomposition above, it is clear that any general unitary channel can be decomposed as the conjugation of an oriented Pauli channel with a unitary channel V . Characterizing quantum channels is routinely done with quantum tomography which, to date, has yielded powerful methods for faithfully reconstructing process matrices with a tractable num- ber of samples [58, 25]. Tomography methods such as compressed sensing require the preparation of resource-intensive randomized quantum states and measurements. A full reconstruction of all 9 the Kraus operators constituting a quantum operation might not be needed for every task how- ever. The goal of more recent work done in modeling error channels has been to allow for ecient simulation of fault-tolerance [40, 33, 56, 45]. In these works, the authors make approximations of physical error models by eectively projecting the true error channel into an approximation that can be simulated eciently. In [22], the authors perform a coarse-grained averaging of the qubits by applying random Cliord operators and qubit permutations which allow for ecient extraction of channel parameters such as the probability phase- ip errors of any given weight. We specically consider characterizing and exploiting asymmetry of a quantum channel in-situ. We will estimate channel parameters by using the existing error correction apparatus which has the benet of not causing interruptions to ongoing computations. It also allows us to account for noise channels that may be changing over time. The authors of [18] already do this with great success in the presence of dephasing noise followed by an unknown X-axis rotation. The recent experiments of [41] demonstrate a similar method in a system of nine physical qubits implementing the 5-bit repetition code. Again, the authors are able to compensate for time-varying parameters in the noise channel but without the restriction that the channel, or the corrective controls, be identical on all qubits. Like these works, our goal in learning channel parameters will be to improve the performance of error correction. 10 Chapter 2 Continuous measurements with probe feedback 2.1 Control scheme We report now on original work found in two publications [26, 27]. Our aim in these two works was to provide a way to perform the step operators M (x) in a way that was motivated by realistic physical experiments. We considered two dierent ways to construct the step operatorsM (x). In both models the target system interacts with a probe qubit which is measured in lieu of the system itself. The models dier in how the pointer variable x is used to update the control scheme. In model A, which we analyze in this chapter, x is used to tune the stream of probe qubits whereas, in model B (see Chapter 3), x is used to tune the probe-system interaction. We illustrate model A in Figure 2.1. In the continuous-time limit, the feedback loop performed by the circuit is considered to occur instantaneously. The step operators in model A have the form M (x) =h (x)je iH PS j(x)i; (2.1) wherej(x)i is the probe qubit,j (x)i are two orthogonal states associated with the destructive measurement of the detector, and H PS is the interaction Hamiltonian between the probe P and system S. In the following section we characterize precisely, the set of measurementsfM 1 ;M 2 g decom- posable by this model. Before we begin, however, we note that we are aorded one advantage through our random walk construction because the reversibility condition of Eq. (1.3) need not be met exactly. Consider that a classical random walk must take O(N 2 ) steps to converge with 11 Figure 2.1: (Model A) The circuit above performs one step of the stochastic process for a pointer value x. Time ows from left to right. The value of the pointer variable is classical and thus symbolized by a double line. All single lines are quantum states. The probe statej(x)i P is a qubit prepared according to the pointer variable. The probe and systemj i then interact for a short time via the Hamiltonian H PS . The probe is destroyed in the projective measurement fh (x)jg. The measurement result from the detector is used to update the pointer variable from x to x and the procedure is repeated with the new value. xed probability (whereN =X=). This implies that the total walk operator in Eq. (1.4) will ac- cumulate N 2 O() terms, N 2 O( 2 ) terms, and so on. However, since the contribution of N 2 O( 3 ) terms vanishes as! 0 regardless of whether or not the step operators are exactly reversible, we only require that the reversibility condition be met only up to O( 2 ). 2.2 Main result The most general Hamiltonian we can write for the interaction of a qubit probe and an arbitrary quantum system is the following H PS =1 H S +X H X +Y H Y +Z H Z (2.2) where 1, X, Y , Z are the usual Pauli matrices on the probe state P , and H S , H X , H Y , H Z are corresponding Hermitian matrices on the system S. We note that the form of any step operator is now given entirely in terms of the geometry of the probe and the detector states. Viewing these states as vectors on the Bloch sphere, we can summarize them by dening the probe basis below. Denition 9 (Probe basis). For any qubit probej(x)i and projective qubit measurementh j, we dene a real orthonormal basis for the Bloch spheref~ n 1 (x);~ n 2 (x);~ n 3 (x)g. We call this a probe basis if 12 • ~ , the Bloch vector associated with the probe statej(x)i, is no further than from ~ n 1 in Euclidean distance, • ~ n 2 is the Bloch vector associated withj + i, • and ~ n 3 =~ n 1 ~ n 2 . This notation reveals the following constraint between the probe and the detector Bloch vectors, the proof of which can be found in appendix B. Lemma 1 (Probe basis of a weak measurement). Any diusive weak measurement given by model A with a probe basisf~ n 1 (x);~ n 2 (x);~ n 3 (x)g must have ~ n 2 ~ O(). Thus, there always exists an orthonormal basis for the Bloch sphere that approximates the probe and detector states. Recall that for a given interaction Hamiltonian we seek to characterize the weak measurement step operators achievable via any probe feedback loop parametrized by x. However, we can instead x a probe basis for the probe feedback loop and consider a family of interaction Hamiltonians H 0 PS (x) which give rise to the same set of weak measurement step operators. This transformation is performed with the following identications H PS =X H X +Y H Y +Z H Z (2.3) =~ n 1 (x) ~ PH 1 (x) +~ n 2 (x) ~ PH 2 (x) +~ n 3 (x) ~ PH 3 (x) (2.4) where ~ P = [X ; Y ; Z] T and 2 6 6 6 4 H 1 (x) H 2 (x) H 3 (x) 3 7 7 7 5 = 2 6 6 6 4 ~ n 1 (x) ~ n 2 (x) ~ n 3 (x) 3 7 7 7 5 2 6 6 6 4 H X H Y H Z 3 7 7 7 5 : (2.5) We will also later abbreviate the vectors above as ~ H 0 (x) = [H 1 (x); H 2 (x); H 3 (x)] T and ~ H = [H X ; H Y ; H Z ] T . We dene an interaction Hamiltonian in the probe basis as H 0 PS (x) =X H 1 (x) +Y H 2 (x) +Z H 3 (x); (2.6) 13 and this yields an advantageous rewriting of the weak measurement step operators M (x) =h (x)j exp (iH PS )j(x)i (2.7) =hj exp (iH 0 PS (x))j0i +O(): (2.8) In this basis, the detector statesj i areji, the1 eigenstates ofX, and the initial statej(x)i is close toj0i, the +1 eigenstate of Z. This choice also allows us to ignore the 1 P H S term in the general Hamiltonian of Eq. (2.2) sincehj1j0i =hjZj0i and any contribution from H S can be rewritten as part of H 3 . This is the setup used in [26] which yields the following result. Theorem 1. Any continuous measurement with step operators generated by model A, can only realize a 2-outcome measurement of the form M 1 =U 1 ( S + S ?)V; (2.9) M 2 =U 2 p 1 2 S + p 1 2 S ? V; (2.10) where U 1 , U 2 , and V are unitary matrices, S , S ? are projectors onto orthogonal subspaces of the system space, and ;2 (0; 1). In lemma 7 we required that the probe and detector Bloch vectors be orthogonal only up to O(). We will therefore allow the probe states to be perturbed fromj0i in our analysis. This causes an adjustment in our expression for the weak measurement step operators, parametrized by two functions c(x) and (x), j(x)i = cos (c(x))j0i + sin (c(x))e i (x) j1i j0i +c(x)e i (x) j1i = j0i +j(x)i; (2.11) where in the last line we have implicitly denedj(x)i = c(x)e i (x) j1i. Equivalently, this con- tributes a term of O() to our step operators M (x) =hje iH 0 PS (x) j0i +hje iH 0 PS (x) j(x)i: (2.12) 14 Grouping together O(1), O() and O( 2 ) terms in the above expression yields M (x) =hj j0i +j(x)i 1 +ihjH 0 PS (x) j0i +j(x)i 2 2 hjH 0 PS (x) 2 j0i = 1 p 2 + ihjH 0 PS (x)j0i c(x)e i (x) 1 p 2 2 2 hjH 0 PS (x) 2 j0i 2ihjH 0 PS (x)j(x)i = 1 p 2 +M (1) (x) 2 2 M (2) (x); (2.13) where we have implicitly dened M (1) (x) and M (2) (x) to collect the O() and O( 2 ) terms. We can now write the reversibility condition in terms of the above: M (x)M (x) = 1 p 2 +M (1) (x) 2 2 M (2) (x) 1 p 2 +M (1) (x) 2 2 M (2) (x) = 1 2 + p 2 M (1) +M (1) (2.14) 2 2 p 2 2@ x M (1) +M (2) +M (2) 2 p 2M (1) M (1) ; where we have dropped x-dependence in the last line for legibility. First, for the O() term we nd M (1) +M (1) =i p 2hH 0 PS i 0 =i p 2H 3 : We provide the calculations for the following O( 2 ) terms in Appendix A: M (2) +M (2) 2 p 2M (1) M (1) = p 2 2(H 2 2 +H 2 3 ) +i [H 1 H 3 ;H 2 ] ifH 1 ;H 2 g [H 3 ;H 1 ] + 4ce i H 2 : (2.15) 15 Expanding the @ x M (1) term is a bit more complicated since it corresponds to an innitesimal rotation of the probe basis at each value of x. We can dene an axis of rotation on the Bloch sphere with three components ~ (x) = [ 1 (x) ; 2 (x); 3 (x)], so that @ x ~ H 0 (x) = ~ (x) ~ H 0 (x). This implies that 2@ x M (1) = p 2 [i;1;i] ~ ~ H 0 ; (2.16) where we have ignored any term proportional to the identity operator as these automatically satisfy the reversibility condition. Altogether these reductions yield the expression of interest for the reversibility condition, M (x)M (x) = 1 2 +iH 3 2 2 n [i;1;i] ~ ~ H 0 +2(H 2 2 +H 2 3 ) +i [H 1 H 3 ;H 2 ] ifH 1 ;H 2 g [H 3 ;H 1 ] + 4ce i H 2 o : (2.17) Finally, we group terms into four types: constant-Hermitian A, stochastic-Hermitian B (that is, with a factor of), constant-anti-Hermitian i A, and stochastic-anti-Hermitian i B: M (x)M (x) = 1 2 +iH 3 2 2 AB +i Ai B ; (2.18) where A = [ ~ ~ H 0 ] 2 + 4c cos H 2 + 2H 2 2 + 2H 2 3 +i [H 1 ;H 2 ]; B = i [H 2 ;H 3 ]; A = [ ~ ~ H 0 ] 1 + 4c sin H 2 fH 1 ;H 2 g; B = [ ~ ~ H 0 ] 3 i [H 1 (x);H 3 (x)]: If the reversibility condition is to be satised then these, along with the O() term, must each be individually proportional to 1. This can be done either by restrictions on the Hamiltonian terms, or by canceling the terms through unitary pulses applied when the random walk changes direction. 16 To eliminate the H 3 term, we must either set H 3 /1, or perform a weak unitary pulse of the form U 1 = exp (2iH 3 ). As it happens, setting H 3 /1 does not change the analysis that follows and thus we'll assume instead that the experimentalist performs the pulse U 1 at each reversal of the walk direction. Next, we assume that both the A and B term can be eliminated via a series of weak uni- tary pulses of the form U 2 = exp i 2 H where H is some Hermitian operator containing linear combinations and products of H X , H Y , and H Z . This leaves only A and B terms: A = [ ~ ~ H 0 ] 2 + 4c cos H 2 + 2H 2 2 +i [H 1 ;H 2 ] (2.19) B = i [H 2 ;H 3 ]: (2.20) SinceB is traceless it cannot be proportional to the identity and must be set to 0. However, A is not traceless and we must consider a more complicated solution, one where A is equal to 1 for some constant : 3 H 1 1 H 3 + 4c cos H 2 + 2H 2 2 +i [H 1 ;H 2 ] =1: Using lemma 8 in the appendix we nd that 3 H 1 1 H 3 + 4c cos H 2 + 2H 2 2 =1 which also implies that [H 1 ;H 2 ] = 0. Together with the commutation relation fromB, this means that we can now express all Hamiltonian terms in one common diagonal basis: H PS = X j (Xx j (x) +Yy j (x) +Zz j (x))jj(x)ihj(x)j: (2.21) In order to satisfy the condition A/1, we must consider the diagonal components of H 2 as they appear in equation Eq. (2.19): @ x y j (x) =q 0 (x) +q 1 (x)y j (x) +q 2 (x)y 2 j (x); (2.22) 17 whereq 0 (x) = from above,q 1 (x) =4c(x) cos (x), andq 2 (x) = 2. This dierential equation is a special instance of the Riccati equation, the solution to which can be found in [48]. In particular, if any solution, y (1) (x), is known, then the general solution is of the form y j (x) =y (1) (x) + (x) C j R q 2 (x)(x)dx : (2.23) for (x) = exp R 2q 2 (x)y (1) (x) +q 1 (x)dx. The important feature of this solution is that there is only one free parameter C j available to match any boundary condition. To complete the proof, we focus on which instantaneous measurements M 1 ,M 2 are achievable at the end points of a continuous decomposition. First, note that all terms in Eq. (2.21), including the diagonal basisjj(x)ihj(x)j, are assumed to be x-dependent. Consider the unitary U(x) which diagonalizesH 1 (x),H 2 (x), andH 3 (x). Each of these is a linear combination ofH X ,H Y , andH Z , and since they are all linearly independent,U(x) must also diagonalizeH 1 ,H 2 , andH 3 . Whatever unitary does this, however, cannot depend onx, and therefore, the basisjjihjj is not x-dependent. Only they j (x),x j (x), andz j (x) coecients depend onx. This means that every step operator is diagonal in the same basis, and we can write the general form M (x) = 1 p 2 p 2 X j (y j (x)c(x) cos (x))jjihjj + p 2 X j i (z j (x)x j (x)c(x) sin (x))jjihjj: Thus the endpoint measurement operators must also be diagonal in the j basis, and the rst of these has the form M 1 / lim !1 bX=c Y j=0 M + (j) (2.24) / lim !1 bX=c Y j=0 diag (1y j (x) +i (z j (x) +x j (x))) = diag exp Z X 0 y j (x) +i (z j (x) +x j (x))dx !! : 18 where the notation diag () represents a diagonal matrix with entries indexed byj. Bothx j (x) and z j (x) only contribute a total phase to each of the diagonal elements. If we let w (1) j = R X 0 z j (x) + x j (x)dx and W 1 = diag expiw (1) j then M 1 /W 1 diag exp Z X 0 y j (x)dx !! : (2.25) Following a similar procedure, we nd that M 2 /W 2 diag exp Z 0 X y j (x)dx (2.26) withW 2 dened accordingly. Recall however, that these must form a complete measurement, and so they must satisfy b 2 1 M y 1 M 1 +b 2 2 M y 2 M 2 = 1 where b 1;2 are the appropriate normalizations for the measurement operators. This condition restricts the parameter C j . Consider thej th diagonal entry of M y 1 M 1 (up to an overall normalization identical for all j), exp 2Y (X) 2 Z X 0 (x) C j R x 0 q 2 (s)(s)ds dx ! = exp 2Y (X) Z X 0 2(x)dx C j R x 0 2(s)ds ! (2.27) = e 2Y(X) C j C j R X 0 2(s)ds ! ; (2.28) where we have implicitly dened Y (X) = Z X 0 y (1) (x)dx Q(x) = Z x 0 2(s)ds: The equation for measurement completeness in each diagonal component is then 1 = b 2 1 C j e 2Y(X) C j Q(X) + b 2 2 C j e 2Y(X) C j Q(X) (2.29) Multiplying both sides of this equation by (C j Q(X)) (C j Q(X)) we get a quadratic equation in C j which has at most two roots (we restrict our attention to real solutions for C j since y j (x) is real). In fact, even if one were to consider two arbitrary boundaries for the random walk, i.e.: 19 thatX be replaced withX 1 and +X be replaced withX 2 , then the expression forC j would still be quadratic and, again, yield only two possible solutions. Finally, we can group the diagonal elements of M 1 andM 2 in terms of the two possible values of C j and express the entire measurement operators as the linear combination of two orthogonal projectors S and S ?. LetC 1 andC 2 be the two roots of Eq. (2.29), and letS =fj : C j =C 1 g, the projectors are then S = X j2S jjihjj S ? = X j= 2S jjihjj: (2.30) The constants and appearing in the statement of Theorem 1 are = b 1 e Y(X) s C 1 C 1 Q(X) ; (2.31) = b 2 e Y(X) s C 1 C 1 Q(X) : (2.32) To complete the proof of Theorem 1, we must justify the appearance of unitariesU 1 ,U 2 andV . First, the rotationV is simply any rotation that an experimentalist applies to the systemS before beginning the measurement procedure. For this reason, it does not depend on the measurement outcome. On the other hand, U 1 andU 2 are rotations applied to the system after the continuous measurement procedure is complete and do not have to be identical. We can absorb the diagonal unitary matricesW 1 andW 2 , which accumulate over the continuous procedure, into the denitions of U 1 and U 2 , respectively. 2.3 Constraints on weak unitary pulses In the section above we made repeated use of weak unitary pulses to reduce the equations for reversibility. However, some of these were generated by Hamiltonians with products of H 1 , H 2 , and H 3 . Since we aim for our result to apply for a general but xed interaction Hamiltonian H PS or, equivalently, three system Hamiltonian terms H X ,H Y ,H Z , it may be too demanding to assume that a set of unitary pulses generated by their products would also be readily available. For this reason, we now consider satisfying the reversibility condition again, but only allowing weak unitary pulses generated by linear combinations of H X , H Y , and H Z . 20 We examine this more restricted set of solutions to the reversibility condition by reintroducing the constraints on H 1 , H 2 , and H 3 that we removed in the previous section by using a weak unitary pulse generated by their products. In particular, we reintroduce the conditions on A and B that required fH 1 ;H 2 g/1; and [H 1 ;H 3 ]/1: The second of these is already automatically satised. The rst yields the following relationship between the eigenvalues of H 1 and H 2 : x j (x)y j (x) = (x) 8j;x; (2.33) for some (x) independent of j. So far our result has only placed a restriction on the number of distinct singular values that the measurement can have. Here we'll actually be able to prove something about the singular values in the interaction Hamiltonian itself. We dene ~ j = ( (x) j ; (y) j ; (z) j ) as the triplet of the j th eigenvalues of H X , H Y , and H Z , and ~ 0 j (x) = (x j (x);y j (x);z j (x)) as the triplet of the j th eigenvalues of H 1 (x), H 2 (x), and H 3 (x). In the solution from the previous section, each ~ 0 j (x) corresponds to some ~ j via a rotation. This rotation takes each triplet from the original basis, to the probe basis in the same way thatj(x)i andh (x)j were rotated from the original basis to j0i andhj. Since we found only two solutions fory j (x), we must also restrict ~ j to lie in one of two planes in the original basis. Furthermore, the restriction in Eq. (2.33) requires that the vectors ~ 0 j (x) be constrained to lie on one of two lines parallel to z in the probe basis. In the original basis, this restricts all vectors ~ j to lie on one of two parallel lines, l 1 and l 2 . If we have only two assignments for ~ j then any probe basis is possible so long as the resulting y j (x) match those of the solution to the Ricatti equation above. However, with three or more assignments, we can only allow probe bases related by a rotation around the axis parallel to the lines l 1 ,l 2 . The constraint Eq. (2.33) limits this even further and we require that y 1 (x) =y 2 (x), meaning also that ~ j =c j ~ 0 ~ 1 for some constantc j , ~ 0 parallel to the linesl 1 ,l 2 , and arbitrary ~ 1 . This last expression for ~ j completely restricts the interaction Hamiltonian one should use if only pulses generated by linear combination of Hamiltonian terms are available. 21 2.4 Interleaving unitaries In [53] the authors generalize their result for positive measurement operators to general measure- ment operators by taking the polar decomposition of the endpoint measurement operators. In other words, M 1 = V 1 (M y 1 M 1 ) 1=2 , and similarly for M 2 . They then construct a one-parameter family of unitary operationsfV (x)g x that yieldV 1 atx =X,V 2 atx =X and 1 atx = 0. The step operators are rst constructed so as to correspond to the positive operators (M y 1;2 M 1;2 ) 1=2 and padded by unitary operators chosen from the family V (x) as follows M (x) =V (x) ~ M (x)V y (x) (2.34) where ~ M (x) is the step operator for the positive part of the polar decomposition. We will show that in our analysis, padding the step operator with this family of unitary operators is equivalent to a shift in the H 1 (x) term. Consider expanding the ~ M (x) term in Eq. (2.34) in terms of , M (x) = 1 p 2 V (x)V y (x) (2.35) +iV (x)hjH 0 PS (x)j0iV y (x): Recall that V (x) forms a continuous family of unitary operators, and if we let V (x) = e iG(x) , then V (x)V y (x) =1i@ x G(x): (2.36) This means we can summarize Eq. (2.34) as M (x) = 1 p 2 + p 2 n i ~ H 1 +@ x G ~ H 2 +i ~ H 3 o (2.37) where ~ H 1;2;3 =VH 1;2;3 V y . If we make the identication H 0 PS (x) =X H 1 +@ x G +Y H 2 +Z H 3 for the interaction Hamiltonian in the probe basis, then we've recovered the form of the step operators that were used in Theorem 1 but with a small modication. Namely, an experimentalist 22 now has the power to introduce a shift to the H 1 term which contributes directly to the unitary terms U 1;2 that appear in the endpoint measurements M 1;2 . It is important to note, however, that all of the derivations of the reversibility condition still follow and we have not aected the singular value decomposition of M 1;2 . 2.5 Applications 2.5.1 A qubit-to-qubit example While the result of Theorem 1 restricts the class of measurements that can be achieved with constant interaction Hamiltonians, model A is sucient to realize any 2-outcome measurement on a qubit. This result complements existing work on qubit-to-qubit open-loop feedback control [4, 19]. A generalized diagonal measurement on a qubit takes the form M 1 =W 1 2 4 0 0 3 5 (2.38) M 2 =W 2 2 4 p 1 2 0 0 p 1 2 3 5 (2.39) whereW 1 andW 2 are unitary matrices. We will use the interaction Hamiltonian Z P Z S in this example. Expressing the interaction Hamiltonian in the probe basis yields H 0 PS (x) =X (n x 3 (x)Z) +Y (n y 3 (x)Z) +Z (n z 3 (x)Z) The step operators take the form M (x) = 1 p 2 p 2 e 4 R c(x)dx Zc(x)1 in x 3 (x)Z p 2 : (2.40) where c(x) is a factor resulting from a warping of the probe basis [26]. If we choose the probe basis warping c(x) to be c(x) = 1 2 (tanh (xa) + tanh (xb)); (2.41) 23 then we recover the endpoint operators M 1 /W 1 2 4 e R X 0 tanh(xa)dx 0 0 e R X 0 tanh(xb)dx 3 5 (2.42) M 2 /W 2 2 4 e R 0 X tanh(xa)dx 0 0 e R 0 X tanh(xb)dx 3 5 : (2.43) where W 1 = e iZ and W 2 = e iZ for some value resulting from the integration of n x 3 (x). An appropriate choice of a and b will yield the desired generalized diagonal measurement: a = ln s tanhX + (2 1) tanhX (2 1) ; (2.44) b = ln s tanhX + (2 1) tanhX (2 1) : (2.45) The following choice of probe basis corresponds to the values of ~ n 3 described above: 2 6 6 6 4 ~ n 1 ~ n 2 ~ n 3 3 7 7 7 5 = 2 6 6 6 4 0 c(x) p 1c(x) 2 0 p 1c(x) 2 c(x) 1 0 0 3 7 7 7 5 (2.46) Figure 2.2 shows a simulation of this scheme for = 0:8, = 0:2 where the initial state of the system qubit isj i =j+i. It is worthwhile noting that the scheme described here also generalizes to the entire class of measurements derived in Theorem 1. In particular, if we choose the interaction Hamiltonian H PS =Z P ( S S ?); (2.47) then the exact same choices of probe and detector states result in operators M 1 , M 2 almost identical to those of Eq. (2.38) and Eq. (2.39) but with the values and copied into the diagonal entries corresponding to spaces S andS ? respectively. The unitaries V , U 1 , and U 2 can simply be applied appropriately before and after the continuous procedure. 24 −1 0 1 −1 0 1 −1 −0.5 0 0.5 1 X Y Z (a) 0 1 2 3 4 x 10 4 −6 −4 −2 0 2 4 6 Step no. Walk position (and bounds) (b) 0 1 2 3 4 x 10 4 0 0.2 0.4 0.6 0.8 1 Step no. State amplitudes (c) Figure 2.2: (a) At the beginning of the process the system qubit is in thej+i state, indicated by the green vector. The continuous measurement procedure causes the state to walk along the blue curve on the surface of the sphere, sometimes reversing direction and doubling back along it. At the end of the process, the system qubit reaches the state M 2 j+i=p 2 . (b) The random walk undertaken by the pointer variable x, illustrated by the blue line, ends when the value of x reaches either of the boundaries illustrated by the red lines. (c) The amplitudes of the state evolve towards their post-measurement values. 25 Chapter 3 Continuous measurements with Hamiltonian feedback 3.1 Control scheme Model B, illustrated in Figure 3.1, follows a very similar structure to model A but thex-dependence is now in the interaction Hamiltonian H PS (x) instead of in the probe and detector states. The step operators in model B have the form M (x) =h je iH PS (x) ji; (3.1) whereji andj i are xed for the entire decomposition. Figure 3.1: (Model B) At each timestep, we perform a weak measurement by preparing the probe ji and tuning the interaction Hamiltonian H PS (x) based on a pointer variable x. The system and probe interact for a short time and the probe is measured in an orthogonal detector basis h j. 26 3.2 Main result In the previous section we required that the probe state, the orthogonal quantum states of the detector, and the eigenstates of the interaction Hamiltonian, have mutually orthonormal repre- sentations on the Bloch sphere. For this reason, we choose the interaction Hamiltonian to be H PS (x) =Y P ^ "(x), the probe state to bej0i, and the detector states to behj. The operator acting on the system S is ^ " and is dened to be an x-dependent linear combination of d constant Hamiltonian terms, ^ "(x) = d X i=0 p i (x)H i : (3.2) The step operators of Figure 3.1 then become M (x) 1 p 2 I p 2 ^ "(x) 2 2 p 2 ^ " 2 (x): (3.3) The reversibility condition can now be rewritten in terms of ^ "(x). Collecting terms by orders of yields M (x)M (x) = I 2 + 2 2 @ x ^ "(x) 2^ " 2 (x) +O( 3 ): Let (x) be the proportionality constant in Eq. (1.3). We nd that the reversibility condition reduces to @ x ^ "(x) = 2^ " 2 (x) +(x)I: (3.4) In the derivations that follow, we will ignore the (x)I term, as it will not change the class of measurements that satisfy the reversibility equation. (In practice, the term can be reintroduced to help nd bounded solutions.) Consider the set of controls that appear in Eq. (3.2). Without loss of generality, we can always assume that H 0 = I since the action of I is equivalent to an overall phase on the probe system. The reversibility condition can then be rewritten as d X k=0 @ x p k (x)H k = d X i;j=0 p i (x)p j (x) 1 2 fH i ;H j g: (3.5) 27 wheref;g is the anti-commutator. It will be useful to introduce the tensor k ij for expressing the action of the anti-commutator on the matrices H i . In particular, 1 2 fH i ;H j g = n(n1)=2 X k=0 k ij H k : (3.6) We choose the matrices H i for i > d such that they form a basis forH n (C), the space of all n-dimensional complex Hermitian matrices. We will use (k) to denote the matrix resulting from xing the index k. The reversibility equation Eq. (3.4) can then be read as 8 < : @ x p k =~ p T (k) ~ p 0kd 0 =~ p T (k) ~ p d<k: (3.7) We now present the main result of [27] which characterizes solutions to the above equations. Theorem 2 (Main result). A continuous measurement using qubit probes and closed-loop feedback on the interaction Hamiltonian (as in Fig. 4.1) can realize any measurementfM 1 ;M 2 g of the form M 1 = S(V) M l=1 U y l 0 @ rank(B l ) M i=1 (l) i (l) i 1 A U l ; (3.8) where M 2 = (IM y 1 M 1 ) 1=2 is diagonal in the same basis. The parameters (l) i are real and contained in (0; 1) and (l) i is a projector onto 1, 2, or 4 basis states. The proof of this theorem is given in the three lemmas that follow. Let us denoteF = spanfH i g so that ^ "2F. We prove the following lemma about solutions to Eq. (3.7). Lemma 2. Any solution ^ "(x) to Eq. (3.7) must lie entirely in V, a subspace of F that is closed under anti-commutation. Proof. We note that ifF is already closed under anti-commutation, then the reversibility equation reduces to an initial value problem in terms of the control coecients ~ p(x) at x = 0. However if F is not closed under anti-commutation, then we must characterize the set of vectors ~ p such that Eq. (3.7) is satised. To do so, consider choosing any k >d and solving the associated equation 28 ~ p T (k) ~ p = 0. Note that the matrix (k) is symmetric and denes a quadratic space overR d . Every quadratic space admits a Witt decomposition [50] which in our case is (k) ;R d = N M i=0 W i V 0 V 0 : (3.9) In the above, W i are hyperbolic planes, V 0 is the nullspace of (k) , and V 0 is an anisotropic subspace ofR d . Solutions to~ x T W i ~ x = 0 are spanf[1; 1]g[ spanf[1;1]g. Additionally, there are no vectors which satisfy~ x T V 0 ~ x = 0 for the anisotropic subspaceV 0 . LetT (k) be the isomorphism of (k) ;R d to I d ;R d and ~ p =T (k) ~ q. Then possible solutions to ~ p T (k) ~ p = 0 must lie in V =T (k) N M i=0 spanf[1;x i ]gV 0 ! (3.10) for a xed choice of x i =1. To fully solve Eq. (3.7) we must now recurse the above procedure. At each step we restrict ~ p to lie in the subspace V dened by a choice of x i . We then dene a new matrix basis for the controls restricted to V and generate a new set of (k) matrices. We then choose a new k and decompose V using (k) . Since the order in which the k are chosen will aect the form of V , it is also important to enumerate all sequences of choices of k and x i . This procedure terminates when the vector space of Hermitian matrices V formed from V is closed under anti-commutation. Furthermore, since the Witt decomposition is unique (up to isometries of V 0 ), we can guarantee that this procedure lists all closed subspaces contained in F. It remains only to show that if~ p(0)2V for a particular sequence of choices ofk andx i , that~ p(x) will remain in the same subspace for all other values of x. This follows directly, however, from the fact that if ^ "(x)2V then ^ " 2 (x)2V and so @ x ^ "2V. Lemma 2 establishes that in order to solve the reversibility equation, one must use a set of controls whose span is closed under anti-commutation. The proof of the lemma also includes an implicit algorithm for nding closed subspaces given a set of Hermitian matrices. The next lemma gives the structure of the subspaces enumerated by lemma 2. 29 Lemma 3. The ^ "(x) operator has the form ^ "(x) = S(V) M l=1 U l (x)D l (x)U y l (x): (3.11) where S(V) is the number of simple components of the algebra V (with anti-commutation as its product), and D l (x) andU l (x) correspond to the l th simple component and are given by Table 3.1. Proof. We begin by identifying V as a nite-dimensional Jordan algebra. Every such algebra accepts a Wedderburn-type decomposition [3, 54], V = S(V) M l=1 B l ; (3.12) whereS(V) is the number of simple componentsB l ofV. A classication of all nite-dimensional simple Jordan algebras was given by Jordan, von Neumann, and Wigner [39]. The three types of Jordan algebras that can be found in our decomposition are the self-adjoint real, complex, and quaternionic matrices. The isomorphism in Eq. (3.12) leaves a lot of freedom in terms of how to represent each of these simple components by Hamiltonian terms. We summarize the possible representations in Table 3.1. (Note that the exceptional Albert algebra is absent, since octonions do not have a matrix representation overR orC [2]). SinceV can be written as a direct sum, we can also write ^ " = S(V) M l=1 ^ " l (x): (3.13) Each operator in the direct sum can, in turn, be diagonalized to yield the form in the statement of the lemma. Before we proceed to the nal lemma that will complete our main result, we will give a few illustrative examples of the last three representations found in Table 3.1. First, we consider the 30 BlockB l D l (x) U l (x) Dimension H n (R) diag (R n ) SO(n) n(n 1)=2 H n (C) diag (R n ) SU(n) n 2 H n (C) =H 2n (R) diag (R n ) I 2 SO(n) SO(2) n 2 H n (H) =H 2n (C) diag (R n ) I 2 SU(n) SU(2) 2n 2 n H n (H) =H 4n (R) diag (R n ) I 4 SO(n) SO(4) 2n 2 n Table 3.1: We list all rank-n representations of Jordan algebras that can be embedded into a span of Hermitian matrices. The third representation corresponds to the 2-dimensional embedding of C intoR. The fourth and fth representations correspond to 2- and 4-dimensional embeddings of H intoC andR. The notation diag (R n ) refers to the space of n-dimensional diagonal matrices. matrix algebra resulting from the 2-dimensional embedding of C into R, i.e. H n (C) =H 2n (R). Let the coecients of a matrix inH n (C) be u jk =a jk +ib jk , then 2 6 6 6 6 4 u 00 ::: u 0n . . . . . . . . . u n0 ::: u nn 3 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 a 00 b 00 ::: a 0n b 0n b 00 a 00 ::: b 0n a 0n . . . . . . . . . . . . . . . a n0 b n0 ::: a nn b nn b n0 a n0 ::: b nn a nn 3 7 7 7 7 7 7 7 7 7 7 5 (3.14) describes the embeddingH n (C) =H 2n (R). For the fourth and fth representations in Table 3.1, we replace each quaternionic elementh =a +b ^ i +c ^ j +d ^ k by one of the following two submatrices. ForH 2n (C) we use h jk = 2 4 a jk +ib jk c jk +id jk c jk +id jk a jk ib jk 3 5 ; (3.15) and forH 4n (R) we use h jk = 2 6 6 6 6 6 6 4 a jk b jk c jk d jk b jk a jk d jk c jk c jk d jk a jk b jk d jk c jk b jk a jk 3 7 7 7 7 7 7 5 : (3.16) 31 Note that up to this point, we've ignored the term(x)I in the reversibility equation Eq. (3.4). We were able to ignore it in lemma 2 because the matrix (0) that corresponds to H 0 =I never appears as one of the matrices we use during the recursive part of the proof. We were also able to ignore it in lemma 3 since all of the simple Jordan algebras listed above always contain a full set of rank-1 idempotents. We will now reintroduce the term (x)I as it will play an important role in regularizing the behavior of the dierential equations in lemma 4. Lemma 4. The ^ "(x) operator and the total walk operatorM(x) are simultaneously diagonalizable. Proof. We begin by noting that Eq. (3.4) can be solved for individual blocks ^ " l (x), yielding @ x U l (x)D l (x)U y l (x) = 2 U l (x)D l (x)U y l (x) 2 +(x)I l : where I l is the identity on the block (l). A few simple manipulations result in the following equivalent expression (for clarity, we've omitted the x-dependence) @ x U l D l U y l +U l @ x D l U y l +U l D l @ x U y l = 2U l D 2 l U y l +I l : Applying U y l and U l from the left and the right gives U y l @ x U l D l +@ x D l +D l @ x U y l U l = 2D 2 l +I l : Since U l is a unitary matrix we can write it as the exponent of a Hermitian matrix G l and we note that U y l @ x U l =i@ x G l . This reduces the above equation to i [@ x G l ;D l ] +@ x D l = 2D 2 l +I l : (3.17) The entries of the commutator term are ([@ x G l ;D l ]) ij =@ x g (l) ij d (l) i d (l) j (3.18) 32 from which we can infer that the diagonal entries of the commutator term are 0 if d (l) i 6= d (l) j . Thus, the equation for any d (l) i (x) reduces to @ x d (l) i (x) = d (l) i (x) 2 +(x) (3.19) which is a special case of the Ricatti rst-order non-linear dierential equation. Without loss of generality however, we can simply consider the case where (x) is xed at a strictly positive real constant . Although a dierent choice of (x) will ultimately lead to dierent solutions for the functionsd (l) i (x), this will not expand the class of measurements possible with our scheme. Thus, xing (x) =, Eq. (3.19) has the solution p tanh p xc (l) i : (3.20) This form immediately implies that for anyi;j such thatc (l) i 6=c (l) j ,g (l) ij is constant. Furthermore, we can even prove that for other solutions where (x) has a more general form, g (l) ij is constant when d (l) i 6= d (l) j . This is because the general solution d (l) i (x) to the special Ricatti equation has only one free parameter c (l) i , d (l) i (x) =f(x) + e 2 R f(x)dx c (l) i R e 2 R f(x)dx dx : (3.21) where f(x) is a known solution to Eq. (3.19). It can be seen, from this general solution that d (l) i =d (l) j only in the case where c (l) i =c (l) j , just as with the tanh solution. Finally, for the case of (x) = Eq. (3.17) has the solution 8 > > > < > > > : d (l) i (x) = p tanh p xc (l) i 8i; g (l) ij (x) =g (l) ij (0) 8i;j : c (l) i 6=c (l) j ; g (l) ij (x) =g (l) ij (x) 8i;j : c (l) i =c (l) j : We note that in the cases where c (l) i = c (l) j , g (l) ij need not be constant. However, in these cases, the x-dependent sub-block of G l (x) is acting on a sub-block of D l (x) that is proportional to the identity. Thus, this freedom in G l (x) does not aect the form of ^ ", or of M 1 , M 2 . 33 We now turn our attention to the total walk operator given in Eq. (1.4) which obeys the following dierential equation (up to a normalization factor): @ x M(x) =^ "(x)M(x): (3.22) We can write M(x) in the diagonal basis of ^ "(x) by introducing the operator N(x) = 0 @ S(V) M k=1 U y l 1 A M(x) 0 @ S(V) M l=1 U l 1 A : (3.23) Eq. (3.22) can then be rewritten as @ x N(x) = S(V) M l=1 D l (x)N(x)i S(V) M l=1 [@ x G l (x);N l (x)]: (3.24) Note that sinceM(0) =I thenN(0) =I and so the commutator term above disappears for all x. This immediately imples thatN(x) must be diagonal and so the total walk operator and the ^ "(x) operator are diagonal in the same basis. We now complete our proof of Theorem 2. Proof. Recall that the number of distinct diagonal entries possible inD l (x) is rank (B l ). However, each distinct entry can appear 1, 2, or 4 times depending on the particular representation from Table 3.1. Using lemma 4 we can plug our solution for D l (x) into Eq. (3.24) to nd that the diagonal entries of N(x) are (l) i (x) = exp Z x 0 p tanh p yc (l) i dy : (3.25) The total walk operator M(x) must then be M(x)/U y l 0 @ rank(B l ) M i=1 (l) i (x) (l) i 1 A U l : (3.26) The endpoint operatorsM 1 andM 2 are proportional toM(X) andM(X). Their diagonal entries are (l) i , which after renormalization approach 0 when c (l) i !1 and 1 when c (l) i !1. 34 Note that in theorem 2 the eigenvalues of M 1 and M 2 are restricted to lie in the open set (0; 1), not the closed set [0; 1]. This is a consequence of the reversibility condition at the points x =X andx =X +. At these points, setting any eigenvalue of the total walk operator to 0 would be eectively a projection, which is an irreversible operation for the random walk. However we can approach arbitrarily close to any such projective measurement. Consider the following corollary. Corollary 1 (Spectrum of the measurement). Given the ability to perform any unitary trans- formations directly before and after the continuous process of theorem 2, one can continuously decompose any measurement with P S(V) l=1 rank (B l ) distinct singular values. Proof. The endpoint measurement operatorsM 1 ,M 2 in theorem 2 can have up to P S(V) l=1 rank (B l ) distinct eigenvalues. We can decompose any pair of general endpoint operatorsM 1 ,M 2 using their polar decompositions M i =W i (M y i M i ) 1=2 . Then, we can use a procedure like that of Figure 4.1 to measure the positive Hermitian operators (M y i M i ) 1=2 and subsequently applyW i depending on the measurement result. To allow for direct comparisons between models A and B, we can count the number of distinct singular values in M 1 and M 2 as realized by both models. In model A, we can only realize the 2 distinct singular values associated with the two orthogonal projectors. In model B, however, we can realize up to P S(V) l=1 rank (B l ) singular values. 3.3 Applications We now illustrate how to make use of Theorem 2 in a few low-dimensional examples. 3.3.1 Two-qubit example Consider the following control terms consisting of Pauli operators on two qubits S =fII;ZI;IZ;ZZ;XXg: (3.27) Note that this set is not closed under anti-commutation sincefXX;ZZg=2 = YY . We will use a simplied version of the implicit algorithm found in lemma 2 to nd algebras contained in S. 35 S S 1 ZZ ZI S 2 ZZ IZ S 3 XX Figure 3.2: The resulting control sets as the search for closed algebras progressively removes elements of S. Each node is an attempt to solve Eq. (3.7) and the labels on the edges are the elements removed from S as a result. Note that strictly speaking, the optimal algorithm in lemma 2 nds all algebras contained in F = spanfSg and our example below will ignore possible linear combinations of terms or any rewriting of the control set in a new basis. Nonetheless, we can demonstrate the branching nature of the search for closed algebras as we descend into smaller sets. The result of the search yields the following three subsets closed under anti-commutation. S 1 =fII;IZ;XXg (3.28) S 2 =fII;ZI;XXg (3.29) S 3 =fII;ZI;IZ;ZZg: (3.30) The identity II along with any one of the Pauli operators in S is also a closed algebra but since these are smaller than these above, we do not list them. The process of nding S 1 , S 2 , and S 3 is illustrated in Figure 3.2. 36 3.3.2 Non-orthogonal states example Consider two quantum statesj1i andj2i such thath1j2i =a witha2R. We choose the following set of control terms H 0 =I H 1 =j1ih1j H 2 =j2ih2j H 3 =j1ih2j +j2ih1j S =fH 0 ;H 1 ;H 2 ;H 3 g One can check thatF = spanfSg is closed under anti-commutation. However, which semi-simple Jordan algebra is F a representation of? To reveal the structure, consider rewriting F in the following basis H 0 0 =H 1 (3.31) H 0 1 = a 2 H 1 +H 2 aH 3 1a 2 (3.32) H 0 2 = 2aH 1 +H 3 p 1a 2 (3.33) H 0 3 =H 0 H 1 +H 2 aH 3 1a 2 (3.34) In this form, when we calculate the productf;g=2, we nd that H 0 0 H 0 0 =H 0 0 H 0 2 H 0 0 =H 0 2 H 0 3 H 0 0 = 0 H 0 0 H 0 1 = 0 H 0 2 H 0 1 =H 0 2 H 0 3 H 0 1 = 0 H 0 1 H 0 1 =H 0 1 H 0 2 H 0 2 =H 0 0 +H 0 1 H 0 3 H 0 2 = 0 H 0 3 H 0 3 =H 0 3 37 From these relations it is clear to see that H 0 0 , H 0 1 , and H 0 3 are idempotents and that F = spanfH 0 0 ;H 0 1 ;H 0 2 g spanfH 0 3 g (3.35) =H 2 (R)R: (3.36) 3.3.3 Revealing algebraic structure For all of the structure that theorem 2 reveals, it does not explicitly resolve the most natural question that might occur to a scientist trying to engineer a continuous measurement decomposi- tion. In particular, given a control set F =fH i g closed under the productf;g=2, which Jordan algebra do the H i represent? As it turns out, we can reveal the semi-simple structure of F via a simultaneous block diagonalization of the matrices H i . An excellent algorithm for doing so was found by [47] and it applies exactly to the question we've outlined here. The algorithm follows three simple steps. First, sample a random linear combination of the matrices H i H = d X i=0 u i H i u i unif [0; 1]: (3.37) Next, diagonalizeH =Q T DQ whereD is a diagonal matrix of eigenvaluesf k g with multiplicities fm k g. Let Q k be the basis of the eigenspace of k . Now, dene an equivalence class as follows jj 0 () 9i : Q T j H i Q j 06= 0: (3.38) That is to say, if any control termH i has a non-zero o-diagonal elements in the [j;j 0 ] block, then the eigenspacesj andj 0 are grouped into a single block. This check proceeds for allj and alli and results a partition of the eigenvectors in Q. The partition represents a direct-sum decomposition of the Hilbert space on which theH i act and, therefore, a semi-simple decomposition of the Jordan algebraF. Note that this technique requires complete descriptions of all H i as matrices and such a description may not necessarily be available or even possible to obtain accurately. Furthermore, once we've identied the semi-simple blocks using this method, we still cannot automatically identify the exact simple Jordan algebras contained therein. For example, consider a 24-dimensional Hilbert space which contains 36 control terms forming a closed Jordan algebra. 38 0 0 0 0 (a) 0 0 0 0 0 0 0 0 (b) 0 0 0 0 0 0 0 0 (c) 0 0 0 0 0 0 0 0 0 0 0 0 (d) Figure 3.3: In the diagrams above we've indicated with black blocks the non-zero elements of all H i following the simultaneous block diagonalization procedure. (a) All matrices in the Jordan algebra of the non-orthogonal states example 3.3.2 can be written in this form in the appropriate basis. (b), (c), (d) Block diagonal forms of the three Jordan subalgebras S 1 , S 2 , and S 3 found in section 3.3.1. These could represent eitherH 8 (R) I 3 orH 6 (C) I 4 , both of which are rank 36. In order to distinguish between the two, the exact form of the control terms needs to be examined more closely. 39 Chapter 4 In-situ noise estimation for error correction 4.1 Control scheme We now present a model for reducing the rate of uncorrectable errors in two dierent parameterized noise channels. Our model of feedback control will be to apply a coherent rotation ^ U t to the physical qubits and to modify our stabilizer measurements by the same unitary as shown in Figure 4.1. Specically, for stabilizer generators g s , g s ! ^ U t n g s ^ U y t n : (4.1) In the single-parameter case we describe below, we will use this construction to eectively rotate a dephasing channel into a bit- ip channel. Later when we examine a multi-parameter model, this will counteract the unitary parts U of an oriented Pauli channel. We will approximate the rate of uncorrectable errors by counting those errors whose weight exceeds the code distance although in practice it may be possible to improve on our results by considering correctable errors of all weights. The uncorrectable error rate for the asymmetric code [[n;k;d X =d Z ]] is p fail = X T n w x ;w y ;w z p wx x p wy y p wz x (1p) 1w (4.2) whereT =fw x ;w y ;w z : w x +w y t x ;w z +w y t z g, t i = (d i 1)=2, p = p x +p y +p z , and w =w x +w y +w z . 40 ˆ θ t ρ θ t A Λ θ C B ˆ θ t+1 ρ θ t+1 Figure 4.1: Our model of feedback control for optimal use of the asymmetric code uses an online estimate of the parametrized noise channel to modify the codespace and stabilizers. (A) We rotate all qubits in the codespace by ^ U t . (B) Stabilizer error correction using the modied stabi- lizers of Eq. (4.1). (C) The estimator updates the form of the stabilizer measurements. Following the syndrome extraction step, the estimator also updates the parameter estimate, symbolized by the two directions of the ow of information. Quantum information and classical information are dierentiated with single and double lines respectively. 4.2 Single-parameter estimation We begin with a channel which is a single-parameter generalization of dephasing noise. The action of the channel on one qubit can be described by a single Kraus operator, A () () = (1p) +pA ()A (): (4.3) where A() =e iY Ze iY : (4.4) We do not know a priori the parameter and thus the eective channel, as perceived by our measurements of the stabilizer generators, is that of under the Pauli twirl approximation [40]: P [ ] () = (1p) +p cos 2 XX +p sin 2 ZZ: (4.5) The goal of the protocol below is to recover the parameter and substitute the stabilizer operators in the standard basisg s with new stabilizers that are aligned to exploit the asymmetry of the noise channel and maximize the lifetime of the code. In particular, we will apply ^ U t = e i ^ Y to the code space and the stabilizers where ^ is our estimate of recovered after each round of error correction. 41 4.2.1 Fixed dephasing angle Our rst case, and the one where we will be able to demonstrate the most gain in code lifetime, is to consider an angle that is xed in time. We use a Bayesian estimator that begins with uniform belief about the orientation channel p( 0 ;) = 1= for 0 2 [0;). Every round of standard error correction will diagnose and correct a total of w x bit- ip errors and w z phase- ip errors. We update our belief about as follows: p( t+1 jw x ;w z ; ^ t ;) = p(w x ;w z j t+1 ; ^ t ;)p( t+1 ; ^ t ;) R p(w x ;w z j t+1 ; ^ t ;)p( t+1 ; ^ t ;)d t+1 (4.6) where t+1 is the random variable resulting from the update, ^ t is the angle to which the stabilizers were congured in the previous round of error correction, and p(w x ;w z j t ; ^ t ;) = n w x ;w z p wx+wz (1p) nwxwz cos 2wx t+1 ^ t sin 2wz t+1 ^ t : (4.7) After each update, we choose the new alignment of the stabilizers to be the maximum likelihood estimator of : ^ t = argmax t p( t jw x ;w z ; ^ t1 ;): (4.8) This update rule has a particular advantage we can exploit to reduce the complexity of our estimator going forward. Note that when w x 0 and w z = 0, ^ t = ^ t1 , which follows from the fact that cos 2 t+1 ^ t has a single maximum on the interval [0;) and is symmetric around it. Thus, we can perform our updates to the distribution of t in bulk whenever we encounter a correctable Z error by multiplying the distribution by the function f( t+1 ;n x ; ^ t ;) = cos 2nx t+1 ^ t sin 2 t+1 ^ t : (4.9) 42 where n x is the number of X errors observed since the previous update, and renormalizing. This yields an advantage for our simulation as well. Since we need only update ^ t on single Z errors (as the [[15; 1; 7=3]] code fails for w z > 1), we can sample the time until the next Z error and retroactively sample the number of X errors to have occurred in the intermittent time (accounting for uncorrectable X errors as well). This proves especially useful when our estimator is very close to the optimal value since in this case Z errors happen very infrequently and the error rate is dominated by O(p 4 ) terms. Furthermore, the distribution p( t ) cannot be stored in memory exactly and must be dis- cretized. Let p( t ) be dened for the midpoints of the cells [0;) = N [ j=1 j N ; (j + 1) N : (4.10) Thus, the minimum error between and the maximum likelihood estimate ^ t is=2N. This means that the rate ofZ errors can be, at best, suppressed top sin 2 (=2N). Recall, however, that we do not need fully suppress Z errors in order to take advantage of an asymmetric code. For very low eective Z error-rates, the uncorrectable error rate in Eq. (4.2) is governed by uncorrectable X errors. If p z =cp tx=tz x for some c> 0 then p fail = c n t z + 1 + n t x + 1 p tx x +O p tx+1 x (4.11) where t x = 3 and t z = 1 for the [[15; 1; 7=3]] code. We can expect this very low rate only when p z = cp 2 x which is possible when sin 2 (=2N) = O(p) or N = O(1=p). Thus, we can only make optimal use of our estimate of when we take N =O(1=p) partitions of [0;). The results of our simulations in Figure 4.2 show that for a xed dephasing angle, our technique not only improves the code lifetime by a constant factor, but also eectively increases the code distance. The mean lifetime of our adaptive [[15; 1; 7=3]] code agrees with a power law of p fail = O p 3:99 yielding an \eective" code distance of 6:98. The next smallest CSS code that could yield a similar scaling is the [[23; 1; 7]] code but our adaptive technique even outperforms this code by a constant factor thanks to the smaller number of physical qubits used. 43 4.2.2 Drifting dephasing angle We now address the version of the previous problem with a parameter that is drifting in time. Consider a dephasing angle that at each time-step evolves via random Brownian motion, t+1 = t +u with uN (0; 2 ): (4.12) We can incorporate our knowledge of this drift into the estimator from the xed angle case by convolving, at each time-step, our belief about p( t ) with the distribution of one step of the Brownian motion. That is, following every update in Eq. (4.6) we also apply p 0 ( t+1 ) =p( t+1 ) 1 exp 2 t+1 =2 2 : (4.13) We then take ^ t+1 to be the maximum likelihood estimate with respect to p 0 ( t+1 ) instead. Unlike our simulations of the xed angle case, we cannot make use of the retroactive X error sampling that allowed us to simulate code lifetimes in the range of 10 17 cycles and therefore we must simulate every update straightforwardly. However, as Figure 4.2 demonstrates, our estimator retains the property that the maximum likelihood estimate ^ t only moves following Z errors. Figure 4.2 shows that using this modied estimator still yields a constant factor improvement to the lifetime of the [[15; 1; 7=3]] code but the lifetimes no longer scale with O(1=p 4 ), the rate of uncorrectable X errors. Instead, the drift in the channel eectively causes a constant but suppressedZ error rate since the drift in t guarantees we cannot stay in the optimal conguration for very long. This fact is also re ected in our estimator by the convolution step which widens any otherwise \sharp" (i.e.: certain) belief about ^ t . From our simulations, the constant factor gain to the code lifetime for 2 = 0:01 is 6:33. 4.3 Multi-parameter estimation Although the single-angle dephasing model from the previous section is useful for studying the limits to which we can exploit a code's asymmetry, it is not very general and we now turn our 44 10 -5 10 -4 10 -3 10 -2 Dephasing channel error rate p 10 1 10 5 10 9 10 13 10 17 10 21 Lifetime [cycles] A B C D E (a) (b) Figure 4.2: (a) Performance of the adaptive stabilizers for the [[15; 1; 7=3]] shortened Reed-Muller code in the presence of xed dephasing noise. (A) Mean lifetime of 5000 samples (100 samples for p = 10 5 ) measured in terms of error correction cycles. (B) Mean lifetime of 200 samples in the presence of drifting dephasing noise ( 2 = 0:01) measured in terms of error correction cycles. (C) Mode of the lifetime of the [[15; 1; 7=3]] code without adaptive stabilizers. (D) Optimal lifetime of the [[15; 1; 7=3]] code given perfect a priori knowledge of. (E) Expected lifetime of the [[23; 1; 7]] code. (b) One run of our Bayesian estimator for drifting according to Brownian motion 2 = 0:03 and an error rate p = 0:003.) 45 attention to the case of oriented Pauli channels. Recall that the Bloch matrix for such a channel can be written as a contraction in some non-standard basis, M = (1 2p)I + 2pQ T 2 6 6 6 4 k 1 0 0 0 k 2 0 0 0 k 3 3 7 7 7 5 Q (4.14) for some Q2 SO(3). We choose this form for the Bloch matrix in order to highlight our true objective for estimatingM, i.e.: to rotate our codespace and stabilizers in a way that the channel appears Pauli and therefore optimizes our use of the asymmetric code. Letk i be the eccentricities of the oriented Pauli channel such that k 1 +k 2 +k 3 = 1, p be the total error rate, and Q be the orientation of the channel. Note that this channel, unlike the one in the previous section, can give rise toY errors. These can be corrected by any CSS code as simultaneous X andZ errors on the same qubit. Finally, we let A be the matrix such that M = (1 2p)I + 2pA. Note that we do not need knowledge of M in order to design the control unitary ^ U t for this scheme but rather knowledge of A is sucient to align the largest value of k i with the X axis. In other words, the optimal choice of ^ U t is invariant with respect to the total error rate p (although estimating p can also be done eciently [29]). If we let ^ Q t be the real orthonormal matrix associated with the unitary ^ U t , then the eect of this rotation on the eective Bloch matrix is M = (1 2p)I + 2p ^ Q t A ^ Q T t and it is clear that we should choose ^ Q t to be the matrix that diagonalizes A and orders the eigenvalues such that k 1 k 3 k 2 whereby Z errors are favored over Y errors and X errors favored to both. The performance of oriented Pauli channels with k i > 0 is limited since even in their optimal orientation, they exhibit non-zeroZ andY error rates. These channels cannot yield improvements to the eective code distance like those in the dephasing case but can yield improvements to the coecient of the uncorrectable error rate (and thereby the code lifetime). Let C opt be the constant factor improvement to the lifetime yielded by the [[15; 1; 7=3]] code for an oriented Pauli channel given perfect a priori knowledge of the optimal orientation Q opt . C opt then is given by the ratio of the expected lifetimes C opt = E [t(Q opt )] R E [t(Q)]dQ (4.15) 46 whereE [t(Q)] is the lifetime at orientationQ given by 1=p fail (Q), the uncorrectable error rate. We take R dQ to be an integral over the Haar measure forSO(3). Fork 1 ;k 2 ;k 3 p, the uncorrectable error is dominated by ZZ, YY , and ZY errors. The approximation becomes p fail (Q) = 15 0; 2; 0 p 2 y + 15 0; 0; 2 p 2 z + 15 0; 1; 1 p y p z = 105(p y +p z ) 2 = 105(pp x ) 2 (4.16) The X error rate is p x = p~ e T 1 QDQ T ~ e 1 where ~ e i are the standard basis vectors and D is the diagonal matrix of k i . Note that for Q with columns ~ q i the X error rate reduces to p~ q T 1 D~ q 1 in which ~ q 1 follows a uniform distribution on SO(3) given by, ~ q = 2 6 6 6 4 u p 1u 2 sin (2v) p 1u 2 cos (2v) 3 7 7 7 5 8 < : u unif [1; 1]; v unif [0; 1]: (4.17) And we can thus derive the distribution on k x k x =k 1 u 2 +k 2 (1u 2 ) sin 2 2v +k 3 (1u 2 ) cos 2 2v: (4.18) Thus, in order to take the Haar integral needed for the average lifetime, we must calculate Z E [t(Q)]dQ = Z 1 1 Z 1 0 1 105p 2 (1k x ) 2 dvdu (4.19) Z 1 1 1 105p 2 (1k 1 u 2 k 3 (1u 2 )) 2 du (4.20) = Z 1 1 1 105p 2 ((1k 3 ) (k 1 k 3 )u 2 ) 2 du (4.21) = tanh 1 p (k 1 k 3 )=(1k 3 ) 210p 2 (1k 3 ) p (1k 3 )(k 1 k 3 ) + 1 210p 2 (1k 3 )(1k 1 ) (4.22) 47 where in the second line we've used that k 3 k 2 . For the channel of eccentricities (0:7; 0:2; 0:1) the optimal and average lifetime become E [t(Q opt )] = 11:11 105p 2 (4.23) Z E [t(Q)]dQ 2:71 105p 2 : (4.24) Thus, the optimal improvement to the code lifetime that we can expect is C opt 4:01. A similar calculation for the [[31; 6; 7=5]] code yields C opt 9:34. 4.3.1 Non-degenerate grids for Bayesian inference We will use Bayesian inference to estimate the relevant parameters of the matrixA. Note, however, that were we to learn the rates ofX,Y andZ errors, we could only determine the matrixA up to its diagonal elements. This is a problem acknowledged by [51] and [18]. In [51] the authors propose \toggling" the codespace to estimate o-diagonal elements of the Bloch matrix by rotating the codespace and pre-processing the encoded state. Although their method is applicable to much more general multi-qubit process matrices, it cannot be performed in-situ without incurring signicant costs by moving the codespace away from the optimum. This problem manifests itself for the Bayesian inference method when we try to estimate the posterior distribution on A by subdividing the parameter space into an even grid. Consider that any error E is equally likely to have been caused by the two Bloch matrices G and H if their diagonal elements match, i.e.: Pr 8 > > > < > > > : E 2 6 6 6 4 g 11 ? ? ? g 22 ? ? ? g 33 3 7 7 7 5 9 > > > = > > > ; = Pr 8 > > > < > > > : E 2 6 6 6 4 h 11 ? ? ? h 22 ? ? ? h 33 3 7 7 7 5 9 > > > = > > > ; (4.25) even though the diering o-diagonal elements could indicate very dierent orientations of the channel. Regularly spaced grids over a parametrization of the matrix A will have a large number of such degeneracies and the maximum likelihood estimator will not be able to distinguish between them. 48 The solution we propose is to sample the parameter space randomly and track posterior prob- abilities for a eld of N \particles" in the parameter space of A. Of course, even a randomly sampled parameter space has a non-zero probability of including channels whose posteriors are degenerate in the sense above but this probability depends on the numerical precision of our simulations and is extremely unlikely even for very high values of N. We will sample each particle X in our randomized grid as follows. First, we will sample eccentricities x 1 , x 2 , x 3 according to the distribution p (x 1 ) = 1 for x 1 2 [0; 1] (4.26) p (x 2 jx 1 ) = 1=(1x 1 ) for x 2 2 [0; 1x 1 ] (4.27) and x 3 = 1x 1 x 2 . We take D X to be the diagonal matrix of x 1 , x 2 , and x 3 . Note that the average channel in our grid will have eccentricities (1=2; 1=4; 1=4) which means that our prior distribution assumes some asymmetry in the channel. Next, we sample an orthonormal basis Q X from the Haar measure over SO(3) (for which ecient methods exist [42]). Finally, we let our particle be X =Q T X D X Q X (4.28) We now relate the number of particlesN to the optimal performance of our asymmetric codes. We do this in two parts: rst, by bounding the minimum distance of a xed channel A to the closest element in a randomized grid, and second by bounding the rates of uncorrectable errors using the minimum distance of the grid. The rst part is summarized in the following Lemma, the proof of which can be found in Appendix B. Lemma 5 (Particle number versus minimum distance). LetX i beN i.i.d copies of randomly ori- ented Pauli channels according to (4.28) and letA be a xed channel with eccentricities (a 1 ;a 2 ;a 3 ). Then we have that Pr n min i kX i Ak 2 <" o N " 5 2 14 3 3 a 2 1 a 2 a 3 (4.29) wherekYk 2 = p Tr [Y T Y ] is the Frobenius norm. 49 As an example, for the oriented Pauli channel with eccentricities k 1 = 0:7, k 2 = 0:2, and k 3 = 0:1 and 30; 000 particles, we can expect PrfZ <"g 6:92" 5 : (4.30) The estimate we found for the code performance in Eq. (4.15) assumes complete and exact knowl- edge of the channel matrix A. What we've found in the derivation above, however, is a bound on our ability to learn an approximation X of the matrix A. As such, the lifetime of our code will not be bounded by the optimal performance of Eq. (4.15) but rather by the quality of this approximation. We now derive the relationship between the Frobenius norm distance between X and A and the code performance. Let x 11 be the rst diagonal element of X, then we know that jx 11 k 1 jkXAk 2 =) j1x 11 jj1k 1 j +kXAk 2 : (4.31) We identify the lifetime coecient C as the ratio of p fail for random and optimal orientations which can be bounded from below by C (2=3) tz (1k 1 +kXAk 2 ) tz : (4.32) We call this lower bound the expected performance. 4.4 Numerical results In Figure 4.3 we graph the results of 1000 trials of our estimation method for the [[15; 1; 7=3]] and the [[31; 6; 7=5]] codes at N = 2500 and N = 30; 000. We also plot the expected performance as a function of the Frobenius norm using our bound from above. Recall that the expected performance serves as a lower bound and our results indicate that it is reasonably tight. Note that the distribution of lifetimes for the [[31; 6; 7=5]] code is signicantly more uniform than for [[15; 1; 7=3]] suggesting the larger code's lifetime does not follow the geometric distribution. We suspect this behavior can be explained in the following way: the 31-qubit code fails at a rate of 50 Code Number of particles Improvement to lifetime C Frobenius error at time of failure Average Frobenius error [[15; 1; 7=3]] 2500 2.04 0.37 0.33 [[15; 1; 7=3]] 30; 000 2.17 0.41 0.32 [[31; 6; 7=5]] 2500 3.17 0.20 0.17 [[31; 6; 7=5]] 30; 000 3.68 0.24 0.18 Table 4.1: Average gains made to the code lifetime from numerical simulations of 1000 samples at each error rate. The average Frobenius error is calculated over the entire life of the estimator. p 3 for weight 3Z errors and thus the average number of updates to the posterior at the expected time of failure scales with 1=p 3 . The 15-qubit code on the other hand, fails at a rate of p 2 and so has, on average, learned a worse estimate of A than the 31-qubit code at the expected time of failure. This suggests that the code distance has a secondary eect beyond simply decreasing the baseline rate of uncorrectable errors. Our data suggest that larger codes benet proportionally more from our adaptive technique as they are able to learn the channel more accurately before the probability of an uncorrectable error becomes appreciably high. In Table 4.1 we summarize the improvement to the lifetime and the Frobenius error at the time of failure of each code. In Figure 4.4 we illustrate that the minimum Frobenius distance of a random channel does indeed follow a power law of N 1=5 but our bound from Eq. (4.29) overestimates the true grid spacing by a signicant factor. In the same gure we demonstrate the decay of the improvement to the code lifetime with decreasing particle number N. Surprisingly, our method still yields some benet with only 10 particles. This is likely due to the fact that when sampling random channels and random grid points, the probability of generating a grid point close to A is still high enough that some trials have this property and inherit disproportionately long lifetimes, thereby raising the average. 51 0 1 2 3 4 5 6 Normalized lifetime (adaptive lifetime / non-adaptive lifetime) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 MLE error (Frobenius norm) Average performance Expected performance p=0.002 p=0.005 p=0.01 [[15, 1, 7/3]] code (2500 particles) Adaptive Non-adaptive (a) 0 1 2 3 4 5 6 Normalized lifetime (adaptive lifetime / non-adaptive lifetime) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 MLE error (Frobenius norm) Average performance Expected performance p=0.002 p=0.005 p=0.01 [[15, 1, 7/3]] code (30,000 particles) Adaptive Non-adaptive (b) 0 2 4 6 8 10 Normalized lifetime (adaptive lifetime / non-adaptive lifetime) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 MLE error (Frobenius norm) Average performance Expected performance p=0.005 p=0.01 [[31, 6, 7/5]] code (2500 particles) Adaptive Non-adaptive (c) 0 2 4 6 8 10 Normalized lifetime (adaptive lifetime / non-adaptive lifetime) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 MLE error (Frobenius norm) Average performance Expected performance p=0.005 p=0.01 [[31, 6, 7/5]] code (30,000 particles) Adaptive Non-adaptive (d) Figure 4.3: Performance of the adaptive stabilizers for the [[15; 1; 7=3]] shortened Reed-Muller code and the [[31; 6; 7=5]] code with 2500 and 30; 000 particles in the presence of a xed unital channel of eccentricities (0:7; 0:2; 0:1) and an unknown orientation. 52 10 1 10 2 10 3 10 4 10 5 Number of particles 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Average lifetime improvement factor 0.0 0.1 0.2 0.3 0.4 0.5 Average min. Frobenius norm Lifetime improvement [[15, 1, 7/3]] Lifetime improvement [[31, 6, 7/5]] Min. distance Figure 4.4: Eect of particle number on lifetime factor of the [[15; 1; 7=3]] and [[31; 6; 7=5]] codes and on the average Frobenius norm distance to the nearest grid element for a random channel of eccentricity (0:7; 0:2; 0:1). The dotted lines represent the standard error around the mean. 4.5 Discussion By adaptively encoding information in asymmetric error correcting codes we've demonstrated an in-situ method for exploiting asymmetric noise channels. Our method provides a constant factor improvement to the code lifetime in all cases and in the case of xed dephasing noise at an unknown axis even results in a higher \eective" code distance. Since our control unitaries on the physical qubits are functions of only the syndrome statistics, our method is also able to track noise parameters that drift in time. It is interesting to note that in the case of oriented Pauli channels, the performance of our code is dominated by our ability to learn the direction of the largest channel eccentricity. This allows us to align our code in a way that matches this eccentricity to X errors, which our code is best equipped to correct. The gains we make by learning the orientation of the second largest eccentricity are much less impactful since our code's ability to correct Y errors is only slightly worse than for Z errors. In addition to our method of adaptively rotating the codespace and stabilizers, one might also consider re-encoding the logical quantum state as information about the noise channel becomes more rened. For example, if the noise channel is discovered to be highly asymmetric for a 53 sustained period of time, then one can convert the stabilizer code into one that tolerates fewer Z andY errors and thereby save on the total number of physical qubits and stabilizer measurements. Several works [64, 10, 35] already demonstrate how to convert between these codes fault-tolerantly. Furthermore, one might consider correcting for some types of errors less often, as is done in [23] in the presence of a strongly biased channel. It would seem that our method could also extend to unital channels in general. In this case, we would model the process matrixM not with an eigenvalue decomposition but with a singular value decomposition and our task would be to estimate both the left and right bases ofM. Equivalently, we could retain our oriented Pauli channel and concatenate it with another unitary channel. This is similar to [18] where the authors consider a Pauli channel in the standard basis concatenated with a unitary rotation. In both cases, a straightforward application of our randomized grid would incur a further " 3 scaling to the bound on the particle number in Eq. (4.29), signicantly driving up the minimum number of particles needed to yield lifetime improvements. Finally, the most natural extension of our method would be to consider separate noise channels and separate control unitaries for each qubit in the code. This case can be addressed by simply running parallel channel estimators for each qubit with updates applied only for errors that occur on that qubit. Note that in this case the learning rate drops by 1=n on average and each estimator converges much more slowly. Nonetheless it is likely that the parameters of each channel would be correlated based on the qubit topology, an assumption which could be used to decrease the number of free parameters needed. 54 References [1] C. Ahn, A. C. Doherty, and A. J. Landahl. Continuous quantum error correction via quantum feedback control. Phys. Rev. A, 65:042301, Mar 2002. [2] A. A. Albert. On a certain algebra of quantum mechanics. Annals of Mathematics, 35(1):pp. 65{73, 1934. [3] A. A. Albert. The wedderburn principal theorem for jordan algebras. Annals of Mathematics, 48(1):pp. 1{7, 1947. [4] F. Albertini and D. D'Alessandro. Algebraic conditions for indirect controllability in quantum coherent feedback schemes. In Control Conference (ECC), 2013 European, pages 2701{2706, July 2013. [5] P. Aliferis and J. Preskill. Fault-tolerant quantum computation against biased noise. Phys. Rev. A, 78:052331, Nov 2008. [6] S. A. Aly. Asymmetric and Symmetric Subsystem BCH Codes and Beyond. ArXiv e-prints, Mar. 2008. [7] S. A. Aly and A. Ashikhmin. Nonbinary quantum cyclic and subsystem codes over asymmetrically-decohered quantum channels. In Information Theory (ITW 2010, Cairo), 2010 IEEE Information Theory Workshop on, pages 1{5, Jan 2010. [8] H. Amini, M. Mirrahimi, and P. Rouchon. Stabilization of a delayed quantum system: The photon box case-study. Automatic Control, IEEE Transactions on, 57(8):1918{1930, Aug 2012. [9] H. Amini, R. A. Somaraju, I. Dotsenko, C. Sayrin, M. Mirrahimi, and P. Rouchon. Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays. Automatica, 49(9):2683 { 2692, 2013. [10] J. T. Anderson, G. Duclos-Cianci, and D. Poulin. Fault-tolerant conversion between the steane and reed-muller quantum codes. Physical Review Letters, 113(8):080501, Aug. 2014. [11] I. Bengtsson and K. Zyczkowski. Geometry of quantum states: an introduction to quantum entanglement. Cambridge University Press, 2007. [12] A. Blais, R.-S. Huang, A. Wallra, S. M. Girvin, and R. J. Schoelkopf. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum compu- tation. Phys. Rev. A, 69:062320, Jun 2004. 55 [13] D. Braun, O. Giraud, I. Nechita, C. Pellegrini, and M. Znidari c. A universal set of qubit quantum channels. Journal of Physics A: Mathematical and Theoretical, 47(13):135302, 2014. [14] T. A. Brun. A simple model of quantum trajectories. American Journal of Physics, 70:719{ 737, July 2002. [15] T. A. Brun, Y.-C. Zheng, K.-C. Hsu, J. Job, and C.-Y. Lai. Teleportation-based Fault-tolerant Quantum Computation in Multi-qubit Large Block Codes. ArXiv e-prints, Apr. 2015. [16] M. Brune, S. Haroche, V. Lefevre, J. M. Raimond, and N. Zagury. Quantum nondemolition measurement of small photon numbers by rydberg-atom phase-sensitive detection. Phys. Rev. Lett., 65:976{979, Aug 1990. [17] J. Clarke and F. K. Wilhelm. Superconducting quantum bits. Nature, 453(7198):1031{1042, Jun 2008. [18] J. Combes, C. Ferrie, C. Cesare, M. Tiersch, G. J. Milburn, H. J. Briegel, and C. M. Caves. In-situ characterization of quantum devices with error correction. ArXiv e-prints, May 2014. [19] D. D'Alessandro and R. Romano. Indirect controllability of quantum systems; a study of two interacting quantum bits. Automatic Control, IEEE Transactions on, 57(8):2009{2020, Aug 2012. [20] I. Dotsenko, J. Bernu, S. Del eglise, C. Sayrin, M. Brune, J. M. Raimond, S. Haroche, M. Mir- rahimi, and P. Rouchon. The quantum Zeno eect and quantum feedback in cavity QED. Physica Scripta, 2010(T140):014004+, Sept. 2010. [21] J. Dressel, T. A. Brun, and A. N. Korotkov. Notes on implementing generalized measurements with superconducting qubits. ArXiv e-prints, Dec. 2013. [22] J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. La amme. Symmetrized characterization of noisy quantum processes. Science, 317(5846):1893{1896, 2007. [23] Z. W. E. Evans, A. M. Stephens, J. H. Cole, and L. C. L. Hollenberg. Error correction optimisation in the presence of X/Z asymmetry. ArXiv e-prints, Sept. 2007. [24] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum Computation by Adiabatic Evolution. eprint arXiv:quant-ph/0001106, Jan. 2000. [25] S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert. Quantum tomography via compressed sensing: error bounds, sample complexity and ecient estimators. New Journal of Physics, 14(9):095022, 2012. [26] J. Florjanczyk and T. A. Brun. Continuous decomposition of quantum measurements via qubit-probe feedback. Phys. Rev. A, 90:032102, Sep 2014. [27] J. Florjanczyk and T. A. Brun. Continuous decomposition of quantum measurements via hamiltonian feedback. Phys. Rev. A, 92:062113, Dec 2015. [28] A. Fujiwara and P. Algoet. One-to-one parametrization of quantum channels. Phys. Rev. A, 59:3290{3294, May 1999. 56 [29] Y. Fujiwara. Instantaneous quantum channel estimation during quantum information pro- cessing. arXiv preprint arXiv:1405.6267, 2014. [30] S. Garnerone, P. Zanardi, and D. A. Lidar. Adiabatic quantum algorithm for search engine ranking. Phys. Rev. Lett., 108:230506, Jun 2012. [31] D. Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997. [32] L. K. Grover. A fast quantum mechanical algorithm for database search. Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, pages 212{219, 1996. [33] M. Guti errez, L. Svec, A. Vargo, and K. R. Brown. Approximation of realistic errors by cliord channels and pauli measurements. Phys. Rev. A, 87:030302, Mar 2013. [34] S. Habib, K. Jacobs, and H. Mabuchi. Quantum feedback control. Los Alamos Science, 27:126, 2002. [35] C. D. Hill, A. G. Fowler, D. S. Wang, and L. C. L. Hollenberg. Fault-tolerant quantum error correction code conversion. Quantum Info. Comput., 13(5-6):439{451, May 2013. [36] K.-C. Hsu and T. A. Brun. Method for quantum-jump continuous-time quantum error cor- rection. Phys. Rev. A, 93:022321, Feb 2016. [37] L. Ioe and M. M ezard. Asymmetric quantum error-correcting codes. Phys. Rev. A, 75:032345, Mar 2007. [38] A. Jordan and A. Korotkov. Uncollapsing the wavefunction by undoing quantum measure- ments. Contemporary Physics, 51:125{147, Mar. 2010. [39] P. Jordan, J. v. Neumann, and E. Wigner. On an algebraic generalization of the quantum mechanical formalism. Annals of Mathematics, 35(1):pp. 29{64, 1934. [40] A. Katabarwa and M. R. Geller. Logical error rate in the pauli twirling approximation. Scientic Reports, 5:14670, 2015. [41] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jerey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Lucero, M. Neeley, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, and J. M. Martinis. Scalable in-situ qubit calibration during repetitive error detection. ArXiv e-prints, Mar. 2016. [42] D. Kirk, editor. Graphics Gems III. Academic Press Professional, Inc., San Diego, CA, USA, 1992. [43] G. G. La Guardia. Asymmetric quantum codes: new codes from old. Quantum Information Processing, 12(8):2771{2790, 2013. [44] D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett., 81:2594{2597, Sep 1998. [45] E. Magesan, D. Puzzuoli, C. E. Granade, and D. G. Cory. Modeling quantum noise for ecient testing of fault-tolerant circuits. Phys. Rev. A, 87:012324, Jan 2013. 57 [46] K. McCrimmon. A Taste of Jordan Algebras. Universitext. Springer New York, 2006. [47] K. Murota, Y. Kanno, M. Kojima, and S. Kojima. A numerical algorithm for block-diagonal decomposition of matrix $$f*g$$ -algebras with application to semidenite programming. Japan Journal of Industrial and Applied Mathematics, 27(1):125{160, 2010. [48] G. M. Murphy. Ordinary dierential equations and their solutions. Van Nostrand, Princeton, N.J., rst edition, 1960. [49] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. [50] T. O'Meara. Introduction to Quadratic Forms. Classics in Mathematics. Springer Berlin Heidelberg, 2012. [51] S. Omkar, R. Srikanth, and S. Banerjee. Characterization of quantum dynamics using quan- tum error correction. Phys. Rev. A, 91:012324, Jan 2015. [52] O. Oreshkov. Continuous-time quantum error correction. ArXiv e-prints, Nov. 2013. [53] O. Oreshkov and T. A. Brun. Weak measurements are universal. Physical Review Letters, 95(11):110409, 2005. [54] A. J. Penico. The Wedderburn principal theorem for Jordan algebras. Trans. Amer. Math. Soc., 70:404{420, 1951. [55] D. P erez-Garc a, M. M. Wolf, D. Petz, and M. B. Ruskai. Contractivity of positive and trace-preserving maps under lp norms. Journal of Mathematical Physics, 47(8), 2006. [56] D. Puzzuoli, C. Granade, H. Haas, B. Criger, E. Magesan, and D. G. Cory. Tractable simulation of error correction with honest approximations to realistic fault models. Phys. Rev. A, 89:022306, Feb 2014. [57] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer. Elucidating Reaction Mecha- nisms on Quantum Computers. ArXiv e-prints, May 2016. [58] A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov. Compressed sensing quantum process tomography for supercon- ducting quantum gates. Phys. Rev. B, 90:144504, Oct 2014. [59] D. Rugar, R. Budakian, H. Mamin, and B. Chui. Single spin detection by magnetic resonance force microscopy. Nature, 430:329, 2004. [60] P. K. Sarvepalli, A. Klappenecker, and M. R otteler. Asymmetric quantum codes: construc- tions, bounds and performance. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465(2105):1645{1672, 2009. [61] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.-M. Raimond, and S. Haroche. Real-time quantum feedback prepares and stabilizes photon number states. Nature, 477(7362):73{77, Sep 2011. [62] J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar, B. E. A. Saleh, and M. C. Teich. Theory of light detection in the presence of feedback. J. Opt. Soc. Am. B, 4(10):1604{1620, Oct 1987. 58 [63] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Review, 41(2):303{332, 1999. [64] A. M. Stephens, Z. W. E. Evans, S. J. Devitt, and L. C. L. Hollenberg. Asymmetric quantum error correction via code conversion. Phys. Rev. A, 77:062335, Jun 2008. [65] S. Taghavi, R. L. Kosut, and D. A. Lidar. Channel-optimized quantum error correction. IEEE Transactions on Information Theory, 56(3):1461{1473, March 2010. [66] M. Varbanov and T. A. Brun. Decomposing generalized measurements into continuous stochastic processes. Phys. Rev. A, 76:032104, Sep 2007. [67] Y. Yamamoto, N. Imoto, and S. Machida. Amplitude squeezing in a semiconductor laser using quantum nondemolition measurement and negative feedback. Phys. Rev. A, 33:3243{3261, May 1986. 59 Appendix A Calculation of the O( 2 ) terms of the reversibilty condition Starting with Eq. (2.14), we can write O( 2 ) in three parts, rst M (2) +M (2) = p 2h(H 0 PS ) 2 i 0 2 p 2cie i h0jH 0 PS j1i: Next, M (1) M (1) =hjH 0 PS j0ihjH 0 PS j0i (A.1) cie i p 2 (hjH 0 PS j0ihjH 0 PS j0i) c 2 e 2i 1 2 = hjH 0 PS j0ihjH 0 PS j0icie i h1jH 0 PS j0i c 2 e 2i 1 2 : We can always discard terms proportional to 1. Grouping together the last two calculations, we get M (2) +M (2) 2 p 2M (1) M (1) = p 2h(H 0 PS ) 2 i 0 2 p 2cie i (h0jH 0 PS j1ih1jH 0 PS j0i) +2 p 2hjH 0 PS j0ihjH 0 PS j0i = p 2h(H 0 PS ) 2 i 0 + 4 p 2ce i H 2 +2 p 2hjH 0 PS j0ihjH 0 PS j0i: (A.2) 60 The two terms above, still expressed as functions of H 0 PS , can be expanded as follows. First, note that (H 0 PS ) 2 = 1 H 2 1 +H 2 2 +H 2 3 +X i [H 2 ;H 3 ] +Y i [H 3 ;H 1 ] +Z i [H 1 ;H 2 ]; (A.3) implying that h(H 0 PS ) 2 i 0 =H 2 1 +H 2 2 +H 2 3 +i [H 1 ;H 2 ]: (A.4) Next note that hjH 0 PS j0i = 1 p 2 (H 1 iH 2 +H 3 ); (A.5) which yields hjH 0 PS j0ihjH 0 PS j0i (A.6) = 1 2 (H 1 +iH 2 H 3 ) (H 1 +iH 2 H 3 ) = 1 2 (H 1 +iH 2 ) 2 +H 2 3 [H 3 ;H 1 ]i [H 3 ;H 2 ] : Putting this all together, we can now write Eq. (2.15). 61 Appendix B Miscellaneous lemmas Lemma 6 (Oriented Pauli channels as contractions). Given the oriented Pauli channel M (Def. 8), we can always write the process matrix M as M = (1 2p)I + 2pQ T U 2 6 6 6 4 k 1 0 0 0 k 2 0 0 0 k 3 3 7 7 7 5 Q U : (B.1) Proof. Recall that M = U y D U . Consider the action of D on a qubit state D () = (1p) +p x XX +p y YY +p z ZZ (B.2) = (1p) I +~ r~ 2 +p x I +X~ r~ X 2 +p y I +Y~ r~ Y 2 +p z I +Z~ r~ Z 2 (B.3) = I 2 + (1p) ~ r~ 2 + (p x p y p z ) r 1 x 2 (B.4) + (p x +p y p z ) r 2 y 2 + (p x p y +p z ) r 3 z 2 (B.5) = I 2 + (1 2p) ~ r~ 2 + 2p x r 1 x + 2p y r 2 y + 2p z r 3 z 2 (B.6) = I + 0 B B B @ 2 6 6 6 4 (1 2p)I + 2p 2 6 6 6 4 k 1 0 0 0 k 2 0 0 0 k 3 3 7 7 7 5 3 7 7 7 5 ~ r 1 C C C A ~ 2 ; (B.7) where p x = pk 1 , p y = pk 2 and p z = pk 3 . It remains only to identify that U and U y act as transformations Q U and Q U y of~ r before and after the channel D to complete the proof. 62 Lemma 7 (Probe basis of a weak measurement). Any diusive weak measurement given by a model A with a probe basisf~ n 1 (x);~ n 2 (x);~ n 3 (x)g must have ~ n 2 ~ O(). Thus, an orthonormal basis for the Bloch sphere that approximates the probe and detector always exists. Proof. Recall that for a probe feedback loop we can expand the operator in orders of , M (x) = h je iH PS ji = h j (1 +iH PS )ji h ji1 +O(): (B.8) A diusive weak measurement must always obtain both results with nearly equal probability (up to O()). The probability of each result on a quantum stateji is p = hjM y (x)M (x)ji jh jij 2 hji +O() = 1 2 +O(); (B.9) which in turn means thatjh jij 1= p 2+O(). In the Bloch vector representation, this implies that~ n 2 ~ O(). Lemma 8 (Commutator identity). For any Hermitian operators ^ O and ^ A, if i h ^ O; ^ A i / ^ O then ^ O = 0. Proof. First, let us express ^ A and ^ O in a basis where ^ A is diagonal, i.e.: ^ A = P i a i jiihij with a i 2R and ^ O = P jk o jk jjihkj. This makes our equation i X jk o jk jjihkj ^ Ai ^ A X jk o jk jjihkj = X jk o jk jjihkj; for some constant . Expanding the operator ^ A yields i X ijk (a i o jk jjihkjiihija i o jk jiihijjihkj) = X jk o jk jjihkj; 63 and this, in turn, reduces to i X jk (a k o jk jjihkja j o jk jjihkj) = X jk o jk jjihkj: This implies that for all j;k we have (a k a j )io jk =o jk and we nd that Refo jk g =(a k a j )Imfo jk g as well as Refo jk g = Imfo jk g=(a k a j ), leading to a contradiction. The only valid solution remaining is o jk = 0 for all j;k exactly. Lemma 9 (Particle number versus minimum distance). Let X i be N i.i.d copies of random oriented Pauli channel channels according to (4.28) and letA be a xed channel with eccentricities (a 1 ;a 2 ;a 3 ). Then we have that Pr n min i kX i Ak 2 <" o N " 5 2 14 3 3 a 2 1 a 2 a 3 (B.10) wherekYk 2 = p Tr [Y T Y ] is the Frobenius norm. Proof. Consider the variable Z min i kX i Ak 2 . The cumulative distribution function of Z is PrfZ <"g = 1 Y i (1 PrfkX i Ak 2 <"g) (B.11) = 1 (1 PrfkXAk 2 <"g) N ; (B.12) and let (X) =kXAk 2 (B.13) = Q T A XQ A D A 2 (B.14) = Q T XA D X Q XA D A 2 ; (B.15) where in the second line we've conjugated both matrices by Q A and Q XA = Q X Q A is also Haar random since the Haar measure is invariant under left or right conjugation. Let E D = 64 fjD X D A j" D g, i.e.: the event in which the eigenvalues of X andA are similar. Conditioning on this event we see that, Prf(X)"g = Prf(X)"jE D g PrfE D g (B.16) + Prf(X)"j:E D g Prf:E D g (B.17) Prf(X)"jE D g PrfE D g: (B.18) Thus, we can consider only those channels X for which E D is true. Let ~ q i be the column vectors of Q XA and let x i and a i be the diagonals of D X and D A respectively. From E D we know that jx i a i j" D . The Frobenius norm can then be simplied to be Q T XA D X Q XA D A 2 = 3 X i=1 x i ~ q i ~ q T i a i ~ e i ~ e T i 2 (B.19) 3 X i=1 x i ~ q i ~ q T i a i ~ e i ~ e T i 2 (B.20) 3 X i=1 a i ~ q i ~ q T i ~ e i ~ e T i 2 +" D (B.21) 3 X i=1 (2a i k~ q i ~ e i k 2 +" D ): (B.22) Let A i =f2a i k~ q i ~ e i k 2 +" D <"=3g then we must have that Prf(X)<"jE D g PrfA 1 \A 2 \A 3 g (B.23) = PrfA 1 g PrfA 2 jA 1 g PrfA 3 jA 1 \A 2 g: (B.24) Note that the marginal distribution of each ~ q i is uniform over the unit sphere and can be con- structed using [42], ~ q = 2 6 6 6 4 u p 1u 2 sin (2v) p 1u 2 cos (2v) 3 7 7 7 5 8 < : u unif [1; 1]; v unif [0; 1]: (B.25) 65 Thus, for ~ q 1 to lie within " of~ e 1 in the Frobenius norm, we need only that u2 1" 2 =2; 1 and for u uniformly distributed over [1; 1] this means that Prfk~ q 1 ~ e 1 k 2 "g =" 2 =2: (B.26) Next, the vector~ q 2 is sampled uniformly from the equator of vectors orthogonal to~ q 1 . Given that k~ q 1 ~ e 1 k", we are guaranteed that~ e 2 lies in the band of latitude" around this equator. Thus, the probability of ~ q 2 lying within " of~ e 2 is given by the ratio of the area of the circle of radius " on the unit sphere to that of the band. For " 1 this is, Prfk~ q 2 ~ e 2 k"g = R " 0 2rdr R 1+" 1" 2rdr = " 2 : (B.27) Finally, given that~ q 1 and~ q 2 each lie within" of~ e 1 and~ e 2 respectively, the vector ~ q 3 lies within" of~ e 3 with probability 1=4. Altogether, Prfk~ q 1 ~ e 1 k 2 "g =" 2 =2 =) PrfA 1 g = 1 8a 2 1 ("=3" D ) 2 ; Prfk~ q 2 ~ e 2 k 2 "g ="=2 =) PrfA 2 jA 1 g = 1 4a 2 ("=3" D ); Prfk~ q 3 ~ e 3 k 2 "g = 1=4 =) PrfA 3 jA 1 \A 2 g = 1 2a 3 (1=4" D :) If we choose " D ="=6 then we have, altogether that Prf(X)<"jE D g " 3 2 12 3 3 a 2 1 a 2 a 3 : (B.28) It now remains to bound PrfE D g to nish the proof. The event E D is characterized by the probabilities, PrfjX 1 a 1 j"g"; (B.29) PrfjX 2 a 2 j"jjX 1 a 1 j"g": (B.30) 66 Finally, ifjX 1 a 1 j" andjX 2 a 2 j", then we are guaranteed thatjX 3 a 3 j 2". Altogether, we can use the chain rule to see that PrfjD X D A j"g ("=2) 2 : (B.31) Thus, we have our bound on the spacing of a random channel X Prf(X)<"g " 5 2 14 3 3 a 2 1 a 2 a 3 : (B.32) In our bound on the minimum distance Eq. (B.12) this becomes PrfZ <"g 1 1 " 5 2 14 3 3 a 2 1 a 2 a 3 N ; (B.33) which for " 1 and N 1 scales as PrfZ <"gN " 5 2 14 3 3 a 2 1 a 2 a 3 : (B.34) Thus, to ensure that a channel in our random grid approximatesA to precision" in the Frobenius norm with probabilityp, we need to populate our random grid with a very large number of particles N = 2 14 3 3 a 2 1 a 2 a 3 " 5 p : (B.35) Lemma 10 (Frobenius distance of outer products). Given~ x;~ y2R n , we can bound the Frobenius distance of their outer product by the Frobenius distance of the vectors themselves, ~ x~ x T ~ y~ y T 2 2k~ x~ yk 2 : (B.36) 67 Proof. Recall that for any real matrix A,kAk 2 = p Tr [AA T ], thus ~ x~ x T ~ y~ y T 2 2 = Tr ~ x~ x T ~ x~ x T ~ x~ x T ~ y~ y T ~ y~ y T ~ x~ x T +~ y~ y T ~ y~ y T (B.37) =h~ x;~ xi 2 +h~ y;~ yi 2 2h~ x;~ yi 2 (B.38) = 2 1h~ x;~ yi 2 (B.39) 4 (1h~ x;~ yi) (B.40) = 2k~ x~ yk 2 2 : (B.41) Lemma 11 (Bound on diagonal matrix elements). LetD A andD B be diagonal matrices of entries a i and b i respectively. Let X = Q T D A Q for some orthonormal matrix Q with columns ~ q i . We have then that j[X] ii b i jkXD B k 2 (B.42) Proof. First, recall the form of the Frobenius norm for any matrix Y kYk 2 = v u u t n X i=1 n X j=1 jy ij j 2 : (B.43) In our case this becomes, kXD B k 2 = v u u t n X i=1 n X j=1 [X] ij b i ij 2 (B.44) q j[X] ii b i j 2 (B.45) =j[X] ii b i j: (B.46) 68
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Quantum feedback control for measurement and error correction
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