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Quantum computation by transport: development and potential implementations
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Quantum computation by transport: development and potential implementations
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Quantum Computation by T ransport: Development and Potential Implementations Nic holas Chancellor University of Southern California 1 T able of Contents P ersonal ac kno wledgemen ts: page 3 In tro duction: page 4 Chapter 1: Lo cal Quenc hes on Heisen b erg Spin Systems: page 8 References: page 31 Chapter 2: T ransp ort b y Degenerate Groundstates: page 33 References: page 61 Chapter 3: A diabatic Quan tum Bus Proto col: page 63 References: page 81 Chapter 4: Holonomic Quan tum Computation b y T ransp ort and Application with Sup er- conducting Flux Qubits: page 83 References: page 100 Summary: page 102 2 Personal Acknowledgements I w ould lik e to thank m y paren ts, Dan and P aula for raising me and instilling an in terest in science in me at an early age. I w ould also lik e to thank m y ancØ Shanelle for her lo v e and supp ort since I met her. In addition I w ould lik e to than m y brother Graham and his wife Ashley for b elieving in me and for their supp ort during college and graduate sc ho ol. As w ell I w ould lik e to thank Bob, Lida, Sarah and Laura W ac hel for taking me in to there home for a large p ortion of m y graduate studies. I w ould also lik e to thank m y extended family , m y grandmothers Bonnie and T ess, m y aun t Theresa, m y cousin Ry an, m y uncle Bill and aun t Nazerina, m y uncle Mik e and m y aun t Nancy , and m y cousin Corrine for there supp ort. In addition I w ould lik e to ac kno wledge the p ositiv e impact on m y life from the brothers of the Gamma Kappa c hapter of Phi Gamma Delta, p ergØ brothers. I w ould also lik e to thank man y p eople who ha v e help ed academically throughout m y academic career, in particular m y advisor Stephan Haas, who has b een an excellen t advisor and a go o d friend. A c kno wledgemen ts of academic con tributions are done on a c hapter b y c hapter basis and can b e found near the end of the c hapter. All of the c hapters are based on published or submitted pap ers, and the references to these pap ers can b e also found in the c hapters. 3 Introduction This thesis deals with eects on an tiferromagnetic Heisen b erg spin c hains and clusters whic h can b e used for univ ersal holonomic quan tum computing. I will discuss in detail an ar- c hitecture for a Holonomic quan tum computer, and ho w to it ma y b e implemen ted with sup erconducting ux qubits (see [1](c h. 4)). In addition to demonstrating this arc hitecture, this thesis will also shed ligh t on the b eha vior of these c hains and the eects whic h mak e suc h an arc hitecture p ossible. These eects include en tanglemen t whic h allo ws disturbances that are excited b y a quenc h to tra v el an unlimited distance, ev en in a gapp ed system. I will also examine anomalous equilibration b eha vior whic h can b e seen in these systems. Holonomic quan tum computation (HQC) w as conceiv ed and sho wn to b e univ ersal b y Zanardi and Rasetti [5](c h. 4) who form ulated it in terms of a non-ab elian Berry phase. HQC is considered to b e an app ealing metho d for ac hieving fault toleran t quan tum computing b ecause of its geometrical nature and b ecause it can b e implemen ted adiabatically . Therefore it has all of the adv an tages of adiabatic quan tum computation [6](c h. 4). Although man y implemen tations of holonomic and geometric quan tum computation are adiabatic, there are examples whic h are not [7, 8](c h. 4). Unlik e most other implemen tations of HQC, the one whic h w e prop ose in this thesis do es not require us to explicitly consider curv ature of a degenerate ground state manifold within a complex pro jectiv e space. Although suc h metho ds could in principle b e applied to the designs giv en here, they are not necessary . The arc hitecture prop osed in this thesis relies on real space t wists p erformed on a Hamiltonian whic h initially p erforms transp ort with a trivial (non-ab elian) Berry phase. Because these t wists ha v e a real space ph ysical in terpretation, the eects they will ha v e on transp orted qubits can b e inferred without considering the geometry of the underlying Hilb ert space. I fo cus on an adiabatic transp ort proto col whic h in v olv es the slo w attac hmen t and remo v al of spins from an an tiferromagnetic Heisen b erg spin c hain or cluster. There are other trans- p ort proto cols whic h could also b e used, most notably the non-adiabatic proto col discussed in [811](c h. 4). The reason that this thesis fo cuses on adiabatic transp ort proto cols is that the arc hitecture prop osed in this thesis could p oten tially b e implemen ted with a sup ercon- ducting ux qubit circuit whic h only faithfully repro duces the lo w energy degrees of freedom of a spin Hamiltonian, and therefore is not appropriate for non-adiabatic computation. 4 Sup erconducting ux qubits are a p opular arc hitecture for implemen ting scalable adia- batic quan tum computing [26](c h. 4), and therefore are a natural c hoice for designing a scalable holonomic quan tum computer. An additional adv an tage of the use of sup erconduct- ing ux qubits is that the designs tend to ha v e spatially extended qubits and a high degree of connectivit y[17](c h. 4). The large spatial exten t of the qubits means that a design could b e implemen ted in whic h a qubit w ould only need to b e transferred across a small n um b er of spins to b e mo v ed from one lo cation in a computer to an y other arbitrary lo cation. There has b een recen t exp erimen tal w ork in v olving quan tum annealing to degenerate ground state manifolds using curren tly a v ailable sup erconducting ux qubit hardw are[18](c h. 4). It w as demonstrated exp erimen tally that signatures of quan tum b eha viors can b e ob- serv ed in the nal state within a degenerate ground state manifold. This pro vides an indi- cation that a ground state manifold can b e pro duced accurately enough on the D-w a v e 1 quan tum annealing pro cessor that quan tum eects dominate o v er classical eects and de- sign inaccuracies. Although the arc hitecture prop osed here cannot b e implemen ted on the hardw are used in [18](c h. 4), this exp erimen t do es pro vide pro of of principle for the use of degenerate manifolds in sup erconducting ux qubit systems. The adiabatic transp ort proto col whic h I prop ose is discussed extensiv ely in c hapter 3 of this thesis whic h is based on a separately published w ork [1](c h. 3). This proto col relies only on the tendency of an tiferromagnetic systems to rep el excess p olarization, and is therefore quite v ersatile. This proto col do es fail for certain highly frustrated regimes of parameter space, but w orks w ell in other regimes. This thesis examines sev eral v ariations of this transp ort proto col including sequen tial v ersus sim ultaneous uncoupling as w ell as metho ds whic h in v olv e sim ultaneous v ariations of the J2 coupling parameter so that the easily prepared nature of the ground state at the so called Ma jumdar-Ghosh p oin t [15](c h. 3) can b e exploited. I also discuss the eect of making the coupling anisotropic, b y using a XYZ or XXZ Heisen b erg mo del instead. This proto col w as motiv ated b y a phenomenology in whic h transp ort can b e ac hiev ed o v er long distances in a system whic h is gapp ed but has a degenerate ground state. It is sho wn in c hapter 2 of this thesis whic h is based on separately published w ork [1](c h. 2) that a ground state degeneracy arising from either top ological eects or from particle hole symmetry can b e used to send disturbances whic h carry information and can lead to equilibration throughout a spin c hain system. This in v estigation mak es hea vy use of 5 quan tum information related quan tities to elucidate the dynamics of a man y b o dy system, as has also b een done successfully in man y other w orks [49](c h. 2). Also the in terest in this phenomenology ma y extend b ey ond the quan tum computer design prop osed in c hapter 4 of this thesis, for example a J 1 -J 2 Heisen b erg spin c hain can b e realized with cold trapp ed atoms [10, 11](c h. 2). An extension of this system has also b een iden tied as b eing imp ortan t for other applications in quan tum computing [15]. Thesis Structure This thesis primarily examines a metho d of ac hieving univ ersal holonomic quan tum com- putation b y using transp ort proto cols to transp ort qubits do wn spin c hains with unitary t wists. This thesis ho w ev er is not in tended to simply summarize these results, whic h ha v e b een published elsewhere [1] (c h. 4). I will also also summarize the transp ort proto col whic h is used, as w ell as to illuminate man y ideas related to the dev elopmen t of this arc hitecture as w ell as examining the p oten tial of implemen tation using sup erconducting qubits. The rst c hapter of this thesis summarizes the phenomenology of J 1 -J 2 Heisen b erg spin c hains whic h are lo cally quenc hed, with a strong fo cus on equilibration. The sub ject of equilibration of closed quan tum systems, whic h is a primary fo cus of the rst c hapter is an in teresting topic in its o wn righ t. This c hapter is included b ecause it giv es insigh ts in to the ric h w orld ph ysics of lo cal manipulations of spin c hains on whic h the design giv en in this thesis is based. This rst c hapter pro vides groundw ork for the pro ceeding c hapters whic h discuss transp ort of information b y exploiting a degenerate ground state. The second c hapter con tin ues the study of J 1 -J 2 Heisen b erg spin c hains under lo cal quenc hes. This c hapter, ho w ev er, shifts its fo cus a w a y from equilibration and instead fo cuses on the transfer of information and p olarization using a degenerate ground state manifold. In this pap er studies of quenc hed systems la y the groundw ork for the next c hapter whic h discusses a kind of transp ort whic h can b e ac hiev ed adiabatically . The third c hapter la ys out an adiabatic transp ort proto col for transp orting a qubit of information through an an tiferomagnetically coupled Heisen b erg spin c hain. In this c hapter I explore the eect of frustration b y examining the proto col on a J 1 -J 2 spin c hain. I also discuss the eect of making the coupling anisotropic, b y using a XYZ or XXZ Heisen b erg mo del instead. 6 The fourth c hapter is divided in to 3 sections. The rst section discusses ho w the spin c hains used in the third c hapter can b e t wisted to pro duce arbitrary single qubit gates. This section giv es a list of the t wists necessary for a univ ersal set of single qubit gates, as w ell as la ying out the metho d for calculating the t wist necessary for an arbitrary gate. The second section demonstrates ho w a con trolled not gate can b e implemen ted with a cluster of 8 spins. The third section suggests a metho d of implemen ting the necessary Heisen b erg systems with sup erconducting ux qubits. A summary of the design itself, and the related data are all con tained in the third and fourth c hapters. The rst t w o c hapters pro vide con text for ho w this design came ab out. 7 Chapter 1: Local Quenches on Heisenberg Spin Systems This c hapter is based on the pap er [1]. In this c hapter, W e study the long-time equilibration b eha vior follo wing a lo cal quenc h using a frustrated quan tum spin c hain as an example of a fully in teracting closed quan tum system. Sp ecically w e examine the statistics of time series of the Losc hmidt ec ho, the trace distance of the time ev olv ed lo cal densit y matrix to its a v erage state and the lo cal magnetization. Dep ending on the quenc h parameters, the equi- libration statistics of these quan tities sho w features of go o d or p o or equilibration, indicated b y Gaussian and exp onen tial or bistable distribution functions of the linear quan tities. This pro vides insigh t in to the univ ersalities and the ric hness of equilibration of complex closed quan tum systems. I. INTR ODUCTION Equilibration b eha vior of closed quan tum systems is a topic of recen tly gro wing in terest. As opp osed to equilibration in an op en quan tum system, for example coupled to a bath at a certain temp erature, a closed quan tum system do es not only not encoun ter dissipation and conserv e energy - lik e ev en a classical system w ould. But its time ev olution is unitary , ini- tially b eing in a pure state it therefore remains pure for all time. Equilibration under these constrain ts is m uc h less straigh t forw ard and in some cases, esp ecially for nite systems, migh t ev en fail completely . Recen t n umerical and theoretical studies ha v e tak en approac hes to study equilibration com- patible with the unitary ev olution of closed quan tum systems. Some approac hes include b ounds related to the Levi lemma [2, 3], through t ypicalit y argumen ts based on treating w a v e functions as random states [4, 5] as w ell as analytical and n umerical approac hes to time series statics [68]. Motiv ated b y our in terest in the equilibration prop erties of isolated quan tum systems follo wing a quenc h - more sp ecically the ev olution statistics of quan tum information quan tities suc h as the Losc hmidt ec ho and the lo cal trace distance - w e p erform n umerical studies of nite-size systems. As a represen tativ e example, w e examine J 1 -J 2 - quan tum-spin-c hains with N spin-1=2 degrees of freedom, p erio dic b oundary conditions and 8 a lo cal magnetic eld term, H (J 1 ;J 2 ;h) = J 1 N X j=1 ~ S j ~ S j+1 +J 2 N X j=1 ~ S j ~ S j+2 h N 0 X j=1 S z j : (1) Here J 1 and J 2 denote nearest-neigh b or and next-nearest-neigh b or Heisen b erg couplings. ~ S j and S z j are the spin op erators of the corresp onding spin j . F or simplicit y J 1 is c hosen to b e one. J 2 v aries from quenc h to quenc h, but remains constan t within a giv en quenc h. Instead, the magnetic eld term in the Hamiltonian is used to p erform lo cal quenc hes on a subset of N 0 adjacen t spins of the form H(h)!H 0 (h 0 ). Starting in the ground statej 0 i of H(h) at t = 0, after the quenc h the system ev olv es according to H 0 (h 0 ). In this set up the lo cal magnetic eld term in tro duces a p erturbation, whic h do es not comm ute with the rst t w o terms of equation (1). Moreo v er it breaks translational symmetry , th us it allo ws one to generate more complex excitations. [10] [11] Ev en for nite h the Hamiltonian giv en b y equation (1) preserv es the total magnetization M = 1=N P N j=1 S z j , sinceM comm utes with H . H therefore splits in to 2N + 1 indep enden t sectors. This reduces the actual system size from 2 N eigenstates to only N Nup eigenstates, where N up =N(M + 1=2) is the n um b er of spins p oin ting up. The largest of these sectors is the sector, where M = 0 or N up =N=2, whic h is also the sector of the ground state of H for zero or small h. T o address the quenc h n umerically w e th us calculate the ground state energy in eac h of the total magnetization sectors of the initial Hamiltonian H(h) using Lanczos diagonalization. W e then only k eep the ground state of the sector with the lo w est ground energy and diago- nalize the ev olution Hamiltonian H 0 (h 0 ) in this sector. This is done through iterativ e Lanczos diagonlization, calculating the rst 500 h undred lo w est energy eigenstates. [12] T o simplify the notation when talking ab out eigenstates or eigenstate expansions in the follo wing w e alw a ys refer to only this sector. Unitary evolution after a quench A sudden c hange of a system parameter pulls the system, originally in the ground state of the Hamiltonian H(h) in to an out of equilibrium state of the new Hamiltonian H 0 (h 0 ). The state of the system then ev olv es according to H 0 . The time ev olv ed densit y matrix is giv en b y (t) =U y t 0 U t , where 0 =j 0 ih 0 j denotes its initial state and U t = exp(iH 0 t) the time ev olution op erator. Note that since the ev olution 9 is unitary this state is alw a ys pure, i. e. its purit y P [(t)] = Tr[(t) 2 ] = 1. Equilibration in the long time b eha vior of some observ able O can then b e studied, b y lo oking at the time series of its exp ectation v alue O(t) = Tr[(t)O]. One can expand this exp ectation v alue in the energy basis of the Hamiltonian H 0 O(t) = X n 0 ;m 0 c n 0c m 0 exp [i (E n 0E m 0)t]O n 0 m 0; (2) wherec n 0 =hn 0 j 0 i. As one can see from equation (2) in a nite system all exp ectation v alues are rapidly oscillating functions o v er time, namely they are trigonometric p olynomials in the energy dierences. The long time b eha vior of an observ able O is giv en b y its probabilit y distribution function P (o)(oO(t)) t . Equilibration of suc h an observ able can then b e dened in terms of concen tration of its probabilit y distribution function. A simple recip e for a concen tration result is the signal to noise ratio O t = p Var(O), where O t lim T!1 1 T T 0 O(t)dt (3) denotes the time a v erage of the exp ectation v alue O(t) and Var(O) its v ariance. Since the trace in the exp ectation v alue comm utes with the time a v erage, one obtains O t = Tr[(t) t O]. If w e assume the system to b e non-degenerate, the time a v erage on the densit y matrix simply leads to a dephasing of the oscillating o-diagonal terms in its expansion in the eigenstate basis (t) = P n 0 ;m 0 c n 0c m 0 exp[i(E n 0 E m 0)t]jm 0 ihn 0 j. The a v erage of the o-diagonal terms v anishes, what remains unc hanged are the diagonal terms = P n 0 p n 0jn 0 ihn 0 j, where p n 0 =c n 0c n 0 . This state is not an y more pure, but describ es an ensem ble, whic h is often called the dephased state. If the v ariance of an observ able O is v ery small, P (o) is p eak ed around its mean, O is equilibrated and pro vides a useful ensem ble. This is generically the case for systems in the large size limit. Ho w ev er if the system is nite the distributions of generic observ ables are less narro w. In this regime it b ecomes in teresting to study probabilit y distribution functions. As sho wn in the follo wing n umerical calculations, in some cases one encoun ters simple distribution functions suc h as Gaussian or exp onen tial distributions. The former is completely dened b y its rst t w o comm ulan ts, i. e. its mean and its v ariance. The later is already dened b y its rst comm ulan t. Because of their simple structure and their generalit y - Gaussian distribution functions are obtained, where ev er the cen tral limit theorem applies - they indicate a straigh t forw ard w a y of equilibration 10 b eha vior in a nite system. In other cases one obtains more complex, sp ecically bistable distribution functions. Usually accompanied b y larger spreads they indicate the lac k of a smo oth equilibration. Intr o duction of the studie d quantities T o study the equilibration b eha vior follo wing a quenc h w e fo cus on four quan tities. W e lo ok at global as w ell as lo cal quan tities. The rst t w o are global: the energy probabilit y distribution of the initial state p n 0jh 0 jn 0 ij 2 (4) describ es the relativ e w eigh t of the elemen tary excitations caused b y the quenc h, and the Losc hmidt ec ho L(t)jh 0 jexp (iH 0 (h 0 )t)j 0 ij 2 (5) is the probabilit y of nding the system in its initial state j 0 i at a giv en time t after the quenc h. It can b e seen as some sort of memory of the initial state left in the system after the quenc h. In recen t publications ([6] and [7]) the Losc hmidt ec ho has b een iden tied as a useful quan tit y to study the equilibration of generic closed quan tum systems after a quenc h. It only dep ends on the initial ground state and its energy probabilit y distribution in the eigen basis of the ev olving Hamiltonian L(t) = X n 0 ;m 0 p n 0p m 0 exp [i (E n 0E m 0)t] (6) = X n 0 p 2 n 0 + 2 X n 0 <m 0 p n 0p m 0 cos[(E n 0E m 0)t] (7) Its mean is equal to the purit y of the dephased state, whic h denes its eectiv e dimension, as in tro duced in [3]: L t = X n p 2 n = Tr h t 2 i =P t 1 d eff (8) The t w o other quan tities w e examine, are lo calized in the subsystem S of the spins 1 to N 0 , whic h are (initially) exp osed to the magnetic eld. The remaining spins N 0 + 1 to N are called the en vironmen t E . The rst lo cal quan tit y follo ws the quan tum informational con text of the Losc hmidt ec ho: the lo cal trace distance of the time-ev olv ed densit y matrix S (t) = Tr E j (t)ih (t)j to its time a v erage d S (t) S (t) S t 1 ; (9) 11 where norm 1 is the trace norm giv en b y kOk 1 1 2 Tr p O y O (10) for an observ able O . It describ es the exp erimen tal distinguishabilit y of the time-ev olv ed lo cal state and the a v erage lo cal state. More precisely equation (9) pro vides a b ound for the exp ectation v alues of an y lo cal observ able O giv en its range of eigen v alues min ;:::; max , whic h is pro v ed in [3] for the generic case of a system with non-degenerate energy gaps: [13] Tr[(t)O] Tr[ t O] j max min jd S (t): (11) Moreo v er this b ound holds for the time a v erage on b oth sides of equation (11) and therefore pro vides a b ound to the uctuations of the exp ectation v alues of an y lo cal observ able.[14] Lo- cally , this trace distance denes equilibration in a strong sense. If it w ere almost v anishingly small for all time, this w ould imply that the lo cal system is in p erfect equilibrium. Namely the system w ould alw a ys b e practically indistinguishable from its time a v erage. Globally , equilibration can not b e ac hiev ed in the same strong sense b ecause the trace distance is in v arian t under unitary ev olution, and therefore globally cannot get smaller than its initial v alue. Through the use of the earlier in tro duced eectiv e dimension one can a lo oser but ev en simpler b ound of equation (11) using [3] d S (t) 1 2 s d 02 S d eff = d 0 S 2 q L t ; (12) where d 0 S is the dimension of the subsystem S in our case d 0 S = 2 N 0 andN 0 is the n um b er of spins in the subsystem. Ho w ev er if the system size is nite and the quenc h comparably small, as in the cases discussed here, this b ound b ecomes trivial, e.g. for N 0 = 4 and L t = 0:9 one obtains d S (t) 7:59, but the normalized trace distance b et w een an y t w o densit y matrices is alw a ys less or equal to 1. As the most natural observ able of a spin c hain w e also lo ok at the normalized lo cal magne- tization m S (t) 1 N 0 h (t)j N 0 X j=1 S z j j (t)i: (13) Because of the nite size of the system - the n umerical calculations presen ted here ha v e b een p erformed on 16 spins in a c hain - all of the quan tities are rapidly oscillating functions o v er 12 time. As in equation (7) the Losc hmidt ec ho and in equation (2) the lo cal magnetization are trigonometric p olynomials o v er time, the quan tit y d S is a more complicated nonlinear functional in v olving square ro ots of a trigonometric p olynomial. T o study the long-time b eha vior, instead of lo oking at the actual time series, w e therefore examine their distribution functions P (x) (xO(t)) t , as w ell as their time-a v eraged mean, and their v ariance. Numerically this is done b y diagonalizing the ev olution Hamiltonian in the corresp onding sector, as describ ed earlier and expanding the ev olution in the eigenstate basis. Though this restricts the analysis to relativ ely small system sizes, it allo ws one to calculate quan tities at an y giv en time. Using 400.000 random samples within a time range more than t w o orders of magnitude larger as the smallest gap in the system, w e obtain go o d statistics of these rapidly oscillating time series. F or the calculation of d S this is done b y expansion of the lo cal densit y matrix at eac h sampling time in the basis of spin correlates S 1 1 S N 0 N 0 , where i = 0;x;y;z and S 0 i = 1 . It is imp ortan t to note that b oth the magnetization and distance from the a v erage densit y matrix ha v e in teresting features only when observ ed lo cally and w ould giv e a trivial Dirac delta distribution if observ ed globally . F or the magnetization this is due to the conserv ation of angular momen tum. F or the trace distance this comes from the fact that unitarily ev olving the state in the trace distance from the a v erage is equiv alen t to ev olving b oth states, b ecause the dephased state is stationary under unitary ev olution. Since the trace distance is in v arian t under unitary transformations, the distance globally remains unc hanged. I I. FIELD-ENER GY-DEPENDENCE AND QUENCHES IN DIFFERENT REGIMES The Hamiltonian of the mo del giv en in equation (1) represen ts a fully in teracting system. Nev ertheless, one can nd appro ximations for some regions and ev en exact solutions for some particular p oin ts. W e n umerically calculated the v e lo w est energy lev els of the mo del as a function of the eld h on the four adjacen t spins in a c hain of 16 spins and dieren t ratios of the nearest and next-nearest neigh b or coupling (see Fig. 1). 13 0 0.5 1 1.5 2 2.5 3 3.5 −12 −11 −10 −9 −8 −7 −6 h E J 2 /J 1 =0.5 J 2 /J 1 =0.0 J 2 /J 1 =1.0 Figure 1: Lo w est v e energy lev els of the mo del Hamiltonian H (J 1 J 2 ;h) = J 1 P 16 j=1 ~ S j ~ S j+1 + J 2 P 16 j=1 ~ S j ~ S j+2 h P 4 j=1 S z j as a function of the eld on four adjacen t spins for dieren t ratios of J 2 =J 1 (J 1 = 1). Phase diagr am for h = 0 The phase diagram for the mo del in zero eld and in the large N limit is w ell kno wn and has b een studied for example in [9]. A t zero eld and J 2 =J 1 = 0 the mo del is exactly solv able. In this case w e ha v e a Heisen b erg spin c hain with only nearest- neigh b or coupling, whic h is solv able using the w ell kno wn Bethe ansatz. F or b oth p ositiv e J 1 and J 2 the system is frustrated. F or small v alues of J 2 a gapless an tiferromagnetic phase is presen t. A t J 2 =J 1 = 0:241 a gap op ens up and the system remains gap ed for all nite J 2 but the gap closes in the limit of J 2 <<J 1 , where the mo del is appro ximately describ ed b y t w o w eakly coupled c hains. A t zero eld and J 2 =J 1 = 0:5 is the so-called Ma jumdar-Ghosh p oin t of the J 1 -J 2 -mo del. A t this sp ecic coupling, the ground state of the system can b e determined analytically . Namely , the system has a t w o-fold degeneracy at its minim um energy , consisting of the symmetric and the an tisymmetric sup erp ositions of the t w o pro duct states of nearest-neigh b or singlets. The mo del in nite eld F or small but nite elds - roughly h < 0:3 - w e encoun ter a regime, in whic h the in ter spin coupling still dominates, but the lo cal eld acts as a p erturbation. The energy-eld dep endence here is relativ ely at. Degeneracies whic h o ccur for zero eld are lifted. F or in termediate elds - 0:5<h< 2 - the Heisen b erg coupling and the lo cal magnetic eld comp ete. In this regime w e observ e n umerous lev el crossings. F or large elds - h > 2:5 - the energy lev els are dominated b y the applied magnetic eld, and th us simply decrease linearly with its amplitude. This is caused b y the alignmen t of the 14 aected spins along the eld direction, i. e. m S ! 1=2, th us maximizing the con tribution of the Zeeman energy term, E Zeeman =hN 0 =2. In this regime one can eectiv ely treat the Heisen b erg in teraction as a p erturbation on the eld Hamiltonian for the spins with an applied eld. In fact for large elds the slop e of all the energy lev els in Fig. 1 approac h what one w ould exp ect from a dominating Zeeman term @E @h !N 0 =2 = 2. Ha ving these regimes of the mo del in mind, one could think of v arious quenc h scenarios: Small quenc hes within eac h of the regimes or large quenc hes across dieren t regimes. The simplest are small quenc hes within the regime of large dominating lo cal elds, where the energy lev els in go o d appro ximation simply v ary linearly with the eld amplitude h. As an example of this case, w e quenc hed from an initial eld h = 3:5 to an ev olution eld h 0 = 3 for the three dieren t couplings J 2 =J 1 = 0; 0:5; 1. In all of these cases the Losc hmidt ec ho as w ell as the lo cal magnetization sho w Gaussian distributions with v ery small v ariances, indicating a go o d, straigh tforw ard equilibration. This is exactly the b eha vior exp ected for small quenc hes in regular regimes.[6] The distribution of the quan tit y d s in these cases also resem bles a Gaussian, only exp eriencing a sligh t asymmetry due to the nonlinearit y of the norm. So in regular regimes our example system sho ws go o d equilibration, b oth lo cally and globally . Another set of relativ ely simple quenc hes are those from large elds to zero eld. As the system in this case is strongly p erturb ed, one w ould exp ect n umerous excitations across a wide range of energy . A ccording to L t = 1=d eff (equation (8)) a large n um b er of excitations causes a small Losc hmidt ec ho. More sp ecically suc h quenc hes lead to an exp onen tial distribution of the Losc hmidt ec ho with an a v erage v ery close to zero.[6] In fact w e n umerically observ e suc h a b eha vior when quenc hing from h = 5 to h 0 = 0 for all three couplings. Figure 2 sho ws a go o d example of a quenc h from h = 3 to h 0 = 0 using only nearest neigh b or coupling. In this quenc h w e also obtain single p eak ed and relativ ely narro w distributions of the lo cal magnetization and the quan tit y d S . Results for dieren t couplings are not sho wn here, but v ery similar. This indicates a measure concen tration and exactly the kind of strong lo cal equilibration that is p ossible ev en for closed quan tum systems whic h has b een discussed in [3]. F or small quenc hes in the regime of dominating coupling, w e observ e dieren t t yp es of equilibration and a strong dep endence on the in ter-spin coupling. This is discussed in the 15 0 200 400 600 800 0 0.01 0.02 0.03 n’ p n’ 0 0.05 0.1 0.15 0 50 100 P(L) 0 0.5 1 0 10 20 30 P(d s ) d s −0.1 −0.05 0 0.05 0.1 0 10 20 30 40 m s P(m s ) 0 0.05 0.025 1 10^3 10^1 10^5 L log−linear plot P(L) L Figure 2: Equilibration statistics of a system of 16 spins in a c hain with only nearest neigh- b or Heisen b erg couplings (J 1 = 1;J 2 = 0), quenc hed from an initial conguration with a eld h = 3 in the z-direction applied to four adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. a) sho ws the energy probabilit y distribution p n 0 jh 0 jn 0 ij 2 , b) the corresp ond- ing probabilit y distribution of the Losc hmidt ec ho P (L) = (Ljh 0 j (t)ij 2 ) t , c) the distribu- tion of d S (d S k S (t) S t k 1 ) t and d) the distribution of the normalized lo cal magnetization (m S 1=2Tr[ S (t) P 4 j=1 S z j ]) t . follo wing section. Because of the n umerous lev el crossings b et w een dieren t lev els at dieren t amplitudes of the lo cal eld, it is v ery dicult to obtain the phenomenology of quenc hes within the in termediate regime. F urthermore since dieren t energy lev el crossings o ccur at close eld amplitudes, w e can not separate them n umerically . I I I. NUMERICAL RESUL TS F OR NON-TRIVIAL QUENCHES W e in v estigate quenc hes of the form H(h)!H 0 (h 0 ) on systems of sizes N=12, N=14 and N=16. W e ha v e also p erformed sim ulations with dieren t subsystem sizes (N’=3,4). The results w e presen t here are six represen tativ e examples of c hains with N=16 and a subsystem of four (Quenc hes 1-3, N 0 = 4) or three adjacen t spins (Quenc hes 4-6, N 0 = 3). F or all of them, as n umerically calculated, the starting ground state is lo cated in the M = 0 sector of v anishing total magnetization. 16 0 10 20 30 40 50 0 0.05 0.1 0.15 0.2 n’ p n’ 0.6 0.8 1 0 5 10 P(L) L 0 0.1 0.2 0 10 20 30 P(d s ) d s −0.02 −0.01 0 0.01 0.02 0 25 50 75 100 m s P(m s ) 0 25 50 0 5 x 10 −3 n’ p n’ p 0 =0.86 a) b) c) d) Figure 3: Equilibration statistics of a system of 16 spins in a c hain with equal nearest and next- nearest neigh b or Heisen b erg couplings (J 2 =J 1 = 1), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to four adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 2a) to 2d). A. Quenc hes on four adjacen t spins Quench 1: The rst example is a quenc h on a system of equal next and next-nearest neigh b or couplings (J 1 =J 2 = 1). The system is quenc hed from H(h = 0:2) to H 0 (h 0 = 0). The resulting equilibration statistics are sho wn in Fig. 3. As one w ould exp ect for a small quenc h, the energy probabilit y distribution p n in Fig. 3a) is dominated b y the ground state (p 0 = 0:86). A more in teresting feature is an additional sizeable con tribution of only the rst excited state (the p opulation of the rst excited state is ab out t w o orders of magnitude larger than the p opulation of all the others (p 1 p i ;i> 1)). The existence of t w o dominating mo des leads to a double-p eak ed distribution function of the Losc hmidt ec ho P (L), whic h is clearly observ ed in Fig. 3b). In fact one can completely neglect all the other statesjii fori> 1, treat the mo del as an eectiv e t w o state system and get a go o d appro ximation for the Losc hmidt ec ho using only a constan t and a cosine from equation (7): L(t) p 2 0 +p 2 1 | {z } L t +2p 0 p 1 cos[(E 0 E 1 )t]: (14) The corresp onding probabilit y distribution function can b e calculated analytically: P (l) =L t + 1L t (l 2 l 1 ) p (ll 1 )(l 2 l) ; (15) 17 where l 1 = L t 2p 0 p 1 , l 2 = L t + 2p 0 p 1 giv e the lo w er and upp er edges of the distribution. The result using p 0 and p 1 as n umerically obtained for this quenc h is sho wn as a red line in Fig 3b). It sho ws ho w the system after the quenc h oscillates b et w een to states, one of whic h is close t w o its initial state, the other relativ ely far a w a y . This indicates a lac k of equilibration, that can also b e observ ed in the v ariance, whic h is an order of magnitude higher than in the follo wing quenc hes (see T able I). Note, that although double p eak ed, this distribution should not b e confused with the uni- v ersal double p eak ed distribution of the Losc hmidt ec ho obtained, if there are exactly three non v anishing p n s men tioned in [6]. A b eha vior describing generic quenc hes from a critical p oin t. The mo del studied here is far from the large N limit and moreo v er at J 2 = J 1 = 1 and N!1 the J 1 J 2 mo del is gapp ed. Esp ecially since the ev olution Hamiltonian is translationally in v arian t, one w ould also ex- p ect to see this non-equilibrium b eha vior lo cally . A ccordingly , the statistics of d s in Fig. 3c) sho ws lac k of equilibration. Its distribution function also has a large spread and t w o max- ima, one of whic h sho ws a similar div ergence as the Losc hmidt ec ho. The asymmetry and the lac k of a second p eak are not n umerical artifacts but arise due to the high nonlinearit y of the trace distance, namely its absolute v alue. A simplied example of this b eha vior is giv en in the App endix. It is in teresting to notice that the distribution of the lo cal magnetization is a Gaussian and is not sensitiv e to the eects whic h caused the distributions of the other t w o observ ables to displa y clear signatures of lac k of equilibration. This can b e explained b y the simple fact that the op erator for the lo cal magnetization of four adjacen t spins has zero w eigh ts on the t w o energy states and their cross terms, whic h dominate the energy probabilit y distribution. The fact that this observ able sho ws a Gaussian distribution function with a small v ariance, while others sho w double-p eak ed distributions and large v ariances pro vides a w arning to an y one who ma y try to use univ ersal distributions of some particular observ able to study equilibration. Ev en though ma yb e v ery natural, some observ ables are simply not w ell suited to detect the eects whic h cause suc h double p eak ed distributions, namely those observ ables with v ery small or zero w eigh ts in the lo w energy states. [15] Quench 2: The second quenc h w e sho w is the same quenc h from a eld of h = 0:2 on four adjacen t spins to zero eld, but using only nearest-neigh b or coupling ( J 2 = 0). The energy probabilit y distribution of this quenc h is sho wn in Fig. 4a). In this distribution the 18 0 10 20 30 40 50 0 0.005 0.01 n’ p n’ 0.94 0.96 0.98 1 0 10 20 30 40 P(L) L 0 0.02 0.04 0.06 0.08 0 10 20 30 P(d s ) d s −0.04 −0.02 0 0.02 0.04 0 25 50 m s P(m s ) b) d) c) a) p 0 =0.99 Figure 4: Equilibration statistics of a system of 16 spins in a c hain with only nearest-neigh b or Heisen b erg couplings (J 1 = 1;J 2 = 0), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to four adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 3a) to 3d). ground state is b y far dominating (p 0 = 0:99), the probabilit y of the next highest p opulated state is t w o orders of magnitudes smaller. Ov erall, there still is a concen tration of excitations in the lo w frequencies (n< 20). As indicated b y the distribution of p n 0 , the Losc hmidt ec ho mean of this quenc h is m uc h closer to one and its v ariance is ab out an order of magnitude smaller than with J 2 =J 1 = 1 (Fig. 4b)). Its distribution still sho ws t w o maxima, but they are less pronounced and more Gaussian shap ed. A more general appro ximation of the Losc hmidt Ec ho distribution function as in the previous quenc h, that has only t w o non negligible terms, can b e obtained b y ordering the cosine terms in equation (7) b y the magnitude of their amplitudes p n 0p m 0 , k eeping only the N max largest terms: L(t) = L t + 2 X n 0 <m 0 p n 0p m 0 cos[(E n 0E m 0)t] = L t + 2 N 2 =2N X j=1 A j cos(! j t) L t + 2 Nmax X j=1 A j cos(! j t); (16) where A j = p n 0 j p m 0 j , ! j = E n 0 j E m 0 j and A i < A j for i < j . This appro ximation can then b e used to calculate the corresp onding distribution function for some particular N max , 19 obtaining an appro ximation for the Losc hmidt ec ho distribution function. This is done for all of the follo wing quenc hes. F or the quenc h discussed here, w e use N max = 5. The resulting appro ximate distribution is sho wn as a red line in Fig 4b). T aking the purit y of the original ground state j 0 i in the eigenstate basis of the ev olution Hamiltonian H 0 as a measure, ev en though h and h 0 are the same, the eectiv e quenc h strength is m uc h smaller than in the strongly coupled system with J 2 =J 1 = 1. The distribution of d S in Fig. 4c) lo oks also v ery dieren t from the one of the previous quenc h. Its v ariance is more than v e times smaller, and its relativ ely at maxim um is at a v alue ab out ten times smaller than the p eak in the case J 2 =J 1 = 1. Its asymmetric shap e should again b e explainable with the high non-linearit y of the norm, though w e can not think of a simple calculation as in case of the previous shap e. Both quan tities, the Losc hmidt ec ho and d S indicate a m uc h b etter equilibration. This is somewhat con trary to the b eha vior of the lo cal magnetization in Fig. 3d), whic h is again Gaussian, but sho ws larger rather than a smaller v ariance. 0 20 40 60 0 0.005 0.01 n’ p n’ 0.85 0.9 0.95 1 0 5 10 15 20 25 P(L) L 0 0.1 0.2 0.3 0.4 0 50 100 P(d s ) d s −0.08 −0.04 0 0.04 0.08 0 10 20 30 m s P(m s ) 0.4 0.45 0.425 0 50 100 Zoom d) a) b) c) p 0 =0.96 Figure 5: Equilibration statistics of a system of 16 spins in a c hain with next-nearest neigh b or couplings half as strong as the nearest neigh b or couplings (J 1 = 1;J 2 = 0:5), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to four adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 3a) to 3d). Quench 3: The last quenc h w e w an t to discuss in detail is the sp ecial case of J 2 =J 1 = 1=2. The ev olution system in zero eld is at the earlier men tioned Ma jumdar-Ghosh p oin t of the J 1 J 2 mo del. The system has a t w o-fold degeneracy at its minim um energy , consisting of 20 t w o states, eac h of them b eing a pro duct state of nearest-neigh b or singlets. This degeneracy is lifted in the prequenc h system of a eld h = 0:2 on again four adjacen t spins. As a consequence the system can not b e treated as non-degenerate as it is done in the in tro duction, where w e discuss the time ev olution of observ ables after a quenc h in a generic system. T o tak e suc h degeneracies in to accoun t the corresp onding form ulas ha v e to b e mo died, e. g. includes o-diagonal terms, where E n 0 = E m 0 for n 0 6= m 0 . F urthermore the pro v e for inequalit y (11) giv en b y [3] no more holds, since degenerate systems also break the no degenerate gap condition. Nev er the less n umerically the b ound still holds for the lo cal magnetization (see section IV). Lo oking at p n 0 in Fig. 5a), as in the case of J 2 = 0 the ground state p opulation is b y far dominating. p 0 = 0:96, whic h is a little less than in the previous quenc h. W e also observ e man y more excitations, non of whic h is considerably larger than the others. The eigenstate probabilities are still concen trated at lo w frequencies, but ev en for n 0 > 100 there are some w eak excitations.[16] Notice that quite a few of the lo w est energy states, lik e the rst excited state, are not p opulated at all, but protected b y symmetry . Since the p n 0 are relativ ely widely distributed, and other than the ground state there is no sp ecial state, w e exp ect and n umerically obtain a Gaussian distributed Losc hmidt ec ho, as sho wn in Fig. 5. As in the previous quenc h w e appro ximate the Losc hmidt ec ho using the expansion giv en in equation 16. Only here to obtain a Gaussian shap ed distribution of the righ t width w e ha v e to include the 20 largest terms, i.e. N max = 20. The corresp onding probabilit y distribution is the line sho wn in Fig. 5b). Compared to the quenc h at zero next-nearest neigh b or coupling the Losc hmidt mean is smaller and sho ws a higher v ariance. Compared to the quenc h at J 2 =J 1 the eectiv e quenc h strength still is relativ ely small. So far this indicates relativ ely straigh tforw ard equilibration. Therefore the distribution of the quan tit y d S sho wn in Fig. 5c) is quite surprising. It sho ws the lo w est v ariance of all the quenc hes discussed and a relativ ely smo oth sligh tly asymmetric shap e, still similar to a Gaussian, but its distribution is separated from zero b y a large gap and concen trated around a mean d S = 0:4195. This b eha vior is v ery dieren t from what w e observ ed so far. Since a similar b eha vior is sho wn for small system sizes of 12 and 14 spins in a c hain, this app ears to b e a prop ert y of the sp ecial p oin t J 2 =J 1 = 1=2. Ev en though the ev olution is ob viously not unitary , since this w ould lead to a P (d s ) delta-p eak ed at some xed v alue, the subsystem system circles around its a v erage state in a relativ ely 21 small shell at a large minim um distance. This indicates that it is relativ ely w eakly coupled to the en vironmen t, whic h can b e explained b y a factorizing dimer phase presen t ev en for small lo cal elds h6= 0. In fact, the ground state of the initial Hamiltonian with a nite lo cal eld sho ws a large o v erlap with one of the t w o earlier men tioned pro duct states of nearest-neigh b or dimers, whereas the o v erlap with the other one is eectiv ely zero. Namely it largely o v erlaps with the state, where the t w o inner spins of the four adjacen t spins aected b y the lo cal eld form a singlet and the t w o outer spins eac h form a singlet with the neigh b oring spin in the en vironmen t. This can b e in tuitiv ely understo o d b y the eld sligh tly breaking the outer singlets suc h that the net magnetization on the single spin of the new dimer, whic h is aected b y the eld, b ecomes nonzero, while the singlet in the cen ter is left almost completely undisturb ed. This is conrmed b y considering a subsystem S 0 consisting of only the t w o spins in the cen ter of the eld and c hec king that their ev olution is close to unitary . This is in fact the case: b oth d S 0 = 0:3449 as w ell as the lo cal v on Neumann en trop y S S 0 =Tr( S 0 log 2 S 0) = 2:0620 10 4 remain constan t o v er time. The ev olution of the original subsystem S consisting of all the four spins initially aected b y the eld, can b e understo o d b y com bining the ev olution of the t w o outer and the t w o inner spins. The t w o dimers crossing the edge of the subsystem are en tangled with the en vironmen t, whic h giv esP (d S ) a nite width, but then the in tact singlet in the cen ter of the subsystem ev olv es unitarily , and so the subsystem remains far form its time a v erage. A larged s do es not mean bad equilibration for an y lo cal observ able, but indicates that there is at least one lo cal observ able, whic h equilibrates badly . As in the other quenc hes, the distribution function of the lo cal magnetization is a Gaussian cen tered around zero. Here, in agreemen t with the Losc hmidt ec ho, its v ariance is larger than in the case J 2 =J 1 . B. Quenc hes on three adjacen t spins W e also sho w the equilibration statics of the same three quenc hes just discussed, but with an initial magnetic eld of h = 0:2 on three instead of four adjacen t spins and also referring to these three spins as the subsystem S , i. e. N = 16 and N 0 = 3 (Fig. 6-8). As b efore the system is quenc hed to an ev olution according to the Hamiltonian in zero eld H 0 (h 0 = 0). These quenc hes do not need to b e discussed in the same detail, but indicate that some of the 22 0 10 20 30 40 50 0 0.005 0.01 n’ p n’ 0.94 0.96 0.98 1 0 10 20 30 40 P(L) L 0 0.05 0.1 0 10 20 30 P(d s ) d s −0.01 −0.005 0 0.005 0.01 0 10 20 30 m s P(m s ) p 0 =0.99 Figure 6: Equilibration statistics of a system of 16 spins in a c hain with equal nearest and next- nearest neigh b or Heisen b erg couplings (J 1 =J 2 = 1), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to three adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 3a) to 3d). 0 10 20 30 40 50 0 0.02 0.04 n’ p n’ 0 0.05 0.1 0.15 0.2 0 5 10 15 P(d s ) d s −0.0125 0 0.0125 0 5 10 15 20 m s P(m s ) 0.85 0.9 0.95 1 0 5 10 15 P(L) L p 0 =0.97 Figure 7: Equilibration statistics of a system of 16 spins in a c hain with only nearest-neigh b or Heisen b erg couplings half (J 1 = 1;J 2 = 0), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to three adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 3a) to 3d). observ ed patterns are not restricted to N 0 = 4 and pro vide a few in teresting other features. Also notice that the corresp onding v ariances of the considered quan tities L, d S and m S in the case of N 0 = 3 are m uc h smaller than for N 0 = 4. One reason, wh y w e do not consider quenc hes on only 1 or 2 spins (see T ables I and I I). Quench 4: In the case of J 2 =J 1 = 1 (Fig. 6) one obtains a similar dominance of t w o mo des as in the case of J 2 = J 1 = 1 and N 0 = 4, but with a few con tributions of other 23 0 10 20 30 40 50 0 0.005 0.01 n’ p n’ 0.92 0.94 0.96 0.98 1 0 10 20 30 40 P(L) L 0 0.05 0.1 0.15 0 50 100 150 200 P(d s ) d s −0.01 −0.005 0 0.005 0.01 0 20 40 60 m s P(m s ) p 0 =0.98 Figure 8: Equilibration statistics of a system of 16 spins in a c hain with next-nearest neigh b or Heisen b erg couplings half as strong as the nearest neigh b or couplings (J 1 = 1;J 2 = 0:5), quenc hed from an initial conguration with a eld h = 0:2 in the z-direction applied to three adjacen t spins (denoted b y the sublab el S) to zero eld on all spins. The quan tities sho wn are the same as in Fig. 3a) to 3d). states, that are only one order of magnitude smaller than the smaller of the t w o dominating mo des. The Losc hmidt ec ho distribution accordingly is similarly double p eak ed but more smo oth and with a spread, that is ab out an order of magnitude smaller. T o obtain a go o d result using the appro ximation giv en b y equation (16), w e use N max = 5. The con tribution of these few other states is ev en more visible in the distribution of d S , whic h again sho ws t w o maxima, but is m uc h less spik ed. Note that the lo cal magnetization as opp osed to the quenc h on four spins is in fact double p eak ed, follo wing a distribution function similar to the one of the Losc hmidt ec ho. This can simply b e explained b y the fact that the t w o dominating mo des, namely the ground state and the second excited state in this case ha v e a nite crossterm in the lo cal magnetization. Quench 5: The quenc h using only nearest neigh b or coupling in Fig. 7 sho ws a distribu- tion of the p n 0 , whic h is v ery similar to case of J 2 =J 1 = 1 and N 0 = 3, but the additional non-negligible mo des are concen trated in the lo w lying energy eigenstates as in the case of J 2 = 0;J 1 = 1 and N 0 = 4. The Losc hmidt ec ho is again double p eak ed and appro ximated nicely using N max = 5 in equation (16), but opp osed to the case of J 2 =J 1 = 1 and N 0 = 3 there is a small additional kink in the to outer anks of the distribution. This leads to an additional kink on the righ t end of distribution function of d S , whic h due to the nonlinearit y 24 of the trace distance is in fact m uc h more visible. As in the case of J 2 =J 1 = 1 and N 0 = 3 the magnetization also sho ws t w o maxima, but they are less sharp. Quench 6: The last quenc h sho wn in Fig. 8 is in the case of J 1 = 1;J 2 = 0:5, where the ev olution Hamiltonian in zero eld is at the Ma jumdar-Gosh p oin t. As in the case of N 0 = 4 one observ es a relativ ely large n um b er of non-v anishing mo des and a Gaussian distribution of b oth the Losc hmidt ec ho and the lo cal magnetization. T o obtain the appro ximation sho wn in Fig. 8b) one has to k eep only the ten largest terms of equation (16), i. e. N max = 10. The distribution of d s sho ws a similar gap at small distances as in the case of N 0 = 4, but its mean is smaller and the distribution is highly non symmetric and p eak ed at its minim um distance. J 2 =J 1 L d S m S 10 5 1 0.7506 0.1540 0.6466 0.5 0.9256 0.4195 0.0467 0 0.9727 0.0332 -2.8494 J 2 =J 1 Var(L) Var(d S ) Var(m S ) 1 0.0282 0.0040 0.000075 0.5 0.0037 0.0007 0.00065 0 0.0039 0.0002 0.00028 T able I: Means and V ariances of the Losc hmidt ec ho L =jh 0 j (t)ij 2 , the quan tit y d S =k S (t) S t k 1 and the normalized lo cal magnetization m S = 1=2Tr h S (t) P 4 j=1 S z j i for quenc hes form a eld of h = 0:2 on a Subsystem S consisting of 4 adjacen t spins to zero eld on all 16 spins for dieren t coupling ratios J 2 =J 1 . IV. DISCUSSION OF THE BOUND INDUCED BY THE QUANTITY d S WITH RESPECT TO THE LOCAL MA GNETIZA TION The in terpretation of d S as an upp er b ound for the distinguishabilit y from equation (11) can b e applied to the magnetization in sev eral w a ys, rst o one can lo ok at the closest the magnitude of the lo cal magnetization can b e to d S at an y time. [17] If the maxim um ratio b et w een the magnitude of the lo cal magnetization and the quan tit y d S is 1 it w ould 25 J 2 =J 1 L d S m S 10 5 1 0.9704 0.0376 -1.5295 0.5 0.9571 0.1214 0.4531 0 0.9334 0.0740 -3.9441 J 2 =J 1 Var(L) Var(d S ) Var(m S ) 1 0.00021 0.00028 0.00043 0.5 0.00013 0.000028 0.00019 0 0.0013 0.00094 0.00082 T able I I: Means and V ariances of the Losc hmidt ec ho L =jh 0 j (t)ij 2 , the quan tit y d S =k S (t) S t k 1 and the normalized lo cal magnetization m S 0 = 1=2Tr h S (t) P 4 j=1 S z j i for quenc hes form a eld of h = 0:2 on a Subsystem S consisting of 3 adjacen t spins to zero eld on all 16 spins for dieren t coupling ratios J 2 =J 1 . indicate that, at some time, the lo cal magnetization is among the b est lo cal op erators for distinguishing a lo cal state from the time a v erage, this w ould corresp ond to the uctuations in the subsystem b eing describable as a sup erp osition of classical states all with the same non zero magnetization, at least this instan t. If this ratio is v ery small it sa ys that the magnetization is alw a ys a relativ ely p o or op erator for distinguishing a lo cal state from the time a v erage, meaning that the uctuations ha v e a v ery small net magnetization for all time. F or the four lo cal spins after a quenc h from a small lo cal magnetic eld applied to these four spins to an ev olution Hamiltonian at the Ma jumdar-Ghosh p oin t (Quenc h 3, Section I I I), the maxim um ratio is 0.049, meaning that the net magnetization is nev er a v ery go o d op erator for detecting uctuations in the subsystem and the b ound easily holds ev en though the system is degenerate. This is not surprising b ecause the subsystem w a v e-function is kno wn to con tain a singlet, whic h b y construction has a zero magnetization. If most of the deviation from the time a v erage in this system is caused b y this singlet than the magnetization will nev er b e a go o d observ able to distinguish a state from the time a v erage. F or a quenc h on three lo cal spins again from a small lo cal magnetic eld applied to these three spins to again zero eld in the ev olution Hamiltonian using J 2 = 1, the ratio of the lo cal magnetization to d S is 0.4859, at some time, roughly an order of magnitude higher than in the previous quenc h at the Ma jumdar-Ghosh p oin t. 26 The maxim um ratio for the other quenc hes lies somewhere b et w een these t w o extremes. Instead of lo oking at the maxim um ratio of magnetization to d S one could lo ok at the time a v erage ratio, on a v erage the lo cal magnetization app ears to b e a v ery p o or op erator for distinguishing from the time a v erage state for all quenc hes w e lo ok at. The ratio is of order 10 7 10 8 in all cases. Surprisingly the a v erage of this ratio for the quenc hes to the Ma jumdar-Ghosh p oin t is not alw a ys smaller than for other quenc hes, this can b e explained b y the fact that for most times the lo cal magnetization seems to b e suc h a p o or op erator for distinguishing from the time a v erage that the eect of the singlet in the Ma jumdar-Ghosh system is not immediately ob vious. V. CONCLUSIONS Using a n umerical approac h to study lo cal quenc hes on a fully in teracting system, w e sho w that quan tum informational quan tities suc h as the Losc hmidt ec ho and the trace distance of the time ev olv ed lo cal densit y matrix to its time a v erage pro vide a useful to ol to study equilibration in closed quan tum systems. Ev en in a relativ ely small but complex system w e observ e some of the univ ersal b eha vior of the Losc hmidt ec ho as discussed in [6] and [7], namely Gaussian and exp onen tial shap ed distribution functions. W e also gain some insigh t in ho w these relate to distributions of the quan tit y d S and the lo cal magnetization. W e thereb y sho w that this natural observ able of spin c hains can b e a bad c hoice to study equilibration. Namely , in some cases it indicates smo oth equilibration, whereas the rst t w o quan tities sho w clear non-equilibrium features. W e also observ e that simple quan tities suc h as the mean and the v ariance of the Losc hmidt ec ho or the quan tit y d s pro vide b ounds, but insigh t in the quenc hes sho wn is only giv en b y their distribution functions. Glob al and lo c al e quilibr ation In fact, the long-time b eha vior of the system after a quenc h can dep end on the details of the parameters used. In most cases w e nd agreemen t b et w een the indications of the global quan tit y Losc hmidt ec ho and the quan tit y d S . F or quenc hes within the regime, whic h is dominated b y the lo cal magnetic eld, or for quenc hes from a large lo cal magnetic eld to zero magnetic eld (section I I) b oth quan tities sho w a smo oth equilibration. In the case of a quenc h from a small lo cal eld to zero eld using equal nearest and next-nearest neigh b or Heisen b erg couplings (quenc h 1, section I I I) b oth sho w 27 strong non-equilibrium features. Ho w ev er in some cases the relation b et w een the rst one pro viding a measure for global w eak equilibration and the second one measuring lo cal strong equilibration, can b e more complicated. In the sp ecial case giv en b y a quenc h form a small lo cal eld to zero eld using J 2 =J 1 = 0:5 (quenc h 3 and quenc h 6,section I I I), ev en though the Losc hmidt ec ho indicates a global equilibration in the w eak er sense, the system lo cally sta ys v ery far from its a v erage state for all time. These observ ations demonstrate that ev en though some natural quan tities migh t sho w a Gaussian distribution and migh t therefore ev en b e w ell describ ed b y some statistical en- sem ble, this do es not imply that the system itself and therefore an y reasonable quan tit y is close to equilibrium. W e sho w ed that in some cases the Losc hmidt ec ho and the lo cal trace distance indicate dieren t equilibration b eha viors globally and lo cally and in suc h a w a y cannot b e though t of as con taining the same information, but as pro viding complimen tary insigh ts. Dier ent shap es of e quilibr ation F or the Losc hmidt ec ho w e sho w a simple w a y of almost p erfect appro ximation using only the largest con tributions in the eigenstate expansion giv en b y equation (16). Not only do es this pro vide a nice to ol for a fast and easy calculation, it is also helpful in understanding the phenomenology observ ed. The quenc h from a small eld on four adjacen t spins to zero eld sho wn in Fig. 3 using J 2 = J 1 = 1 pro vides the extreme example of only t w o dominating mo des giving rise to a distribution function with t w o square ro ot div ergences and a large spread. As in Fig. 6, a quenc h starting from a small eld on three adjacen t spins using again J 2 = J 1 = 1, an increased n um b er of non-negligible excitations can lead to a distribution that is still double p eak ed but less div ergen t. The example pro vided in Fig. 4, a quenc h from a small eld on four adjacen t spins using only nearest neigh b or Heisen b erg coupling J 2 = 0;J 1 = 1, sho ws a transien t example of a smo oth transition, to a Gaussian distribution of the Losc hmidt ec ho as it is seen in quenc hes using J 2 = 0:5;J 1 = 1 from a small eld on three or four adjacen t spins (Fig. 8 and 5), nicely appro ximated using 10 or 20 non-negligible terms of p n p m . F urthermore the quenc hes studied here, a large spread of the time series of the Loshmidt ec ho is accompanied with the arise of t w o rather then only a single p eak in its distribution function. Although w e only sho w this for some examples on a sp ecic system and do not pro vide an y further argumen ts, w e do not see a reason, wh y this b eha vior should not b e encoun tered in other systems of similar size, esp ecially in one dimensional spin c hains. 28 App endix Let us assume that w e ha v e a system divided in a subsystem S and an en vironmen t E . The t w o corresp onding basis shall b e giv en b yfjai;jbi;g andfji;ji;g. T o mo del quenc h 1 and to simplify the calculation w e further assume that only t w o eigenstates of the closed system S E con tribute to the full densit y matrix. (A reasonable simplication of the p n distribution in Fig. 3a).) T o mo del the strongly coupled system and to obtain a non-unitary ev olution of the subsystem, w e ha v e to in tro duce en tanglemen t. This can b e done b y taking the to follo wing states: j1i (ja;i +jb;i)= p 2 (17) j2i (ja;ijb;i)= p 2: (18) Note that this is an assumption, whic h mak es the further calculation v ery simple, but here is not ph ysically motiv ated. Using the initial state j 0 i c 1 j1i +c 2 j2i w e compute d s (t) and its distribution. Giv en the energy dierence ! = E 2 E 1 of the t w o eigenstates w e obtain = c 1 c 2 j1ih2je i!t +h:c: (19) T racing out the degrees of freedom of the bath this simplies to S S = Tr B () (20) = c 1 c 2 (jaihajjbihbj)e i!t +h:c:: (21) Calling c 1 c 2 = p p 1 p 2 e i' and using a matrix represen tation w e get S S = p p 1 p 2 cos (!t +') 0 @ 1 0 0 1 1 A (22) W e nally get d S (t) = p p 1 p 2 jcos (!t +')j (23) Its distribution function can b e calculated analytically , giving a simple description of a 29 div ergence similar to what w e ha v e seen n umerically: P (d S ) = lim T!1 1 T T 0 (d S (t)d S )dt (24) = 1 =! =! 0 ( p p 1 p 2 jcos (!t)jd S )dt (25) = 2= p p 1 p 2 d 2 S (26) Note that in this case d S tak es v alues ind S 2 0; p p 1 p 2 , so suc h aP (d S ) has only one square ro ot singularit y at the upp er edge. If w e simply tak e the t w o largest amplitude con tributions of Fig. 3a), w e can estimate the p eak in Fig. 3c) to b e around p 0:86 0:13 = 0:33. Giv en the great simplication this estimate is quite useful. 30 Chapter 1 References [1] N. Chancellor and S. Haas, Ph ys. Rev. B 84, 035130 (2011) [2] S. P op escu, A. J. Short and A. Win ter, Nature 444, 754-758 (2006) [3] N. Linden, S. P op escu, A. J. Short and A. Win ter, Ph ys. Rev. E 79, 061103 (2009) [4] S. Goldstein, J. L. Leb o witz, R. T um ulk a and N Zanghi, Ph ys. Rev. Lett. 96, 050403 (2006) [5] P . Reimann, Ph ys. Rev. Lett. 99, 160404 (2007) [6] L. Camp os V en uti and P . Zanardi, Ph ys. Rev. A 81, 022113 (2010) [7] L. Camp os V en uti and P . Zanardi, Ph ys. Rev. A 81, 032113 (2010) [8] D. Rossini, T. Calarco, V. Gio v annetti, S. Mon tangero and R. F azio, Ph ys. Rev. A 75, 032333 (2007) [9] L.C. K w ek, Y. T ak ahasi and K.W. Cho o, Journal of Ph ysics: Conference Series 143, 012014 (2009). [10] A global eld of the form h P N j=1 S z j comm utes with Heisen b erg couplings. Corresp onding quenc hes w ould b e trivial. [11] Using a lo cal eld of the form h P N 0 j=1 S z j , there is an in teresting symmetry . P erforming a quenc h b y applying an initial eld on N 0 adjacen t spins is eectiv ely the same as applying the same eld on NN 0 adjacen t spins. [12] T o obtain a go o d appro ximation of the full sector w e c hec k ed for the normalization condition P 500 n 0 =1 jh 0 jn 0 ij 2 > 0:9999. [13] i. e. giv en an y four energy eigen v alues E n E m =E l E k implies that ether n =m andk =l or n = l and m = k . As describ ed in [3], this assumption is v ery generic, since it can alw a ys b e ac hiev ed b y in tro ducing an arbitrarily small p erturbation. [14] Note as a p ositiv e quan tit y uctuations can b e b ound against the mean using Mark o v’s in- equalit y . [15] F or a quenc h using a eld on only three adjacen t spins, w e observ ed a similar p n distribution. Only here not the rst, but the second excited state got a large con tribution. This state has a cross term with the ground state in the lo cal magnetization of three adjacen t spins, and hence a double-p eak ed distribution function w as observ ed not only in the Losc hmidt ec ho, but also 31 in the lo cal magnetization. (see Quenc h 4) [16] F or visibilit y reasons these are not sho wn in the graph. But for all the examples giv en here, w e calculated the rst 500 eigenstates of H 0 and their o v erlap with the ground state of H . [17] In the quenc hes discussed here the a v erage of the lo cal magnetization is so close to zero that the oset from the time a v erage is basically already giv en b y the magnetization 32 Chapter 2: T ransport by Degenerate Groundstates This Chapter is based on Propagation of Disturbances in Degenerate Quan tum Systems b y N. Chancellor and S. Haas [1]. Disturbances in gapless quan tum man y-b o dy mo dels are kno wn to tra v el an unlimited distance throughout the system. Here, w e explore this phenomenon in nite clusters with degenerate ground states. The sp ecic mo del studied here is the one-dimensional J1-J2 Heisen b erg Hamiltonian at and close to the Ma jumdar-Ghosh p oin t. Both op en and p erio dic b oundary conditions are considered. Quenc hes are p erformed using a lo cal magnetic eld. The degenerate Ma jumdar-Ghosh ground state allo ws disturbances whic h carry quan tum en tanglemen t to propagate throughout the system, and th us dephase the en tire system within the degenerate subspace. These disturbances can also carry p olarization, but not energy , as all energy is stored lo cally . The lo cal ev olution of the part of the system where energy is stored driv es the rest of the system through long-range en tanglemen t. W e also examine appro ximations for the ground state of this Hamiltonian in the strong eld limit, and study ho w couplings a w a y from the Ma jumdar-Ghosh p oin t aect the propagation of disturbances. W e nd that ev en in the case of appro ximate degeneracy , a disturbance can b e propagated throughout a nite system. VI. INTR ODUCTION This pap er uses quan tum information measures, suc h as en tanglemen t, and trace distance to study quan tum man y b o dy systems. Unlik e ph ysical observ ables, suc h quan tities usually cannot b e directly measured [2], but can giv e an imp ortan t insigh t in to the prop erties of the system. Abstract concepts suc h as quan tum en tanglemen t ha v e b een imp ortan t for almost as long as quan tum mec hanics has existed [3]. The p o w er of these information theoretical quan tities is that they represen t general ideas that can b e applied to an y system whic h can b e considered quan tum. By studying suc h abstract quan tities one can more easily generalize a result for a sp ecic system to more univ ersal b eha vior. Examples of successful application of quan tum information measures to the study of quan tum man y b o dy systems 33 are man y , a few examples are [49]. The sp ecic uses of these quan tities can b e div erse, for example in [5] the authors use the concept of trace distance from an a v eraged densit y matrix to dene a t yp e of quan tum equilibration whic h w ould b e analogous to equilibration in classical thermo dynamics. Similar questions are examined, but with dieren t metho ds, in [6, 7], where the concept of equilibration is used to detect criticalit y in a system. In [4] a quan tit y related to delit y is used to detect quan tum c haos. This c hapter will mak e broad use of suc h quan tum informational quan tities, but will deal with relativ ely few direct observ ables. This is b ecause our in ten tion is to pro vide a study whic h can b e easily related to other quan tum systems, and to quan tum man y b o dy theory in general. The cen tral result of this pap er in v olv es a t yp e of lo cal quenc h whic h can propagate disturbances an unlimited distance in a J1-J2 Heisen b erg spin c hain. The unitary dynam- ics of spin c hains whic h can b e studied through quenc hes can b e realized exp erimen tally with trapp ed cold atoms [10, 11]. Certain sup erconducting qubit arra ys can also pro vide promising ph ysical realizations of spin c hain Hamiltonians [12, 13]. Quenc hes are also im- p ortan t from a theoretical p ersp ectiv e. F or example, quan tum equilibration can b e induced and studied in spin c hains using v arious quenc hes [57, 14]. Certain lo cal quenc hes ha v e also b een prop osed as a w a y to ph ysically measure en tanglemen t en trop y [2]. F urthermore lo cal magnetic eld quenc hes similar to those studied in this pap er ha v e b een used to study en tanglemen t sp ecically in Heisen b erg spin c hains[8], as w ell as other quan tum systems [9]. A generalization of the sp ecic system whic h is studied in this pap er has also b een prop osed as b eing p ossibly useful in quan tum computation [15]. The frustrated spin-1/2 an ti-ferromagnetic Heisen b erg c hain has one of the most pro- tot ypical matrix pro duct ground states, featuring a t w o-fold degeneracy at the so-called Ma jumdar-Ghosh p oin t [16], when the nearest-neigh b or and next-nearest-neigh b or exc hange in tegrals are the same. The Hamiltonian of this system is giv en b y H MG = N X j=1 ~ S j ~ S j+1 + 1 2 ~ S j ~ S j+2 ; (27) where the sum extends o v er N lattice sites, and the t w o terms represen t an ti-ferromagnetic nearest-neigh b or and next-nearest neigh b or Heisen b erg in teractions resp ectiv ely . The 34 ground state of this mo del is exactly kno wn [16], j 1;MG i = N 2 O l=1 (j" 2l1 # 2l ij# 2l1 " 2l i) p 2 ; (28) i.e. the pro duct of nearest-neigh b or spin singlets, assuming an ev en n um b er of lattice sites. F or the case of op en b oundary conditions, this state is unique, whereas for p erio dic b oundary conditions it is t w o-fold degenerate, as the underlying lattice can b e decorated b y the singlet pro duct state in another unique w a y , j 2;MG i = N 2 O l=1 1 p 2 (j" mod N 2l # mod N (2l+1) ij# mod N 2 2l " mod N 2 (2l+1) i): The resulting ground state for the p erio dic system is a sup erp osition, j PB;MG i =aj 1;MG i +bj 2;MG i; (29) where the t w o terms are not automatically orthogonal.[17] Hence, c hanging the b oundary conditions of the Hamiltonian from op en to p erio dic one go es from a unique to a t w o-fold degenerate ground state, th us allo wing us to study the eects of a ground state degeneracy . Lo cal disturbances of this ground state can b e in tro duced b y applying a lo cal magnetic eld h to a subset of N 0 adjacen t spins, H(h;N 0 ) =H MG h N 0 X j=1 S z j ; (30) where without loss of generalit y w e consider the direction of the applied eld to b e along the z-direction. One can tak e adv an tage of the fact that spin p olarization is conserv ed in this system, allo wing one to reduce the complexit y of the problem b y dividing the Hamiltonian in to indep enden t spin sectors, whic h ma y eac h b e diagonalized indep enden tly . These sectors corresp ond to the total p olarization of the system in the z direction, and ma y b e diagonalized indep enden tly . The p olarization sector whic h con tains the global ground state of the system c hanges with eld strength, therefore gures 11, 11,13,17, 18, and 19 all sho w curv es for three dieren t p olarization sectors. Eac h sector is lab eled with the total z p olarization of the en tire spin c hain in that sector, whic h is conserv ed under the action of all Hamiltonians considered in this pap er. F or example in the basis where S z j is diagonal, all of the basis 35 Figure 9: Example of a lo cal eld applied to the Ma jumdar-Ghosh Hamiltonian. states in the L=0 sector will ha v e the same n um b er of spins p oin ting in +z as -z, in the L=-1 state, 2 more spins will b e facing in -z than +z, etc. In this study , w e iden tify sev eral eects induced b y the application of a lo cal magnetic eld, as depicted in Fig. VI. Here w e briey summarize our ndings. Firstly , for sucien tly small eld amplitudes p olarization induced b y the lo cal magnetic eld is stored in the vicinit y of the region to whic h the eld is applied, instead of spreading throughout the en tire system. Only b ey ond a certain threshold eld, i.e. once some of the p olarization in this b oundary region has saturated, can it spread throughout the en tire system. W e argue that this is to b e exp ected b ecause at the Ma jumdar-Ghosh p oin t the energy sp ectrum of the J1-J2 Heisen b erg Hamiltonian is gapp ed. Pro vided that the energy gained from the lo cally applied magnetic eld is small compared to the coupling energy of the spins, an y state whic h k eeps the ma jorit y of spins in a matrix pro duct conguration similar to the zero eld ground state will ha v e a lo w er energy . F or an ev en n um b er of spins in the non-eld region, the system can only accomplish this if the total p olarization of a giv en subsystem far from the eld region is zero. The spins in the eld region align in the direction of the applied eld, th us in turn leading to an excess opp osite p olarization of the spins not directly sub jected to the eld. This induced p olarization is t ypically lo calized near the edge of the eld region. W e will sho w, ho w ev er, that this eect do es not o ccur if the t w o degenerate ground states lie in dieren t p olarization sectors, b ecause in this case the p olarization can spread through the degenerate subspace at no energy p enalt y . W e will also sho w that, for a sucien tly small fraction of the spins sub jected to the eld, there exists at least one state in one of the p olarization sectors whic h lo oks lo cally lik e the zero eld (MPS) ground state far from the eld (Fig. 10). F or the systems studied in this pap er one of these states is alw a ys the ground state.[18] F or the case of p erio dic b oundary conditions, an y state whic h lies lo cally in the degenerate 36 Figure 10: Sk etc h of a t ypical state of the spins when exp osed to a eld. F or all eld strengths studied here, the ground state of at least one total spin sector b eha v es lik e this, and one of these states alw a ys is the global ground state of the system. Ov als represen t en tanglemen t, arro ws indicate spin p olarization. subspace far from the lo cal magnetic eld region will ha v e the minim um lo cal con tribution to the energy . This means that ev en for a system with man y more spins outside of the eld than within it, a disturbance can easily propagate throughout the en tire zero eld region. This pap er is organized as follo ws. In the follo wing section VI I, w e in tro duce the observ- ables on whic h w e fo cus on understanding the eects of a lo cal applied magnetic eld on this man y-b o dy system. The cases of op en and p erio dic b oundary conditions need to b e treated separately . In section VI I, w e then discuss the ph ysics of op en c hains, and in section 4 the phenomena observ ed in p erio dic systems. In section IX, w e consider ho w these results are aected when one departs from the Ma jumdar-Ghosh p oin t in the underlying Hamiltonian. This is follo w ed b y conclusions in section X. VI I. PHYSICAL OBSER V ABLES A. Op en b oundary conditions F or op en b oundary conditions the eld is applied to N’ spins on one end of the c hain. Unless otherwise stated, w e consider nite c hains with a total n um b er of spins, N, p erforming full n umerical diagonalizations of the frustrated Ma jumdar-Ghosh Heisen b erg Hamiltonian. Sev eral observ ables are studied. The rst is the total p olarization outside of the region sub jected to the applied eld. While the total spin p olarization of the c hain is conserv ed, lo cal p olarization is not. This quan tit y is dened as L :N 0 = D j P N j=N 0 +1 S z j j E (31) 37 F urthermore, w e study the trace distance from a singlet state of the t w o spins at the end of the c hain opp osite to the region of the applied magnetic eld, i.e. the spins lo cated at sites N 1 and N . This observ able is dened as d s = 1 2 k s 1 2 (j"#ij#"i)(h"#j +h#"j)k 1 ; (32) where s = Tr :s (j ih j); (33) kOk 1 Tr p O y O: (34) Finally , w e fo cus on the p olarization of the spins at sites N 1 and N , dened the same as in Eq. 31, but with the sum running from N-1 to N. In this pap er the subsystem of the 2 furthest spins will b e lab eled f. This observ able tells ab out whether the p olarization has b een allo w ed to spread to the furthest 2 spins from the eld. T w o dieren t sizes of eld regions are considered, N’=5 and N’=4. The reason that b oth are considered separately is that there are signican t ev en-o dd eects. In this pap er, no actual quenc hes are p erformed in the system with op en b oundary conditions, and all observ ables are giv en for the ground state of a giv en sector. B. P erio dic b oundary conditions F or p erio dic b oundary conditions, the eld is applied to a region of N’ adjacen t spins. In this case, w e are considering c hains with an ev en n um b er of sites. While the observ ables studied in the p erio dic case are dened in analogy to those studied in the op en case, some extra care is necessary . In particular, a complication arises for the trace distance from a singlet for the t w o spins furthest from the eld region. F or p erio dic b oundary conditions, there is no unique c hoice of singlet co v ering for the system. T w o dieren t approac hes to this problem are examined. Firstly , one can consider the distance from the closest of the t w o singlet co v erings for a subsystem, d s;cover =min(k s Tr :s (j 1;NF ih 1;NF j)k 1 ; k s Tr :s (j 2;NF ih 2;NF j)k 1 ): (35) Ho w ev er, this quan tit y has a dra wbac k, i.e. all but a zero measure set of states in the degenerate subspace will ha v e a nite distance to either of these co v erings. An alternativ e 38 approac h is to lo ok at the distance from the closest p oin t in the subspace to the reduced densit y matrix, d s;subspace =min a;b (k s (kaj 1;NF i +bj 2;NF ik 2 ) 2 (Tr[(aj 1;NF i +bj 2;NF i)(a y h 1;NF j +b y h 2;NF j)]k 1 ): (36) This equation app ears as though it can b e further simplied in an ob vious w a y , but remem b er that the t w o w a v e functions are not orthogonal. The norm in the denominator is the usual L2 norm for a v ector. Also in this case the minimization is actually simpler than it lo oks, b y realizing that it can b e reduced to: d sing;subspace =min 01 k s ((1) 1 2 (j"#ij#" i)(h"#j +h#"j) + 1 4 )k 1 where 1 4 is the 4-dimensional iden tit y op erator. While the observ ables presen ted in this section could b e considered as time dep enden t v ariables, in this pap er they are alw a ys studied for the ground state of a giv en p olarization sector. C. Small magnetic eld quenc hes Because of the degeneracy caused b y the p erio dic b oundaries there is another quan tit y whic h is in teresting to lo ok at, relating to a eld quenc h p erformed b y c hanging the mag- netic eld instan taneously and subsequen tly monitoring the time ev olution of the system, esp ecially in regions far from where the lo cal eld is applied. Unitary ev olution giv es the time ev olution of a system follo wing a quenc h at time t = 0, in terms of energies E n , m;n (t) =c m c n exp[{(E n E m )t] ; (37) where c m =hmj i, wherej iis the pre-quenc h ground state of the system. This leads to a denition of the time a v eraged state, m;n =c m c n (E n E m ) : (38) The eld quenc h is p erformed b y taking j (t = 0)i =j 0 i to b e the ground state of a Hamiltonian with a sligh tly stronger eld, H 0 =H P N 0 j=1 S z j . A tt = 0, is instan taneously turned o. In our analysis of the time ev olution, w e will fo cus on the trace distance from the time a v eraged (or dephased) state of the densit y matrix of the t w o spins furthest a w a y from the eld region d av (t) =kTr :s (j (t)ih (t)j) s k 1 : (39) 39 D. Large magnetic eld quenc hes W e also examine the time ev olution due to large lo cal eld quenc hes. Sev eral statistical distributions are studied to understand the ensuing equilibration b eha vior. These quenc hes are p erformed in a regime where quenc hes are sho wn to disturb the en tire system, ev en regions far a w a y from the eld region. A global quan tit y whic h is studied is the Losc hmidt ec ho, a measure of the o v erlap of the time ev olv ed system with the initial state, LE(t) =jh jexp({Ht)j ij 2 : (40) T w o lo cal linear quan tities are examined as w ell. In the region sub jected to the lo cal external eld, the lo cal p olarization is studied. This is simply the exp ectation v alue of the magnetization op erator with resp ect to the lo cal densit y matrix, L N 0(t) = Tr( N 0(t)M): (41) In the region far from the spins where the lo cal magnetic eld is applied, all of the states are exp ected to b e lo cally within the degenerate ground state subspace and therefore ha v e zero magnetization. Therefore, a more appropriate observ able to use is the o v erlap with a singlet state, O s (t) = Tr( s (t)Tr :s (j 1;NF ih 1;NF j)): (42) Finally an imp ortan t non-linear lo cal quan tit y is studied far from the lo cal magnetic eld, the time ev olving distance to the a v erage state, dened b y d s (t) =k s (t) s k 1 : (43) This quan tit y is imp ortan t, as it pro vides a direct measure of equilibration lo cally , and can th us b e used to sho w that the quenc h not only disturbs the system far from the eld, but also that these disturbances can cause equilibration. E. En tanglemen t maps A to ol whic h is used in this pap er for visualizing quan tum states is a map of t w o p oin t en tanglemen t. In these graphics, colors are used to indicate en tanglemen t strength b et w een single spins using v on Neumann en trop y , S VN () Tr( log()); (44) 40 as a measure of en tanglemen t. These graphics consist of arra ys of colored squares where, for o-diagonal elemen ts, the color corresp onds to the en tanglemen t b et w een the t w o spins. The diagonal elemen ts cor- resp ond to the dierence b et w een the maxim um p ossible en trop y on a spin and the actual en trop y . This represen ts the the amoun t of information left ab out a spin after the rest of the system is measured. These maps are created using entMap(i;j) = (1 ij ) ((S VN ( i ) + (S VN ( j )S VN ( ij )) + (45) ij (S VN ( 1 2 1 2 )S VN ( i )) The color scale with the maxim um en tanglemen t normalized to 1 app ears in Fig. 12. It is imp ortan t to note that while these gures can giv e a go o d general impression of en tanglemen t b eha vior of the system, they do not tell the whole story , i.e. they only giv e information ab out t w o-p oin t en tanglemen t. Just b ecause one of these gures sho ws no t w o p oin t en tanglemen t for a pair of spins, this do es not mean that they are not en tangled in a more complicated w a y .[19] Although in principle there is nothing prev en ting one from obtaining en tanglemen t maps for time a v eraged states, in this pap er w e only use this tec hnique to study eigenstates. VI I I. LOCAL MA GNETIC FIELD APPLIED TO MAJUMD AR-GHOSH CHAINS WITH OPEN BOUND ARIES Sub jecting a lo cal region of a Ma jumdar-Ghosh spin c hain to an external magnetic eld forces the exp osed spins to align with the eld. Because of p olarization conserv ation, excess p olarization opp osite to the direction of the eld is generated in the eld-free region of the system. In the sector of zero total spin p olarization (L = 0), and for sucien tly large magnetic eld strengths, this can cause spins far from the eld region to switc h to non- trivial p olarized congurations, whereas for smaller applied elds they remain in a spin singlet pro duct state. In con trast, in p olarization sectors with L6= 0 excess p olarization is trapp ed close to the region where the eld is applied, and singlets are pushed far a w a y from this eld region. This is demonstrated graphically in Fig. 11 where parts (a) and (b) sho w the trace distance of t w o spins far from the lo cally applied magnetic eld from a singlet for elds on an ev en and o dd n um b er of spins resp ectiv ely . P arts (c) and (d) sho w 41 the p olarization stored in the region with no applied magnetic eld v ersus eld strength, again for elds on an ev en and o dd n um b er of spins resp ectiv ely . As Figs. 11(a) and (b) sho w, for lo cal elds applied to regions with b oth an ev en and o dd n um b er of spins, there is alw a ys at least one p olarization sector for whic h singlets are lo cated far a w a y from the eld region. Ev en for relativ ely small nite systems, suc h as the ones studied here, the ground state alw a ys lies in one of these sectors. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 −2 −1 0 1 2 3 L=0 L=−1 L=−2 a) b) c) d) d s d s L ¬ N’ L ¬ N’ h h J add h Figure 11: (a) and (b): T race distance from a singlet state of the t w o spins at the end of the c hain opp osite to the region sub jected to the lo cal magnetic eld. (c) and (d): T otal spin p olarization outside of the region sub jected to the lo cal eld. (a) and (c) are for lo cal elds applied to 4 spins, and (b) and (d) are for lo cal elds applied to 3 spins. On all gures, the solid line is the L=0 sector, dashed lines indicate the L=-1 sector and the dot dashed lines indicate the L=-2 sector. Note that for sucien tly small lo cal elds, the global ground state lies in the L=0 sector, whereas for larger lo cal eld strengths it lies in higher p olarization sectors. In b oth cases the global ground state is lo cally close to the singlet state on spins far from the eld. These plots are all prop erties relating to the ground states of giv en sectors. It is also in teresting to note from Figs. 11(c) and (d) that for a small eld in the L=0 sector, the spins in the eld-free area b eha v es dieren tly , dep ending on whether the lo cal eld is applied to an o dd or to an ev en n um b er of spins. This can b e explained b y the fact that for a eld on an o dd n um b er of spins, the b oundary b et w een the eld and the region 42 with no eld cuts through a singlet, in the original ground state. E.g. the eld gradien t mak es one comp onen t of the singlet more energetically fa v orable than the other. By rotating these t w o spins b et w een the singlet and the classical j#"i state, the ground state can b e adjusted lo cally . When, ho w ev er the eld b oundary is b et w een t w o singlets, a critical lo cal eld strength m ust b e reac hed for an y p olarization to b e transferred from the eld region to the eld-free region as Fig. 11(c) demonstrates. This is b ecause the matrix pro duct state of singlets is still an eigenstate of the Hamiltonian for an y eld strength in this case, and a lev el crossing m ust o ccur b efore the ground state can c hange. [20] Fig. 12 sho ws the en tanglemen t map of a system in the L=-1 global spin sector, with a magnetic eld of h=5J applied on 4 of 16 spins. Fig. 12 suggests that for a range of eld v alues, the distance from a singlet is caused b y frustration from ha ving an eectiv ely o dd n um b er of spins a v ailable in the Ma jumdar-Gosh Hamiltonian. In this case, ho w ev er, the frustration is alleviated b y an in termediate transition region b et w een the eld b eha vior and far from eld b eha vior c hanging its length (at the cost of some energy). [21] A. P olarization eects The w a y the system distributes p olarization dep ends strongly on ev en-o dd eects. T o study the eects of p olarization w e examine Fig. 13 whic h sho ws the dep endence of trace distance from a singlet for spins far from the lo cally applied magnetic eld on the p olarization in the non-eld region in parts (a) and (b) for elds on an ev en and o dd n um b er of spins resp ectiv ely . P arts (c) and (d) sho w the p olarization of the last 2 spins rather than trace distance from a singlet. F rom Figs. 13(c) and (d) one can tell that if the eld is placed on an ev en n um b er of spins, an y p olarization that is in the non-eld region will b e immediately spread, ev en to the furthest spins. In the case where the eld is placed on an o dd n um b er of spins, ho w ev er, a nite amoun t of p olarization can b e sequestered near the b oundary . Figs. 13(a) and (b) sho w that this trend is mirrored in distance from a singlet for far-a w a y spins. The dierences b et w een the ev en-spin and o dd-spin ground state for the zero-eld spin c hain can b e used to explain wh y p olarization sequestration can o ccur in one case and not the other. An y spin 1 2 spin c hain with an o dd n um b er of spins and no applied lo cal eld m ust ha v e a degenerate ground state b ecause the particle-hole dualit y . The degenerate ground states also ha v e dieren t p olarization and, therefore, 2 degenerate ground states with a 43 Figure 12: En tanglemen t map for the L=-1 sector ground state (note that this is not the global ground state) for a Ma jumdar-Ghosh c hain of 16 spins with 4 adjacen t spins, whose p osition is indicated b y the white square, sub jected to a lo cal magnetic eld of strength h=5J . The color scale is normalized to 1 as sho wn. con tin uum of p olarization b et w een L = 1 2 and L = 1 2 are p ossible. This means that for a c hain whic h is eectiv ely o dd, there is no energy p enalt y for b eing an ywhere in this range. This eect allo ws p olarization to b e spread throughout the no-eld region without increasing the energy in that region. P olarization can eectiv ely b e mo v ed through this lo cally degenerate subspace, therefore p olarization sequestration do es not o ccur. Con v ersely , for a spin c hain whic h is eectiv ely ev en, the ground state is unique, and p olarization will tend to b e lo calized in the ground state to a v oid raising the energy of all of the no-eld spins. As Fig. 12 suggests, for certain eld ranges in a giv en sector, the length of the non-eld region of the c hain is eectiv ely o dd. When this happ ens p olarization can b e spread freely throughout the non-eld region, and sequestration do es not o ccur, see Fig. 14. 44 −2 −1 0 1 2 3 −0.2 0 0.2 0.4 0.6 0.8 −1.5 −1 −0.5 0 0.5 1 −0.2 0 0.2 0.4 0.6 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 −2 −1 0 1 2 3 0 0.2 0.4 0.6 0.8 L=0 L=−1 L=−2 c) d) L L polarization sequestration keeps far state close to singlet b) a) d s d s polarization is not able to be trapped near field boundary polarization is trapped near field boundary L ¬ N’ L f L f any polarization immediately starts to destroy singlets L ¬ N’ Figure 13: (a) and (b): T race distance from a singlet state of the t w o spins at the end of the c hain opp osite to the region sub jected to the lo cal magnetic eld v ersus p olarization of the en tire eld free region. (c) and (d): Spin p olarization of the 2 furthest spins v ersus p olarization of the en tire eld free region. Lo cal eld on 4 of 16 spins with p erio dic b oundary conditions (left column). Lo cal eld on 5 of 16 spins with p erio dic b oundary conditions (righ t column). In all parts, the solid line is the L=0 sector, dashed lines indicate the L=-1 sector and the dot dashed lines indicate the L=-2 sector. These plots are all prop erties relating to the ground states of giv en sectors. Field Applied No field Singlets cannot form because of an odd # of spins boundary region Figure 14: Carto on represen tation of the eect whic h prev en ts p olarization sequestration for a eld on an ev en n um b er of spins. 45 Figure 15: The appro ximation used to sim ulate b eha vior with a strong eld. h 0 j app 0 i J add N’=1 0.9976 -0.3323 N’=3 0.9980 -0.3786 N’=5 0.9983 -0.3756 N’=7 0.9987 -0.3706 T able I I I: Statistics considering a eld of h=100 placed on N’ spins, comparing the appro ximate to the actual Hamiltonian. The coupling listed here is the additional coupling added to the 2 closest spins to the eld B. Field Induced Eects F or v ery strong elds, the spins within the eld should ha v e no en tanglemen t with the rest of the system, in lo w energy states. This is b ecause the spins sub jected to the eld will align with the eld. Therefore an eectiv e Hamiltonian whic h acts only on the spins outside of the eld should b e able to describ e the system in lo w energy states. A simple mo del for this Hamiltonian w ould b e to alter the coupling b et w een the t w o spins closest to the eld, with the supp osition that the coupling with the eld spins acts to mediate the in teraction b et w een the t w o spins coupled to them (see Fig. 15). The o v erlap b et w een the kno wn ground state, and the ground state calculated using the appro ximation sho wn in Fig. 15 for dieren t added coupling strengths and dieren t spins in the eld region app ear in Fig. VI I I B. Fig. VI I I B supp orts the claim that this appro ximation w orks fairly w ell in the ground state for a eld on an o dd n um b er of spins. F or n umerical results see table I I I. 46 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 N’=1 N’=3 N’=7 N’=5 <ψ 0 |ψ 0 app > J add Figure 16: Ov erlap b et w een actual ground state, and ground state of a Hamiltonian whic h is applied only to the non-eld spins (tensored with spins opp osing the eld in the eld region), but with a mo died coupling on the t w o spins closest to the eld. X axis is the additional coupling added to the Ma jumdar-Ghosh Hamiltonian. Data w as tak en with h=100, N=16 (op en b oundaries). Dieren t lines are as follo ws solid-N’=1, dashed-N’=3, dotted-N’=5, dot dashed-N’=7. Ev en N’ (not sho wn) are not accurately represen ted b y this mo del. IX. LOCAL MA GNETIC FIELD APPLIED TO MAJUMD AR-GHOSH CHAINS WITH PERIODIC BOUND AR Y CONDITIONS Unlik e op en b oundary conditions, p erio dic b oundaries presen t a case where the unp er- turb ed Hamiltonian has a degenerate ground state. Therefore, the lo cal Hamiltonian for the spins far a w a y from the region sub jected to the lo cal eld will also alw a ys ha v e a degenerate ground state. The complications from this degeneracy add a new series of eects whic h are not observ ed in the op en-b oundary case. These eects are illustrated b y Fig. 17 whic h sho ws in parts (a) and (b) the closest distance from the singlet subspace for the t w o furthest spins from the region of the lo cally applied magnetic eld v ersus p olarization on all non-eld spins for 3 of 20 and 4 of 20 spins in the eld resp ectiv ely . P arts (c) and (d) sho w p olarization on 47 −2 −1 0 1 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0.25 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −2 −1 0 1 2 −0.2 −0.1 0 0.1 0.2 0.3 L=0 L=−1 L=−2 d s a) b) d s c) d) L L system is only near a singlet for very weak field singlets are maintained large amounts of polarization can be trapped on boundary only for weak fields can polarization be captured L ¬ N’ L ¬ N’ L f L f Figure 17: Closest lo cal distance from singlet subspace of 2 furthest spins from the lo cally applied magnetic eld v ersus p olarization on all non-eld spins (top ro w). P olarization on 2 furthest spins v ersus p olarization on all non eld spins (b ottom ro w). Field on 3 of 20 spins with p erio dic b oundary conditions (left column). Field on 4 of 20 spins with p erio dic b oundary conditions (righ t column). On all gures, the solid line is the L=0 sector, dashed lines indicate the L=-1 sector and the dot dashed lines indicate the L=-2 sector, where I call negativ e L to b e in the direction of the eld. These plots are all prop erties relating to the ground states of giv en sectors. the t w o furthest spins from the lo cally applied eld v ersus total p olarization in the non-eld region, again for eld on 3 of 20 and 4 of 20 spins resp ectiv ely . The most immediately ob vious dierence is that if the lo cal magnetic eld is placed on an o dd n um b er of spins, neither spin sequestration nor closeness in trace distance to the singlet subspace for an y spins are observ ed, except for the L=0 subspace in w eak lo cal elds. Figs. 17(a) and (c) sho w the trace distance from a singlet in spins far from the applied magnetic eld and lo cal angular momen tum for spins far from the lo cal magnetic eld resp ectiv ely , 48 b oth v ersus total angular momen tum in the eld free region. F or larger elds, the spins far from the region where the external magnetic eld is applied do not approac h the singlet subspace b ecause of frustration caused b y ha ving an o dd n um b er of spins in the non-eld region. In the case of p erio dic b oundary conditions, the eects of the frustration are stronger than in the case of op en b oundaries. This is b ecause here a c hange in the length of the eld- to-far-from-eld transition region will do nothing to reliev e the frustration b ecause of the symmetry b et w een the t w o eld b oundaries. Regardless of whether the length of one of these regions is o dd or ev en, the total length of transition regions is alw a ys ev en b ecause it is the length of a single transition region m ultiplied b y t w o. A. Eect of lo cal degeneracy on small quenc hes Shifting the fo cus to the case where the external eld is placed on an ev en n um b er of spins, one can consider the eects of no w ha ving a lo cally degenerate ground state, i.e. ha ving a Hamiltonian whic h has a ground state degeneracy when no eld is applied, and therefore is degenerate in a lo cal sense far from the spins with an applied magnetic eld. Fist the ground state can b e studied b y observing Fig. 18, this gure sho ws in parts (a) and (c) the trace distance from the closest singlet co v ering and minim um distance from the manifold of singlet co v erings resp ectiv ely for the t w o furthest spins from the lo cally applied magnetic eld v ersus eld strength, for a eld applied to 4 of 20 spins with p erio dic b oundary conditions. P arts (b) and (d) sho w en tanglemen t maps for a lo cal magnetic eld strength of h=1.3J and h=1.6J resp ectiv ely , again for a eld on 4 of 20 spins with p erio dic b oundaries. Fig. 18(c) indicates that the global ground state of the system is alw a ys close to the singlet subspace far from the eld, ho w ev er 18(a) suggests that around a eld strength of 1.5 the system ma y undergo a switc h b et w een singlet co v erings far from the spins to whic h the eld is applied. Figures 18(b,d) conrm this suspicion b y sho wing that indeed b efore the p eak in 18(a) there are an ev en n um b er of dimers outside of the eld region, while after there are an o dd n um b er of dimers. This indicates that disturbances from the lo cal eld can b e felt far from the spins with an applied eld, but only for a narro w range of eld v alues. One can no w consider the eect of small quenc hes at v arious applied eld strengths on spins far from the eld spins. The results of suc h quenc hes are sho wn in Fig. 19, parts (a) and (b) sho w the trace distance to a v erage for the t w o furthest spins from the lo cally 49 Figure 18: a) T race distance b et w een the t w o spins furthest from the eld and the nearest singlet co v ering (see Eq. 35) v ersus eld for 20 spins with p erio dic b oundary conditions and a eld placed on 4 of the spins. b) En tanglemen t map for 16 spins with a eld of h=1.3J placed on spins 1-4 (indicated b y the white rectangle) with p erio dic b oundary conditions c) Same as (a), but no w with distance to the closest state in the degenerate subspace (see Eq. 36) d) Same as (b) but with a eld of h=1.6J. On all gures, the solid line is the L=0 sector, dashed lines indicate the L=-1 sector and the dot dashed lines indicate the L=-2 sector, where negativ e L is in the direction of the eld. All plots in this gure are for eigenstates of the Hamiltonian. applied magnetic eld, after a quenc h whic h in v olv es a small c hange in eld strength v ersus the strength of that eld for a eld on 3 of 20 and 4 of 20 spins resp ectiv ely with p erio dic b oundary conditions. P arts (c) and (d) sho w the p olarization of the t w o furthest spins from the lo cally applied magnetic eld v ersus eld, and are included to emphasize the imp ortan t role pla y ed b y p olarization in this system. One w ould exp ect that these disturbances can only b e propagated through the still lo cally degenerate ground state subspace of the no-eld Hamiltonian and therefore w ould only ha v e an eect when the co v erings shift. Fig. 19(b) sho ws that in fact a small quenc h do es disturb the system strongly at the p oin t where the 50 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 L=0 L=−1 L=−2 d s c) d) a) h h h h L ¬ N’ L ¬ N’ b) d s Figure 19: Initial trace distance from a v erage (see Eq. 39) for a subsystem far from the elds after a small eld quenc h = 0:001 (top ro w). P olarization on all non-eld spins v ersus eld strength (b ottom ro w). 20 spins with eld placed on 3 of them and p erio dic b oundary conditions (left column). same with eld placed on 4 spins (righ t column). Dotted v ertical lines ha v e b een added to emphasize correlation b et w een the t w o graphs. On all gures, the solid line is the L=0 sector, dashed lines indicate the L=-1 sector and the dot dashed lines indicate the L=-2 sector. Lines at the top are added to sho w whic h spin sector the global ground state is in. The top t w o plots are time a v eraged quan tities from a quenc h, while the b ottom t w o gures are prop erties of the ground state of eac h sector. co v erings switc h. The other t w o p eaks in Fig. 19(b) are less relev an t b ecause they o ccur in the ground state of a spin sector, but not in the global ground state of the system. Also none of the quenc h disturbances whic h o ccur far from the spins with an applied eld o ccur in the global ground state in Fig. 19(a), demonstrating another dierence caused b y ev en- o dd eects. This is to b e exp ected, b ecause the the t w o degenerate ground states of an o dd length Ma jumdar-Gosh c hain lie in dieren t p olarization sectors and therefore cannot exhibit lev el repulsion, at least lo cally , in the region far from the applied magnetic eld. 51 Figure 20: Carto on of eld quenc h for p erio dic b oundaries. Note also that Fig. 19 suggests that there is a strong correlation b et w een p olarization outside of the subsystem where a magnetic eld is applied and quenc h disturbance to the far spins, in the sense that when the quenc h has a strong eect, there is a rapid c hange in p olarization in the non-eld region. The con v erse ho w ev er is not supp orted b y this gure. This demonstrates than p olarization pla ys a strong role in the global b eha vior of this system. The same energy argumen ts used in the static case for b eha vior of spins far from the spins with an externally-applied magnetic eld should b e usable as a dynamical argumen t. A nite lo cal eld can only in tro duce a nite amoun t of energy in to the system. Therefore only states whic h lie sucien tly close in energy to the ground state can b e accessed in an y signican t w a y . F or a large enough system, all of the lo w energy states will ha v e to b e lo cally close to the ground state for most of the spins far from the lo cally-applied magnetic eld, therefore lo cally , far from the eld, spins can only b e disturb ed within the degenerate subspace. Put another w a y , in gapp ed systems the eects of a lo cal quenc h ha v e to b e lo calized, unless there exists a lo cally degenerate subspace far from the region where the quenc h is applied. When suc h a subspace exists it ma y b e able to transp ort conserv ed c harges, quan tum en tanglemen t and dynamical disturbances an unlimited distance a w a y from the disturbance site. A lo cally degenerate ground state can b e though t of as a sp ecial symmetry whic h allo ws transp ort of information and c harges (but not energy) with no losses throughout the part of a system far from the quenc h. [22] Long-range en tanglemen t allo ws a part of the system whic h lies en tirely in a degenerate subspace to ha v e its ev olution driv en b y lo cal ev olution far a w a y , see Fig. 20. 52 B. Large quenc hes No w that it is established that a disturbance will b e able to b e propagated throughout the en tire system from a small quenc h, one can p erform a large quenc h from h=1.6 to h=1.3 for a lo cal eld applied to 4 adjacen t spins of 20 total spins with p erio dic b oundary conditions. One can then examine the time statistics of v arious prop erties of the system. These statistics are sho wn in Fig. 21, in this gure part (a) is the trace distance of 4 spins far from the lo cally applied magnetic eld from a singlet state, part (b) is the time statistics of the Losc hmidt ec ho of the en tire system, part (c) is the time statistics of the distance from the time a v eraged state for 4 spins far from the lo cally applied eld, and part (d) is the time statistics of the magnetization of the spins sub jected to the eld. These statistics will sho w the abilit y of the system to equilibrate, ev en lo cally for spins far from the spins where the lo cal magnetic eld is applied. In the case studied here, the system only equilibrates p o orly , ev en in the global sense, not surprisingly , p o or equilibration is also sho wn in lo cal observ ables b oth close to and far from the spins with an applied magnetic eld. The double-p eak ed pattern of equilibration seen here is t ypical of small systems, see [5] (see c h. 1 starting on page 8), and is th us consisten t with the theory that although the system itself is rather large [23], the actual ev olution is only taking place on a few spins in or near the region of externally applied eld, the rest of the system is simply b eing drug along b y long range en tanglemen t with these spins. As Fig. 21(c) demonstrates, ev en though the dynamics is driv en b y long range en tanglemen t with far a w a y spins, a subsystem of spins is still able to b e pushed to w ard equilibration in the trace distance sense. The fact that there is no lo cal energy dierence do es not seem to in terfere at all with equilibration of these spins. The trace distance from the a v erage is observ ed quite close to zero at some times, unlik e in similar quenc hes p erformed at the Ma jumdar-Ghosh p oin t in [5]. This is b ecause an undisturb ed singlet somewhere in the region b eing observ ed w ould yield a large distance from the a v erage at all times as sho wn in [5] where the quenc h did not cause a c hange in singlet co v erings. In the case w e are observing, where the co v erings switc h, there are no undisturb ed singlets in the region a w a y from the eld spins. Although the equilibration is globally p o or in this system, there are no signs that equi- libration via long-range en tanglemen t through a lo cally degenerate subspace is an y less eectiv e than direct equilibration of the spins to whic h the eld is applied. The data from 53 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 0.05 0.1 0.15 0.2 0.25 0 100 200 300 400 500 600 700 0 0.05 0.1 0.15 0.2 0 100 200 300 400 500 600 −1.6 −1.4 −1.2 −1 0 100 200 300 400 500 600 x x P LE x x P L N’ P O s P D s c) d) a) b) Figure 21: Equilibration statistics for N=20, N’=4 with a quenc h from h (i) =1.6 to h (f) = 1:3. a) Time statistics of the trace distance of 4 spins far from the lo cally applied magnetic eld from a singlet state (Eq. 42). b) Time statistics of the Losc hmidt ec ho (Eq. 40) with an appro ximation based on few frequencies. c) Time statistics of distance from time a v eraged state (Eq. 43) for 4 spins far from the lo cally-applied magnetic eld. d) Time statistics of lo cal magnetization (Eq. 41) of the eld spins. These plots are all time statistics obtained from ev olution. this quenc h therefore indicate that the en tire system can b e equilibrated (at least somewhat) b y a quenc h whic h only aects a v ery small region. In fact a system of an y size should b e able to b e brough t lo cally close to equilibrium in this w a y . Because all states of the far spins lo cally ha v e the same energy , than they cannot aect the time ev olution of the system, therefore the same b eha vior w ould b e exp ected for a spin c hain of an y sucien tly long (ev en) length. X. OTHER COUPLING STRENGTHS One can no w ask what w ould happ en if the coupling w ere c hanged suc h that the system w as no longer using the Ma jumdar-Ghosh Hamiltonian, but allo w ed the next nearest neigh- 54 Figure 22: Initial trace distance to a v erage for far spins after a small eld quenc h, larger distances are ligh ter, smaller distances are dark er. T race distance is plotted on a logarithmic scale, con tour lines (red) are included for clarit y . Data using 20 spins with p erio dic b oundaries in the L=-1 sector. b or coupling to tak e on arbitrary v alues, see Eq. 46. This study is done with 20 spins and p erio dic b oundary conditions, with a lo cal magnetic eld on 4 adjacen t spins. H J2 = N X j=1 ~ S j ~ S j+1 +J 2 N X j=1 ~ S j ~ S j+2 (46) Small eld quenc hes can b e considered on this new Hamiltonian exactly in the same w a y they can b e considered for the Ma jumdar-Ghosh Hamiltonian, the results app ear in Fig. 22 whic h sho ws the initial trace distance from a v erage for a small eld quenc h v ersus coupling and eld strength. This data sho ws that for a wide range of coupling near the Ma jumdar- Ghosh p oin t, small eld quenc hes can drastically aect spins v ery far from the spins with an applied magnetic eld at sp ecic eld strengths. Ho w ev er, when the eld strength is ev en tually dieren t enough, these p eaks broaden out and disapp ear (note logarithmic scale in Fig. 22). The basic b eha vior seen previously in this pap er holds for a wide range of couplings, where the ground state is no longer degenerate. 55 F or an innite system one w ould exp ect that, far spins from the lo cal magnetic eld could not b e disturb ed b y a lo cal eld quenc h unless either the system is gapless or there exists a degenerate ground state. F or a nite system, this w ould only b e necessarily true if the gap b et w een the ground state and rst excited state is sucien tly large compared to the energy in tro duced b y the applied lo cal magnetic eld, in whic h case the eld will b e unable to in tro duce enough energy to aect the en tire system. In this case the order of magnitude of the energy whic h the eld in tro duces can b e estimated b y simply m ultiplying the eld strength b y the n um b er of spins it is applied to. Because b oth of the quan tities are of order 1, one w ould exp ect that the energy in tro duced w ould also b e of order 1. The energy gap in the system whic h will b e used for this calculation can b e determined b y exact diagonalization. The energy of the gap b et w een the ground state and rst excited state of this system are sho wn in Fig. 23 part (c) whic h sho ws the gap energy v ersus coupling at zero applied eld, part (a) sho ws the initial distance from the a v erage for 4 spins far from the lo cally applied magnetic eld after a large quenc h (within the L = 1 2 sector), part (b) sho ws the lo cal trace distance from the nearest singlet co v ering for spins far from the lo cal eld in the ground state of the L = 1 2 sector v ersus eld strength and coupling, and part (d) is the same as (b), but with trace distance from the nearest state in the ground state manifold. It can b e seen from Fig. 23(c) that the gap energy is at most of order 0.1, therefore, one w ould exp ect that for the en tire range of couplings, the far spins could b e disturb ed b y the lo cal eld. The results seen in Fig. 22 are as exp ected, ho w ev er if the system size w ere increased to innit y , one w ould exp ect that in the gapp ed region for J 2 &0.25, the p eaks in the distance w ould ha v e to disapp ear except for exactly at J 2 =0.5, or an y other p oin t with a degenerate ground state. T w en t y spins, ho w ev er, is still to o small a system for c hanges in the coupling to destro y the abilit y to dephase far spins with a lo cal eld, in other w ords the system can b e considered to ha v e an appro ximately degenerate (wrt. the energy scale asso ciated with the eld) ground state for all v alues of J 2 , the next nearest neigh b or coupling. One can no w ask whether the eects seen in Fig. 22 a w a y from the Ma jumdar-Ghosh p oin t are also caused b y some kind of shift in singlet co v ering. T o answ er this question, one can compare Fig. 23(b) to Fig. 23(d) and notice that where the p eaks in Fig. 22 are lo cated, the trace distance from either co v ering tends to b e relativ ely large, but the distance from the subspace tends to b e relativ ely small. This indicates that mo v emen t within the singlet 56 Figure 23: a) Initial distance from a v erage for 4 spins far from the lo cally applied eld for a large quenc h from h=2 to h=1 for the L=-1 sector. b) distance of far spins from nearest singlet co v ering (Eq. 35) v ersus h and J 2 , L=-1 sector, color scale same as for en tanglemen t maps, but normalized to largest v alue. c) gap b et w een ground state energy and rst excited state, for dieren t J 2 and h=0. d) distance from singlet subspace for far spins (Eq. 36) v ersus h and J 2 , L=-1 sector, color scale same as for en tanglemen t maps, but normalized to largest v alue. All plot except for (a) are static quan tities relating to eigenstates. subspace is the cause of m uc h of the disturbance in the far spins. Also in teresting to note is that for a signican t p ortion of the couplings, the spins far from the lo cally applied magnetic eld are closest to the singlet subspace when the small eld quenc hes ha v e the most eect on far spins. It app ears that ev en at man y couplings a w a y from the Ma jumdar-Ghosh p oin t, the mo del of switc hing b et w een co v erings as a w a y to spread a disturbance throughout the system is accurate. In fact for man y v alues of J 2 , the system app ears to mo v e in to the singlet subspace for a narro w range of elds only when the co v ering c hange o ccurs. F or J 2 & 0:6 this mo del seems to break do wn, but it is still at least relev an t for a large range of J 2 . Although not directly related to the quenc h, it is in teresting to note that ab o v e a certain lo cal magnetic eld strength the spins far from the eld seem to lie on the singlet sup erp osition manifold for a fairly large range of coupling strengths near Ma jumdar-Ghosh 57 coupling, as w ell as a narro w strip b et w een J 2 = 0:6 and J 2 = 0:8, the reason for this is not kno wn. The results of a large lo cal magnetic eld quenc h o v er v arying J 2 as sho wn in Fig.23(a) simply helps to underscore what has already b een noted ab out c hanging coupling not b eing an eectiv e w a y of prev en ting disturbances from propagating throughout the system at this system size. Not only do the large quenc hes ha v e a signican t eect on far spins from the lo cal magnetic eld for all coupling strengths, but the exp ected trend of decreasing quenc h eect with increasing gap is not visible in an y denitiv e w a y , indicating that, not only is the energy scale of the gap (Fig. 23(c)) to o small to b e the dominating factor in the quenc h eectiv eness, it seems to not ev en pla y a v ery signican t role. This result is consisten t with the previous energy scale argumen t, the energy scale asso ciated with the eld is alw a ys at least an order of magnitude larger than the gap b et w een the rst 2 eigenstates. XI. CONCLUSIONS In systems with degenerate ground states, quan tum en tanglemen t, disturbances, and c harges can propagate freely , as long as the quenc h crosses b et w een pre and p ost quenc h ground states whic h are lo cally dieren t from eac h other far a w a y from the region aected b y the lo cal quenc h. This eect is dieren t and indep enden t from gapless excitations, and has b een demonstrated to o ccur in a gapp ed system. Unlik e in gapless systems where excitations carry an arbitrarily small amoun t of energy far from the quenc h, these excitations store all energy lo cally near the quenc h, and ev olution far a w a y is driv en solely b y long-range en tanglemen t. The lo cal energy far from the region aected b y a lo cal quenc h Hamiltonian is exactly zero in these systems, not arbitrarily small. T o allo w a c harge to b e propagated through a degenerate subspace, the t w o degenerate ground states m ust ha v e dieren t lo cal exp ectation v alues for said c harge far from the region aected b y a lo cal quenc h Hamiltonian. An eectiv ely o dd spin c hain far from the eld is allo w ed to propagate p olarization throughout the far region for example. Again, in suc h a case, long range en tanglemen t can propagate the c harge, but do es not propagate an y energy far from the eld. In cases where t w o degenerate ground states with dieren t exp ectation v alues for a c harge far from the region where the quenc h is applied do not exist, the c harge can b ecome lo cally trapp ed in part of the system. In the case of the Ma jumdar-Ghosh 58 Hamiltonian, the p olarization is trapp ed near the b oundary of the lo cal magnetic eld region. A lo cally unique ground state (in a gapp ed system) means that c harges, as w ell as disturbances, are conned after a lo cal quenc h. Energy argumen ts prev en t a disturbance from tra v eling throughout the system and also therefore forbid c harges from mo ving outside of a small area. In the system studied here, a large lo cal magnetic eld causes the spins within the eld to b ecome eectiv ely ’xed’, facing in the direction of the eld in the ground state. An appro ximation whic h do es not include these spins directly but includes an eectiv e mo du- lation in coupling b et w een the t w o spins neigh b oring the eld can faithfully repro duce the ground state when an o dd n um b er of spins remain. F or the case that an ev en n um b er of spins are left out of the eld region, this simple appro ximation fails. W e b eliev e that the eectiv e transition region b et w een the eld and non-eld region consists of an o dd n um b er of spins, and that this ground state cannot b e faithfully repro duced in this w a y b ecause of o dd length frustration eects. F or Ma jumdar-Ghosh spin c hains with p erio dic b oundaries, with a lo cal magnetic eld on some ev en n um b er of spins, there exists a range of elds where a small eld quenc h can propagate a disturbance through the en tire system using long range en tanglemen t. This disturbance is propagated lo cally through the degenerate subspace of the lo cal ground state. In this case, this range of elds is relativ ely narro w. A quenc h across this en tire range do es not only cause equilibration near the eld, but also mo v es the far spins to w ards equilibration, within the lo cally degenerate subspace. Study of systems with dieren t next nearest neigh b or coupling indicate that the basic eect whic h causes disturbances to b e propagated to far spins can, at least for small enough systems, b e extended a w a y from the Ma jumdar-Ghosh p oin t. F or nite systems, if the gap b et w een the ground state and the rst excited state is small enough, the same eect whic h w as describ ed here for degenerate systems can also b e applied to systems where the rst 2 states are close in energy . In other w ords, under the righ t conditions, an appro ximate degeneracy will w ork in place of an exact degeneracy . F or a system of 20 spins an y v alue of J 2 b et w een 0 and 1 still allo ws the far spins to b e signican tly aected b y the eld. W e strongly susp ect that for the v alues of J 2 where the Hamiltonian is gapp ed and for whic h a degenerate ground state do es not exist, spins far from a lo cal magnetic eld applied to a few spins cannot b e aected in the large system limit. 59 A c kno wledgemen ts This Chapter is based on Propagation of Disturbances in Degenerate Quan tum Systems b y N. Chancellor and S. Haas [1]. The authors of that pap er w ould lik e to thank N. T o- bias Jacobson for his assistance in editing. W e w ould also lik e to thank S. Garnerone, B. Normand, and M. Diez for v aluable discussions. The n umerical computations w ere carried out on the Univ ersit y of Southern California high p erformance sup ercomputer cluster. The w ork in the original pap er w as supp orted b y NSF gran ts: PHY-803304,DMR- 0804914. 60 Chapter 2 References [1] N. Chancellor and S. Haas, Ph ys. Rev. B 84, 035130 (2011) [2] J. Cardy , arXiv:1012.5116v1 [cond-mat.stat-mec h] (2010) [3] A. Einstein, B. P o dolsky , and N. Rosen, Ph ys. Rev. 47, 777 (1935). [4] N. T. Jacobson, P . Giorda and P . Zanardi, Ph ys. Rev. E 82, 056204 (2010) [5] M. Diez, N. Chancellor, L. Camp os V en uti, S. Haas and P . Zanardi, Lo cal Quenc hes in F rus- trated Quan tum Spin Chains: Global vs Subsystem Equilibration, arXiv:1007.0041v1 [quan t- ph] [6] L. Camp os V en uti and P . Zanardi, Ph ys. Rev. A 81, 022113 (2010). [7] L. Camp os V en uti and P . Zanardi, Ph ys. Rev. A 81, 032113 (2010). [8] Jie Ren and Shiqun Zh u, Ph ys. Rev. A 81, 014302 (2010) [9] Benjamin Hsu, Eytan Grosfeld, and Eduardo F radkin, Ph ys. Rev. B 80, 235412 (2009) [10] L. M. Duan, E. Demler, and M. D. Lukin, Ph ys. Rev. Lett. 91, 090402 (2003). [11] D.P orras and J.I. Cirac, Laser Ph ysics, V ol. 15, No. 1, 2005, pp. 8894. [12] Y. Makhlin, G. Sc hn, A. Shnirman, Rev. Mo d. Ph ys. 73, 357 (2001) [13] L. S. Levito v, T. P . Orlando, J. B. Ma jer and J. E. Mo oij, arXiv:cond-mat/0108266v2 [cond- mat.mes-hall] (2001) [14] Da vide Rossini, Sei Suzuki, Giusepp e Mussardo, Giusepp e E. San toro, and Alessandro Silv a, Ph ys. Rev. B 82, 144302 (2010). [15] Rahel Heule, C. Bruder, Daniel Burgarth and Vladimir M. Sto jano vi¢, arXiv:1010.5715v1 [quan t-ph] (2010) [16] C K Ma jumdar 1970 J. Ph ys. C: Solid State Ph ys. 3 911 [17] Ho w ev er, one can orthogonalize them, still preserving the symmetry under a ! b; 2l! mod N (2l + 1), b y adding a term(a N N 2 k=1 j" 2k1 # 2k i p 2 +b N N 2 k=1 j# 2k1 " 2k i p 2 ). [18] The only case where this is not observ ed is for the case of p erio dic b oundary conditions with a lo cal eld applied to an ev en n um b er of spins. In this case these spins orien t lo cally lik e in the frustrated ground state of a Ma jumdar-Ghosh c hain with an o dd n um b er of spins. [19] An example of this w ould b e to consider a sup erp osition of the t w o singlet co v erings. An y 61 non-adjacen t spins will ha v e exactly zero t w o-p oin t en tanglemen t (see Eq. 45). Ho w ev er, an y set of t w o pairs of adjacen t spins will ha v e a nite en tanglemen t b et w een them. T o see this, consider the case where one set of t w o spins is measured to b e b oth in the up direction. If there is an o dd n um b er of spins b et w een the t w o spin pairs, this forces the other pair to b e a singlet, regardless of the distance b et w een the pairs. [20] Sho wing that this happ ens consists of demonstrating that the singlet co v ering is an eigenstate of an y eld whic h do es not c hange within a singlet. This can b e seen b y realizing that the eld op erator on the rst spin of the singlet will giv e the zero magnetization triplet state 1 p 2 (j"#i +j#"i), while the op erator on the second spin will giv e the same but with a negativ e sign. Th us the t w o will cancel making the singlet co v ering an eigenstate with a zero eigen v alue. [21] The closeness to the b oundary ma y also b e a factor in the trace distance of the last 2 spins from a singlet, the fact that no en tanglemen t can cross the op en b oundary ma y cause spins close to it to assume more lo calized states, ho w ev er while this eect could mak e nite distances smaller, it should not b e able to mak e the trace distance (virtually) zero as it is in man y parts of Fig. 11(a). [22] Note that the excitations whic h tra v el through a lo cally degenerate ground state are not the same as gapless excitations, whic h lo cally carry an arbitrarily small amoun t of energy . The parts of these excitations whic h exist far from the region aected b y the lo cal quenc h Hamiltonian carry exactly zero energy lo cally . [23] One could argue that 20 spins is not suc h a large system, but it has b een sho wn that the spins far from the ev olution are alw a ys lo cally in the degenerate subspace. A dding more far spins and making the system in to one whic h all of the readers w ould agree w ould b e large (for example making the system size 100,000 spins) w ould not eect the dynamics, and the double p eak ed pattern w ould remain. 62 Chapter 3: Adiabatic Quantum Bus Protocol This Chapter is based on the pap er Using the J1-J2 Quan tum Spin Chain as an A diabatic Quan tum Data Bus b y N. Chancellor and S. Haas [1]. This c hapter in v estigates n umerically a phenomenon whic h can b e used to transp ort a single qubit do wn a J1-J2 Heisen b erg spin c hain using a quan tum adiabatic pro cess. The motiv ation for in v estigating suc h pro cesses comes from the idea that this metho d of trans- p ort could p oten tially b e used as a means of sending data to v arious parts of a quan tum computer made of articial spins, and that this metho d could tak e adv an tage of the easily prepared ground state at the so called Ma jumdar-Ghosh p oin t. W e examine sev eral anneal- ing proto cols for this pro cess and nd similar results for all of them. The annealing pro cess w orks w ell up to a critical frustration threshold. There is also a brief section examining what other mo dels this proto col could b e used for, examining its use in the XXZ and XYZ mo dels. In tro duction The abilit y to send data from one part of a computer to another accurately and quic kly is an essen tial feature in virtually an y design. The use of articial spin clusters in quan- tum computing has b een of gro wing in terest. There is an implemen tation whic h has b een demonstrated using sup erconducting ux qubits[26]. This pap er demonstrates an eectiv e and scalable w a y of sending arbitrary qubit states along a spin c hain with Heisen b erg t yp e coupling using quan tum annealing. Assuming one could implemen t a Hamiltonian whic h follo ws this mo del, for example using the metho ds prop osed in [7] using coupled ca vities, this system design could b e used for a data bus whic h transp orts quan tum states to dieren t sections of a quan tum computer system. F or instance, the proto cols discussed in this pap er could p oten tially b e used to mo v e states from memory to a system of quan tum gates in an implemen tation of the circuit mo del. There has already b een signican t w ork done on the sub ject of quan tum data buses using spin c hains, [811]. Ho w ev er these w orks dier signican tly from the metho d prop osed in 63 this pap er in that the enco ded qubit is not transmitted through a degenerate ground state manifold, but through excitations of the Hamiltonian. This pap er in v estigates a metho d of using qubits as an in termediate bus for the transfer of quan tum information. This metho d can b e compared to another metho d whic h is that of pulses [12], where a Hamiltonian is applied to a system for a p erio d of time to p erform a giv en op eration. In the case of information transfer this op eration is usually a sw ap. Unlik e the metho d of using pulses, this metho d of using qubits do es not require precise timing to insure that the correct op eration is p erformed. The metho d of using a spin c hain Hamiltonian as a data bus also means that one do es not need to either b e able to address an y pair of qubits in the system or p erform m ultiple op erations to transfer an arbitrary qubit. The pulse metho d do es ha v e the adv an tage that ev ery in termediate spin can b e used as quan tum memory . Ho w ev er this is at the cost of the increased complexit y of using dynamic quan tum ev olution in excited states, and the requiremen t of precise timing. The adiabatic quan tum bus metho d also has the adv an tage that, as in an y adiabatic quan tum pro cess, only the lo w est energy parts of Hamiltonian need to b e faithfully realized b y the implemen tation metho d. F or example, a Hamiltonian whic h actually has an innite n um b er of excited states on eac h spin, but where only the lo w energy states whic h act lik e a spin 1 2 Heisen b erg system, con tribute to the ground state w ould b e p erfectly acceptable to use as an adiabatic quan tum bus without mo dication. But the higher energy states ma y cause issues using a metho d suc h as pulses. This general feature of adiabatic quan tum pro cesses suc h as the one illustrated in this pap er mak es them more v ersatile than their non-adiabatic coun terparts. The eect w e will examine exploits the SU(2) symmetry of the Heisen b erg Hamiltonian and uses the ground state degeneracy created b y this symmetry in a c hain with an o dd n um b er of spins. It has already b een demonstrated [13] (see c h. 2 starting on page 33) that disturbances can b e sen t an unlimited distance along suc h c hains b ecause of their degenerate ground state. This c hapter go es a step further and actually demonstrates ho w a sp ecic state can b e transp orted across the c hain using a quan tum annealing proto col. F urther in v estigation will also b e pro vided in to application of this metho d to systems suc h as the XYZ spin c hain whic h only ha v e a Z 2 symmetry . 64 Setup The mo del w e consider is the J1-J2 Heisen b erg spin c hain with op en b oundaries, H = N1 X n=1 J n 1 ~ n ~ n+1 + N2 X n=1 J n 2 ~ n ~ n+2 : (47) This mo del has SU(2) symmetry , whic h is expressed b y the Hamiltonian b eing blo c k diagonal, suc h that there are N+1 blo c ks eac h with N k states. Eac h blo c k represen ts all of the states with a giv en n um b er, k, of up spins. If the n um b er of spins in the mo del is o dd, then the additional symmetry under a ip of the spins in the z direction, i.e. z ! z implies that all states of the Hamiltonian ha v e a t w ofold energy degeneracy . In the an ti- ferromagnetic case, ( J 1 ; J 2 > 0 ) the ground state manifold consists of one state from the k=o or( N 2 ) and one from the k=ceil( N 2 ) sector. A simple example of this w ould b e taking a system with 5 spins, the ground state w ould b e t w ofold degenerate and w ould span the k=2 and k=3 sectors. One can no w consider an initial Hamiltonian of the form of Eq. 47 where the couplings are the ones giv en b y J n 1 = 8 > < > : J n;init 1 n<N 1 0 n =N 1 ; (48) J n 2 = 8 > < > : J n;init 2 n<N 2 0 n =N 2 : (49) The general condition on J n;init 1 and J n;init 2 is that the coupling is predominan tly an ti- ferromagnetic ev erywhere and that eac h spin is coupled to the others b y at least one non zero J. F or simplicit y this pap er considers only J n;init 1 = 1 and J n;init 2 = J init 2 . This ground state manifold consists of the tensor pro duct of the (unique) ground state of the c hain of length N-1 with the Nth spin in an arbitrary state, a state in this manifold is of the from giv en b y j init i =j N1 0 ij init i; (50) wherej N1 0 i is the ground state of the spin c hain of length N-1 andj init iis an arbitrary single spin state. One can no w consider the same Hamiltonian, but with n! (Nn) + 1 . 65 This Hamiltonian also has the form of Eq. 47, but with couplings J n 1 = 8 > < > : J n;final 1 n> 1 0 n = 1 ; (51) J n 2 = 8 > < > : J n;final 2 n> 2 0 n = 2 : (52) The general condition on J n;final 1 and J n;final 2 is that the coupling is predominan tly an ti- ferromagnetic ev erywhere and that eac h spin is coupled to the others b y at least one non-zero J. F or simplicit y this pap er considers only J n;final 1 = 1 and J n;final 2 =J final 2 . A state in the ground state manifold is no w giv en b y j final i =j final ij N1 0 i; (53) wherej final i is an arbitrary single spin state. One can no w consider a quan tum annealing pro cess with describ ed b y H(t; ) = A(t; ) H init + B(t; ) H nal ; (54) where H init is 47 with the conditions giv en in 48 and 49 and H nal is 47 with the conditions giv en in 51 and 52. Also A and B follo w the conditions A(t 0; ) = 1; (55) B(t 0; ) = 0; (56) A(t ; ) = 0; (57) B(t ; ) = 1: (58) F or all v alues of A and B the SU(2) symmetry is preserv ed. Therefore the Hamiltonian remains blo c k diagonal at all times. The symmetry of the Hamiltonian under z ! z is also preserv ed at all times. This implies that the ground-state degeneracy (as w ell as the t w ofold degeneracy of all states) is preserv ed. The blo c k diagonal structure implies that there will b e no exc hange of amplitude b et w een spin sectors during the annealing pro cess, while the degeneracy implies that no relativ e phase can b e acquired b et w een the states in 66 Figure 24: Carto on represen tation of a pro cess where a spin is joined to the c hain, then the spin on the opp osite end is remo v ed. Note that this is only one sp ecic example of man y p ossible pro cesses for transp orting a qubit. the k=o or( N 2 ) and the k=ceil( N 2 ) sector. F rom the com bination of these t w o conditions one can see that as long as one anneals slo wly enough with H(t; ) [16] one can start with a state of the form giv en in Eq. 50 and reac h a nal state in the form Eq. 53 where j fin i = exp({')j init i, and ' is an irrelev an t phase. One sp ecic example of suc h an annealing proto col to transp ort a spin is giv en in Fig. 24. A dv an tages The use of the J1-J2 Heisen b erg c hain for transp ort b y quan tum annealing has sev eral adv an tages. First the mo del with uniform coupling is gapp ed for J 2 J 1 & 0:25 [14]. This suggests that within the adiabatic ev olution pro cess, at least lo cally , the system should b eha v e as a gapp ed system in this regime, as long as global eects suc h as o dd length frustration do not cause problems. It is imp ortan t to note that ev en the largest system size considered here is far from the thermo dynamic limit. One should note, ho w ev er, that giv en the connectivit y sc hemes of adiabatic quan tum c hips already in existence [6], one ma y only need to transp ort a qubit state a few spins to get it to an y part of the system. F urther evidence of fa v orable scaling comes from [13] (see c h. 2 starting on page 33) 67 whic h demonstrates that disturbances can tra v el an unlimited distance in the presence of a degenerate ground state, ev en in a gapp ed system. F urthermore, [13] suggests that these disturbances can carry en tanglemen t, p olarization, and quan tum information. The transp ort b y annealing giv en here is a sp ecic example of ho w this eect can b e tak en adv an tage of. Another adv an tage of the use of the J1-J2 Heisen b erg Hamiltonian is the existence of the so called Ma jumdar-Ghosh p oin t [15] ( J 2 J 1 = 0:5). A t this p oin t the ground state (with an ev en n um b er of spins) has the simple form of a matrix pro duct of singlets. Due to this fact the system should b e relativ ely easy to prepare. The system is also gapp ed at the Ma jumdar- Ghosh p oin t, making the Ma jumdar-Ghosh Heisen b erg Hamiltonian, an ideal system for transp ort b y quan tum annealing and the ideal candidate for building an adiabatic quan tum data bus. Although this pap er fo cuses on the J1-J2 Heisen b erg mo del, it should b e noted that this same annealing sc heme should w ork with an y pattern of coupling in the in termediate spins (i.e. J1-J2-J3)[17]. One w ould also exp ect this sc heme to w ork in mo dels where the SU(2) symmetry is brok en but there is a remaining Z 2 symmetry suc h as the XYZ or XY mo del, again with arbitrary patterns of coupling. Note ho w ev er that this metho d will not w ork in the Ising mo del, b ecause although there is a Z 2 symmetry , the Hamiltonian lac ks terms to exc hange qubits b et w een sites b ecause it is diagonal in the computational basis. XI I. PR OOF OF PRINCIPLE None of the argumen ts so far ha v e giv en m uc h illumination to the dicult y or ease of annealing within the sector. While w e ha v e discussed that transp ort of a qubit state is p ossible in principle b y annealing, w e ha v e not y et sho wn that the annealing pro cess is fast enough to b e practical. F or this w e turn to n umerics. F or the purp oses of this pap er w e will consider the annealing time, , required to reac h a giv en xed delit y , F (), with the true nal ground state, F () =j * fin j 0 dtH(t;)j init + j: (59) The J1-J2 Heisen b erg mo del is not an analytically solv ed mo del, at least for nite v alues of J 2 , so n umerical metho ds m ust b e used in this calculation. One can rst consider one 68 Figure 25: Coupling constan t (t;) from Eq.60 and Eq. 62 v ersus t . part of the annealing pro cess, in whic h a single spin is joined to a ev en length J1-J2 spin c hain, using b oth J 1 and J 2 couplings whic h are linearly increased to equal v alues of those used in the rest of the c hain [18], H(t;) = N2 X n=1 J 1 ~ n ~ n+1 + N3 X n=1 J 2 ~ n ~ n+2 +(t;)(J 1 ~ N1 ~ N +J 2 ~ N2 ~ N ); (60) (t;) = 8 > > > > < > > > > : 0 t 0 t 0<t< 1 t : As sho wn in Fig. 26, the annealing time required b ecomes large and highly sensitiv e to small v ariations for larger v alues of J 2 . Also the b eha vior seems to get w orse in this regime as system size is increased, and is p o or at the Ma jumdar-Ghosh p oin t [19]. As a further demonstration of the scaling with annealing time v ersus J 2 , one can plot the annealing time v ersus system size, as w e ha v e done in Fig. 27. This gure sho ws p olynomial or ev en sub p olynomial scaling for small v alues of J 2 , but than sho ws strongly non-monotonic b eha vior for stronger coupling. It is imp ortan t to note ho w ev er that ev en the longest c hain length considered here is probably far from the innite system limit, and this data ma y not 69 Figure 26: Annealing time required to reac h a 90% delit y with the true ground state within one of the t w o largest spin sectors of the Hamiltonian vs. J 2 , with J 1 set to unit y . One can see that for larger v alues of J 2 the annealing time b eha v es unpredictably . The annealing time also scales p o orly with system size close to the Ma jumdar-Ghosh p oin t. Figure 27: Scaling of annealing time to ac hiev e 90% nal ground state delit y (in units of in v erse Hamiltonian energy) v ersus length of c hain on a log-log plot. 70 Figure 28: Plots of gap for joining a single spin to an ev en length J1-J2 Heisen b erg spin c hain. F or densit y plots ligh ter colors indicate larger gap. a) gap v ersus in Eq. 60 and J 2 for 15 total spins d) Gap v ersus J 2 with = 1 b e trust w orth y for making predictions for scaling as the c hain length approac hes the innite system limit. By examining the gap one can hop e to gain insigh t in to the underlying cause of the b eha vior of annealing time curv es. As Figs. 28(a) and (b) sho w, the b eha vior of the annealing time curv es is reected b y the presence of what app ear to b e true crossings [20] for the o dd length spin c hain with uniform coupling. Fig. 28(b) sho ws the gap for an o dd length spin c hain and seems to conrm the presence of p oin ts with v ery small gap with uniform coupling forJ 2 ab o v e 0.5. Figs. 26 and 28 together sho w that, at least at the length scales considered here, there are go o d annealing paths for joining a single spin to an ev en length c hain. Ho w ev er, the simplest metho d of taking adv an tage of the simple ground-state w a v efunction at the Ma jumdar-Ghosh p oin t is not optimal. F ortunately there are man y other p ossible options to tak e adv an tage of the easily prepared ground state and hop efully a v oid the regions of small gap found here. XI I I. D YNAMICALL Y TUNING J2 One metho d to a v oid regions of small gap while still taking adv an tage of the Ma jumdar- Ghosh p oin t w ould b e to start at the Ma jumdar-Ghosh p oin t and then dynamically reduce the v alue of J 2 during the annealing pro cess, a simple w a y of doing this w ould b e to use the Hamiltonian in Eq. 61. 71 Figure 29: In this annealing proto col not only is a spin coupled to the c hain, but J 2 is also c hanged dynamically . H(t;) = N2 X n=1 J 1 ~ n ~ n+1 + N3 X n=1 J 2 (t;)~ n ~ n+2 +(t;)(J 1 ~ N1 ~ N +J 2 (t;)~ N2 ~ N ); (61) (t;) = 8 > > > > < > > > > : 0 t 0 t 0<t< 1 t ; J 2 (t;) = 8 > > > > < > > > > : 0:5 t 0 0:5 + t (J 2f 0:5) 0<t< J 2f t : Fig. 30 sho ws that taking adv an tage of the easily prepared ground state at the Ma jumdar- Ghosh p oin t do es in fact w ork, and the curv es in this gure are strikingly similar to those in Fig. 26. This similarit y is to b e exp ected b ecause Fig. 28 demonstrates that the gap is the smallest where the spin is completely joined. Hence this part of the pro cess should dominate the annealing time. It is reasonable to argue that b ecause the regions of phase space whic h are visited are the same in the uncoupling pro cess as coupling, the b eha vior of the system during the uncoupling 72 Figure 30: Annealing time required to reac h a 90% delit y with the true ground state within one of the t w o largest spin sectors of the Hamiltonian with dynamical coupling starting at J 2 =0.5 and linearly c hanging to J 2f while also joining a spin to the c hain, with J 1 set to unit y throughout the pro cess. Notice that this gure is qualitativ ely and quan titativ ely v ery similar to Fig. 26. pro cess is determined b y the gaps sho wn in Fig. 28, and therefore the annealing times for the uncoupling pro cess should b e at least qualitativ ely similar to those giv en in Fig. 26. One adv an tage to the uncoupling pro cess is that unlik e the coupling pro cess, the need is not as strong to end in an easily prepared state. The only reason one ma y ha v e to w an t to end in the Ma jumdar-Ghosh p oin t is as an error c hec k. The spins in the c hain can b e measured after the end of the pro cess to ensure that no error has o ccurred [21]. Fig. 31 sho ws the time required to uncouple a spin from the c hain, not surprisingly this gure lo oks v ery similar to Fig. 26 whic h is the coupling pro cess. Note that in this system the Hamiltonian is simply Eq. 60 with t ! (1 t ) . As exp ected, except for one curv e where a n umerical error made some p oin ts unable to plot one can see from Fig. 32 that the uncoupling pro cess also requires roughly the same time as the coupling pro cess for dynamically tuned J 2 . Note that the Hamiltonian for this pro cess is simply Eq. 61 with t ! (1 t ) and J 2f !J 2i . 73 Figure 31: Annealing time required to reac h a 90% Fidelit y with the true ground state for uncoupling pro cess within one of the t w o largest spin sectors of the Hamiltonian vs. J 2 with J 1 set to unit y . One can see that this gure is v ery similar to Fig. 26 as one w ould exp ect b ecause it is simply the time rev ersed v ersion of that pro cess. XIV. SIMUL T ANEOUS UNCOUPLING AND COUPLING Because man y of the issues encoun tered with the coupling proto col seem to relate to o dd-spin frustration, it ma y b e reasonable to consider sim ultaneously coupling one qubit to the c hain while uncoupling the other. The Hamiltonian in this case is giv en in Eq. 62. H(t;) = N2 X n=1 J 1 ~ n ~ n+1 + (62) N3 X n=1 J 2 (t;)~ n ~ n+2 +(t;)((J 1 ~ N1 ~ N +J 2 ~ N2 ~ N ) (J 1 ~ 1 ~ 2 +J 2 ~ 1 ~ 3 )); (t;) = 8 > > > > < > > > > : 0 t 0 t 0<t< 1 t : 74 Figure 32: Annealing time required to reac h a 90% delit y with the true ground state for uncoupling pro cess within one of the t w o largest spin sectors of the Hamiltonian vs. initial J 2i with a nal J 2 at the Ma jumdar-Ghosh p oin t with J 1 set to unit y . This gure is v ery similar to Fig. 30 as one w ould exp ect, b ecause it is simply the time rev ersed v ersion of that pro cess. Figure 33: Plots of gap for sim ultaneously joining a single spin to an ev en length J1-J2 Heisen b erg spin c hain and unjoining a spin from the other end. F or densit y plots ligh ter colors indicate larger gap. a) gap v ersus from Eq. 62 and J2 for 17 total spins b) Gap v ersus J 2 with = 0:5. Fig. 33 sho ws the gaps for v arious system sizes for the pro cess where the couplings are turned on and o sim ultaneously . This pro cess do es not seem to a v oid the area of lo w gap for J 2 & 0:5 seen in Fig. 28. Ho w ev er b y comparing Fig. 33 d) and Fig. 28 d) one can see that it app ears that the pro cess of sim ultaneous uncoupling and coupling is c haracterized b y a v oided crossings rather than true crossings [22]. 75 Figure 34: Annealing time required to reac h a 90% Fidelit y with the true ground state for com bined coupling and uncoupling pro cess within one of the t w o largest spin sectors of the Hamiltonian vs. J 2 with J 1 set to unit y . Fig. 34 sho ws the time required for annealing pro cesses with for the com bined coupling and uncoupling pro cess, the results are consisten t with what one w ould exp ect from lo oking at Fig. 33, and conrm that the annealing time also tends to b e v ery long and v ary a lot for larger v alues of J 2 . XV. REQUIREMENTS F OR USE AS AN ADIABA TIC QUANTUM BUS It is no w useful to consider a broader class of mo dels that ma y b e used as adiabatic quan tum buses, as in general the full SU(2) symmetry of the Heisen b erg Hamiltonian is not required. The requiremen ts for a spin c hain (or net w ork) Hamiltonian to b e usable as an adiabatic quan tum bus are as follo w: 1. The ground state m ust b e at least 2 fold degenerate, and the ground state manifold m ust b e able to enco de a qubit. In this pap er this is ac hiev ed b y ha ving at least a Z 2 symmetry , and an o dd n um b er of spins, but there ma y b e other w a ys. 2. The Hamiltonian (or at least the lo w energy states) m ust b e predominan tly an ti- 76 ferromagnetic in nature. This guaran tees that the enco ded qubit will b e excluded from the larger spin c hain (or net w ork) when a single qubit is remo v ed. 3. The Hamiltonian m ust con tain terms whic h p erform exc hanges b et w een sites. This excludes mo dels suc h as the Ising mo del whic h, although it has the required symmetry , cannot b e used a quan tum bus b ecause its Hamiltonian is diagonal in the computa- tional basis 4. One m ust b e able to slo wly couple in a spin with an arbitrary state on one end of the c hain (net w ork) and also to slo wly remo v e coupling on the other end. More con trol ma y impro v e p erformance, but is not necessary . 5. Annealing paths in parameter space m ust not con tain true crossings. This is a general requiremen t for adiabatic quan tum computing. XVI. XXZ AND XYZ MODEL As previously men tioned, the full SU(2) symmetry of the Heisen b erg Hamiltonian is not required. The Hamiltonian m ust only ha v e a Z 2 symmetry to enco de and transp ort one qubit of information. In this section w e will briey examine t w o other p ossibilities: the XXZ mo del, where the SU(2) symmetry is brok en, but the blo c k diagonal structure imparted b y this symmetry remains, and the XYZ mo del where only the blo c k diagonal structure of a Z 2 symmetry is presen t. As one can see from Fig. 35, the XXZ mo del can b e used as an adiabatic quan tum data bus. There is a regime where this system outp erforms the XXX Heisen b erg mo del for Z/X b et w een 1 and roughly 2. This is to b e exp ected b ecause adding additional coupling in the z direction ma y serv e to op en the gap b et w een the the ground-state manifold and the next excited state. The increasing time as the z coupling is increased further can b e explained b ecause the system w ould b eha v e lik e an Ising mo del in the limit of Z X 1 . One can further examine the b eha vior of an XYZ mo del as an adiabatic quan tum spin bus. F or this purp ose w e consider the quan tum bus proto col p erformed on the follo wing normalized XYZ Hamiltonian 77 Figure 35: Annealing time to reac h 90% delit y on using the adiabatic quan tum bus proto col on an XXZ spin c hain v ersus the ratio of X and Z coupling strengths note that Z/X=0 is an XX mo del while Z/X=1 is a J1 Heisen b erg spin c hain. This data w as obtained with joining and disconnecting of spins o ccurring sim ultaneously . H XYZ (;N) =C N1 X i=1 x i x i+1 + (1 + ) y i y i+1 + (1 + 2) z i z i+1 ; (63) where the normalization is C = p 3 p 1 + (1 + ) 2 + (1 + 2) 2 : One can no w examine the p erformance of this Hamiltonian for dieren t v alues of , noting that H XYZ (0;N) is simply the J1 Heisen b erg spin c hain of length N. As Fig. 36 sho ws, a sligh t adv an tage can b e gained b y using an XYZ mo del rather than a simple Heisen b erg c hain. Fig. 36 also seems to suggest that the b enet gained is relativ ely indep enden t of c hain length. 78 Figure 36: Plot of fractional dierence from annealing time for an c hain with small (Heisen b erg c hain). This data is for the adiabatic quan tum bus proto col p erformed on a c hain of the form eq. 63 with spins b eing attac hed and remo v ed sim ultaneously . XVI I. OTHER PR OTOCOLS So far w e ha v e only in v estigated a small subset of the p ossible annealing proto cols whic h meet the criteria giv en in the in tro duction. F or example the XY spin c hain should also ha v e and easily prepared ground state and ma y b e easier to exp erimen tally realize [2]. One could also try to examine the case of dynamically tuning the y and or z direction coupling and starting out at the Ma jumdar-Ghosh p oin t but using mo died coupling in the y and z directions with an XYZ mo del to a v oid lo w gap regions. One could also try to c hange the coupling sc heme to a v oid the lo w gap region, b y either randomly or systematically mo difying the coupling b et w een in termediate spins, if this is done dynamically , one can still tak e adv an tage of the Ma jumdar-Ghosh p oin t. This tec hnique could also b e used in conjunction with an y of the ideas in the previous paragraph. This c hapter is in tended only to pro vide pro of of principle for this metho d and is b y no means an exhaustiv e searc h of all p ossible proto cols. 79 XVI I I. CONCLUSIONS W e ha v e demonstrated ho w a J1-J2 Heisen b erg spin c hain can b e used to transp ort a qubit state adiabatically . W e ha v e also sho wn that man y extensions of this Hamiltonian; suc h as dieren t coupling sc hemes or the XY or XYZ mo del whic h ha v e only a Z 2 symmetry , will also b e able to b e used to transp ort a qubit [23]. W e ha v e found that for v alues of high frustration, transp ort b y quan tum annealing do es not w ork v ery w ell. W e ha v e also demonstrated that this do es not prev en t us from exploiting the easily prepared ground state at the Ma jumdar- Ghosh p oin t. W e ha v e giv en some examples of p ossible annealing proto cols in this pap er, but ha v e really only in v estigated a v ery small section of a v ast space of p ossible proto cols for transp ortation of quan tum states b y annealing. A c kno wledgemen ts This Chapter is based on the pap er Using the J1-J2 Quan tum Spin Chain as an A dia- batic Quan tum Data Bus b y N. Chancellor and S. Haas [1]. The n umerical computations w ere carried out on the Univ ersit y of Southern California high p erformance sup ercomputer cluster. This researc h in the original pap er w as partially supp orted b y the AR O MURI gran t W911NF-11-1-0268. 80 Chapter 3 References [1] N. Chancellor and S. Haas 2012 New J. Ph ys. 14 095025 (2012). [2] M. W. Johnson et al. Nature 473, 194-198 (12 Ma y 2011) [3] R. Harris et al. Ph ysical Review B 81, 134510 (2010) [4] A. P erdomo et al. Ph ysical Review A 78, 012320 (2008) [5] S. H. W. v an der Plo eg et al. IEEE T rans. App. Sup ercond. 17, 113 (2006) [6] R. Harris et al. Ph ys. Rev. B 82, 024511 (2010) [7] Z. Chen, Z. Zhou, X. Zhou, X. Zhou, and G. Guo, Ph ys. Rev. A 81, 022303 (2010) [8] L. Banc hi, T. J. G. Ap ollaro, A. Cuccoli, R. V aia, and P . V errucc hi, Ph ys. Rev. A 82, 052321 (2010) [9] L Banc hi, T J G Ap ollaro, A Cuccoli, R V aia, and P V errucc hi, New Journal of Ph ysics 13 (2011) 123006 [10] T. J. G. Ap ollaro, L. Banc hi, A. Cuccoli, R. V aia, P . V errucc hi, arXiv:1203.5516v1 [quan t-ph] 2012 [11] L. Banc hi, A. Ba y at, P aola V errucc hi, and Sougato Bose, Ph ys. Rev. Lett. 106, 140501 (2011) [12] J. Fitzsimmons and J. T w amley , Ph ys. Rev. Lett. 97, 090502 (2006) [13] N. Chancellor and S. Haas, Ph ys. Rev. B 84, 035130 (2011) [14] Ph ys. Rev. B 52, 6581 - 6587 (1995) [15] C K Ma jumdar, J. Ph ys. C: Solid State Ph ys. 3 911 (1970) [16] T ec hnically one m ust giv e the additional condition that there is no true crossing within the spin sectors on the annealing path. [17] A t least this should w ork for small systems. In the con tin uum limit man y of these systems ma y b ecome gapless, so that quan tum annealing cannot b e eectiv ely p erformed. Also one ma y b e able to construct certain pathological cases with paths whic h pass through true crossings. [18] Note that this Hamiltonian (and all other annealing Hamiltonians in this pap er) can b e rewrit- ten in the form of 54. Ho w ev er it is m uc h more compact not to write the unc hanging parts of the Hamiltonian t wice. [19] A t least for xed coupling, the case of dynamically c hanging coupling will b e considered later. 81 [20] Strictly sp eaking nothing in this pap er has demonstrated them to b e true crossings, they could just b e close a v oided crossings, it do es not matter for the purp oses of this c hapter. [21] F or example if t w o spins whic h should b e in a singlet together ended up b eing measured to b e facing in the same direction than the annealing pro cess w ould ha v e failed. [22] This statemen t is based on the fact that the gap do es not ha v e a cusp when plotted on a log scale. Strictly sp eaking this just sho ws that there is not a true crossing at the line where the t w o couplings are equal. [23] Assuming there is not a true crossing along the annealing path, the coupling m ust also b e (at least predominately) an ti-ferromagnetic so that the excess spin do es not b ecome trapp ed in the larger spin c hain. 82 Chapter 4: Holonomic Quantum Computation by T ransport and Application with Superconducting Flux Qubits This c hapter is based on [1]. In this c hapter w e examine the use of an adiabatic quan tum data transfer proto col to build a univ ersal quan tum computer. Single qubit gates are realized b y using a bus proto col to transfer qubits of information do wn a spin c hain with a unitary t wist. This t wist arises from altered couplings on the c hain corresp onding to unitary rotations p erformed on one region of the c hain. W e sho w ho w a con trolled NOT gate can b e realized b y using a con trol qubit with Ising t yp e coupling. The metho d discussed here can b e extended to non-adiabatic quan tum bus proto cols. W e also examine the p oten tial of realizing suc h a quan tum computer b y using sup erconducting ux qubits. In tro duction It has recen tly b een demonstrated ho w an op en-ended an tiferromagnetic Heisen b erg spin c hain can b e used as an adiabatic quan tum data bus [2]. This data bus tak es adv an tage of an tiferromagnetic couplings to transfer qubits of information adiabatically . First a single qubit, enco ded in a single spin is joined to an ev en length Heisen b erg spin c hain slo wly enough suc h that the adiabatic theorem applies. Then a single spin on the other end of the c hain is separated, again slo wly enough for the adiabatic theorem to apply . As long as the in teractions b et w een the spins on the c hain are predominately an tiferromagnetic, the qubit will b e successfully transferred from one end of the c hain to the other. This proto col is illustrated in Fig. 37. An tiferromagnetic spin clusters ha v e b een studied for there p oten tial usefulness in quan tum computing in other con texts, for example in Refs. [3, 4]. This pap er demonstrates ho w b y applying particular unitary op erations to spin c hains for single qubit gates, and b y using a sp ecic spin net w ork for a CNOT gate, one can ac hiev e 83 Figure 37: Carto on of an adiabatic quan tum bus proto col for the Heisen b erg spin c hain [2]. A spin with the enco ded qubit is connected to one end of an ev en length an tiferromagnetic c hain. Afterw ards, the spin on the opp osite end is remo v ed adiabatically . As long as the c hain in terac- tions are predominately an tiferromagnetic and the adiabatic theorem is satised the qubit will b e transferred. univ ersal holonomic quan tum computation. This metho d uses op en-lo op holonomies, mean- ing that the Hamiltonian is not necessarily returned to the same state after the adiabatic pro cess. The metho ds used here can also b e extended to non-adiabatic implemen tations of geometrical quan tum computing. Holonomic quan tum computation (HQC) w as conceiv ed and sho wn to b e univ ersal b y Zanardi and Rasetti [5] and w as form ulated in terms of a non-ab elian Berry phase. HQC is considered to b e an app ealing metho d for ac hieving fault toleran t quan tum computing b ecause of its geometrical nature and b ecause it can b e implemen ted adiabatically , and therefore has all of the adv an tages of adiabatic quan tum computation [6]. Although man y implemen tations of holonomic and geometric quan tum computation are adiabatic, there are examples whic h are not [7, 8]. Other prop osed arc hitectures for holonomic quan tum computation use a v ariet y of arc hi- tectures, including sup erconducting systems with Josephson junctions [9]. F urther examples prop ose using quan tum dots [8, 10]. Single molecule magnets ha v e b een another system of in terest [8]. A recen t prop osal has also b een made for using Holonomies whic h in v olv e at- tac hing and remo ving a spin from a spin 1 c hain[11]. This arc hitecture, although it lo oks sup ercially v ery similar to ours, p erforms computations lo cally , at the site of the spin rather than b y transp ort as our’s do es. Ref. [11] also prop oses an implemen tation on ultracold 84 p olar molecules. Another in teresting prop osal in v olv es using a quan tum wire with a t wisted cluster state Hamiltonian[12]. This prop osal is similar to ours, but implemen ts the t wist in a fundamen tally dieren t w a y . Most other approac hes to HQC in v olv e building a system, and explicitly calculating the holonomies caused b y v arious manipulations of the system. In our prop osal w e start with a pro cess whic h has a trivial Berry phase[27]. Real space t wists are then p erformed on the spin c hain used in this pro cess. Unlik e most examples of HQC, all results can b e deriv ed without explicitly considering the curv ature of the underlying manifold of states, all single qubit gates result from the same underlying Hamiltonian with basis rotations applied to it. The mathematical dierences of this approac h from others aords us the adv an tage that the sp ectrum of the underlying Hamiltonian is the same for all t wists, meaning that, b y construction, all single qubit gates can b e implemen ted in a w a y whic h requires the same annealing time to reac h a giv en accuracy . This arc hitecture has the adv an tage that the only op eration it ev er requires to b e adiabatically p erformed is the joining or remo v al of a spin from a c hain or cluster. The nature of the t wists used here also means that a non-adiabatic transp ort proto col could b e used instead, and univ ersal computation w ould still b e ac hiev ed. This c hapter also outlines an implemen tation of the necessary comp onen ts of this design using sup erconducting ux qubits. Sup erconducting ux qubits are a p opular arc hitecture for implemen ting scalable adiabatic quan tum computing [26], and therefore are a natural c hoice for designing a scalable holonomic quan tum computer. An additional adv an tage of the use of sup erconducting ux qubits is that the designs tend to ha v e spatially extended qubits and a high degree of connectivit y[17]. The large spatial exten t of the qubits means that a design could b e implemen ted in whic h a qubit w ould only need to b e transferred across a small n um b er of spins to b e mo v ed from one lo cation in a computer to an y other arbitrary lo cation. F or this reason it is only necessary that the transp ort proto col b e ecien t for short c hains, as has already b een demonstrated in [2], rather than in the thermo dynamic limit. There has b een recen t exp erimen tal w ork in v olving quan tum annealing to degenerate ground state manifolds using curren tly a v ailable sup erconducting ux qubit hardw are[18]. In this pap er it w as demonstrated exp erimen tally that signatures of quan tum b eha viors can b e observ ed in the nal state within a degenerate ground state manifold. This pro vides an indication that a ground state manifold can b e pro duced accurately enough on curren t 85 hardw are that quan tum eects dominate o v er classical eects and design inaccuracies. Al- though the arc hitecture prop osed here cannot b e implemen ted on the hardw are used in [18], this exp erimen t do es pro vide pro of of principle for the use of degenerate manifolds in sup erconducting ux qubit systems. While it is not the main fo cus of this c hapter, w e w ould also lik e to p oin t out that there are other p oten tial metho ds of implemen ting this arc hitecture. One example of suc h an implemen tation w ould b e to use a coupled ca vities sc heme similar to the one explored in [19]. F or suc h an implemen tation long spin c hains ma y b e required, and as a result prop erties in the thermo dynamic limit ma y b e imp ortan t. F or suc h an implemen tation, the arc hitecture giv en in this pap er could easily b e generalized to a J1-J2 spin c hain with J 2 J 1 & 0:25 whic h is kno w to b e gap ed in the thermo dynamic limit[20]. In this case, another option w ould b e to implemen t the arc hitecture non-adiabatically using the metho ds describ ed in [811]. XIX. SINGLE Q-BIT GA TES A. The T wisted Spin Chain Consider initially an an tiferromagnetic Heisen b erg spin c hain. It has b een sho wn that suc h a c hain can act as a quan tum data bus, b oth adiabatically[2] and b y using the dynamics of its excitations [811]. The initial Hamiltonian is giv en b y H = N1 X i=1 ~ i ~ i+1 = N1 X i=1 ( x i x i+1 + y i y i+1 + z i z i+1 ): (64) No w imagine that one inserts a t wist in to the spin c hain b y applying a lo cal unitary transformation of the form x;y;z! x 0 ;y 0 ;z 0 on N 0 = NL spins, where x 0 ;y 0 ;z 0 are all m utually orthogonal to eac h other. This yields a new Hamiltonian of the form H t wist = L1 X i=1 ~ i ~ i+1 +~ L ~ 0 L+1 + N1 X j=L+1 ~ 0 j ~ 0 j+1 : (65) Suc h a t wist do es not eect the sp ectrum of the Hamiltonian, and therefore the dynamics of the adiabatic quan tum bus proto col, or other quan tum bus proto cols whic h ma y mak e use of the unitary dynamics of the Hamiltonian. It is imp ortan t to note, ho w ev er, that after transfer across the c hain, the spin will b e rotated in to the x 0 ;y 0 ;z 0 basis. As w e will 86 Figure 38: Illustration of a single unitary gate implemen ted b y adiabatic transp ort on a t wisted c hain. demonstrate later, transfer through this t wisted spin c hain can p erform an y desired unitary rotation on the qubit b eing transferred, and th us can b e used to implemen t an y single qubit gate, see Fig. 38. One should note that while in this example w e consider a simple Heisen b erg spin c hain, gates can b e implemen ted in this w a y on an XYZ spin c hain or a J1-J2 spin c hain, or other sucien tly complex quan tum spin Hamiltonians [28]. Figuring out whic h t wist to use to p erform a giv en gate can b e done easily and will b e illustrated in the next section. B. Example: Implemen ting a Hadamard Gate The lo cal t wist to implemen t the Hadamard gate is x ! z , y ! y and z ! x . This t wist can b e calculated without dicult y , for details see Sec. XXIV A. One can therefore conclude that the Hadamard gate can b e implemen ted b y p erforming the quan tum bus proto col on the Hamiltonian H Hadamard = N 0 1 X i=1 ~ i ~ i+1 + x N 0 z N 0 +1 + (66) x N 0 z N 0 +1 y N 0 y N 0 +1 + N1 X j=N 0 +1 ~ j ~ j+1 C. Other Single Qubit Gates One can p erform similar t wists to implemen t an y giv en single qubit gate. The calculation to nd x 0 ;y 0 ;z 0 in Eq. 65 for other gates can b e p erformed in the same w a y as the one in the 87 Gate Name Matrix x 0 y 0 z 0 Hadamard 1 p 2 0 @ 1 1 1 1 1 A z y x 8 0 @ 1 0 0 exp({ 4 ) 1 A 1 2 ( x + y ) 1 2 ( y x ) z phase 0 @ 1 0 0 { 1 A y x z NOT a 0 @ 0 1 1 0 1 A x y z a This gate is needed for the construction of the CNOT T able IV: T wists for implemen ting v arious single qubit gates previous section for the Hadamard. T able IV sho ws ho w to implemen t single qubit gates. These gates are sucien t to p erform an arbitrary unitary op eration on a single spin. It is sho wn in [25] that an y unitary rotation can b e appro ximated to arbitrary precision with the gates giv en in table IV. W e ha v e already sho wn ho w to build an adiabatic quan tum bus to mo v e qubit states to arbitrary lo cations in the system. Next w e discuss that a CNOT gate can b e implemen ted under this arc hitecture. Then w e ha v e demonstrated a univ ersal quan tum computer. XX. IMPLEMENT A TION OF THE CONTR OLLED NOT GA TE A. CNOT design Let us no w turn our atten tion to the implemen tation of a con trolled NOT (CNOT) using an adiabatic quan tum bus proto col. In Fig. 39a) w e sho w a design for suc h a gate. The time dep enden t Hamiltonian for this gate is 88 H CNOT (t;h;t fin ) =(t;t fin )~ in (~ 1 +~ 2 +~ 3 +~ 4 ) +~ a (~ 1 +~ 2 +~ 3 +~ 4 ) +h(( z 1 z 2 )(1 z c ) + ( z 3 z 4 )(1 + z c )) + (1(t;t fin ))(~ 0 out (~ 1 +~ 2 )) +~ out (~ 3 +~ 4 ); (67) (t;t fin ) = 8 > > > > < > > > > : 0 t< 0 t t fin 0tt fin 1 t>t fin : Here in refers to the spin whic h is the input spin, where the target qubit is initially enco ded; out refers to the spin to whic h the target qubit is transferred to, c refers to the con trol qubit, a to an ancilla to mak e the n um b er of in termediate spins o dd. The other 4 spins are assigned n um b ers 1-4. ~ 0 out refers to a NOT t wist b eing p erformed on these P auli matrices, see T ab. IV. This gate op erates b y ha ving 2 c hannels through whic h a qubit of information can pass. One c hannel, consisting of spins 3 and 4, allo ws the information to pass through the gate unaltered, while another c hannel, consisting of spins 1 and 2 p erforms a t wist on the qubit as it tra v els though the gate. The con trol spin c con trols though whic h c hannel the information tra v els. The con trol spin is connected with Ising t yp e coupling to spins 1-4 in suc h a w a y that when the con trol spin is up the external eld on spins 1 and 2 cancels with the eect of the Ising b ond with spin c b ecause ( 1 2 < z c >) = 0, and the information can easily pass though these spins. On the other hand ( 1 2 + < z c >) = 1. So spins 3 and 4 b oth ha v e an eectiv e magnetic eld of 2h. F or sucien tly large h these spins are frozen in the direction of the eld and will therefore not b e able to transp ort an y information. As w e sho w in Fig. 39b) the net eect is that information all tra v els though spins 1 and 2, and therefore a NOT t wist is p erformed. In the case where the spin c is in the do wn direction, information will instead b e allo w ed to tra v el though spins 3 and 4 and blo c k ed on spins 1 and 2. Therefore in that case the gate acts trivially on the qubit. An y state of the con trol spin c can b e expressed as j c i = aj"i +bj#i where a and b are complex n um b ers. With an arbitrary input state j in i = j"i +j#i w e ha v e b efore the gatej init i =j in i j c i = aj""i +bj"#i +aj#"i +bj##i. After the gate is p erformed, the nal state b ecomes j fin i = aj#"i +bj"#i +aj""i +bj##i. F rom these general states w e see that 89 Figure 39: a) Design of a CNOT gate whic h uses the adiabatic data bus proto col. Note that one could replace the NOT op eration with an y other single qubit unitary . b) CNOT system with con trol cen tral spin up, executes a NOT t wist on target spin under quan tum bus proto col. See T ab. V for the meaning of v arious sym b ols. Lab els are based on Eq.67. j fin i = 0 B B B B B @ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 C C C C C A j init i; (68) whic h is the denition of a con trolled NOT gate [25]. 90 Sym b ol Meaning spin 1 2 con trol spin spin 1 2 w orking spin initial AF Heisen b erg b ond (w eak) nal AF Heisen b erg b ond (w eak) xed AF Heisen b erg b ond (w eak) t wist to p erform NOT op eration xed F Ising b ond (strong) xed AF Ising b ond (strong) xed up magnetic eld (strong) xed do wn magnetic eld (strong) T able V: Legend of sym b ols used in Figs. 39. B. P erformance of Con trolled NOT Gate W e need to test ho w this design for a CNOT gate p erforms b ecause w e cannot rely on previous w ork to sho w that the qubit is actually transferred accurately . The t w o free parameters in Eq. 67 are the strength of the Ising b onds and the elds whic h w e denote b y h, and the time for the proto col to b e p erformed, t fin : W e no w examine whether this Hamiltonian actually implemen ts a CNOT gate eectiv ely for reasonable v alues of h and t fin . T o test this w e need to answ er 2 questions. Firstly , is the system close enough to the adiabatic limit for reasonable v alues of t fin ? Secondly , is the desired eect of sh utting o one p ossible path for the information ac hiev ed for reasonable v alues of h? T o answ er these questions, w e examine the o v erlap of the nal output state (nal state of the out qubit in Fig. 39) with the exp ected output state from a con trolled NOT gate (Fig.40). Note that b ecause the case where a NOT gate is p erformed, and the case where the gate acts trivially are related b y a simple unitary transformation on the Hamiltonian, acceptable p erformance in one of these cases implies acceptable p erformance 91 Figure 40: Measures of p erformance of the CNOT gate a) 1-delit y of the output spin v ersus t fin for h=10 b) 1-delit y of the output spin v ersus h for t fin =10 c) output delit y for initial up spin with a NOT p erformed v ersus h and t fin ligh t is larger (more p ositiv e), dark is smaller (more negativ e) d) gap v ersus t t fin for v arious v alues of h. in the other. A v eraging o v er dieren t initial states is therefore unnecessary as it w ould yield the exact same result as an y particular c hoice of con trol and input states. Fig. 40 demonstrates the eectiv eness of this gate. Fig. 40 a) sho ws that for a mo derate eld and Ising b ond strength the gate can b e made to p erform w ell as long as the annealing time is sucien t. This gure also sho ws that the gate can con tin ually b e made more eectiv e b y running it longer without ha ving to raise h. The oscillations in the delit y are related to the time scale of small excitations pro duced during the annealing pro cess. In man y applications one ma y ha v e enough con trol o v er t fin that the annealing time can b e c hosen in a w a y that the pro cess lies near one of the lo cal minima of error sho wn in Fig. 40 a). Fig. 40 b) sho ws the eect of h on delit y for an annealing time of 10 (in units of in v erse Heisen b erg couplings). In this gure one can see that increasing h is ineectiv e at impro ving p erformance ab o v e a certain v alue. This indicates that at this p oin t the eld is already eectiv ely completely blo c king one path that the information transp ort can tak e. Fig. 40 c) sho ws the com bined eects of h and t fin on the output p olarization. It is consisten t with the conclusions w e ha v e reac hed from a) and b). Finally , Fig. 40 d) sho ws that the system 92 gap remains quite large throughout the pro cess. It also demonstrates that b ey ond a certain v alue of h the gap do es not increase signican tly with c hanging h, whic h is consisten t with the picture of one path b eing completely closed to information transfer. It is in teresting to note that increasing h increases the gap throughout the pro cess. XXI. IMPLEMENT A TION USING SUPER CONDUCTING FLUX QUBITS T o build an implemen tation of this holonomic arc hitecture based on sup erconducting ux qubits one needs to design circuits that implemen t Heisen b erg spins and the appropriate gate couplings b et w een spins. F ortunately signican t w ork has already b een done, for example in Refs. [26], to w ards the design of couplers for sup erconducting ux qubits. Ho w ev er since the previously discussed sc hemes w ere based on Ising spin systems, w e still need to establish a metho d for designing circuits whic h em ulate Heisen b erg spins. It is in teresting to p oin t out that this computational arc hitecture w orks with the limited connectivit y of the designs prop osed in Refs. [26]. Sp ecically the CNOT gate w e ha v e prop osed ts in a single 2x4 c himera lattice cell lik e the one used in Ref. [17]. A. Flux Qubit Motiv ation: It has b een sho wn in [14] that a qubit whic h b eha v es lik e an Ising spin can b e constructed from a Hamiltonian of the form H Ising = 2 X n=1 ( Q n 2C n +U n n x n 2 )U q cos( 1 1 ) cos( 2 2 ); (69) where the applied uxes x n act as eectiv e magnetic elds, altering the shap e of a p o- ten tial w ell for the system in a w a y that mimics a spin constrained to mo v e in a plane. The constan ts simply act to scale the eect of the ux. T o construct a Heisen b erg qubit one needs to add a third direction, leading to a Hamiltonian of the form H Heis: = 3 X n=1 ( Q n 2C n +U n n x n 2 ) (70) U q cos( 1 1 ) cos( 2 2 ) cos( 3 3 ): 93 Suc h a Hamiltonian w ould allo w the shap e of the p oten tial w ell to b e c hanged along 3 directions and w ould therefore mimic a Heisen b erg spin rather than an Ising spin. Previously prop osed designs also only couple qubits along one direction. If a new t yp e of coupler w ere added to the curren tly implemen ted circuits whic h couple the spins in the y direction in addition the z direction, then an XY mo del could b e implemen ted. T o implemen t an XYZ Heisen b erg mo del, one needs to b oth design qubits whic h are not constrained to lie in a plane and build 3 t yp es of couplers, one for eac h direction in space. XXI I. FLUX QUBIT DESIGN: First consider a CCJJ (Comp ound-Comp ound Josephson Junction) circuit as dened in [14]. The eectiv e Hamiltonian of this circuit is giv en b y [14] H = X n ( Q n 2C n +U n n x n 2 )U q eff cos( q 0 q ); (71) where n2fq;cjj;l;rg. Note that there are more indeces than in [14] b ecause w e do not ha v e the condition that L = x L or R = x R . The rst t w o terms of the Hamiltonian in Eq.71 are not imp ortan t for what w e are trying to demonstrate here [14]. The denition of all of these terms can b e found in Eq. B4b-f in [14], and in Sec. XXIV B of this pap er. Let us mak e the simplifying assumption that all of the critical curren ts are equal for all junctions. In practice there is v ariabilit y in junction fabrication, but this error can b e comp ensated b y building a CCCJJ device (see Fig. 41). Let us also assume that our circuit is designed in suc h a w a y that w e can inductiv ely couple the left and righ t lo op to eac h other v ery strongly suc h that y L = R and x y x L = x R . These assumptions cause the equations to simplify greatly (see Sec. XXIV B), yielding eff = + cos( ccjj 2 ); (72) + 2 L = 2 R ; L(R) = 4L q I c 0 cos( y 2 ): (73) 94 Figure 41: One can build a CCCJJ b y replacing ev ery Josephson Junction with a pair of parallel junctions in the CCJJ. By con trolling the ux in an y of the smallest lo ops ( nc ) one can eectiv ely c hange the critical curren t of the junction pair and comp ensate for man ufacturing errors. A similar example with a CJJ and a CCJJ can b e found in [14]. This leads to an eectiv e Hamiltonian of the form H = X n ( Q n 2C n +U n n x n 2 )U q + cos( ccjj 2 ) cos( q ) (74) When w e substitute in + from Eq. 73, this Hamiltonian b ecomes H = X n ( Q n 2C n +U n n x n 2 ) (75) U q 8L q I c 0 cos( y 2 ) cos( ccjj 2 ) cos( q ); whic h is of the form giv en in Eq. 70. The corresp onding circuit is sho wn in Fig. 42. XXI I I. CONCLUSIONS W e ha v e demonstrated an arc hitecture for a univ ersal quan tum computer using Heisen- b erg spin c hains and clusters. This arc hitecture has the adv an tage that it can b e imple- men ted adiabatically and therefore has all of the adv an tages of adiabatic quan tum comput- ing. It has already b een demonstrated in [2] that the single qubit gates and data bus used in this computer can b e implemen ted with high delit y for reasonable annealing time. W e further demonstrate that a con trolled NOT gate can b e implemen ted at high delit y with a reasonably short annealing time and reasonable Hamiltonian parameters. 95 Figure 42: Design using strong inductiv e coupling b et w een small lo ops, where uxes mimic v arious magnetic elds applied to a spin. Note that this design assumes that all Josephson junctions are iden tical. In addition to suggesting an arc hitecture, w e also prop ose a design to ph ysically realize this arc hitecture. W e suggest a metho d for building sup erconducting ux qubit systems whic h mo del the lo w energy degrees of freedom of a Heisen b erg mo del. Because the arc hi- tecture w e prop ose can b e implemen ted adiabatically , only the lo w energy degrees of freedom need to b e repro duced. W e c ho ose a sup erconducting ux qubit implemen tation b ecause of the exp erimen tal success of these systems in p erforming non-univ ersal adiabatic quan tum computing with an Ising spin glass mo del. F urthermore it has b een sho wn that these Ising spin glass mo dels can b e realized accurately enough that the degeneracy of the ground state manifolds is not brok en. The univ ersal Heisen b erg spin based computer w ould represen t a signican t impro v emen t o v er the curren t non-univ ersal Ising systems b ecause it w ould allo w these computers to implemen t imp ortan t algorithms whic h the Ising spin glass system has not b een able to, suc h as Shors algorithm for factoring large n um b ers [26]. 96 A c kno wledgmen ts The authors w ould lik e to thank P . Zanardi, I. Marvian, S. Boixo, D. A. Lidar, L. C. V en uti, and S. T ak ehashi for helpful con v ersations. Some n umerical calculations w ere p er- formed on the USC high p erformance computing cluster. This researc h is partially supp orted b y the AR O MURI gran t W911NF-11-1-0268. XXIV. APPENDIX A. Calculation of t wist for Hadamard gate Sho wing ho w to implemen t an y giv en single spin gate using this metho d is straigh tforw ard. T ak e for example the Hadamard gateH, Hj i = 1 p 2 0 @ 1 1 1 1 1 A j i: (76) W e no w consider the action ofH on the eigen v ectors of the P auli spin matrices, rst for x : x + = 1 p 2 0 @ 1 1 1 A !x 0 + =Hx + = 0 @ 1 0 1 A =z + x = 1 p 2 0 @ 1 1 1 A !x 0 =Hx = 0 @ 0 1 1 A =z Similarly for z : z + = 0 @ 1 0 1 A !z 0 + =Hz + = 1 p 2 0 @ 1 1 1 A =x + z = 0 @ 0 1 1 A !z 0 =Hz = 1 p 2 0 @ 1 1 1 A =x And for y : y + = 1 p 2 0 @ 1 { 1 A !y 0 + =Hy + 97 = 1 2 0 @ 1 +{ { 1 1 A = { + 1 p 2 0 @ 1 { 1 A =y exp({) y = 1 p 2 0 @ 1 { 1 A !y 0 =Hy = 1 2 0 @ 1 +{ { 1 1 A = { 1 p 2 0 @ 1 { 1 A =y + exp({) In this case the phase factor ( = 4 ) is irrelev an t b ecause of the o v erall U(1) symmetry . B. Detailed discussion of simplifying assumptions for sup erconducting ux qubits The last termU q in Eq. 71 is a constan t whic h is not relev an t for this discussion. Ho w ev er the other constan ts in this term are relev an t and are dened as follo w (Eq. B4b-f in [14]) eff = + cos( 2 ) s 1 + ( + tan( 2 )) 2 ; (77) 0 q = 0 L + 0 R 2 + 0 ; (78) ccjj ( 0 L 0 R ); (79) 0 arctan( + tan( 2 )); (80) L R : (81) Here w e need the additional denitions: L(R) = L(R);+ cos( L(R) 2 ) s 1 + ( L(R); L(R);+ tan( L(R) 2 )) 2 ; (82) 0 L(R) = arctan( L(R); L(R);+ tan( L(R) 2 )); (83) 98 L(R); = 2L q (I 1(3) I 2(4) ) 0 : (84) Let us mak e the simplifying assumption that all of the critical curren ts are equal, I 1 = I 2 = I 3 = I 4 . In practice there is v ariabilit y in junction fabrication, but this error can b e comp ensated b y building a CCCJJ device (see Fig. 41). This assumption causes the equations to simplify greatly b ecause L(R); ! 0, whic h has the consequence that 0 L(R) ! 0 and ! ccjj . W e can further assume that in our design that L = R , this additional assumption causes = 0 and 0 ! 0, whic h in turn causes 0 q ! 0. After this simplication w e no w ha v e eff = + cos( ccjj 2 ); (85) L(R) = 4L q I c 0 cos( L(R) 2 ): (86) 99 Chapter 4 References [1] N. Chancellor and S. Haas, Ph ys. Rev. A 87, 042321 (2013). [2] N. Chancellor and S. Haas 2012 New J. Ph ys. 14 095025 (2012). [3] F. Meier, J. Levy , D. Loss, Ph ys. Rev. B 68, 134417 (2003). [4] S. C. Benjamin and S. Bose, Ph ys. Rev. Lett. 90, 247901 (2003). [5] P . Zanardi and M. Rasetti, Ph ys. Lett. A264 (1999) 94-99 [6] J. P ac hos and P . Zanardi, In t. J. Mo d. Ph ys. B15 (2001) 1257-1286 [7] G. F. Xu, J. Zhang, D. M. T ong, Erik Sjo qvist, L. C. K w ek, Ph ysical Review Letters 109, 170501 (2012) [8] V ahid Azimi Mousolou, Carlo M. Canali, Erik Sjqvist arXiv:1209.3645 (2012) [9] L. F aoro, J. Siew ert, and R. F azio Ph ys. Rev. Lett. 90, 028301 (2003) [10] P . Solinas, P . Zanardi, N. Zanghi, F. Rossi, arXiv:quan t-ph 0301090v1 (2003) [11] J. M. Renes, A. Miy ak e, G. K. Brennen, S. D. Bartlett arXiv:1103.5076 (2011) [12] D. Bacon, S. T. Flammia, G. M. Crosswhite, arXiv:1207.2769 (2012) [13] M. W. Johnson et al. Nature 473, 194-198 (12 Ma y 2011) [14] R. Harris et al. Ph ysical Review B 81, 134510 (2010) [15] A. P erdomo et al. Ph ysical Review A 78, 012320 (2008) [16] S. H. W. v an der Plo eg et al. IEEE T rans. App. Sup ercond. 17, 113 (2006) [17] R. Harris et al. Ph ys. Rev. B 82, 024511 (2010) [18] S. Boixo, T. Albash, F. M. Sp edalieri, N. Chancellor, D. A. Lidar arXiv:1212.1739 (2012) [19] Z. Chen, Z. Zhou, X. Zhou, X. Zhou, and G. Guo, Ph ys. Rev. A 81, 022303 (2010) [20] R. Chitra, S. P ati, H. R. Krishnam urth y , D. Sen, Diptiman, S. Ramasesha Ph ys. Rev. B 52, 6581 - 6587 (1995) [21] L. Banc hi, T. J. G. Ap ollaro, A. Cuccoli, R. V aia, and P . V errucc hi, Ph ys. Rev. A 82, 052321 (2010) [22] L Banc hi, T J G Ap ollaro, A Cuccoli, R V aia, and P V errucc hi, New Journal of Ph ysics 13 (2011) 123006 [23] T. J. G. Ap ollaro, L. Banc hi, A. Cuccoli, R. V aia, P . V errucc hi, arXiv:1203.5516v1 [quan t-ph] 100 2012 [24] L. Banc hi, A. Ba y at, P aola V errucc hi, and Sougato Bose, Ph ys. Rev. Lett. 106, 140501 (2011) [25] Nielsen, Mic hael A. & Ch uang, Isaac L. (2000), Quan tum Computation and Quan tum Infor- mation, Cam bridge Univ ersit y Press [26] N. Chancellor and S. Haas, Ph ys. Rev. B 84, 035130 (2011) [27] Here I mean trivial if the adiabatic bus proto col w ere p erformed t wice, once to transp ort the spin to another site and once to return it to its original lo cation. [28] This argumen t breaks do wn in general if one considers cases where there is not coupling in all three spin directions, for example in an XY mo del. 101 Summary In this thesis I put forw ard a design for a univ ersal holonomic quan tum computer based on an adiabatic transp ort proto col using Heisen b erg spin c hains and clusters. This w ork is based on four published pap ers, eac h of whic h pro vide a dieren t piece of motiv ation and bac kground for the main topic of this thesis. In c hapter 1 I discussed the equilibration of nite spin c hains sub ject to lo cal quenc hes, whic h pro vides imp ortan t bac kground on the t yp es of systems examined in later c hapters. This section also pro vides a a v or of the in teresting ph ysics of nite spin clusters, of whic h this thesis only pro vides a small glimpse. In c hapter 2 I sho w ed a sp ecic example of a t yp e of phenomenology in spin c hains whic h can propagate disturbances an unlimited distance ev en in a gapp ed system b y taking adv an tage of a ground state degeneracy . The eects sho wn here pro vide an underlying ph ysical explanation of the eects exp osited in the for the adiabatic implemen tation of the prop osed arc hitecture. In c hapter 3 I demonstrated sp ecically ho w an adiabatic transp ort proto col can b e implemen ted whic h tak es adv an tage of the eects observ ed in the previous c hapter. This transp ort proto col is imp ortan t for the holonomic quan tum computing arc hitecture discussed in the next c hapter. In c hapter 4 I established an arc hitecture for holonomic quan tum computing based on transp ort though t wisted Heisen b erg c hains. This arc hitecture can b e implemen ted either adiabatically using the metho ds prop osed in the previous c hapter or non-adiabatically . This c hapter also outlined ho w the adiabatic implemen tation of this arc hitecture ma y b e con- structed from sup erconducting ux qubits. This thesis not only sho ws a viable design for a univ ersal quan tum computer and prop oses ho w it migh t b e built, it also pro vides con text of the ideas relating to the design and bac kground for where these ideas came from. 102
Abstract (if available)
Abstract
This thesis deals with e ffects on antiferromagnetic Heisenberg spin chains and clusters which can be used for universal holonomic quantum computing. I will discuss in detail an architecture for a Holonomic quantum computer, and how to it may be implemented with superconducting flux qubits (see [1](ch. 4)). In addition to demonstrating this architecture, this thesis will also shed light on the behavior of these chains and the eff ects which make such an architecture possible. These eff ects include entanglement which allows disturbances that are excited by a quench to travel an unlimited distance, even in a gapped system. I will also examine anomalous equilibration behavior which can be seen in these systems.
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Creator
Chancellor, Nicholas
(author)
Core Title
Quantum computation by transport: development and potential implementations
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
05/28/2013
Defense Date
05/27/2013
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