Controlling qubit arrays with anisotropic XXZ Heisenberg interaction by acting on a single qubit
Rahel Heule, C. Bruder, Daniel Burgarth, Vladimir M. Stojanovic
aa r X i v : . [ qu a n t - ph ] O c t EPJ manuscript No. (will be inserted by the editor)
Controlling qubit arrays with anisotropic
X X Z
Heisenberginteraction by acting on a single qubit
Rahel Heule , C. Bruder , Daniel Burgarth , and Vladimir M. Stojanovi´c Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Institute for Mathematical Sciences, Imperial College London, SW7 2PG, United Kingdom QOLS, The Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2BW, United KingdomReceived: date / Revised version: date
Abstract.
We investigate anisotropic
XXZ
Heisenberg spin-1 / √ SWAP).We find the minimal times for realizing different gates depending on the anisotropy parameter ∆ of themodel, showing that the shortest among these gate times are achieved for particular values of ∆ largerthan unity. To study the influence of possible imperfections in anticipated experimental realizations ofqubit arrays, we analyze the robustness of the obtained results for the gate fidelities to random variationsin the control-field amplitudes and finite rise time of the pulses. Finally, we discuss the implications of ourstudy for superconducting charge-qubit arrays. PACS.
Coherent control of quantum systems is one of the prereq-uisites for quantum information processing. While alreadysimple arguments lead to the conclusion that almost anycoupled quantum system can be controlled in principle [1],the mathematical foundations of the subject are based onthe notion of controllability and formulated using the lan-guage of Lie algebras [2]. In particular, a system is com-pletely controllable if its internal dynamics governed byexternal fields can give rise to an arbitrary unitary trans-formation in the Hilbert space of the system [3]. Both statecontrol and the more general operator control have beenimplemented in a variety of systems [4].Recent quantum control studies have focused their at-tention on interacting systems. A familiar example is fur-nished by spin chains, systems that can be used as databuses [5] for state- [6,7,8,9] and entanglement transfer [10].In such systems, “always-on” interactions between the con-stituents (typically nearest neighbors) allow for a globalcontrol of the system dynamics by manipulating only asmall subsystem, in the extreme case a single spin. Themain question is then what is the smallest possible sub-system of a given system that one needs to act upon to en-sure the complete controllability, or, at least, the ability toperform certain pre-determined unitary transformations.This is the central idea behind the local-control approach. The fact that the local-control approach can be ad-vantageous in interacting systems provides an incentivefor identifying minimal controlling resources that guaran-tee controllability in particular classes of systems. Quiterecently, several Lie-algebraic results pertaining to localcontrol of spin chains have been obtained [11,12,13,14,15]. For example, it was demonstrated that acting only onone of the end spins of an
XXZ -Heisenberg spin chainensures complete controllability of the chain [12]. Adopt-ing the last result as our point of departure, in this paperwe investigate the feasibility of local operator control inqubit arrays modeled as spin-1 / XXZ coupling. The main motiva-tion stems from the relevance of the
XXZ -case for imple-mentations of Josephson-junction based superconductingqubit arrays [17,18,19].We determine piecewise-constant control fields, actingonly on the first spin in the chain, which lead to the highestpossible fidelities for a selected set of quantum logic oper-ations: the spin-flip (NOT) of the last spin in the chain, aswell as the controlled-NOT (CNOT) and the square-root-of-SWAP ( √ SWAP) gates applied to the last two spins.We optimize the gate fidelities with respect to the control-field amplitudes for three-spin chains. We then carry out a
Rahel Heule et al.: Controlling qubit arrays with anisotropic
XXZ
Heisenberg interaction by acting on a single qubit sensitivity analysis, i.e., discuss the robustness of the ob-tained results with respect to random errors in the controlfields, as well as finite rise/decay-times for control-fieldamplitudes. The present work is concerned with
XXZ -Heisenberg spin chains and our conclusions apply to anyphysical realization of qubit arrays with this type of cou-pling [17,18].
The total Hamiltonian of a Heisenberg spin-1 / N s reads H ( t ) = H + H c ( t ) , (1)where H = J N s − X i =1 ( S i,x S i +1 ,x + S i,y S i +1 ,y + ∆S i,z S i +1 ,z ) , (2)is a XXZ
Heisenberg part with anisotropy ∆ , and H c ( t ) = h x ( t ) S x + h y ( t ) S y (3)a Zeeman-like control part, with control fields h x ( t ), h y ( t )acting only on the first spin. In what follows, we will alsoemploy control Hamiltonians with fields in the x and z -directions. Whether the XXZ spin chain under consider-ation is ferromagnetic or antiferromagnetic is not crucialhere, as we are concerned with operator control; aspectssuch as, for example, the different nature of the groundstates in the two cases (separable vs. entangled) wouldonly be consequential for issues related to, e.g., state con-trol or entanglement transfer. For definiteness, we will as-sume that
J > ∆ >
0. It is useful to recall thatthe one-dimensional
XXZ model has an antiferromag-netic ground state for ∆ ≥
1, a ferromagnetic one for ∆ < −
1, while for the intermediate values of ∆ it is char-acterized by a critical gapless (quasi-long-range ordered)phase [20].For convenience, we hereafter set ~ = 1 and, in ad-dition, express all frequencies and control fields in unitsof the coupling strength J . Consequently, all times in theproblem are expressed in units of 1 /J .Since implementing control fields with a complex timedependence is difficult, we resort to piecewise-constantones according to the following scheme. At t = 0 westart acting on the first spin of the chain with an x con-trol pulse of amplitude h x, , which is kept constant until t = T . Thus the system is governed by the Hamiltonian H x, ≡ H + h x, S x . We then apply a y pulse with theamplitude h y, (Hamiltonian H y, ≡ H + h y, S y ) overthe next interval of length T , etc. This sequence repeatsuntil N t pulses are carried out at t = t f ≡ N t T . The fulltime evolution is described by U ( t f ) = U y,N t / U x,N t / . . . U y, U x, , (4)where U x,i ≡ e − iH x,i T and U y,i ≡ e − iH y,i T are the respec-tive time-evolution operators corresponding to H x,i and H y,i , which can be evaluated using their spectral form. Our control objectives (target unitary operations) areboth one-qubit gates, such as the spin-flip (NOT) opera-tion on the last spin of the chain X N s := ⊗ ⊗ . . . ⊗ ⊗ X ( X being the Pauli matrix), and some entangling two-qubit gates. For instance, CNOT N s := ⊗ ⊗ . . . ⊗ ⊗ CNOT performs the controlled-NOT operation onthe last two qubits in the chain. Similarly, √ SWAP N s := ⊗ ⊗ . . . ⊗ ⊗ √ SWAP performs the √ SWAP operationon the same pair of qubits.Unlike in many other control studies [11], which makeuse of single-excitation subspaces, we retain the full Hilbertspace of the system. This puts constraints on the systemsize that can be treated within our framework. In whatfollows, we discuss three-spin chains.
In Ref. [12] a very general graph-infection criterion wasproven, which – as a special case – guarantees the completecontrollability of
XXZ
Heisenberg spin chains throughacting on one end spin. The more conventional approachesfor proving complete controllability entail finding the di-mension of the relevant dynamical Lie algebra, a task forwhich special algorithms have been developed [21]. In thepresent problem, such an algebra is generated by the skew-Hermitian traceless operators {− iH , − iS x , − iS y } andhas dimension d −
1, where d ≡ N s is the dimension ofthe Hilbert space of the system. Being generated by trace-less operators, this algebra is then isomorphic to su ( d ),the Lie algebra associated with the special unitary group SU ( d ) [22].Setting aside the issue of complete controllability, onemight be interested to know if some particular unitaryoperations – on an otherwise not completely controllablesystem [23,24] – are possible with an even smaller degreeof manipulation, e.g.., a control field only in one direction.For such operations, equation (4) goes over into U ( t f ) = U x,N t . . . U x, . For example, the X N s and √ SWAP N s gatesrequire only a control field in the x direction. To demon-strate this for X N s , let L x be the dynamical Lie algebragenerated by − iH and − iS x , a subalgebra of su ( d ) withdimension 30 in a three-spin XXZ chain (note that thecounterpart of this algebra in the isotropic-coupling casehas smaller dimension, namely 18). For showing that X N s belongs to the connected Lie subgroup e L x of SU ( d ) itsuffices to find an element A ∈ L x such that X N s = e A .Using the repeated commutators of the generators of L x ,it can be demonstrated that X N s is an element of this al-gebra. X N s is both unitary and Hermitian, implying that X N s = . It is then easy to show that A = − i π X N s ,an element of L x , fulfills e A = − iX N s . Therefore, X N s isreachable using only an x control field. Recalling that the √ SWAP gate on two qubits is given by [25] √ SWAP = e i π e − i π ( X ⊗ X + Y ⊗ Y + Z ⊗ Z ) , (5)the reachability of √ SWAP N s using an x control onlyreadily follows from the fact that ⊗ ⊗ . . . ⊗ ⊗ ( X ⊗ X + Y ⊗ Y + Z ⊗ Z ) is an element of L x . ahel Heule et al.: Controlling qubit arrays with anisotropic XXZ
Heisenberg interaction by acting on a single qubit 3 - h y (cid:144) J h x (cid:144) J - X CNOT H a LH b L time (units of 1/J) Fig. 1.
Optimal control sequences for ∆ = 5 realizing the (a) X and (b) CNOT gates with fidelity higher than 0.999. In this section our goal is to find control fields leading tooptimal fidelities for a chosen set of quantum gates, with aparticular emphasis on minimal times needed for realizingdifferent gates depending on the anisotropy ∆ .In quantum operator control, the figure of merit is thegate fidelity F ( t f ) = 1 d (cid:12)(cid:12) tr (cid:2) U † ( t f ) U target (cid:3)(cid:12)(cid:12) , (6)where U ( t f ) is the time-evolution operator of the sys-tem at time t = t f (Eq. (4)) and U target stands for thequantum gate that we want to realize. We perform opti-mization, i.e., maximize the gate fidelity with respect tothe N t control-field amplitudes, for varying number ( N t )and durations ( T ) of pulses (hence different total evolu-tion times t f ). We make use of a quasi-Newton methoddue to Broyden, Fletcher, Goldfarb, and Shanno (BFGS-algorithm) [26]. It should be stressed that, much like otheroptimization approaches, this algorithm ensures only con-vergence to a local maximum. Therefore, to determine aglobally-optimal sequence of control-field amplitudes (fora given target gate and given value of ∆ ) we ought to re-peat the optimization process for a number of different ini-tial guesses for these amplitudes. We generate these initialguesses using a uniform random number generator [26].An alternative to fixing the pulse durations and maxi-mizing over the control-field amplitudes would be to keepthe amplitudes constant and treat the pulse durations asvariable control parameters. However, we choose optimiza-tion over the control-field amplitudes since this approach Table 1.
Minimal times (in units of J − ) needed to reachfidelities higher than 0 .
999 for the relevant gates in the x - y control case. The corresponding values in the x - z control caseare given in the brackets. ∆ X CNOT √ SWAP allows us to easily fix t f and determine its minimal valuefor implementing the desired gate for any fixed value ofthe parameter ∆ .The obtained results for the gate fidelities have the fol-lowing two salient features. Firstly, for fixed parameters ofthe model and fixed total evolution time t f , the fidelity forany given gate can increase significantly with increasing N t (or, equivalently, decreasing T ). In other words, morerapid switching leads to higher fidelities. For instance, inthe case of the CNOT gate with ∆ = 1 . t f = 30,for N t = 10 , , , , , ,
70 we obtain the respectivefidelities F = 0 . , . , . , . , . , − − , − − . Secondly, for each gate there exists a minimal valueof t f (i.e., minimal gate time), below which fidelities closeto unity cannot be reached regardless of the value of N t .The obtained minimal gate times for different values of ∆ in the x - y ( x - z ) control cases are given in Table 1. Ap-parently, there exists an optimal value of ∆ which cor- Rahel Heule et al.: Controlling qubit arrays with anisotropic
XXZ
Heisenberg interaction by acting on a single qubit responds to the shortest among these times. For the X and CNOT gates, for example, these values are around ∆ = 5. The corresponding optimal sequences of x and y control pulses for the X and CNOT gates are shownin Figures 1(a) and 1(b), respectively. Since ideal steplikepulses cannot be realized in practice, in Ref. [16] we alsostudied frequency-filtered control fields and showed thatsufficiently high fidelities can still be retained.In Table 2 the minimal times are given for the X and √ SWAP gates realized using only an x control field. It isinteresting to compare these minimal times to the abovecase with both x and y (or x and z ) controls. For smallvalues of ∆ (with the exception of ∆ = 1) the minimaltimes for realizing the X gate in the x -only control caseare significantly longer than their counterparts in the x - y ( x - z ) case. In contrast, for larger ∆ these times becomemore and more similar. Finally, for ∆ ≥
11 the minimaltimes in the x -only control case are even shorter than inthe x - y and x - z cases. Since x -only control is easier toimplement, this surprising observation provides an addi-tional argument for using x -only control in the regime ofinterest for superconducting charge qubits.As is well known [25], the √ SWAP gate on two qubitsis naturally implemented by the isotropic Heisenberg Hamil-tonian after a time τ = π/ ≈ .
57. As can be seen inTable 1, the minimal √ SWAP -gate times indeed seem tocorrespond to ∆ ≈ Table 2.
Minimal times (in units of J − ) needed to reachfidelities higher than 0 .
999 for the relevant gates using only an x control field. ∆ X √ SWAP ∆ X √ SWAP N t =60 N t =70 N t =80 N t =90 N t =100 F /J Fig. 2. (Color online) Average fidelity versus half-width ( δ ) forthe √ SWAP gate with ∆ = 1 . t f = 60. √ SWAP gate performs the √ SWAP operation on the lasttwo qubits (leaving the state of the first qubit unchanged)while the Heisenberg Hamiltonian of equation (2) also con-tains the interaction between the first two qubits. Thus wecan conclude that the role of control fields in this case isto counteract the effect of the free evolution of the firstqubit governed by H .Generally speaking, the minimal gate times can in prin-ciple be found based on the time-optimal unitary opera-tion formalism put forward by Carlini and co-workers [27].This method requires solving a system of coupled nonlin-ear equations for Lagrange multipliers resulting from thequantum brachistochrone equation. In practice, extractingminimal times for different quantum gates in this way isfeasible only when the time evolution of the total Hamilto-nian of the system is as simple as to allow for an analyticalsolution of these equations. This is possible, for instance,when this Hamiltonian has a block-diagonal form in thecomputational basis, where each block commutes with it-self at different times. In the problem at hand this is notthe case, therefore an alternative strategy for finding min-imal times is required. In the following, we analyze the sensitivity of the fidelityto random errors in the control-field amplitudes, as wellas to a finite rise time.The random errors in control-field amplitudes are as-sumed to follow a uniform distribution of half-width δ . Forgiven δ , we generate a large sample of N ∼ F = P Ni =1 F i /N , where the F i are fidelitiesfor specific realizations of the random field, versus δ forthe gates of interest and varying values of ∆ . ahel Heule et al.: Controlling qubit arrays with anisotropic XXZ
Heisenberg interaction by acting on a single qubit 5
In our previous work [16], using the isotropic Heisen-berg model ( ∆ = 1) as an example, it was demonstratedthat the shape of the fidelity decay curves ( ¯ F vs. δ ) de-pends on the number of control pulses N t and their length T . Provided that the system satisfies the conditions forcomplete controllability, the saturation regime of the av-erage fidelity sets in for δ & J . The universal saturationvalue is 1 /d , where d is the dimension of the Hilbert spaceof the system. Importantly, for fixed t f = N t T , the av-erage fidelity is closer to the intrinsic (in the absence ofrandom errors) optimal values for larger N t (faster switch-ing), this being a consequence of general properties ofsystems that exhibit competition between the resonance-and relaxation-type behavior [16]. Therefore, more rapidswitching leads not only to higher intrinsic fidelities inthe absence of randomness (recall section 4), but also ren-ders these fidelities less sensitive to random errors. This isa manifestation of an intrinsic robustness of the system.Figure 2 illustrates that these features are also present inthe anisotropic XXZ case.The sensitivity to random errors in the control-fieldamplitudes depending on the anisotropy ∆ is illustratedin Figure 3. As can be inferred from this figure, for larger ∆ the system is less sensitive to random errors.Another unavoidable source of imperfections in qubit-array realizations is the finite rise time of the control fields.Instead of a stepwise behavior, experimental control fields h j,n ( j = x, y ; n = 1 , . . . , N t /
2) are expected to have afinite rise/decay time τ . Figure 4 shows the dependence ofthe fidelity on the finite rise time. For larger values of ∆ ,the fidelities of optimal control sequences seem to be lessaffected by the finite rise time.The central result of this section is that values of ∆ > =1 =2 =3 =4 =5 =6 =7 =8 F /J Fig. 3. (Color online) Average fidelity versus half-width ( δ ) ofrandom-noise distribution for optimal control sequences with N t = 70 and T = 1 corresponding to the CNOT gate. Our results are of direct relevance to superconductingqubit arrays [6]. One-dimensional Josephson arrays of ca-pacitively coupled superconducting islands can be describedas
XXZ
Heisenberg spin-1 / XY -part of Hamiltonian is characterized by a nearest-neighbor interaction, whereas the Z -part will also havecoupling contributions beyond nearest neighbors. How-ever, by properly choosing the junction capacitances andthe capacitance of each island to the back gate of the struc-ture, the Z -part will also be approximately of nearest-neighbor type. The correspondence between the parame-ters of the Josephson array and the spin chain is as fol-lows: the Josephson energy E J of the junctions couplingthe islands corresponds to the exchange coupling constant J of the spin system and can be controlled by a magneticfield if we assume that the coupling junctions are realizedas SQUIDs. The parameter ∆J of the spin system corre-sponds to the charging energy E C , i.e., the anisotropy pa-rameter ∆ corresponds to E C /E J . Values of ∆ like thosestudied in Tables 1 and 2 can be experimentally realized.Finally, the first island should form a charge qubit, andthe control field h z corresponds to the gate voltage, while h x and h y play the role of the Josephson energy of thischarge qubit. Our study shows that, in principle, arbitraryquantum algorithms can be realized on a one-dimensionalJosephson array by controlling only the first island in thearray.In summary, we have shown that local control of thefirst spin of an anisotropic XXZ
Heisenberg spin-1 / √ SWAP). We have found the min- F /T Fig. 4. (Color online) Illustration of sensitivity to finite risetime for the X gate. The optimal control sequences used cor-respond to N t = 70 and T = 1. Rahel Heule et al.: Controlling qubit arrays with anisotropic XXZ
Heisenberg interaction by acting on a single qubit imal times for realizing different gates depending on theanisotropy parameter ∆ of the model, showing that theshortest among these gate times are achieved for particu-lar values of ∆ larger than unity. Another surprising re-sult was that in the regime of interest for superconduct-ing charge qubits, the minimal times in the simpler x -onlycontrol case can be even shorter than in the x - y and x - z control cases. We have also analyzed the sensitivity ofthe obtained results for the gate fidelities to random vari-ations in the control-field amplitudes and finite rise timeof the pulses. Our results are independent of a partic-ular experimental realization of the XXZ chain, yet, asuperconducting Josephson array would be a particularlyappealing candidate. Our investigation paves the way forfuture studies, involving more sophisticated control strate-gies [30,31].
We would like to thank R. Fazio for discussions. This work wasfinancially supported by EU project SOLID, the EPSRC grantEP/F043678/1, the Swiss NSF, and the NCCR Nanoscience.
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