Converting a real quantum bath to an effective classical noise
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Converting a real quantum spin bath to an effective classical noise acting on a centralspin
Wayne M. Witzel, Kevin Young, and Sankar Das Sarma Sandia National Laboratories, Albuquerque, New Mexico 87185 USA Sandia National Laboratories, Livermore, California 94550 USA University of Maryland, College Park, Maryland 20742-4111, USA
We present a cluster expansion method for approximating quantum spin-bath dynamics in terms ofa classical Gaussian stochastic process. The cluster expansion produces the two-point correlationfunction of the approximate classical bath, permitting rapid evaluation of noise-mitigating quantumcontrol strategies without resorting to computationally intensive dynamical decoupling models. Ourapproximation is valid for the wide class of models possessing negligible back-action and nearly-Gaussian noise. We study several instances of the central spin decoherence problem in which thecentral spin and randomly-located bath spins are alike and dipolarly coupled. For various pulsesequences, we compare the coherence echo decay computed explicitly quantum mechanically versusthose computed using our approximate classical model, and obtain agreement in most, but not all,cases. We demonstrate the utility of these classical noise models by efficiently searching for the4-pulse sequences that maximally mitigate decoherence in each of these cases, a computationallyexpensive task in the explicit quantum model.
PACS numbers: 03.65.Yz; 03.67.Pp 76.30.-v; 76.60.Lz;
Spin echo decay, which operationally defines the de-phasing/decoherence time T , is an important measurecharacterizing the viability of prospective qubit realiza-tions. For solid-state spin qubits, this echo decay maybe attributed to the dynamical flip-flopping of impu-rity spins surrounding the central qubit spin. Clusterexpansion techniques [1–8], which simulate the micro-scopic quantum dynamics of the spin-bath system, haveproven exceptionally reliable in quantitatively reproduc-ing and predicting measured spin echo decays [6, 7, 9–13] and in the evaluation of dynamical decoupling strate-gies [14–19], which extend the coherence time of the cen-tral spin through the application of precisely timed pulsesequences. These studies, however, are computationallyintensive and must be repeated for each pulse sequenceunder consideration.In contrast, the echo decay of a quantum spin sub-jected to classical Gaussian noise may be computed ex-tremely efficiently in terms of filter functions and thenoise correlation function [20–22]. This efficiency fa-cilitates, for example, optimal control calculations thatwould be computationally intractable on a fully quantummodel. Such considerations lead us to question underwhat circumstances a quantum spin bath Hamiltonian,consisting of many interacting impurity spins, may bewell approximated as a classical stochastic noise.A semiclassical stochastic noise model should well ap-proximate the dynamics of a fully quantum model if twoconditions are met: (i) the bath dynamics are indepen-dent of the central spin state, i.e. back action effects areinsignificant, so the bath effects on the central spin ap-pear classical and (ii) the effective noise is approximatelyGaussian, so is characterized completely in terms of itstwo-point correlation function, and is therefore amenable to filter function techniques. In this paper, we shall con-sider a central spin decoherence problem in which thecentral spin and randomly-located bath spins are alikeand dipolarly coupled, such as an electron spin in an elec-tron spin bath. The large, identical spin bath possessedby this model is very likely to satisfy both of the aboveconditions: the first because the bath and central spinsare alike, so the state of the central spin is unlikely todrastically affect the bath dynamics; the second becausethe spin bath resembles as a large collection of two-levelfluctuators which combine to yield Gaussian statistics,as implied by the central limit theorem. This model hasbeen extensively studied with cluster expansion methodsin previous work [8], and evidence [8, 23–26] suggeststhat the bath effects are well represented as a classicalOrnstein-Uhlenbeck (O-U) stochastic process.As the only stochastic process which is Markovian,Gaussian and stationary, O-U noise represents an ide-alized approximation to the quantum dynamics. In thispaper we extend this semiclassical approximation, gener-alizing the classical stochastic process so that its correla-tion function matches that of the fully quantum model,which we compute using a modified cluster expansion.We then compare the resulting spin echo decays againstthose computed directly with a fully quantum mechani-cal treatment. Remarkable agreement is shown for mostinstances of the randomly distributed bath spins. Af-ter demonstrating the broad validity of the semiclassicalmodels, we further illustrate their utility through the con-struction of optimally noise-mitigating pulse sequences.The free evolution Hamiltonian of our problem is:ˆ H = X i µ B g i B i ˆ S zi + µ B X j>i g i g j ˆ S i · D ( R i − R j ) · ˆ S j , (1)written in atomic units ( ~ = 1 and 1 / πǫ = 1) whereˆ S i are spin operators for the spin-1/2 particles, µ B is theBohr magneton, g i is the g -factor of the i th electron, B i isthe externally applied magnetic field at each electron site,and D ( r ) is a tensor to characterize dipolar interactionsand is defined by D α,β ( r ) = (cid:20) δ αβ − r α r β / r r (cid:21) , (2)with α, β = x, y, z . δ αβ is the Kronecker delta and r α is the α vector component of r . Our convention is toindex the central spin as i = 0. We shall investigate anumber of randomly generated spatial configurations ofelectron spins at average concentration 10 cm − andwith g i = 2. We assume the limit in which B i is largeand equal amongst the bath spins but not necessarily thecentral spin [31], permitting a secular approximation ofthe interbath dynamics. That is, processes must conservethe net polarization of the bath spins; bath spins mayflip-flop with each other but not the central spin. Sucha situation may arise for an addressable qubit tuned offresonance with the bath spins. Under the secular approx-imation in the rotating frame, the effective Hamiltonianis ˆ H eff = X i,j> b i,j ˆ S + i ˆ S − j − X i,j b i,j ˆ S zi ˆ S zj . (3)where b i,j = − g i g j µ B ~ (1 − θ ij ) / (4 R ij ), θ ij is thepolar angle of the displacement vector connecting spin i and spin j with respect to the ˆ z unit vector (the directionof applied magnetic field), and R ij is its length.We are interested in computing the spin echo de-cay, which is proportional to the expectation value ofthe coherence operator on the central spin as a func-tion of time, h σ + ( t ) i = Tr ( σ +0 U ( t ) ρ (0) U ( t ) † ), wherethe initial state of the bath is taken to be thermal atsome relevant temperature. In what follows, we presenta method for approximating this expectation value byconstructing an effective semiclassical Hamiltonian forthe central spin ˆ H cl ( t ) = µ B g B ( t ) ˆ S z , where the effec-tive magnetic field B ( t ) is a classical stochastic variablewhose action approximates that of the quantum opera-tor ˆ B z = − P j> b ,j ˆ S zj . We permit the system to besubject to π -pulse control about the x axis. In a tog-gling frame [21, 22], each π pulse causes the effectivefield felt by the central spin to flip, leading to a Hamilto-nian ˜ H ( t ) = µ B g y ( t ) B ( t ) ˆ S z , where y ( t ) = ± π pulse is applied. If the fluctuatingfield is Gaussian [27], then the coherence decay may becalculated directly as h σ + ( t ) i = exp (cid:16) − µ B g R t du C ( u ) F t ( u ) (cid:17) , (4) F t ( u ) = R t − uu dv y (cid:0) v + u (cid:1) y (cid:0) v − u (cid:1) . (5)where F t ( u ) is a time-domain filter function[22] de-scribing the action of the control pulses and C ( t ) = h B ( t ) B (0) i is the correlation function of the effectiveclassical field. We choose this correlation function tobe equal to that of the fully quantum model, C Q ( t ) = D ˆ B z ( t ) ˆ B z (0) E , with ˆ B z ( t ) the operator in the Heisen-berg picture. The filter function is efficiently computable,so knowledge of the correlation function is sufficient torapidly determine the coherence remaining in the systemafter any sequence of π pulses. We compute the quantumcorrelation function by using a variant[8] of the clustercorrelation expansion [4, 5] (CCE).The original CCE assumed the coherence decay (orany observable quantity), L = h σ + ( t ) i , could be decom-posed as a product of contributions, L = Q S ˜ L S , fromeach subset, S , of bath spins. The modified contribu-tions, ˜ L S , are then defined implicitly through the rela-tion, ˜ L S = L S / Q C⊂S ˜ L C , where the product is takenover all subsets, C of the set S of bath spins (we shallrefer to subsets of n bath spins as n -clusters ). Each ofthe unmodified contributions L S may be computed byexactly solving the dynamics of a system of bath spins S much smaller than the original problem. By decompos-ing the observable in this manner, the solution may besuccessively approximated by including relevant clustersof increasingly large size. A Dyson series expansion [28]implies that only small clusters should be relevant to theshort-time dynamics, with clusters of increasing size be-coming more important with increasing time. This small-cluster approximation has been quite successful, showingremarkable agreement with experimental data in a num-ber of previous studies [9–13]. Reference [8] refined theperformance of the CCE by providing heuristics to selectand evaluate a subset of the clusters that are more likelyto contribute (e.g., ones that are strongly interacting witheach other), a strategy we employ in this work. We notethat the CCE is related to a linked clusters perturbationexpansion [29] but is more convenient to evaluate in anautomated way.Unfortunately, for a sparse bath of like spins, the CCEsuffers numerical instability issues in the evaluation of˜ L S due to the occasional division by small numbers. Thephysical system we consider here can be particularly vul-nerable to this problem because the decoherence rate isstrongly dependent on the initial state of the bath. Con-sider, for example, a completely polarized bath in whichthere are no flip-flopping spins and therefore no nontrivialdecoherence. Other states exhibit the opposite extreme.Because of this diversity, for times at which the expectedcoherence has not yet decayed significantly, there may ex-ist spin configurations for which the expected coherenceis zero, implying the ˜ L S formula will involve a divisionby zero. In Ref. [8] we presented as a solution to thisproblem a highly technical variation of the CCE that wecalled interlaced spin averaging, in which each evaluationof ˜ L S was averaged over bath spin states in a relativelyefficient manner (which is the fairly technical aspect), − − − − − − A − − − − − − − − B − − − − − − − C − − − − − − − − D − − − − − − − E . . . . . . time (s) − − − − − F − − − time (s) - − - − - − - − - - - - C Q ( t ) − C Q ( ) ABCDEF
FIG. 1: (Color online) Relative correlation function calcu-lation results, C Q ( t ) − C Q (0) in (rad / s) , for the five casesof bath spatial configurations labeled A-F studied in Ref. [8]with linear scales (left) and logarithmic scales (right). The C Q (0) values are 55594, 14 .
3, 59 .
8, 19 .
1, 5 .
93, and 287(rad / s) , respectively. Black dashed curves are O-U typeof the form A exp ( − Bt ) + C , loosely fitting the calculationsover some respective time ranges. Thick (and colored) solid,dashed, and dotted curves on the right plot are 2-cluster, 3-cluster, and 4-cluster results, respectively. thus removing the numerical instability. Here we presenta far simpler solution in which we formulate L as a sum of contributions rather than a product. Thus, L S = X C⊆S ˜ L S , ˜ L S = L S − X C⊂S ˜ L C . (6)We also redefine L as h σ + ( t ) i − t = 0 [32]. As in the original CCE, thisis exact in the limit that all cluster contributions are in-cluded (but without any division by zero pathology). Wefind the convergence behavior is different but comparableto the multiplicative version with interlaced spin averag-ing.To directly compute correlation functions, we employthis additive form of the cluster expansion Eq. (6). Theonly change is taking the quantity of interest to be therelative [33] correlation function L = C Q ( t ) − C Q (0). Weestimate the average over initial spin states of the bathby taking random samples of up/down product statesand compute the cluster expansion each initial spin stateseparately. In Fig. 1, we show results of these relative correlation function calculations for five cases of differentrandom spatial locations of bath spins, the same cases la-beled A-F in Ref. [8]. The right plot displays all casestogether with 2-cluster, 3-cluster, and 4-cluster results.Each case demonstrates convergence in time with respectto cluster size. Short time dynamics are dominated by2-cluster contributions, with higher-order contributionsbecoming necessary with increasing time. Case C illus-trates this most clearly.This figure also makes comparisons with exponential-like decay of O-U correlation functions. In Fig. 6 ofRef. [8], we fit Hahn echo decay results of these cases withthe form exp ( − t ) as a confirmation of the O-U noise ap-proximation. These were good fits on the time scale ofthe initial substantial echo decay (out to the first 25% to50% of the decay). With exception to case A, which is anunusual case as we shall see, the O-U noise approxima-tion fits our correlation function results well on the samecorresponding time scales. For short times, however, wesee in the right plot that the correlation functions are ofthe form α − βt , as predicted by perturbation theory.While this correlation function is well defined as aquantum mechanical expectation value and we believethe cluster expansion is working well to successively ap-proximate this quantity, whether or not the approximat-ing classical model is sufficient for calculating echo decaysis a separate question. As discussed earlier, our semiclas-sical approximation relies on two assumptions: (i) min-imal back-action and (ii) approximately Gaussian noise.To verify that these assumptions hold, we compare theecho decays computed from the correlation function toecho decays computed directly using the CCE in Fig. 2.We do this on cases A-F using Uhrig dynamical decou-pling (UDD)[30] sequences with 1-4 pulses (1-pulse UDDis a Hahn echo).For case A, we only see agreement for the Hahn echosequence in Fig. 2. For all other cases, we see excellentagreement for all pulse sequences. The failure of the semi-classical model for case A implies a breakdown of one ofthe above assumptions. In the absence of back-action,our echo decay results should scale in a simple manneras we reduce the central-spin gyromagnetic ratio. Thatis what we find, implying that it is the Gaussian noise as-sumption that is violated in case A. Indeed, by looking atthe distribution of correlation function contributions fordifferent initial spin states at various times, we see thatcase A exhibits multiple peaks. The other cases typicallyexhibit single peaks well approximated as Gaussian. Fig-ure 1 of Ref. [8] reveals that case A, by happenstance, hasone particular bath spin that has a conspicuously stronginteraction with the central spin and we find that the dy-namics of this spin dominates the noise, which takes onnon-Gaussian random telegraph character [27]. This spininteracts strongly with two other nearby bath spins andthis small system dominates the initial decoherence of thecentral spin. The central limit theorem does not apply − − − − − − − A − − − − − − − − | h σ + ( t ) i | B − − − − − − − C − − − − − − − − | h σ + ( t ) i | D − − − time (s) − − − − − − − E − − − time (s) − − − − − − − − | h σ + ( t ) i | F FIG. 2: (Color online) Comparison of echo decays (plotted as1 − Echo versus total pulse sequence time on a log-log scale)calculated directly using the CCE versus via the correlationfunctions from Fig. 1 for cases A-F. The solid black curves aredirect calculation results. The yellow dashed curves are de-rived from correlation functions. Both include up to 4-clustercontributions. Each case shows results for 1- to 4- pulse UDDsequences (UDD1 is a Hahn echo). Initial coherence behaviorimproves with an increased number of pulses; that is, the 1-4pulse curves are seen left to right. Additionally for cases B-Fwe show echo decay for optimized 4-pulse sequences that per-form slightly better than UDD4; the red dashed curves are thecorrelation function derived results for these pulse sequences,in excellent agreement with direct calculations in solid black.Dotted black curves show direct 2-cluster results; we find thatcorrections from larger clusters are important, even at shorttimes, for some of the multipulse sequences. and the noise cannot be approximated as Gaussian, thusmaking case A a rather nongeneric special situation. Asa quick test, the Gaussianity assumption may be verifiedby computing four -point correlation functions and com-puting the deviation from the Wick’s theorem prediction.We performed a statistical study of about 900 and 750random instances respectively for Hahn echo and UDD4which shows a general correlation between atypical in-stances (like case A) and the presence of particularlystrongly coupled (nearby) bath spin. Figure 3 shows scat-ter plots of relative discrepancies of echo infidelity versusmaximum coupling to the central spin. Relative discrep-ancies are comparisons between correlation function re-sults and direct CCE evaluations of the echo infidelities(1- echo). Instances where the maximum coupling be-tween central and bath spins is small will likely have noisedominated by many similar bath spins and will tend tobe Gaussian. The non-Gaussian cases are ones in which maximum coupling constant to central spin (Hz) -10 -10 -10 -1 -1 r e l a t i v e d i s c r e p a n c y o f e c h o i n f i d e li t y estimated−expectedmin(estimated,expected) UDD4Hahn c o un t s FIG. 3: Scatter plot of the extreme values for the rela-tive discrepancy of echo infidelity (1 - echo) estimates ver-sus the maximum coupling constant to the central spin.Hahn echo (UDD1) and UDD4 cases are represented fromabout 900 and 750 random spatial configurations respec-tively. The relative discrepancy is computed as (estimated -expected)/min(estimated, expected) where “estimated” usesthe correlation function and “expected” uses the direct CCEapproach. Positive values correspond with pessimistic esti-mates. Relative discrepancies shown are the two extremes(per spatial configuration) over 100 logarithmically spacedtime points between 1 ms and 0 . − where the numerical precision is suspect. Theseare plotted on a symmetric positive and negative logarithmicscale with a linear region between ± .
1. The top figure showscumulative counts of instances versus maximum coupling con-stants. The right figure shows cumulative probability distri-butions of relative discrepancies. the noise is dominated by a small number of spins andthus more likely to have a large maximum coupling. In-deed, we find that discrepancies increase in magnitude asthe maximum coupling increases, confirming that this isa good signature of the degree to which the noise can beexpected to be Gaussian. The Hahn echo is much betterbehaved indicating that the atypical behavior tends tomanifest itself with increasing dynamical decoupling aswe saw in case A above (see Fig. 2). Also note that thecorrelation function estimates tend, strongly, to error onthe side of being pessimistic (positive relative discrepan-cies). These results over a large number of random in-stances were computed using an automated convergencestrategy for selecting clusters with a convergence crite-ria of 10% variation in the logarithm of the contribution,per cluster size, to the echo infidelity. Because some in-stances failed to converge at the end of the allotted 48 − − time (s) . . . . . . P u l s e T i m e F r a c t i on s B CD E F
UDDCPMG
FIG. 4: (Color online) Optimal, symmetric, 4-pulse refocus-ing sequences for cases B-F found efficiently using the corre-lation functions of Fig. 1. The curves indicate the fractionaltimes of π pulses as a function of the total pulse sequencetime. For comparision, we show the 4-pulse UDD and Carr-Purcell-Meiboom-Gill sequence as dashed and dotted lines re-spectively. hour run time on 128 processors, there may be some biaswith these “harder” instances being underrepresented.In most cases the correlation function is an adequateand convenient characterization of the noise. Where itis valid to use, we can use it to compute the echo decayfor any pulse sequence very efficiently. We offer a simpledemonstration of this by optimizing over the symmetric,4-pulse, refocusing sequences (parameterized by a singlenumber). We show these optimal sequences as a func-tion of total pulse sequence time for cases B-F in Fig. 4.We leave out the nongeneric case A since the correlationfunction is not a sufficient characterization of its noise.In Fig. 2 we see that these optimal sequences performslightly better than UDD4. While the improvement isnot substantial, it demonstrates the utility of a correla-tion function description of the noise.In conclusion, we have presented a method for deriv-ing a semiclassical correlation function directly from amicroscopic quantum spin-bath model using a cluster ex-pansion approach. We have applied this approach to theproblem of a dipolarly-interacting system of sparse, likespins finding that the correlation function is an adequatecharacterization of the noise for most, but not all, cases ofthis problem. Some nongeneric instances are dominatedby small system dynamics producing non-Gaussian noise.We estimated the statistical likelihood for an instanceto be atypical and revealed a correlation between non-Gaussian noise behavior and the presence of a bath spinthat has a particularly strong interaction with the cen-tral spin. We establish that correlation functions, whereapplicable, can be used to efficiently find optimal pulsesequences. With this tool, we can learn about the differ-ence in the performance of generic pulse sequences ver-sus optimized pulse sequences that are tailored to spe-cific qubit environments. Our explicit conversion of aquantum bath Hamiltonian to an effective classical noisedescription allows for a tremendous improvement in theefficiency of evaluating various control pulse sequences to preserve system coherence, enabling optimized quantumerror correction protocols for spin qubit architectures.We thank and acknowledge Rogerio de Sousa, RobinBlume-Kohout, Toby Jacobson, Erik Nielsen, RickMuller, Malcolm Carroll, and particularly LukaszCywi´nski for valuable discussions and contributions. Wefurther acknowledge the IARPA QCS program whosesupport for WMW and KY initiated our line of inquiry.Sandia National Laboratories is a multi-program labo-ratory managed and operated by Sandia Corporation, awholly owned subsidiary of Lockheed Martin Corpora-tion, for the U.S. Department of Energys National Nu-clear Security Administration under contract DE-AC04-94AL85000. The work at the University of Maryland issupported by LPS-CMTC and IARPA. [1] W. M. Witzel, R. de Sousa, and S. 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