Fidelity Decay as an Efficient Indicator of Quantum Chaos

4 Pages·0.18 MB·

by  Joseph Emerson

Fidelity Decay as an Efficient Indicator of Quantum Chaos
Joseph Emerson,1,∗ Yaakov S. Weinstein,1 Seth Lloyd,2 and D.G. Cory1
1Department of Nuclear Engineering
2Department of Mechanical Engineering
Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139
(Dated: February 1, 2008)
Recentworkhasconnectedthetypeoffidelitydecayinperturbedquantummodelstothepresence
of chaos in the associated classical models. We demonstrate that a system’s rate of fidelity decay
underrepeatedperturbationsmaybemeasuredefficientlyonaquantuminformationprocessor,and
analyze the conditions under which this indicator is a reliable probe of quantum chaos and related
statistical propertiesoftheunperturbedsystem. Thetypeandrateofthedecayarenotdependent
on the eigenvalue statistics of the unperturbed system, but depend on the system’s eigenvector
3 statisticsintheeigenbasisoftheperturbationoperator. Forrandomeigenvectorstatisticsthedecay
0 is exponential with a rate fixed precisely by the variance of the perturbation’s energy spectrum.
0 Hence,evenclassically regularmodelscanexhibitanexponentialfidelitydecayundergenericquan-
2 tumperturbations. Theseresultsclarifywhichperturbationscandistinguishclassically regularand
chaotic quantumsystems.
n
a
PACSnumbers: 05.45.Mt,03.67.Lx
J

2

Over the last two decades a great deal of insight has RMT statistics for the system eigenvector components,
3
been achieved regarding the manifestations of chaos and expressedin the eigenbasis of the perturbation operator,
v
9 complexityinquantumsystems. Weareinterestedinthe are sufficient to produce the characteristic exponential
9 problem of identifying such signatures in the context of decay. These observations are checked in the case of a
0 quantumsimulationonaquantuminformationprocessor dynamical model, where we demonstrate that an expo-
7
(QIP).ItisknownthataQIPenablesefficientsimulation nential fidelity decay can arise not only for a classically
0
ofthedynamicsofawideclassofquantumsystems[1,2]. chaoticsystem,butalsoforclassicallyregularsystemun-
2
0 In the case of quantum chaos models, quantum simula- deragenericchoiceofperturbation. Ouranalysisimplies
/ tion provides an exponential speedup over direct classi- specificrestrictionsthatmustbeimposedonthechoiceof
h
cal simulation [3, 4, 5]. Recently, the quantum baker’s applied perturbation operator in order to extract useful
p
- maphasbeenimplementedusinganuclearmagneticres- information about the statistical signatures of quantum
nt onanceQIP[6]. Thesedevelopmentshighlighttheimpor- chaos in the unperturbed system.
a tance of devising efficient QIP methods for the measure- The aim of this Letter is to characterize certain static
u mentofquantumchaossignaturesandrelatedproperties properties of a unitary map, U, by observing the rate
q
of complex quantum systems. of divergence between a fiducial Hilbert space vector
:
v evolved under this map, |ψ (n)i = Un|ψ(0)i, and the
Perhaps the most established signature of quantum u
i
X chaos is given by the (nearly) universal correspondence same initial vector evolved under this map but sub-
ject also to a sequence of small perturbations |ψ (n)i =
r between the eigenvalue [7] and eigenvector [8] statistics p
a (U U)n|ψ(0)i, where U = exp(−iVδ) is some unspeci-
of quantized classically chaotic systems and those of the p p
fied perturbation operator and n denotes the number of
canonical ensembles of random matrix theory (RMT).
iterations. The fidelity,
Unfortunately direct detection of these spectral signa-
tures is algorithmically inefficient by any known tech-
O(n)=|hψ (n)|ψ (n)i|2, (1)
nique. However, following the original observation of u p
Peres [9], some recent work has demonstrated that, un-
provides a natural indicator of this divergence. The
der sufficiently strong perturbation, the fidelity exhibits
value of O(n) may be determined by an efficient al-
acharacteristicexponentialinthecaseofclassicalchaotic
gorithm on a QIP as follows. We start by preparing
systems[10,11,12,13]. Below,wedemonstratethatthis
charecteristicdecaymaybemeasuredbyanefficientalgo- the fiducial state |ψ(0)i = Uo|0i⊗nq, where nq is the
number of qubits required to span the system’s Hilbert
rithim, and analyze in detail how the observationof this
space N = 2nq. As shown below, the choice of fidu-
decay may be applied as an indicator of canonical RMT
cial state is not critical and the computational basis
statistics (quantum chaos) in the unperturbed system.
states, which are simplest to implement, are a conve-
By first considering RMT models we show that nientset. After applyingthe sequence(U†)n(U U)n,the
p
Wigner-Dysonfluctuationsinthesystemeigenvaluespec- system register contains a final state |ψ(n)i. The circuit
trum are not necessary to produce this characteristic implementation of U requires only Poly(n ) operations
q
decay. More importantly, we find that the canonical for the simulation of a wide class of quantum systems
2
to arbitrary accuracy [1, 2, 3, 4, 5]. Here we observe AsafirsttestoftheLDOS/FGRframeworkinthecase
that |hψ (n)|ψ (n)i| = |hψ(n)|ψ(0)i| = |hψ |0i|, where of the qubit perturbation (5) we evaluate the fidelity de-
p u f
the state |ψ i is obtained by time-reversing the initial cayforamapU =U drawnfromthecircularunitary
f CUE
state preparation |ψ i = U†|ψ(n)i. The magnitude of ensemble (CUE). These matrices form an established
f o
O(n) is then determined from sampling the population model for classically fully chaotic time-periodic systems
of the state |0i. The entire algorithm therefore scales as (without additionalsymmetries) since these systems (al-
Poly(n ). most always) exhibit the same characteristic (Wigner-
q
Recently, Jacquod and coworkers [11] observed that Dyson)spectralfluctuationsandeigenvectorstatisticsas
for |ψ(0)i = |v i, an initial eigenstate of U with eigen- the CUE [20]. The randomness of the system eigenvec-
o
phaseφ ,thefidelityrelatestothelocaldensityofstates tors enables a system-independent estimate of the rate
o
(LDOS), η(φ −φ′ )=|hv |v′ i|2,via Fouriertransform, Γ of the FGR decay. Since the components of random
o m o m
eigenstatesaredistributeduniformlyoverthebasisstates
anduncorrelatedwiththedistributionofeigenvalues,the
O(n)=| η(φo−φ′m)exp(−i(φo−φ′m)n)|2. (2) second moment of the matrix elements Vmn may be di-
Xm rectly evaluated,
In the above |vm′ i are eigenstates of the perturbed map Vm2n =λ2/N (6)
U U|v′ i = exp(−iφ′ )|v′ i. These observations locate
p m m m
the decay of O(n) in an existing theoretical framework. assuming λ = 0 and where λ2 = N−1 N λ2 denotes
In the non-perturbative regime σ/∆ >∼ 1, where σ2 = thevarianceoftheeigenvaluesofV. AsParie=s1ulti,therate
δ2V2 denotes a typicaloff-diagonalmatrixelement and oftheFGRdecayisdeterminedbytheeigenvaluesofthe
mn
∆ is the average level spacing, previous work suggests perturbation,
thatwhen the perturbed systemis complex the LDOS is
typically Lorentzian [14, 15], Γ=δ2λ2, (7)
Γ where we have used ∆ = 2π/N. For the qubit pertur-
η(φ −φ′ )∝ (3)
o m (φ −φ′ )2+(Γ/2)2 bation (5) the variance of the eigenvalues has a simple
o m
form,
withwidthΓ=2πσ2/∆determinedbytheFermigolden
rule(FGR).From(2)and(3)oneexpectstheexponential λ2 = 1 nq 2k−nq 2Cnq, (8)
decay, N (cid:18) 2 (cid:19) k
Xk=0
O(n)≃exp(−Γn). (4) wherethe Cnq arebinomialcoefficients. UsingourRMT
k
estimate (7), for n =10, the rate is,
q
The onset of the exponential decay (4) has been con-
firmed recently in a few classically chaotic models [11, Γ=2.50δ2. (9)
12], though the rate is not always given by the golden
rule [10, 16]. However, in the case of integrable U the While a CUE map may be generated on a QIP using
situationis lessclearsince under someperturbationsthe the gate decomposition devised in Ref. [21], for our nu-
LDOS is known to take on a Lorentzian shape [15, 17]. merical study we construct U = UCUE directly from the
Below, we examine which statistical properties of the eigenvectors of a random Hermitian matrix. Since com-
unperturbedsystemleadto the FGRdecay,andhowthis putational basis states are easiest to prepare in the QIP
characteristicdecaydependsonthepropertiesoftheper- setting,weconsiderthefidelitydecayforbothsinglecom-
turbation operator. This approach is motivated by the putational basis states and averages over 50 such states.
contextofQIPsimulation,inwhichtheeigenbasisofthis The behaviour of the fidelity decay for a matrix typical
perturbation may be mapped onto an arbitrary basis of of CUE is displayed in Fig. 1. The three perturbation
the simulated system U [18]. Our choice of perturbation values displayed in the figure are chosen near the onset
eigenvaluestructureismotivatedfromthe perspectiveof ofthenon-perturbativeregime(δ >0.1)anditisevident
quantum control studies. Specifically, we consider that the fidelity decay even for individual computational
basisstatesexhibitsFGRdecay(4)attheexpectedrate.
U =Πnq exp(−iδσj/2), (5) TheFGRdecaypersistsforatime-scaleΓ−1log(N)until
p j=1 z
saturation at a time-average that decreases as 1/N.
where σ is the usualPaulimatrix and(5)therefore cor- WenextconsiderO(n)fortheGaussianunitaryensem-
z
responds to a collective rotation of all the qubits by an ble (GUE) in order to clarify the relationship between
angleδ. Eq.(5)isamodelofcoherentfar-fielderrors[19], the FGR decay and the distinct statistical features of
and for this type of error model a better understanding RMT that represent signatures of quantum chaos. The
of the fidelity decay is a subject of intrinsic interest. GUE consists of Hermitian matrices with independent
3
100 100
1
10−1 10−1 P(s)0.5
0
O(n) O(n) 0 2s 4
10−2 10−2
10−3 10−3
0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80
n n
FIG. 1: Fidelity decay for UCUE averaged over 50 compu- FIG. 2: Fidelity decay for UGUE averaged over 50 computa-
tational basis states (dash lines) is in excellent agreement tional basis states with δ = 0.3 for τ = 0.001 (dashed line),
with the golden rule decay (4) and the RMT rate (9) for τ =0.01(dottedline)andτ =0.1and100(chainlines),com-
δ = (0.1,0.2,0.4) (solid lines top to bottom). Chain lines pared to the FGR/RMT prediction (solid line). Inset: Spac-
show the fidelity decay for two typical computational basis ingdistributionforUGUE forτ =100comparedtoPoissonian
states in thecase δ=0.1. (solid line) and Wigner-Dyson (dashed line) distributions.
elements drawn randomly with respect to the unique kicked top, which is an exemplary dynamical model of
unitarily-invariantmeasure[20]. GUEformstherelevant quantum chaos [20, 22]. The kicked top is a unitary
RMTmodelfortheimportantclassofchaoticorcomplex map UQKT = exp(−iπJy/2)exp(−ikJz2/j) acting on the
autonomous Hamiltonian systems that are unrestricted Hilbert space of dimension N = 2j +1 associated with
byanyadditionalsymmetries. Wemayexaminethe sen- an irreducible representation of the angular momentum
sitivity to perturbations for the GUE by constructing operator J~. In previous fidelity decay and LDOS stud-
the unitary operator U = exp(−iH τ), where τ ies the choice of perturbation has usually been tied to a
GUE GUE
is a time-delay between perturbations. We consider the physicalcoordinate of the system U. We firstfollow this
same perturbation as for the CUE case. For sufficiently conventionandidentifytheeigenbasisoftheperturbation
smallτ thepropagatorapproachesidentityandtheover- (5) with the eigenbasis |mji of the system coordinate Jz
lap decay is dominated by the perturbation operator, (where mj = {j,...,−j}). In Fig. 3 we compare the fi-
O(n) = |hexp(−iδV)i|2 +O(δ2τ2n2). This behaviour is delity decay for the chaotic and regular regimes of the
demonstrated in Fig. 2 for τ = 0.001 and τ = 0.01 and kickedtop for averagesover 50 initial computational ba-
with δ = 0.3. For larger values of τ the fidelity decay sisstates. The fidelity decayforthe chaotictop(k =12)
under U obeys the FGR with the RMT rate (9). iswelldescribedbytheFGRprediction(4)andtheRMT
GUE
rate (9), whereas the regular top (k =1) shows a slower
The importantpoint is that for sufficiently largeτ the
non-exponentialdecayrate. Similarly,ifweassociatethe
eigenphasesof U become spreadpseudo-randomlyin
GUE
perturbation eigenbasis with the basis of the J coordi-
the interval [0,2π). Under these conditions the eigen- y
nate, the fidelity decay for the chaotic top remains in
phases of the map U exhibit the Poissonian spec-
GUE
agreementwith the RMT rate and the regulartop again
tralfluctuationsthatarecharacteristicofclassicallyinte-
exhibits non-exponential decay, though in this case with
grable (time-periodic) systems. We checked the nearest-
a faster decay than the RMT rate (9). However,we now
neighbor spacing distribution of U and found that
GUE
demonstratethat anexponentialdecay atthe RMT rate
forτ =100the statisticsarein excellentagreementwith
arisesevenfortheregularkickedtopwhenthequbitper-
the Poissonian distribution P(s) ∝ exp(−s) (see inset
turbation(5)isdiagonalinagenericbasisrelativetothe
to Fig. 2). However, the eigenvectors of U are ran-
GUE
eigenbasis of U . Specifically, we leave the perturba-
dom(byconstruction)andindependentofτ (forfiniteτ). QKT
tion eigenvalue spectrum unchanged but set
From these observations it is clear that the presence of
Wigner-Dyson spectral fluctuations in the implemented U =T Π exp(−iδσj/2) T−1, (10)
U, which comprises the only basis-independent criterion p j z
(cid:2) (cid:3)
ofquantumchaos,isnotactuallynecessaryfor the onset where T is drawn from CUE. As demonstrated in the
of exponential (FGR) decay at the rate (7). This sug- insettoFig.3,underthistypeofperturbationthefidelity
gests that it is the RMT statistics of the eigenvectors of decay for the regular top is indistinguishable from that
U that lead to the FGR decay with the RMT rate. ofthechaotictopandisveryaccuratelydescribedbythe
We next consider the fidelity decay for the quantum FGR at the RMT rate (9).
4
sitivelyonthe RMT statistics ofthe systemeigenvectors
100 100
inthe eigenbasisofthe appliedperturbation. Hence, the
fidelity decayfor both classicallyregularandchaotic dy-
10−2
namicalsystemsisgivenbytheFGRunderallbutasmall
10−1 subset of unitary perturbation operators. In the case
50 100 150 200 250 300
of classical models, we conclude that the fidelity decay
n)
O( providesa reliableindicatorofRMT statistics (quantum
chaos)in the unperturbed system only when the applied
10−2
perturbation is restricted to the subset of perturbations
that commute with a classical coordinate.
We are grateful to K. Zyczkowski for suggesting the
10−3
relevance of the eigenvector statistics, Ph. Jacquod,
0 50 100 150 200 250 300 E.M. Fortunato, C. Ramanathan, and T.F. Havel for
n
helpful discussions, and DARPA and the NSF for finan-
FIG. 3: Decay of O(n) for the kicked top in chaotic regime cial support.
(k = 12) averaged over 50 computational basis states, with
the perturbation eigenbasis mapped to the eigenbases of Jz
(dashlines)andJy (chainlines) comparedtotheFGRdecay
(solid lines) at the RMT rate (9) for δ = (0.1,0.3) (top to
bottom). Lines with circles and squares are for the regular ∗ Corresponding author: [email protected]
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