Reconstructing the ideal results of a perturbed analog quantum simulator

0.56 MB·

by  Iris Schwenk

Reconstructing the ideal results of a perturbed analog quantum simulator
Iris Schwenk,1 Jan-Michael Reiner,1 Sebastian Zanker,1 Lin Tian,2 Juha Leppäkangas,1 and Michael Marthaler1
1Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
2School of Natural Sciences, University of California, Merced, California 95343, USA
(Dated: February 24, 2017)
Well controlled quantum systems can potentially be used as quantum simulators. However, a
quantum simulator is inevitably perturbed by coupling to additional degrees of freedom. This con-
stitutes a major roadblock to useful quantum simulations. So far there are only limited means
to understand the effect of perturbation on the results of quantum simulation. Here, we present
a method which, in certain circumstances, allows for the reconstruction of the ideal result from
measurements on a perturbed quantum simulator. We consider extracting the value of correlator
(cid:104)Oˆi(t)Oˆj(0)(cid:105)fromthesimulatedsystem,whereOˆi aretheoperatorswhichcouplethesystemtoits
7 environment. Theidealcorrelatorcanbestraightforwardlyreconstructedbyusingstatisticalknowl-
1 edgeoftheenvironment,ifanyn-timecorrelatorofoperatorsOˆi oftheidealsystemcanbewritten
0 as products of two-time correlators. We give an approach to verify the validity of this assumption
2 experimentally by additional measurements on the perturbed quantum simulator. The proposed
method can allow for reliable quantum simulations with systems subjected to environmental noise
b
without adding an overhead to the quantum system.
e
F
PACSnumbers: 03.67.Pp,03.67.Lx,85.25.Cp
3
2
I. INTRODUCTION AND CENTRAL RESULTS measurement to extract a time-ordered correlation func-
]
tion (Green’s function),
h
p Today we posses in principle the full knowledge to de-
t- scribe all processes of interest in a wide range of fields, iGS0(t) = (cid:104)TOˆ(t)Oˆ(0)(cid:105)0 (1)
n
such as chemistry, biology and solid state physics. In = (cid:104)0|TeiHStOˆe−iHStOˆ|0(cid:105) ,
a
all these fields a truly microscopic description is possible
u
q using quantum mechanics. However, it is also well un- where T is the time-ordering operator. The index S0 in-
[ derstood that in practice full quantum mechanical sim- dicatesthatweareconsideringtheidealGreen’sfunction
ulations of even modestly-sized systems are impossible.1 oftheunperturbedHamiltonianH ,withoutcouplingto
2 S
v To efficiently study quantum problems, we need to use additionaldegreesoffreedom,and|0(cid:105)isthegroundstate
3 other,wellcontrolledquantummechanicalsystems.2–5 In of H (zero-temperature limit). We start our analysis
S
8 recent years unprecedented direct control over quantum fromthissimpleexampleandlaterinSec.IVextendthe
6 systems has been achieved.6–10 Precise experiments in theory to multiple operators Oˆi, and to finite tempera-
2 the quantum regime have been performed using atomic tures. We consider time-ordered Green’s functions, since
0 systems,11–13 superconducting qubits,14–18 photonic cir- these are in general connected to numerous quantities of
1. cuits,19–21 and nuclear spins.22,23 Larger systems have interestinexperiments,suchasheatorelectrictransport
0 been demonstrated using trapped ions24,25 and the equi- coefficients. There are several proposals which describe
7 libration of interacting bosons has been studied in cold methods to measure the relevant correlators in the con-
1 gases.26,27 textofanalogquantumsimulation.44–47Thus,weassume
:
v Apromisingapproachtounderstandquantumsystems that Green’s functions play a central role in extracting
Xi isanalogquantumsimulation,28 wherethegoalistocre- results from a quantum simulator.
ateanartificialsystemwithaHamiltonianthatisequiv- However, if we want to use measurements on a quan-
r
a alent to the system we intend to study. Apart from tum simulator to study the properties of an ideal Hamil-
quantum simulations using cold gases29,30 and trapped tonian, the key challenge remains: What is the role of
ions,31,32 there are many proposals for analog quantum errors and imperfections of the artificial system in a real
simulationwithsuperconductingcircuits,33–36 exploiting measurement?5,48–50 Usually we quantify the influence
the controllability of superconducting systems, which in of external degrees of freedom by comparing measure-
principle allows the creation of a large class of Hamilto- ments to theoretical predictions. However, by definition,
nians. While most current superconducting systems are for quantum simulation it should not be possible to pre-
relatively small,37,38 larger networks of superconducting dicttheresult;neitheranalyticallynornumericallyusing
non-linear elements are now being explored.39–41 Other classical computers. The approach we introduce in this
architectures for analog quantum simulation have also paper is based on connecting the ideal Green’s function,
been investigated.42,43 Eq. (1), to the perturbed Green’s function we measure
In this article, we study an analog quantum simula- using a quantum simulator. We consider Green’s func-
tor with the ideal Hamiltonian H . To understand the tionswhereOˆ isalsotheoperatorbywhichthequantum
S
propertiesofthesimulatedsystem,wewouldliketousea simulatorcouplestoadditionaldegreesoffreedom(which
2
cause the errors). This restricts the generality of the ap-
proach, but in reality it is actually very likely that the
same mechanism which connects the system to its bath
alsoallowsforthereadoutofthesystem. Soitisreason-
Perturbed Simulator Bath
able to assume that this is one of the Green’s functions
simulator
to which we have an easy access in experiments.
G (t) G (t) G (t)
SB S0 B0
We show that under specific conditions it is in fact
possible to extract the ideal correlator of the operator Oˆ
evenfromaperturbedsystem. Oneingredientinourap-
proach is a good statistical knowledge of the additional Thermal environment
degrees of freedom which act on H . This assumption is
S
justified,forexample,foraquantumsimulatorbuildfrom
Figure1: Thequantumsimulatoriscoupledtoaperturbative
tunable qubits, where qubits can be decoupled and the
bath. The simulator-bath system is coupled weakly to an
propertiesofthebathsofindividualqubitscanbeprobed
environment that establishes thermal equilibrium. For each
byestablishedspectroscopicalmethods. Apartfromthis, sub-component of the system we define a free correlator: the
only one assumption is necessary about the properties of ideal correlator of the simulator iG (t) = (cid:104)TOˆ(t)Oˆ(0)(cid:105) as
S0 0
the ideal correlators. We need that any n-time correla- definedinEq.(1)andthefreecorrelatorofthebathiG (t)=
B0
tion function can be expressed as products of two-time (cid:104)TXˆ(t)Xˆ(0)(cid:105) . The full correlator iG (t) = (cid:104)TOˆ(t)Oˆ(0)(cid:105)
0 SB
correlationfunctions. Thisconditionwillbediscussedin accounts for the coupling in the full Hamiltonian, Eq. (2).
moredetailinSec.IA.Inthepresentpaper, wedescribe
this method assuming Oˆ and the additional degrees of
freedomarebosonic,butthemethodcanalsobedirectly be used to expand the full Green’s function GSB(ω)
transfered to fermionic operators Oˆ and fermionic baths. in terms of the ideal Green’s functions GS0(ω) and
G (ω).54 But to apply these techniques there is one
B0
key assumption that is absolutely crucial: Wick’s the-
orem needs to apply in some form. Using this theorem
A. Principal idea
it is possible to connect a single correlator of 2n opera-
tors with n two-time correlators. Wick’s theorem for the
We start by presenting a simple example of our ap- system operator Oˆ takes the form
proach,whereweshowhowtoextracttheidealproperties
from an imperfect simulator in equilibrium. In Sec. IV, (cid:104)TOˆ(t )Oˆ(t )...Oˆ(t )Oˆ(t )(cid:105) (3)
1 2 n−1 n 0
we extend this result to more general situations.
=(cid:104)TOˆ(t )Oˆ(t )(cid:105) (cid:104)TOˆ(t )...Oˆ(t )Oˆ(t )(cid:105)
The full system we consider can be described by the 1 2 0 3 n−1 n 0
Hamiltonian, +(cid:104)TOˆ(t1)Oˆ(t3)(cid:105)0(cid:104)TOˆ(t2)...Oˆ(tn−1)Oˆ(tn)(cid:105)0
+...+(cid:104)TOˆ(t )Oˆ(t )(cid:105) (cid:104)TOˆ(t )...Oˆ(t )(cid:105) .
H =H +H +H , H =OˆXˆ . (2) 1 n 0 2 n−1 0
S C B C
This relation can be applied repeatedly until only two-
Here the ideal Hamiltonian of the simulator HS is cou- time correlators remain. For the bath operator Xˆ it is
pled via the Hamiltonian H to the additional degrees natural to assume that Wick’s theorem applies, in ac-
C
of freedom contained in the bath Hamiltonian H . The cordancewithnumeroussystem-bathdescriptions. How-
B
system operator in HC is Oˆ, which is the same as what ever,forthesystemoperatorOˆ thisisnotingeneraltrue.
we used to define the ideal correlator in Eq. (1), and the A well known case, where Eq. (3) holds is if the system
bath operator is Xˆ. H canbedescribedasasystemofnon-interactingquasi-
S
The bath can usually be described by a set of bosonic particles and Oˆ can be written as a linear combination
modesandweassumethatthefreecorrelatorofthebath oftheannihilationandcreationoperatorsofthesequasi-
G (t) is known, for example, from spectroscopic mea- particles. More generally, Eq. (3) is valid if the fluctua-
B0
surements. ForthedefinitionofallrelevantGreen’sfunc- tions of Oˆ(t) have a Gaussian distribution. The expan-
tionsseeFig.1anditscaption. InSec.IV,wegiveamore sion of n-time correlators in pair and higher correlators
precise definition. hasbeenstudiedextensivelyforspinsystems55–57andde-
The total system described by H is in thermal equi- viationsfromGaussianstatisticshavebeenstudiedinthe
librium. It should be emphasized that if coupling to the field of full-counting statistics.58–61 From relatively gen-
thermal bath is not infinitely weak it cannot be assumed eral considerations, such as the central limit theorem,62
that the only result of this coupling is the creation of we expect that fluctuations become more Gaussian as
equilibrium.51–53 In the main part of this paper we focus the system size increases, which is also the most inter-
on the situation at zero temperature and in Sec. IVD esting limit for a quantum simulator. However, in some
extend our method to finite temperatures. systems non-Gaussian fluctuations are known to persist
We want to connect the spectral function of the bath even at large system size63,64 or become even size inde-
to the properties of the perturbed quantum simulator. pendent.65 Therefore, in Sec. II we discuss, how Eq. (3)
Standard many-body physics techniques exist which can can be checked, to some extend, by making appropriate
3
measurements on the perturbed quantum simulator. With(cid:104)...(cid:105) and(cid:104)...(cid:105),werefertocorrelatorsforwhichwe
0
Assuming Eq. (3) holds, we find an exact relation be- assume Wick’s theorem to be exactly valid. The index 0
tween the Green’s functions, indicates that the system is considered without pertur-
bation by the bath. In contrast to this, (cid:104)...(cid:105) ((cid:104)...(cid:105) )
F 0,F
GSB(ω)=GS0(ω)+GS0(ω)GB0(ω)GSB(ω). (4) describethe(un)perturbedcorrelatorsincludingthecor-
rections to Wick’s theorem. In this paper we consider
This is the well-known Dyson equation that defines the
corrections up to first order in G .
4
total Green’s function as a function of the free system
Withmeasurementsonthequantumsimulatorwehave
and bath Green’s functions. access to n-time correlators (cid:104)...(cid:105) of Oˆ. Measuring two-
F
and four-time correlators,
(cid:88)
B. Central result (cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) − (cid:104)TOˆ Oˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) = , (7)
1 2 3 4 F a b F c d F
3perm.
From Eq. (4) we see that the perturbed quantum sim- a,b,c,d
∈{1,2,3,4}
ulator can be used to find the correlator of the unper-
we get access to the quantity,
turbed simulator G (ω) as long as we know the free
S0
Green’s function of the bath G (ω), since
B0 = + +
G (ω)
G (ω)= SB . (5) + +... (8)
S0
1+G (ω)G (ω)
B0 SB
+ +... , (9)
This states the central idea of this paper in the simplest
form. To derive Eq. (5) we use an important assump-
wherethethincrossrepresentsthecorrectionG andthe
4
tion: that Wick’s theorem in the form in Eq. (3) applies
sinuous lines stand for the bath correlation function (see
for the system operator Oˆ. This condition will be dis- table I). The central result here is that the correction to
cussed in more detail in Sec. II, where we also show how the perturbed two-time correlator can be expressed as
to extract the lowest order correction to this result from
the perturbed simulator. Apart from this the quality of (cid:104)TOˆ Oˆ (cid:105) =(cid:104)TOˆ Oˆ (cid:105)+ . (10)
1 2 F 1 2
the reconstruction is also restricted by the precision of
the knowledge of the correlators, which is the subject Eqs. (7) and (10) show that it is possible to estimate the
of Sec. III. In particular, we presume that the proper- deviation from Wick’s theorem by measuring the two-
ties of the bath are measured independently of the sys- and four-time correlators and combining the measured
tem, which will be discussed more detailed in Sec. IV. In result with our knowledge of the bath correlator. This
Sec. IV, we also consider the case where multiple baths allows to check whether the assumption of Wick’s the-
couple to system via operators Oˆi and extend the recon- orem is justified and the result of the reconstruction is
struction method to finite temperatures. reliable.
II. VERIFYING WICK’S THEOREM III. IMPERFECT KNOWLEDGE
The validity of Wick’s theorem for the system oper- A fundamental prerequisite for the reconstruction of
ator Oˆ is crucial for the derivation of Eq. (5). But for the unperturbed correlator is the knowledge of the per-
non-trivialsystemswecannotingeneralpredictifWick’s turbed correlator of the system GSB and the correlator
theoremholds. Therefore,wedescribeamethodtoverify of the bath GB0. In reality, we will not receive these
the validity of Wick’s theorem using the quantum simu- quantities with full accuracy. In this section, we address
latoritself. AdetailedderivationisgiveninAppendixB. the question how imperfect knowledge affects the recon-
We introduce the lowest-order correction to Wick’s struction of the ideal Green’s function.
theorem G (t ,t ,t ,t ):
4 1 2 3 4
G (t ,t ,t ,t ) A. Bath correlator
4 1 2 3 4
=(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) −(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) (6)
1 2 3 4 0,F 1 2 3 4 0 We consider a variation of the Green’s function of the
(cid:88)
=(cid:104)TOˆ1Oˆ2Oˆ3Oˆ4(cid:105)0,F − (cid:104)TOˆaOˆb(cid:105)0(cid:104)TOˆcOˆd(cid:105)0 , bath GB0(ω)+δGB0(ω). With this Green’s function, we
reconstruct the correlator of the simulator using Eq. (5)
3perm.
a,b,c,d with
∈{1,2,3,4}
G (ω)
G˜ (ω)= SB .
where we make use of the abbreviation Oˆ =Oˆ(t ). The S0 1+G (ω)G (ω)+δG (ω)G (ω)
i i B0 SB B0 SB
summation runs over all indistinguishable permutations. (11)
4
For |δG (ω)|(cid:28)|G−1(ω)+G (ω)|, we find of the form
B0 SB B0
G˜ (ω)≈G (ω)[1−G (ω)δG (ω)]. (12) (cid:88)N
S0 S0 S0 B0 H = λ OˆiXˆi. (16)
C B
Hence, the impact of δG (ω) is large at the peaks of i=1
B0
G (ω). The influence of δG (ω) is independent of the
S0 B0 The system and bath variables satisfy the commutation
value of GB0(ω). This means that the quality of the relation [Xˆi,H ] = [Oˆi,H ] = [Xˆi,Xˆj] = 0. We have
reconstruction is defined by the absolute error δG (ω) S B
B0 nowN systemoperatorsOˆiwhichcouplethesystemtoN
only.
bathsviathecorrespondingbathoperatorsXˆi. Wehave
introducedthedimensionlessconstantλ ∈{0,1},which
B
allows us to define the free and perturbed correlators in
B. Full system correlator
a more rigorous way (see Table I).
To perform the reconstruction of the unperturbed
Fora deviation ofthe full system correlator G (ω)+
SB Green’s function of the system, we need to character-
δG (ω), we have
SB ize the properties of the baths independently of the sys-
tem.66 This assumption is justified, for example, for a
G (ω)+δG (ω)
G˜S0(ω)= SB SB . large network of superconducting flux qubits coupled in
1+G (ω)G (ω)+G (ω)δG (ω)
B0 SB B0 SB a 2D-structure to simulate a spin system. Such sys-
(13)
tems have been realized with up to 1000 qubits.40,41
For |δG (ω)|(cid:28)|G−1(ω)+G (ω)|, we find
SB B0 SB The ideal Hamiltonian in this case would be, e.g., H =
S
G˜S0(ω)≈GS0(ω)(cid:18)1+ GS0(ω) δGSB(ω)(cid:19) . (14) p21a(cid:80)raimheitσexirs+w(cid:80)hiicjhJidjσefiziσnzej.tHheermeohdiealnudnJdiejrairneveasdtjiugsattaiobnle,
G (ω) G (ω)
SB SB and σi are the Pauli matrices acting on qubit i. The
k
The ratio of G (ω) and G (ω) implies that the vari- qubits are coupled to individual baths, whose bath cor-
S0 SB
ation of the full system correlator GS0(ω) is large at relators (cid:104)Xˆi(t)Xˆi(0)(cid:105)0 are known relatively well, as esti-
the peaks of this function. In contrast to the varia- mated in Ref. [68]. From a multitude of similar experi-
tion of the bath correlator in Eq. (12), the relative error ments we know that the system operator that couples to
δG (ω)/G (ω) enters here. thebathcorrespondstoOˆi =σi. Thus, forsuchaquan-
SB SB z
In addition, this equation shows the limit of our re- tumsimulatorthecharacterizationofthebathcorrelator
construction method. Consider the limit of large cou- is possible independently of the properties of the simu-
pling of the bath to the system. Eq. (5) is still valid. lator. Furthermore, the applicability of Wick’s theorem
Butareconstructionisnotanymorepossible, ifthebath hasbeenstudiedbroadly55–57 incontextofspinsystems.
widens the peaks of G (ω) significantly. In this case Devices such as large networks of superconducting flux
S0
G (ω)/G (ω) (cid:29) 1 at the peaks. Therefore even a qubits coupled in a 2D-structure can also be tuned into
S0 SB
smallrelativeerrorinthemeasurementofG (ω)makes alternative regimes, e.g., into a weakly nonlinear regime
SB
the reconstruction of G (ω) practically impossible. where proposals exist on how to use such devices for the
S0
simulationofvibronictransitions.67 Inthislimit,theap-
plicationofWick’stheoremwouldalsobemorestraight-
IV. FULL MODEL AND DISCUSSION forward.
A. Extended Model
B. The full Green’s function
In this section, we extend the model to a more gen-
eral scenario and discuss the derivation of our results in In Eq. (1), we introduced the Green’s function of the
detail. To make our model more realistic, we consider system without coupling to external degrees of freedom.
multiple baths. In practice, a system consisting of N In this section, we consider the Green’s function GSB of
coupled qubits or resonators arranged in a certain two- the system coupled to its bath in matrix form with the
dimensional geometry, does not couple to a single bath. elements
Instead, weconsiderasystemwithmultipleindependent
bathsH =(cid:80)N Hi with[Hi ,Hj]=0andasimilarly GiSjB(t)=−i(cid:104)TOˆi(t)Oˆj(0)(cid:105) , (17)
B i=1 B B B
adjusted coupling term. The full Hamiltonian can now where(cid:104)...(cid:105)isanexpectationvalueofthegroundstateof
be written in the form thefullsystem. UsingthestandardtechniqueforGreen’s
functions at zero temperature, we expand Gij (t) in or-
H =HS +HC +(cid:88)N HBi . (15) dgievresnobfyHHC. =ThHeref+or(cid:80)e, tHheiz.erWotehdoerfidneertHhaemStBiimltoeneivaonluis-
i=1 0 S i B
tion,
The coupling H between the system and the additional
C
degrees of freedom contained in (cid:80) Hi is assumed to be S (t)=e−iHt, (18)
i B λB
5
and transform all operators Aˆ into the appropriate pic- D. Extension to finite temperatures
ture using the definition
The diagrammatic expansion in Sec. IVC can also be
Aˆ(t)=S−1(t)AˆS (t). (19)
λB λB appliedtotheMatsubaraGreen’sfunctionsGM,X,which
areconnectedtotheretardedGreen’sfunctionsforfinite
For unperturbed correlators (cid:104)...(cid:105)0 this transformation temperaturesGR. Thisisawaytoextendthismethodto
with SλB=0(t) = e−iH0t defines operators in the in- systems in thermXal equilibrium. The analogue of Eq. (5)
teraction picture. While λB = 1 denotes the full time for finite temperatures is given by
evolutionintheHeisenbergpicturefortheperturbedcor-
relators (cid:104)...(cid:105). The full Green’s function can be written GR (iω )
in the form GSR0(iωn)= 1+GR (SiBω )GnR (iω ). (26)
B0 n SB n
(cid:104)TS(∞)Oˆi(t)Oˆj(0)(cid:105)
Gij (t)=−i 0 , (20) BelowweintroducetheMatsubaraGreen’sfunctionsand
SB (cid:104)TS(∞)(cid:105)0 explain the connection to the spectral function.
with the time evolution operator
S(∞)=Te−i(cid:82)−∞∞dtHC(t), (21) 1. Expansion in imaginary time
Asweconsiderthewholesystemtobeinthermalequi-
where we use the coupling Hamiltonian in the interac-
librium, it is reasonable to use the standard Matsubara
tion picture. We introduce the Fourier transform of the
Green’sfunctionmethod. Therefore,wedefinetheimag-
Green’s function
inary time τ = it where we require 0 < τ < β. The
(cid:90)∞ Matsubara Green’s function equivalent to Eq. (17) is
Gij(ω)= dteiωtGij(t). (22)
X X
Gij (τ)=−(cid:104)TOˆi(τ)Oˆj(0)(cid:105), (27)
−∞ M,SB
where T is the time-ordering operator for τ. In the case
of finite temperatures, (cid:104)...(cid:105) refers to the equilibrium ex-
C. Diagrammatic expansion
pectationvalueTr(1e−βH...),withZ =Tr(e−βH). The
Z
time evolution in imaginary time is given by
We show now the diagrammatic expansion that leads
to expressions such as Eq. (4) if Wick’s theorem is valid U (τ)=e−Hτ. (28)
for the coupling operators. All relevant correlators and λB
their diagrammatic representations are shown in Table I
WetransformalloperatorsAˆintotheappropriatepicture
and the interaction term is shown in Table II.
in imaginary time using the definition
Using an expansion of S(∞) in H we can directly
C
show the connection between the Green’s function of the Aˆ(τ)=U−1(τ)AˆU (τ). (29)
simulatorperturbedbyabathGij andtheunperturbed λB λB
SB
ideal Green’s functions,
The full correlator can be written in the form
= + + +... (cid:104)TU(β)Oˆi(τ)Oˆj(0)(cid:105)
G (τ)=− 0 , (30)
= + ( + ...) M,SB (cid:104)TU(β)(cid:105)
0
= + . (23)
with evolution operator
Here, all disconnected diagrams are canceled by the vac-
uumdiagramsin(cid:104)TS(∞)(cid:105) (seeAppendixA).Therefore U(τ)=Te−(cid:82)0τdτ(cid:48)HC,I(τ(cid:48)). (31)
0
we can write the Dyson equation in matrix form as
As for zero temperature, all disconnected diagrams are
G (ω)=G (ω)+G (ω)G (ω)G (ω). (24) canceled by the factor (cid:104)TU(β)(cid:105) , the so-called vacuum
SB S0 S0 B0 SB 0
diagrams.
If all Green’s functions Gij (ω) and Gij (ω) are known, The correlator in imaginary time is periodic in τ with
SB B0
this equation can be solved for G : period β. It is convenient to transform it to frequency
S0
space using the discrete Fourier transform,
G (ω)=G (ω)[1+G (ω)G (ω)]−1 . (25)
S0 SB B0 SB
1 (cid:88)
Gij (τ)= Gij (ω )e−iωnτ (32)
Ifwereducethesystemtoasingle-bathsituation,thisre- M,X β M,X n
n
sulttransformstoEq.(5). Itconnectstheidealcorrelator
in Eq. (1) to quantities which can be readily measured. with the Matsubara frequencies ω =2πn/β.
n
6
Green’s function Matrix form Diagram Definition
Gij (t)=−i(cid:104)TOˆi(t)Oˆj(0)(cid:105) [G ] =Gij Full correlator of the system operators, including the effects of
SB SB ij SB
the bath. (λ =1)
B
Gij (t)=−i(cid:104)TOˆi(t)Oˆj(0)(cid:105) [G ] =Gij Free correlator of the system operators, without the effects of
S0 0 S0 ij S0
the bath. (λ =0)
B
Gij (t)=−i(cid:104)TXˆi(t)Xˆj(0)(cid:105) [G ] =Gij Free correlator of the bath, without the effects of the system.
B0 0 B0 ij B0
(λ =0)
B
Table I: Summary of all relevant correlators and their diagrammatic representation.
2. Connecting a real time correlator to the Matsubara Describing the Matsubara Green’s function in terms of
Green’s function the spectral function shows a connection to the retarded
Green’s function for finite temperatures GR,
P
Now we discuss the connection of the Matsubara
Gij (ω )=GR,ij(iω ), ω >0, (38)
Green’s function to measurable quantities like the spec- M,P n P n n
tral function or correlators. As an example we focus on with P ∈ {B0,S0,SB}. This requires an analytic con-
the Green’s function of the bath. tinuation of GR in the complex plane. Via the spectral
P
We define the correlation function function we can derive the Kramers-Kronig relation,
(cid:16) (cid:17)
Ci(t)= (cid:104)Xˆi(t)Xˆi(0)(cid:105) −(cid:104)Xˆi(0)Xˆi(t)(cid:105) θ(t). (33) Gij(ω)=ReGR,ij(ω)+i(1+2n¯(ω))ImGR,ij(ω), (39)
0 0 P P P
with n¯(ω) = (eβω − 1)−1. Starting from the Matsub-
The eigenstates of the bath are given by |n(cid:105), with
araGreen’sfunctionweobtaininformationaboutthere-
H |n(cid:105) = E |n(cid:105). This allows us to rewrite the correla-
B n tardedGreen’sfunctionatthepointsiω . Wewouldlike
tor, n
to have the ideal Green’s function GR , i.e. the spectral
S0
θ(t)(cid:88) function,forthecompleterealaxis. Thiscanbeachieved
Ci(t)= |(cid:104)n|Xˆi|m(cid:105)|2ei(En−Em)t(e−βEn−e−βEm), byusingnumericalmethodslikethePadéapproximation
Z
B nm approach.69,70However,itshouldbeemphasizedthatthe
(34)
numerical transformation of a Green’s function at the
with the partition function ZB = Tr(e−βHB). The real Matsubarafrequenciestotherealaxisisstillanon-trivial
part of the Fourier transform of the correlator gives us
problem and an active research field.71
the spectral function
1 (cid:18)(cid:90) ∞ (cid:19)
Ai(ω) = Re dteiωtCi(t) (35) E. Model system: chain of resonators with
π
−∞ individual baths
1 (cid:88)
= |(cid:104)n|Xˆi|m(cid:105)|2
ZB In this section, we give an explicit example of our
nm
method and particularly of the validity of Eq. (25). We
(e−βEm −e−βEn)δ[ω−(E −E )] .
n m consider a system of coupled harmonic oscillators,
Apart from a factor −1, the imaginary part of the re-
traelradteidonGfurenecnt’isonfuCnci(ttio),nsiGnBRc0e(t) is equivalent to the cor- HS =(cid:88)N (cid:18)21mωr2qj2+ 21mp2j + m2Ω2(qj+1−qj)2(cid:19) ,
j=1
(40)
GRii(t)=−i(cid:104)[Xˆi(t),Xˆi(0)](cid:105) θ(t). (36)
B0 0 where N is the number of resonators, m refers to the
Assuming that Ai(ω) has been measured, the retarded mass,ωr istheeigenfrequencyofanuncoupledresonator,
Green’s function GR (ω) of the bath can be calculated and Ω describes the coupling between neighboring oscil-
B0 lators. We assume periodic boundary conditions. For a
using
system of coupled resonators, Wick’s theorem stated in
(cid:90) ∞ Ai(ω ) Eq. (3) is clearly valid. Here we show the validity of our
GR,ii(ω)= dω 1 . (37)
B0 1ω−ω +i0 previously derived results. We validate our results for
−∞ 1
the connection between the ideal and perturbed correla-
tors by using the quantum regression theorem,72 (QRT).
Interaction Diagram Definition While the system of bare coupled resonators would not
(cid:80)N Oˆ Xˆi Interaction between bath and make for a good quantum simulator, proposals exist for
i=1 i
system. modeling the Bose-Hubbard model using coupled non-
linear resonators.73 Similarly, limiting cases from non-
Table II: Each circle represents a term of the expansion in interactingbosonstohard-corebosonshavebeenstudied
HC. in the context of analog quantum simulation.74
7
Weassumethateachoftheresonatorsiscoupledtoan where n¯k =(eβΩk−1)−1. Assuming the spectral density
individual bosonic bath, of the bath to be smooth, we find the effective rates
(cid:88) (cid:88)
H = OˆjXˆj, H = ω¯(j)b(j)†b(j), (41) 1
C B m m m Γ = J(Ω ), (50)
k k
j j,m 2mΩk
(cid:88)
with Oˆj =q , and Xˆj = t(j)(b(j)†+b(j)).
j m m m where the prefactor (2mω )−1 arises from Oˆi =
m √ r
2mω −1(d† +d ) and ωr is a result of the transition
We assume the baths to be identical, i.e., r j j Ωk
from d†+d to a† +a . In accordance with the assump-
j j k k
ω¯(j) =ω¯ , t(j) =t , (42) tions used for the Lindblad equation, Eq. (37) reduces
m m m m
to
but independent
1
iGR,ij(ω)≈δ sign(ω)Ji(|ω|). (51)
(cid:104)Xˆj1(t )Xˆj2(t )(cid:105) =0 for j (cid:54)=j . (43) B0 ij2
1 2 0 1 2
Diagonalizing the system Hamiltonian results in FortheLindbladequationtobevalid,someassumptions
have to be made about the spectral density of the bath.
(cid:114)
HS =(cid:88)Ωka†kak, with Ωk = (cid:104)2Ωsin(kϕ20)(cid:105)2+ωr2, equWatitiohnthfoerQaRnTa,rbthiteraLriyndobpleardattoerrmAˆsafunldfilalltlhke72following
k
(44)
(cid:104) (cid:105) Γ (cid:104) (cid:105)
where ϕ = 2π. The connection of annihilation and cre- Tr a LAˆ =−(iΩ + k)Tr a Aˆ . (52)
0 N k k 2 k
ation operators of system eigenstates, a and a†, to the
k k
original operators has the form For t>0 we get
(cid:114)
qj = 2m1ω (d†j +dj), (45) (cid:104)Aˆ(t0)ak(t+t0)(cid:105)=e−iΩkte−Γ2kt(cid:104)Aˆ(t0)ak(t0)(cid:105) , (53)
r
dj = 2√1N k(cid:88)N=1(cid:34)e−ikjϕ0(cid:32)(cid:114)ΩωKr −(cid:114)ΩωKr (cid:33)a†k (cid:104)(cid:104)Aaˆk((tt0)+a†kt0(t)A+ˆ(tt00))(cid:105)(cid:105)==ee−+iiΩΩkkttee−−ΓΓ22kktt(cid:104)(cid:104)Aaˆk((tt00))aA†kˆ((tt00))(cid:105)(cid:105) ,, ((5545))
+ eijkϕ0(cid:32)(cid:114) ωr +(cid:114)ΩK(cid:33)a (cid:35) . (46) (cid:104)a†k(t+t0)Aˆ(t0)(cid:105)=e+iΩkte−Γ2kt(cid:104)a†k(t0)Aˆ(t0)(cid:105) . (56)
k
Ω ω
K r
The stationary solution of the Lindblad equation is pro-
We consider finite temperatures. Therefore, the spectral portional to e−β(cid:80)kΩka†kak. Using this we calculate the
density of the bath is given by initial values for (cid:104)a(†)(t )a(†)(t )(cid:105) and find
k 0 k 0
1
Ai(ω)≈ sign(ω)Ji(|ω|), (47) (cid:104)ak(t1)ak(cid:48)(t2)(cid:105)=0, (57)
2π
with Ji(ω)=J(ω)=2π(cid:80) t2 δ(ω−ω¯ ). (cid:104)a†k(t1)ak(cid:48)(t2)(cid:105)=δk,k(cid:48)n¯keiΩk(t1−t2)e−Γ2k|t1−t2|, (58)
To compare Eq. (25) to cmormrelators camlculated using a (cid:104)ak(t1)a†k(cid:48)(t2)(cid:105)=δk,k(cid:48)(n¯k+1)eiΩk(t1−t2)e−Γ2k|t1−t2|,
master-equation approach, we calculate the full Green’s (59)
ftuhnecdtiyonnamGMji1cj,sS2BofuthsiengfutllhseyQstRemT.tToobethaipspernodxiwmeataeslsyudmee- (cid:104)a†k(t1)a†k(cid:48)(t2)(cid:105)=0. (60)
scribed by the Lindblad equation,
A direct calculation of the free correlators results in
ρ˙(t)=Lρ(t), (48)
(cid:104)ak(t1)ak(cid:48)(t2)(cid:105)0 =0, (61)
with the Lindblad terms (cid:104)a†k(t1)ak(cid:48)(t2)(cid:105)0 =δk,k(cid:48)n¯keiΩk(t1−t2), (62)
Lρ=−i[HS,ρ] (cid:104)ak(t1)a†k(cid:48)(t2)(cid:105)0 =δk,k(cid:48)(n¯k+1)eiΩk(t1−t2), (63)
+(cid:88)N Γk(n¯ +1)(cid:16)2a ρa† −a†a ρ−ρa†a (cid:17) (cid:104)a†k(t1)a†k(cid:48)(t2)(cid:105)0 =0. (64)
2 k k k k k k k
k=1 FromthisresultswecalculatetheretardedGreen’sfunc-
+(cid:88)N Γ2kn¯k(cid:16)2a†kρak−aka†kρ−ρaka†k(cid:17) , (49) tfoiornms.GSRW0,ji1tjh2(ta)n,GaSRn,aj1lyj2t(ict)caonndtipneuraftoiromntahnedFoEuqr.ie(r3t8r)anwse-
k=1 finally arrive at the Matsubara Green’s functions for
8
ω >0: are left with
n
N 1
GMj1j,S2O(ωn)=N1 (cid:88)2m1Ω GMj1j,B2O(ωn ≈Ωk)≈δj1,j22J(Ωk), (71)
k
k=1
(cid:104) (cid:105) FromacomparisontoEq.(51),weconcludethatEq.(25)
× e−ik(j1−j2)ϕ0n¯ −eik(j1−j2)ϕ0(n¯ +1)
k k holds for this example. For an ohmic spectral density
(cid:18) (cid:19)
1 1 and with Ω → iω the Matsubara Green’s function of
× − , k
iω +Ω +i0 iω −Ω +i0 the bath coincides with Eq. (51).
n k n k
(65)
V. CONCLUSIONS
N
1 (cid:88) 1
Gj1j2 (ω )=
M,SB n N 2mΩ
k The main result we present in this paper is twofold.
k=1
(cid:104) (cid:105) On the one hand, we introduce a method that can be
× e−ik(j1−j2)ϕ0n¯ −eik(j1−j2)ϕ0(n¯ +1)
k k used to reconstruct certain unperturbed (ideal) Green’s
(cid:32) (cid:33) functions from the perturbed ones, measured by a quan-
1 1
× − . tum simulator coupled to additional degrees of freedom.
iω +Ω +iΓk iω −Ω +iΓk
n k 2 n k 2 To achieve this, we assume that any n-time correlator of
(66) the coupling operator of the ideal system can be written
as a product of two-time correlators. This is known as
TocalculatethebathGreen’sfunctionusingEq.(25)we
Wick’s theorem. On the other hand, we explain how to
introduce the following transformation
verify this assumption by a measurement. Furthermore,
we assume a good knowledge of the bath correlators to
(cid:88)
GMk ,S0(ωn)= GMj1j,S2Oeik(j1−j2)kϕ0 perform the reconstruction. In particular, we presume
j1,j2 that these correlators are measured independently, when
(cid:18) (cid:19)
N 1 1 not coupled to the ideal system. We also clarify how im-
= − ,
2mΩ iω −Ω +i0 iω +Ω +i0 perfect measurements of the bath and of the full corre-
k n k n k
(67) lator affect the reconstruction. For example, in the case
of strong coupling to the bath our result is still valid,
(cid:88)
GMk ,SB(ωn)= GMj1j,S2Beik(j1−j2)kϕ0 but the reconstruction fails even in the presence of small
j1,j2 noise during the measurement.
(cid:32) (cid:33)
N 1 1 Presently, the applicability of analog quantum simula-
= − .tion is severely restricted, since the influence of sources
2mΩk iωn−Ωk+iΓ2k iωn+Ωk+iΓ2k oferrorsisnotwellunderstood. Theapproachpresented
(68)
in this paper leads the way to quantify and even cor-
rect errors in quantum simulation. Since the reconstruc-
With this Eq. (25) results in
tion method is based on classical post processing, this
method helps to make the results of quantum simula-
Gk (ω )=Gk (ω )+Gk (ω )Gk (ω )
M,SB n M,S0 n M,S0 n M,SB n tionreliablewithoutaddinganoverheadtothequantum
×(cid:88)Gjj (ω ) 1 . (69) system. Therefore, the promising potential of quantum
M,BO n N2 simulation, to yield interesting results even using small
j
quantum systems, remains.
In the Lindblad equation we take into account the spec-
tral density of the bath at Ω . Since the bath Green’s
k
functiondependsonthespectraldensityofthebath,the
Acknowledgments
relation is true for ω ≈ Ω . Using the assumption of
n k
identical and independent baths we arrive at
The authors thank Daniel Mendler, Christian Kar-
(cid:18) Γ (cid:19) lewski and Gerd Schön for enlightening discussions. I.
Gj1j2 (ω ≈Ω )≈δ mΓ Ω + k . (70)
M,BO n k j1,j2 k k 4 S. acknowledges financial support by Friedrich-Ebert-
Stiftung. L.T.issupportedbytheNationalScienceFoun-
In the limit of small coupling to the bath, Γ (cid:28) Ω , we dation under Award No. NSF-DMR-0956064.
k k
9
Appendix A: Disconnected Diagrams
In this section, we explain how the so-called vacuum diagrams (cid:104)TS(∞)(cid:105) cancel the disconnected diagrams in the
0
free two-time correlator (cid:104)TS(∞)Oˆ (t)Oˆ (0)(cid:105) . To shorten the equations we use Aˆ as an abbreviation for Aˆ(t ). For
I I 0 i i
simplicity we base our discussion on a coupling Hamiltonian of the form H =OˆXˆ. It is straight forward to extend
C
this calculations on the full model described in Sec. (IV). The vacuum diagrams are given by
∞ ∞
(cid:90) (cid:90)
(cid:88) 1 (cid:88)
(cid:104)TS(∞)(cid:105) = (−i)n dt ··· dt (cid:104)TOˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) = V , (A1)
0 n! 1 n 1 n 0 1 n 0 n
n −∞ −∞ n
where we assume
(cid:104)Oˆ (t)(cid:105) =0, (cid:104)Xˆ (t)(cid:105) =0, (A2)
I 0 I 0
so that terms with n being an odd number are zero. We have introduced V , the vacuum diagrams of order n. Now
n
we elaborate the connection between the free correlator and the vacuum diagrams. The free two-time correlator is
given by
∞ ∞
(cid:90) (cid:90)
(cid:88) 1
(cid:104)TS(∞)Oˆ Oˆ (cid:105) = (−i)n dt ··· dt (cid:104)TOˆ Oˆ Oˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) . (A3)
a b 0 n! 1 n a b 1 n 0 1 n 0
n −∞ −∞
From this we apply Wick’s theorem and take out the two-time correlators which form a connected diagram and
recombine the surplus correlators in a higher order correlator. There are n! possibilities to choose m vertices out
(n−m)!
of n. Therefore a connected diagram with m vertices occurs n! times
(n−m)!
∞ ∞
(cid:90) (cid:90) n
(cid:88) 1 (cid:88)
(cid:104)TS(∞)Oˆ Oˆ (cid:105) = (−i)n dt ... dt (cid:104)TOˆ Oˆ (cid:105) (cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) ...(cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105)
a b 0 n! 1 n a 1 0 1 2 0 2 3 0 m−1 m 0 m b 0
n −∞ −∞ m
· n! (cid:104)TOˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) . (A4)
(n−m)! m+1 n 0 m+1 n 0
By resorting the factors we can identify the vacuum diagrams of order n−m,
∞ ∞
n (cid:90) (cid:90)
(cid:88)(cid:88)
(cid:104)TS(∞)Oˆ Oˆ (cid:105) = (−i)m dt ... dt (cid:104)TOˆ Oˆ (cid:105) (cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) ...(cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105)
a b 0 1 m a 1 0 1 2 0 2 3 0 m−1 m 0 m b 0
n m −∞ −∞
(cid:124) (cid:123)(cid:122) (cid:125)
=Cma,b
=Vn−m
(cid:122) (cid:125)(cid:124) (cid:123)
∞ ∞
(cid:90) (cid:90)
· 1 (−i)n−m dt ... dt (cid:104)TOˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) , (A5)
(n−m)! m+1 n m+1 n 0 m+1 n 0
−∞ −∞
and find the connected diagrams of order m, which we will call Ca,b, with Ca,b = (cid:104)TOˆ Oˆ (cid:105) . One can factor out
m 0 a b 0
(cid:104)TS(∞)(cid:105) by using the Cauchy product formula
0
∞ n ∞ ∞ ∞
(cid:88)(cid:88) (cid:88) (cid:88) (cid:88)
(cid:104)TS(∞)Oˆ Oˆ (cid:105) = Ca,bV = Ca,b V =(cid:104)TS(∞)(cid:105) Ca,b. (A6)
a b 0 m n−m m n 0 m
n m m n m
This means, that the vacuum diagrams cancel all disconnected diagrams, i.e.,
(cid:104)TS(∞)Oˆ Oˆ (cid:105)
a b 0 = + + +... . (A7)
(cid:104)TS(∞)(cid:105)
0
10
Appendix B: Four-time correlator
In this section, we consider a system where Wick’s Theorem is not exactly valid. The goal is to derive Eqs. (7)
and (10), in order to quantify the deviation from Wick’s theorem. We define the lowest order correction to Wick’s
Theorem as G (t ,t ,t ,t ),
4 1 2 3 4
(cid:88)
G (t ,t ,t ,t )=(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) −(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) =(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105) − (cid:104)TOˆ Oˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) , (B1)
4 1 2 3 4 1 2 3 4 0,F 1 2 3 4 0 1 2 3 4 0,F a b 0 c d 0
3perm.
a,b,c,d
wherethesummationrunsoverallthreeindistinguishablepermutations. With(cid:104)...(cid:105)((cid:104)...(cid:105) )wereferto(un)perturbed
0
correlatorsforwhichweassumeWick’sTheoremtobeexactlyvalid. Incontrasttothis,(cid:104)...(cid:105) ((cid:104)...(cid:105) )describethe
F 0,F
(un)perturbed correlators including the corrections to Wick’s Theorem. In this paper we only consider the lowest-
order correction to Wick’s Theorem (G ). All higher-order corrections are neglected. To shorten the equations we
4
use the abbreviation G (1,2,3,4)=G (t ,t ,t ,t ). An n-time correlator is then given by
4 4 1 2 3 4
(cid:88) (cid:89)
(cid:104)TOˆ ...Oˆ (cid:105) =(cid:104)TOˆ ...Oˆ (cid:105) + G(α,β,γ,δ)(cid:104)T Oˆ (cid:105) . (B2)
1 n 0,F 1 n 0 k 0
perm. k∈{1,...n}\{α,β,γ,δ}
α,β,γ,δ
Atfirstweshowforthefour-timecorrelatorthatifWick’stheoremisvalidfortheunperturbedcorrelator,itisalso
valid for the perturbed one. We start with,
(cid:104)TS(∞)Oˆ Oˆ Oˆ Oˆ (cid:105)
(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105)= I II III IV 0 (B3)
I II III IV (cid:104)TS(∞)(cid:105)
0
∞
(cid:88)(−i)n (cid:90) ∞ (cid:90) 1
= dt ... dt (cid:104)TOˆ Oˆ Oˆ Oˆ Oˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) . (B4)
n! 1 n(cid:104)TS(∞)(cid:105) I II III IV 1 n 0 1 n 0
n −∞ −∞ 0
WefocusonacouplingHamiltonianoftheformH =OˆXˆ. Weproceedasintheabovesectionandidentifyconnected
C
diagrams Ca,b with m vertices. Such diagrams occur n! times. There are six indistinguishable possibilities to
m (n−m)!
chooseaandb. Outoftheremainingn−moperatorswechooseaconnecteddiagramCc,d withkvertices. Thisoccurs
k
(n−m)! times. AsforexampleCI,II andCI,II form=k areindistinguishablewehaveinfactthreeindistinguishable
(n−m−k)! m k
permutations to take into account
(cid:104)TOˆ Oˆ Oˆ Oˆ (cid:105)
I II III IV
∞ ∞
(cid:88)(−i)n(cid:90) (cid:90) 1 (cid:88) (cid:88)n n!
= dt ... dt (cid:104)TOˆ Oˆ (cid:105) (cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) ...(cid:104)TOˆ Oˆ (cid:105)
n! 1 n(cid:104)TS(∞)(cid:105) (n−m)! a 1 0 1 2 0 2 3 0 m b 0
n −∞ −∞ 0 3perm. m
a,b
n(cid:88)−m (n−m)!
· (cid:104)TOˆ Oˆ (cid:105) (cid:104)TXˆ Xˆ (cid:105) (cid:104)TOˆ Oˆ (cid:105) ...(cid:104)TOˆ Oˆ (cid:105)
(n−m−k)! c m+1 0 m+1 m+2 0 m+2 m+3 0 m+k d 0
k
·(cid:104)TOˆ ...Oˆ (cid:105) (cid:104)TXˆ ...Xˆ (cid:105) (B5)
m+k+1 n 0 m+k+1 n 0
∞ n n−m ∞ ∞ ∞
(cid:88) 1 (cid:88)(cid:88) (cid:88) (cid:88) 1 (cid:88) (cid:88) (cid:88)
= Ca,b Cc,d V = V Ca,b Cc,d (B6)
(cid:104)TS(∞)(cid:105) m k n−m−k (cid:104)TS(∞)(cid:105) n m k
3perm. 0 n m k 3perm. 0 n m k
a,b a,b
(cid:88)
= (cid:104)TOˆ Oˆ (cid:105)(cid:104)TOˆ Oˆ (cid:105) . (B7)
a b c d
3perm.
a,b
The resummation in Eq. (B6) represents the Cauchy product formula for three series followed by an index shift.
Hence, we expressed the full four-time correlator in terms of full two-time correlators.
Now we include the corrections to Wick’s theorem and only consider the lowest-order correction G . We introduce
4
the correction to the normalization (cid:104)TS(∞)(cid:105) ,
0,corr
1 1 1 (cid:18) (cid:104)TS(∞)(cid:105) (cid:19)
= ≈ 1− 0,corr . (B8)
(cid:104)TS(∞)(cid:105) (cid:104)TS(∞)(cid:105) +(cid:104)TS(∞)(cid:105) (cid:104)TS(∞)(cid:105) (cid:104)TS(∞)(cid:105)
0,F 0 0,corr 0 0