Describing Trotterized Time Evolutions on Noisy Quantum Computers via Static Effective Lindbladians_2

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Describing Trotterized Time Evolutions on Noisy Quantum

Computers via Static Eﬀective Lindbladians

Keith R. Fratus, Kirsten Bark, Nicolas Vogt, Juha Lepp¨

akangas, Sebastian Zanker,

Michael Marthaler, and Jan-Michael Reiner

HQS Quantum Simulations GmbH, Rintheimer Straße 23, 76131 Karlsruhe, Germany

We consider the extent to which a noisy

quantum computer is able to simulate the time

evolution of a quantum spin system in a faith-

ful manner. Given a speciﬁc set of assump-

tions regarding the manner in which noise act-

ing on such a device can be modelled at the

circuit level, we show how the eﬀects of noise

can be reinterpreted as a modiﬁcation to the

dynamics of the original system being simu-

lated. In particular, we ﬁnd that this modiﬁ-

cation corresponds to the introduction of static

Lindblad noise terms, which act in addition to

the original unitary dynamics. The form of

these noise terms depends not only on the un-

derlying noise processes occurring on the de-

vice, but also on the original unitary dynam-

ics, as well as the manner in which these dy-

namics are simulated on the device, i.e., the

choice of quantum algorithm. We call this ef-

fectively simulated open quantum system the

noisy algorithm model. Our results are con-

ﬁrmed through numerical analysis.

1 Introduction

Simulating the time evolution of quantum systems is

widely discussed as one of the prime applications of

quantum computers due to the exponential speedup

these devices promise over conventional computers [1–

3]. Current error rates on present-day universal de-

vices, however, prohibit solving more than small-scale

example systems [4–7], while quantum error correc-

tion remains out of reach for the foreseeable future [8–

10]. As a result, research regarding early utilization of

quantum computers often focuses on algorithms with

low circuit depth [11], and on mitigating errors rather

than trying to remove them completely [12,13]. In

this endeavor of enabling useful, near-term quantum

computing, it is crucial to understand the eﬀects that

noise can have on the results of a simulation per-

formed on such a device. While we have investigated

this question already in earlier work for speciﬁc noise

types and quantum systems [14], we present here a

more extensive approach to this problem.

We focus in this work on the time evolution of quan-

tum spin systems – systems described by a Hamil-

tonian in which a number of spin degrees of free-

dom experience few-body interactions among each

other. A wide variety of physical systems are well-

approximated by such a description, but solving quan-

tum spin systems is in general hard, either analyti-

cally or using conventional computers [15,16]. Since

there exists a direct mapping between spin degrees

of freedom and qubits on a quantum device, such a

time-evolution can be implemented in a natural fash-

ion on such a device using the Suzuki-Trotter decom-

position and the natively available gate set [2,17].

However, the presence of noise will result in gate op-

erations which are not faithful representations of their

intended unitary operations, and thus in turn will al-

ter the true time evolution of the quantum register.

Our aim in the present work is to understand how

the eﬀects of noise on such a time evolution can be

interpreted as a modiﬁcation to the dynamics driving

this time evolution. In a separate work [18], we have

argued for the validity of a particular model for how

the eﬀects of noise in a quantum device manifest at

the circuit level. Given such a model, we demonstrate

that these modiﬁcations can be well-approximated

by the introduction of static Lindblad noise terms,

which act in addition to the existing unitary dynam-

ics. The nature of these Lindblad terms depends on

the noise present on the device, but also on the par-

ticular choice of Hamiltonian dynamics, as well as the

manner in which these Hamiltonian dynamics are im-

plemented on the device as a sequence of gate oper-

ations, i.e., the quantum algorithm. For this reason,

we call the resulting eﬀective Lindbladian the noisy

algorithm model.

The outline of this paper is as follows: In Section 2

we discuss the types of spin systems and quantum

algorithms considered in our analysis, and in Section 3

we outline our assumptions regarding the nature of

the noise on the devices we consider. The main results

of our analysis, presented in Section 4, constitute a

method for deriving the noisy algorithm model for a

given quantum circuit, along with some of its general

properties, while a numerical analysis of its accuracy

is given in Section 5. We conclude in Section 6. In

Appendix Awe describe our software implementation

of the presented method, while in Appendix Bwe give

an extensive error analysis of our methods.

arXiv:2210.11371v2 [quant-ph] 18 Dec 2023

2 Time Evolution of Spin Systems

We will restrict ourselves to quantum circuits which

involve the digital simulation of the time evolution of

a quantum system comprised of a set of spin- 1

2degrees

of freedom (or simply “spins”), with the Hamiltonian

H=X

hX.(1)

Each subset Xis limited to a non-extensive number

of spins (in practice two spins at most), and hXde-

scribes the interactions among these spins. For a sys-

tem with kspins, the Hilbert space Hhas dimen-

sion D= 2k. For simplicity, we will assume that

the Hamiltonian is time-independent, although the

generalization of our results to the time-dependent

case is straightforward. A common example of such

a Hamiltonian would be the Transverse-Field Ising

Model with nearest-neighbor interactions,

H=JX

⟨ij⟩

σz

iσz

j+gX

σx

i,(2)

the physics of which depends strongly on the geom-

etry of the underlying lattice. Since the degrees of

freedom in the Hamiltonian are spin-1

2observables, it

is possible to associate each degree of freedom with

a qubit on a quantum device, without the need for

additional transformations (for example, the Jordan-

Wigner transformation in the case of fermionic de-

grees of freedom).

To perform such a digital simulation, the unitary

time evolution is approximated using the usual Trot-

ter expansion [19–21],

U(t) = exp (−iHt) =

n=1

exp (−iHτ)

≈

n=1 Y

exp (−ihXτ)≡

n=1 Y

UX(τ)

(3)

where τ=t/N. Such a Trotter expansion of course

involves some degree of approximation. The true

Hamiltonian being simulated in such a time evolution

can be found using the Baker-Campbell-Hausdorﬀ

(BCH) formula, which to ﬁrst order in the Trotter

step size is given according to

H→Heﬀ =H+δH =X

hX−i

2τX

X<Y

[hX, hY],

(4)

where X < Y when the term hXappears to the left of

hYin the Trotter product. Since the individual terms

hXin the Hamiltonian involve only a small number

of sites, the unitary operators appearing as products

in the Trotterized time evolution can be eﬃciently

simulated using gates natively available on a quantum

device [2,17].

3 Assumptions Regarding Noise in a

Quantum Circuit

Throughout the implementation of a Trotter step, de-

coherence will lead to the accumulation of errors in

our simulation. In order to analyze the eﬀects of

this noise, we must assume a model for how noise

manifests itself at the circuit level. Our assumptions

throughout this work regarding noise will be based

upon a separate work [18] which concludes that the

eﬀects of noise can be accounted for at the level of

individual gates. In particular, we will assume that

a circuit can be modeled as a sequence of noise-free

quantum gates, {G}, with each gate followed by an in-

dividual, discrete decoherence event NG, which leads

to some decoherence of the device register.

The form of the discrete noise following a gate will

generally depend on the particular choice of gate, and

the particular choice of hardware. We will assume

that the form of this noise is known (perhaps as a re-

sult of tomography performed on the relevant device),

and that it remains constant throughout at least a

single run. We will also assume that the noise event

which occurs after a gate aﬀects only those qubits

which are involved in the gate - in other words, the

noise following a gate operation on a set of qubits does

not lead to any entanglement with any of the other

qubits on the device. Note that this framework in-

cludes the noise accumulated on qubits as they idle,

when accounting for the action of the “trivial” gate (in

other words, the action of doing nothing to a qubit).

To account for the eﬀects of noise in our simulation

in concrete terms, we must adopt the notation of the

density matrix, rather than the pure wave function,

which cannot describe the evolution of mixed quan-

tum states. In the strictly unitary case in which the

density matrix remains pure, this replacement can be

made according to

|ψ(t)⟩ → ρ(t) = |ψ(t)⟩⟨ψ(t)|,(5)

so that the dynamics of the density matrix are given

according to

ρ(t) = exp (−iHt)|ψ(0)⟩⟨ψ(0) |exp (+iHt)

= exp (−iHt)ρ(0) exp (+iHt)(6)

Using the identity,

eadXY=eXY e−X;adX≡[X, ·],(7)

the expression for the time-evolution of the density

matrix can be further rewritten as,

ρ(t) = eLHtρ0,(8)

where the Liouvillian super-operator for the time-

evolution of the density matrix is a linear operator

acting on the space of density matrices, given as

LH≡ −iadH=−i[H, ·] (9)

Moving beyond the unitary case, the density matrix

will, in general, no longer remain pure,

Tr ρ2(t)̸= 1.(10)

However, in order to respect the basic statistical in-

terpretation of quantum mechanics, the density ma-

trix must remain a positive semi-deﬁnite matrix with

unit trace. In other words, the time-evolution of the

density matrix must be a completely positive, trace-

preserving (CPTP) map. The most general CPTP

map on the space of density matrices, which is also

linear, Markovian, and time-homogeneous, is gener-

ated by the so-called Lindblad equation [22],

L[ρ] = LH[ρ] + LD[ρ] =

−i[H, ρ] + X

n,m

Γnm AnρA†

m−1

2A†

mAn, ρ(11)

The ﬁrst term in this equation is recognizable as the

original unitary evolution, while the second piece, of-

ten referred to as the Lindblad term, accounts for de-

coherence through noise. The operators {An}repre-

sent a basis for the space of all traceless operators on

H, while the time-independent rate matrix Γmust

be Hermitian and positive semi-deﬁnite in order to

preserve the statistical properties of the density ma-

trix. If we relax the assumption that the rate matrix

is time-independent, such that the dynamics are no

longer time-homogeneous, then the necessary condi-

tions for preserving the statistical interpretation of

the density matrix are still not fully understood [23].

We will not concern ourselves with this case here.

The set of operators {An}is not unique - any ba-

sis of traceless operators is valid. For our purposes,

however, we will restrict ourselves to a basis which is

orthonormal with respect to the usual Frobenius inner

product on matrices,

DTr A†

mAn=δmn.(12)

We ﬁnd this basis to be ideal for stating our results

cleanly, though other choices of normalization are pos-

sible (it is however important to remain consistent in

this choice, since the value of the rate matrix will de-

pend on it). Common choices for such a basis {An}

include the (properly normalized) generalized Gell-

Mann matrices [24], or, since we are working with

collections of qubits, the set of all possible products

of Pauli operators on kqubits,

Pα≡

i=1

σαi

i.(13)

For a given Lindblad term, changing the basis of trace-

less operators will result in a corresponding change in

the form of the rate matrix, according to a transfor-

mation law which can be found in [18].

One particularly relevant choice of (orthonormal)

basis is given by

Li=Xv(n)

iAn.(14)

where the vectors {⃗vi}are the (normalized) eigen-

vectors of the rate matrix (as expressed in the basis

{An}). Because the rate matrix is Hermitian, these

eigenvectors will always constitute a complete basis

with real eigenvalues {γi}, and since the rate matrix

is positive semi-deﬁnite, these eigenvalues will be non-

negative. This choice of basis leads to the more com-

mon diagonal form of the Lindblad term,

LD[ρ] = X

γiLiρL†

i−1

2nL†

iLi, ρo,(15)

The eigenvalues {γi}correspond to the physical decay

rates of the system under the eﬀects of decoherence.

Common examples of Lindblad terms, written in such

a diagonal basis, include independent (single qubit)

damping noise,

Ldamp [ρ] = X

γdamp

jσ+

jρσ−

j−1

2σ−

jσ+

j, ρ,

(16)

independent dephasing noise,

Ldeph [ρ] = 1

γdeph

jσz

jρσz

j−ρ,(17)

and independent depolarizing noise,

Ldepo [ρ] = 1

γdepo

α∈{x,y,z}σα

jρσα

j−ρ.(18)

The rates nγdamp

jo,nγdeph

jo, and nγdepo

joindicate

the characteristic noise strength at each site in the

system, and aside from some conventional numerical

factors, coincide with the eigenvalues of the rate ma-

trix. For example, for an observable OXwhich is a

product of Pauli operators living on the sites in the

set X, evolving under purely independent depolariz-

ing dynamics, we have

dt⟨OX⟩=−

X

j∈X

γdepo

j

⟨OX⟩.(19)

One should note, however, that the conventional

choice of basis for damping noise satisﬁes

DTr σ+

iσ−

j=1

2δij ,(20)

and so care must be taken when properly deﬁning

the rate matrix in this case. More discussion of the

Lindblad equation can be found in [25,26].

With this understanding of the Lindblad equation

in mind, we will assume that the decoherence event

following the application of a gate can be modeled as

Lindblad noise accumulating during a small but ﬁnite

time duration, namely the gate application time,

NG→exp tGLG

N,(21)

where LG

Nrepresents a pure Lindblad noise term (i.e.,

there is no Hamiltonian component). We emphasize

that the form of LG

Nmay diﬀer signiﬁcantly from the

form of the underlying environmental processes acting

directly on the quantum register, and the relationship

between the two will depend upon the precise manner

in which the quantum gate is implemented as a se-

quence of operations on the register. However, under

a reasonable set of assumptions regarding how these

operations act, LG

Ncan always be written in Lindblad

form, at least to ﬁrst order in the noise strength. An

argument for the validity of such a model, as well as

further insight into how LG

Ncan be calculated in spe-

ciﬁc situations, can be found in [18].

Having chosen a model for how the eﬀects of noise

manifest themselves at the level of a quantum circuit,

we now proceed to analyze how this noise alters the

eﬀective model simulated by the circuit.

4 The Noisy Algorithm Model

Over the course of a single Trotter step, the state of

the quantum register will have evolved from some ini-

tial ρ, to some ﬁnal ρ′. In the ideal noise-free case, the

evolution would correspond to the originally desired

Hamiltonian time evolution,

ρ′=eLHτρ(22)

However, the discrete noise terms occurring in the

quantum circuit will result in a time-evolution which

deviates from this ideal case. Our aim in this work

is to describe these eﬀects through an eﬀective time-

evolution operator

ρ′=eLeff τρ;Leﬀ [ρ]≡

−iHeﬀ, ρ+X

n,m

Γeﬀ

nm AnρA†

m−1

2A†

mAn, ρ,

(23)

and thereby interpret the quantum circuit as now per-

forming a simulation of this eﬀective model, the noisy

algorithm model, rather than the original coherent

model. We now proceed to characterize Leﬀ , the prin-

cipal component of the noisy algorithm model.

4.1 Circuits with Native Gates

To begin our analysis, we will assume that all of the

exponential products in the Trotter expansion corre-

spond to natively available gates on the given hard-

ware. In other words, we will assume that the unitary

operators

UX(τ)≡exp (−ihXτ) (24)

are natively available on the hardware as quantum

logic gates (we will relax this assumption shortly).

Written as a super-operator acting on the space of

operators,

UX(τ)≡eLXτ;LX≡ −iadhX=−i[hX,·] (25)

The Hamiltonian term hXwill generally consist of

a term proportional to a product of Pauli operators,

for example,

hX=JXσz

iσz

j,(26)

where sites iand jlive on the domain X. For the

Trotter decomposition to be well-behaved, we must

generally assume that the quantity

ϕX= 2JXτ(27)

is suﬃciently small. Thus, UXcorresponds to the ex-

ponentiation of an argument which is parametrically

small in the quantity ϕ. We will refer to such an op-

eration as a small angle gate (SAG).

All of these gate operations will of course have non-

unitary dissipation terms interspersed among them.

These terms also correspond to the exponentiation of

an argument which is paremetrically small in some

quantity. In this case, however, the small quantity is

the product of the gate time with the characteristic

noise strength,

µX=γXtX.(28)

For example, for independent dephasing noise on each

of the two qubits on Xfollowing the application of the

above gate, the argument of the exponential term will

tXLX

N[ρ] = 1

2γdeph

i(σz

iρσz

i−ρ) + 1

2γdeph

jσz

jρσz

j−ρ

(29)

In general, we will assume that there are some char-

acteristic ϕand µwhich describe the coherent and

incoherent terms throughout the circuit, respectively,

and that they satisfy

µ≪ϕ(30)

Without this assumption, the eﬀects of noise on the

device would be too great to perform any sort of useful

computation.

Despite not being unitary, the noise terms are still

generated through exponentiation of some linear (su-

per) operator, and hence it is still possible to combine

a noise term and a coherent gate term into a single

exponential,

etGLG

NeτLX→eτL,(31)

where τLis given according to the BCH formula.

Since all of the noise and gate terms appearing in the

circuit correspond to the exponentiation of “small”

quantities, to lowest order in the BCH expansion we

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DescribingTrotterizedTimeEvolutionsonNoisyQuantumComputersviaStaticEffectiveLindbladiansKeithR.Fratus,KirstenBark,NicolasVogt,JuhaLepp¨akangas,SebastianZanker,MichaelMarthaler,andJan-MichaelReinerHQSQuantumSimulationsGmbH,RintheimerStraße23,76131Karlsruhe,GermanyWeconsidertheextenttowhichanoisyquant...

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Describing Trotterized Time Evolutions on Noisy Quantum Computers via Static Effective Lindbladians_2

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