A quantum algorithm for solving open system dynamics on quantum computers using noise Juha Lepp akangas Nicolas Vogt Keith R. Fratus Kirsten Bark Jesse A. Vaitkus

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A quantum algorithm for solving open system dynamics on quantum

computers using noise

Juha Lepp¨akangas, Nicolas Vogt, Keith R. Fratus, Kirsten Bark, Jesse A. Vaitkus,

Pascal Stadler, Jan-Michael Reiner, Sebastian Zanker, and Michael Marthaler

HQS Quantum Simulations GmbH, Rintheimer Str. 23, 76131 Karlsruhe, Germany

In this paper we present a quantum algorithm that uses noise as a resource. The goal of our

quantum algorithm is the calculation of operator averages of an open quantum system evolving in

time. Selected low-noise system qubits and noisy bath qubits represent the system and the bath of

the open quantum system. All incoherent qubit noise can be mapped to bath spectral functions.

The form of the spectral functions can be tuned digitally, allowing for the time evolution of a wide

range of open-system models at ﬁnite temperature. We study the feasibility of this approach with a

focus on the solution of the spin-boson model and assume intrinsic qubit noise that is dominated by

damping and dephasing. We ﬁnd that classes of open quantum systems exist where our algorithm

performs very well, even with gate errors as high as 1%. In general the presented algorithm performs

best if the system-bath interactions can be decomposed into native gates.

I. INTRODUCTION

Quantum computers promise a substantial speedup for

solving certain types of numerical tasks, in particular the

simulation of large quantum systems [1]. However, due to

the large error rates and short coherence times of present

quantum computers [2], only small examples have been

demonstrated using digital quantum computing. For use-

ful near-term applications, there is a need for current re-

search to be focused on more eﬃciently exploiting noisy

intermediate-scale quantum (NISQ) computers. In this

paper we present a quantum algorithm which, in fact,

utilizes typical noise in NISQ devices by incorporating it

into the computation itself.

Models for open quantum systems [3,4] (short, open-

system models or OSM) have been developed for the

study of a small number of degrees of freedom, the sys-

tem, interacting with a large environment, the bath. Ac-

cordingly, the use of an open-system model is often called

the system-bath approach. The idea is that certain dy-

namical or steady-state properties of the system can be

the key to understanding the overall behavior of the

full system-environment pair. The bath can be modeled

with less accuracy, making the approach economic. One

widely studied example is the energy transfer involved in

photosynthesis [5], where the system is a ﬁnite number

of local excitations (excitons) and the bath are the vi-

brational modes. The eﬀect of the bath on the system is

described by a spectral function and the eﬀect strongly

depends on its height and form [6]. The limit of a smooth

spectral function (on the scale of the system-bath cou-

plings) can be described by the well-known Lindblad or

Bloch-Redﬁeld master equation.

An open system is characterized by non-unitary time

evolution. Therefore, a quantum algorithm that time

evolves an open-system model on a quantum computer

must implement this in some way. This can be accom-

plished by introducing entanglement with additional (ex-

ternal) qubits and performing measurements on them.

Here, one class of methods implements directly a quan-

tum map between an initial and ﬁnal density matrix of

the system [7–11]. This assumes that the time evolu-

tion is known beforehand. If one must instead solve for

this time evolution, a Trotterization needs to be imple-

mented [12–20]. Most such approaches consider an open-

system model that is described by a Lindblad master

equation. A Trotter time evolution under a Lindblad

master equation can be performed on a quantum com-

puter using the same approach known for closed quan-

tum systems: the system is coupled to the external qubits

at each Trotter step, which are measured and thus cre-

ate non-unitary quantum operations. Without access to

quantum feedback control, a technical challenge here is

the required reset of the external qubits after each Trotter

step during the computation. This can however also be

realized via controlled qubit dissipation [14,15,17,18].

Classical gates can also be implemented using similar ap-

proaches [21].

One attractive way to implement open system time-

evolution on NISQ computers is to directly map intrinsic

qubit noise to noise processes in the model to be simu-

lated [9,10,20,22,23]. Here, noise is no longer an im-

pediment, but is rather a key component of the computa-

tion itself. Such an approach is more intuitive for analog

quantum simulation. However, it can also be developed

for digital quantum simulation; although a Trotter time-

evolution might involve many diﬀerent gate operations,

it is possible to describe the eﬀect of noise in the form of

a static Lindblad master equation [24]. One limitation

of previous noise-utilizing algorithms has been that they

fall in the category of smooth spectral functions.

In this paper, we present a noise-utilizing quantum al-

gorithm that time evolves an open-system model that

has a strongly structured bath. The structured bath is

represented by a ﬁnite number of bath modes subjected

to Lindblad dissipation, a method known from classical

numerical methods [25–30]. On a quantum computer,

low and high noise qubits represent the system and the

bath modes of the open quantum system. Signiﬁcant

variations in qubit coherence times may appear in state-

arXiv:2210.12138v3 [quant-ph] 11 Dec 2023

of-the-art quantum computers [31], and it is typically

possible during calibration to make some qubits better

at the expense of others. In our work, we use this to

our advantage by using the lower quality qubits as bath

qubits for mimicking the eﬀect of a continuous spectral

function. By utilizing the intrinsic noise, we also avoid

the need to introduce additional external qubits and their

reset methods. We give a detailed description of how the

spectral function, as seen by the time-evolved system,

can be made strongly structured, digitally tunable and

to match to the open-system model of interest.

The performance of the proposed algorithm is stud-

ied through numerical simulation on conventional com-

puters. In particular, we study the quality of the re-

sults of the algorithm for a spin coupled strongly to

a broad bosonic mode as well as to a bosonic ohmic

bath. Here, we establish a mapping between the open-

system model and the noisy-algorithm model in the case

of noiseless system-qubits. We also study the solution

for an electron-transport model by representing it using

a generalized spin-boson model. Here, we map also the

system-qubit noise to the model spectral function. Our

central ﬁnding is that the quantum algorithm performs

best if the system-bath interactions can be decomposed

into native gates, for instance the XX Ising interaction

to variable Mølmer-Sørensen (MS) gate or the XX + YY

interaction to variable iSWAP gate. The restriction to

native gates can, however, be lifted for some open-system

models.

The paper is structured as follows. In Sec. II, we in-

troduce the concrete open system model whose noise-

utilizing quantum algorithm will be presented in this pa-

per. In Sec. III, we present the protocol of the bath map-

ping, i.e., the mapping between an open-system model

and a noisy-algorithm model. In Sec. IV, we go through

three practical examples of solving the open-system dy-

namics. The examples were implemented using numerical

simulations on conventional computers. Conclusions and

discussion are given in Sec. V. Many important details

of our approach are left to be presented in the Appen-

dices. In Appendix A, we generalize the approach to also

cover multi-spin systems. In Appendix B, we go through

details of our model of noisy quantum computation. In

Appendix C, we discuss principles of a quantum circuit

optimization, where one tailors the form of the eﬀective

Lindbladian. In Appendix D, we quantify the main er-

ror sources in our approach. Finally, in Appendix E, we

discuss how to map a fermionic open-system model to a

spin-boson model.

II. OPEN-SYSTEM MODEL

While the plethora of physical phenomenon encoun-

tered in nature naturally correspond to a wide range of

potential open system models, we focus here on one con-

crete model of a speciﬁc form. In particular, in its most

general form, the open-system Hamiltonian we consider

can be written

H0=ˆ

HS+ˆ

HB+ˆ

HC,(1)

where ˆ

HSis the Hamiltonian of the system, ˆ

HBof the

bath, and ˆ

HCdescribes their coupling.

In this work, we explicitly consider problems that can

be represented as a two-state system interacting with a

bosonic bath [6] by the so-called spin-boson model:

HS=−ℏ∆

2ˆσz,(2)

HB=X

ℏωkˆ

b†

kˆ

bk,(3)

HC= ˆσxX

2ˆ

b†

k+ˆ

bk.(4)

The model system consists of a single spin which has an

energy-level splitting ℏ∆. The model bath consists of

bosonic modes kwith natural frequencies ωk. The boson

creation and annihilation operators satisfy [ˆ

bk,ˆ

b†

l] = δkl.

The coupling between the system and bath is here chosen

to be transverse [32] and occurs via the bath operator

X=X

vk(ˆ

b†

k+ˆ

bk).(5)

The couplings vkare real numbers (possible phases were

absorbed into the deﬁnitions of the boson operators). A

generalization to the multi-spin case is presented in Ap-

pendix A.

Since the bath Hamiltonian is non-interacting, its free-

evolution statistics are Gaussian (time evolution accord-

ing to ˆ

HB). This means that in the interaction picture,

a trace over the bath degrees of freedom results in an ex-

pression where the bath appears only via two-time corre-

lation functions, whose Fourier Transform is the spectral

function:

S(ω) = Z∞

−∞

dteiωt Dˆ

X(t)ˆ

X(0)E0.(6)

Here ˆ

X(t) = eitˆ

HBˆ

Xe−itˆ

HBand the index 0 refers to an

average according to the free evolution of the bath. The

eﬀect of the bath on the system is then fully determined

by this function.

In thermal equilibrium we have:

S(ω)=2πX

kδ(|ω| − ωk)

1−exp −ℏω

kBTsign(ω).(7)

The temperature controls the symmetry (or lack-thereof)

between the positive (energy absorption by the bath) and

negative (energy emission by the bath) frequencies. The

functional form of the spectral function is important in

the sense that it is not a constant and therefore does not

correspond to just white noise. In particular, this implies

that the bath has a memory, i.e., it is non-Markovian.

The wide applicability of the spin-boson theory is

based on the fact that the bath described by ˆ

HBdoes

not necessarily need to consist microscopically of bosonic

modes, but need only be eﬀectively Gaussian. An exam-

ple of such bath is given in Appendix E, where we derive

a spin-boson model of a spin coupled to an electronic

bath.

III. BATH MAPPING

At the center of our approach is a mapping between an

open-system model and a noisy-algorithm model. The

mapping is visualized in Fig. 1. The open-system model

is ﬁrst represented using a model that includes only spins,

with Hamiltonian ˆ

H, and additional spin broadenings.

The spins are represented by qubits on a quantum com-

puter. The noisy-algorithm model, on the other hand,

describes the Trotter time evolution according to ˆ

Has a

static Lindbladian Leﬀ operating on a time-evolved den-

sity matrix [24]. This indicates how noise added after

each unitary gate appears as non-unitary processes in

the simulated system, see Appendix Band Ref. [24]. The

spin broadenings and the noise described by Leﬀ are set

equal by choosing the Trotter time step τcorrectly. The

full procedure is described in more detail below.

A. Coarse graining

In this part of the mapping, we reduce the number

of bath modes in the open-system model from inﬁnite

to nmodes with broadening. These broad modes are

now called auxiliary boson modes and later they will be

mapped to the (noisy) bath qubits on the quantum com-

puter. As a practical matter, in this step we ﬁt the tar-

get spectral function by nLorentzians. We also write

down a Lindblad master equation that is equivalent to

this coarse-grained spectral function.

1. Coarse graining when only the bath qubits are noisy

In this coarse-graining scheme, the target spectral

function, Eq. (7), is ﬁtted by nLorentzians,

S(ω) =

i=1

κi

(κi/2)2+ (ω−ωi)2.(8)

Here we use the counting index iinstead of k, referring

to the auxiliary modes. The ﬁtting is done by optimiz-

ing the peak areas 2πv2

i, frequencies ωi, and broaden-

ings κi. Later, the broadenings will be mapped to qubit

noise. Therefore, in the ﬁtting, the relative sizes of the

broadenings need to be ﬁxed so that they will be consis-

tent with the eﬀective noise given by the noisy-algorithm

model Leﬀ (Sec. III C). The absolute size (a common pref-

actor) is a free ﬁtting parameter, since it will correspond

to choosing the Trotter time step τ. The coarse graining

results shown in this paper are based on a least-squares

ﬁt, whose details are presented in Appendix A.

The couplings and frequencies obtained in the ﬁtting

correspond to parameters of the Hamiltonian:

H=ˆ

HS+ ˆσx

i=1

2ˆ

b†

i+ˆ

bi+

i=1

ℏωiˆ

b†

iˆ

bi,(9)

and the mode broadenings to damping rates in the Lind-

blad master equation:

ˆρ=i

ℏ[ˆρ, ˆ

H] +

i=1

κiˆ

biˆρˆ

b†

i−1

2nˆ

b†

iˆ

bi,ˆρo,(10)

where ˆρis the density matrix of the spin and the auxiliary

boson modes.

We note that similar coarse-graining approaches have

been presented earlier in the context of representing non-

Markovian master equations using pseudo modes [25–30].

Indeed, our work oﬀers an optimal translation of such ap-

proaches to digital quantum simulation. Similar coarse-

graining approaches have also been presented in analog

quantum simulation of spin-boson models [32–35]. Also

closely related are other exact numerical methods where

one replaces continuous bosonic baths by a set of discrete

modes, such as the hierarchical equations of motion [36–

38].

2. Coarse graining when the system qubit is also noisy

Our coarse graining can also account for decoherence of

a qubit representing the (system) spin. The decoherence

of this system qubit contributes via a background rate in

the spectral function. Here, we proceed as before, except

we instead optimize the spectral function

S(ω) =

i=1

κi

(κi/2)2+ (ω−ωi)2+ 4ℏ2κsystem .(11)

The system noise appears via the rate κsystem, which how-

ever will not be an independent ﬁtting variable. In the

ﬁtting, the overall scale of the variables κis arbitrary,

but the relation between the bath mode broadenings κi

and the system noise κsystem is ﬁxed, reﬂecting the cor-

responding hardware properties. We then insert

κsystem =rκ , (12)

where, for simplicity, we assume homogeneous bath

broadening κ. If the system qubit noise matches the bath

qubit noise, we have r= 1, unless a noise-symmetrization

algorithm is used (Sec. IV C), where we have r= 1/2.

The spectral function of Eq. (11) corresponds to a time

FIG. 1. Bath mapping, i.e., mapping between an open-system model and a noisy-algorithm model. As described in the

main text, the open-system model ˆ

H0is ﬁrst coarse grained to an auxiliary-spin model ˆ

Hand additional spin broadening.

A quantum algorithm time evolves the qubits describing the spins according to ˆ

H. The combined eﬀect of unitary gates

and non-unitary qubit noise is described by an eﬀective Lindbladian Leﬀ. The auxiliary-spin model broadening and the noise

broadening according to Leﬀ are set equal by choosing the Trotter time step correctly. A circuit optimization may be needed

to have Leﬀ in a desired form.

evolution according to the Lindblad master equation

ˆρ=i

ℏ[ˆρ, ˆ

H] +

i=1

κiˆ

biˆρˆ

b†

i−1

2nˆ

b†

iˆ

bi,ˆρo

+κsystem (ˆσxˆρˆσx−ˆρ).(13)

The diﬀerence to Eq. (10) is the term proportional to

κsystem. We see that the system collapse-operator in this

term is ˆσx, since the coupling to the bath is via this op-

erator, see Eq. (9). It follows that the bath mapping in

this scheme is exact if the system-qubit noise operator (in

the corresponding noisy-algorithm model) is also propor-

tional to ˆσx. It should be noted that an exact mapping

can be obtained also in the case of all qubits being sub-

ject to damping, after a noise-symmetrization algorithm,

as shown in Sec. IV C. More generally, the bath mapping

in such a scheme is most likely only an approximation,

when, for example, the system is subjected to depolaris-

ing noise, but can work well for a weak coupling between

the system and the bath. The ﬁtting in the two diﬀerent

schemes is illustrated in Fig. 2.

B. Representing auxiliary bosons by auxiliary spins

Here we write down a Lindblad master equation that

includes only spins and is equivalent with the coarse-

grained spin-boson model. For this, we represent the de-

rived auxiliary boson modes by auxiliary spins. Common

digital encodings [39] cannot be applied, since they do not

map damping of an arbitrary auxiliary spin to single-

boson annihilation, which is the key correspondence in

our algorithm. Instead, we replace bosonic energy oper-

ators ˆ

b†ˆ

bby a sum of auxiliary spin operators ˆσj

+ˆσj

−and

bosonic coupling operators ˆ

b†+ˆ

bby a sum of auxiliary

spin operators ˆσj

x. In other words, in the Hamiltonian ˆ

we switch

b†

iˆ

bi→

j=1

ˆσi,j

+ˆσi,j

−(14)

ˆ

b†

i+ˆ

bi→1

√Ni

j=1

ˆσi,j

x.(15)

This mapping is based on the well-known property that

collective spin operators can behave like bosonic opera-

tors if their excitation numbers stay low, with an error

O(1/N). For a more detailed analysis of this point, see

for example Ref. [40]. The coarse-grained model Hamil-

tonian, represented now by only spins, becomes

H=ˆ

HS+ ˆσx

i=1

j=1

√Ni

ˆσi,j

i=1

ℏωi

j=1

ˆσi,j

+ˆσi,j

−.

(16)

The spectral function as seen by the system, Eq. (8)

or (11), keeps its form, with the summation performed

now over the corresponding auxiliary-spin parameters.

How the ﬁtting and the replacement of the auxiliary

bosons by auxiliary spins is done exactly, is inﬂuenced by

the open-system model as well as the hardware proper-

ties. For example, for sharp model peaks in comparison

to model couplings one may need Ni>1. This choice im-

plicitly assumes that the corresponding bath qubits have

homogeneous decoherence rates, since the auxiliary spins

are mapped later directly to bath qubits. On the other

hand, if signiﬁcant diﬀerences exist in the bath-qubit de-

coherence rates, or if the spectral function ﬁtting is more

important than the bath Gaussianity, one should impose

one-to-one correspondence between the auxiliary-boson

modes and the auxiliary spins, i.e., ﬁx Ni= 1. These

two choices are visualized in Fig. 3.

It is very valuable for the bath mapping that for aux-

iliary spins the spectral-peak broadening can be theoret-

ically not only due to the damping rate γbut also due

... ...

(a) (b)

frequency

spectral function

target

ﬁt

individual aux modes

frequency

target

ﬁt

individual aux modes

system noise

FIG. 2. Fitting of a target spectral function in the two

discussed bath-mapping schemes. (a) The system spin scou-

ples to nauxiliary boson modes b1, ..., bn. The auxiliary

modes further couple to an environment, which leads to mode

broadenings κi. (b) In the second scheme, also the system

spin couples to the environment, which leads to a decoher-

ence rate κsystem. (c) An example of ﬁtting by eight auxiliary

modes without system noise. All modes are assumed to have

the same broadening. (d) An example of ﬁtting the same

spectral function but in the presence of system noise. The

system noise contributes via a constant shift of the ﬁtting

function, leading to a diﬀerent set of optimal auxiliary-mode

parameters.

to the dephasing rate Γ,

κi=γi+ 2Γi≡γ∗

i.(17)

This property is important since it allows one to do the

full mapping procedure also when the bath qubits are

subjected to signiﬁcant dephasing. The corresponding

Lindbladian is (for simplicity considering here Ni= 1)

ˆρ=i

ℏ[ˆρ, ˆ

H] +

i=1

γiˆσi

−ˆρˆσi

+−1

2ˆσi

+ˆσi

−,ˆρ

i=1

Γi

2ˆσi

zˆρˆσi

z−ρ.(18)

The procedure is otherwise the same as for bath qubits

subjected to damping only, except the connection be-

tween the broadenings of the bosonic bath and the hard-

ware qubits is made via Eq. (17). We verify this con-

nection by numerical simulations presented in Sec. IV A.

[The possible system noise is added similarly as in

Eq. (13).]

It should be noted that broadening due to dephasing

alone does not lead to Gaussian equilibrium statistics: a

ﬁnite damping rate γis needed. This is because certain

bath correlations that describe non-bosonic statistics de-

cay in time according to γinstead of γ+ 2Γ.

C. Matching spin broadening with qubit noise

Here we describe how the Lindblad master equation

operating only on spins can be implemented in digital

quantum computing using noise. The key part is the

matching of the spin broadening with the qubit noise.

This step should be done consistently with the coarse

graining (Sec. III A): variations in the noise of qubits

must appear as corresponding relative variations in the

auxiliary-mode broadenings κi. This demand however

does not lead to a self-consistency loop, but rather to a

unique determination of the variations of the auxiliary-

mode broadenings.

In this work, we assume that the qubit noise is predom-

inantly damping and dephasing. The form of the eﬀec-

tive noise can however depend on the choice of gate de-

compositions, which needs to be take into account when

designing the quantum circuits. This is detailed in Ap-

pendices Band C.

In the considered case, the noisy-algorithm model (Ap-

pendix Band Ref. [24]) gives the eﬀective spin damping

and dephasing rates:

γi=Dtgate¯γi

τ,(19)

Γi=Dtgate ¯

Γi

τ,(20)

where Dis the depth of one Trotter-step circuit, tgate is

the physical time needed to perform one gate (assumed

here to be a constant), ¯γiand ¯

Γiare the physical damp-

ing and dephasing rates of the qubit representing the

auxiliary spin i, and τis the chosen Trotter time step.

We assume here that the errors are similar for every gate

and that they act also on qubits that are at rest (idling).

The potential variation in the widths γi+2Γineeds to be

accounted for in the coarse graining, as discussed above.

The contribution from ﬁnite system qubit noise is similar.

Eqs. (19-20) with Eq. (17) can be used to solve the

Trotter time step τ, which was still a free parameter.

This solution corresponds to matching the spin broad-

ening with the qubit noise. For simplicity, let us now

assume that all bath qubits have homogeneous damping

and dephasing rates, and obtain

τ=Dϵ

κ,(21)

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AquantumalgorithmforsolvingopensystemdynamicsonquantumcomputersusingnoiseJuhaLepp¨akangas,NicolasVogt,KeithR.Fratus,KirstenBark,JesseA.Vaitkus,PascalStadler,Jan-MichaelReiner,SebastianZanker,andMichaelMarthalerHQSQuantumSimulationsGmbH,RintheimerStr.23,76131Karlsruhe,GermanyInthispaperwepresentaquan...

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A quantum algorithm for solving open system dynamics on quantum computers using noise Juha Lepp akangas Nicolas Vogt Keith R. Fratus Kirsten Bark Jesse A. Vaitkus

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