HOQST: Hamiltonian Open Quantum System Toolkit

Huo Chena,b,∗, Daniel A. Lidara,b,c,d

aDepartment of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USAbCenter for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA

cDepartment of Chemistry, University of Southern California, Los Angeles, California 90089, USAdDepartment of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA

Abstract

We present an open-source software package called “Hamiltonian Open Quantum System Toolkit" (HOQST), a col-lection of tools for the investigation of open quantum system dynamics in Hamiltonian quantum computing, includingboth quantum annealing and the gate-model of quantum computing. It features the key master equations (MEs) usedin the field, suitable for describing the reduced system dynamics of an arbitrary time-dependent Hamiltonian witheither weak or strong coupling to infinite-dimensional quantum baths. This includes the Redfield ME, the polaron-transformed Redfield ME, the adiabatic ME, the coarse-grained ME, and the universal Lindblad ME. HOQST alsoincludes the stochastic Schrödinger equation with spin-fluctuators. We present an overview of the theories behind thevarious MEs and provide examples to illustrate typical workflows in HOQST. The package is ready to be deployedon high performance computing (HPC) clusters and aimed at providing reliable open-system analysis tools for noisyintermediate-scale quantum (NISQ) devices. To start working with HOQST, users should go to its Github repository:https://github.com/USCqserver/OpenQuantumTools.jl. Detailed information can be found in theREADME file.

Keywords: Hamiltonian quantum computing; Open quantum system; Master equation; Quantum annealing

PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARYProgram Title: Hamiltonian Open Quantum System Toolkit (HOQST)Licensing provisions: MITProgramming language: JuliaNature of problem: Simulation of open quantum system dynamics of systems described by arbitrary time-dependent Hamiltonians,with an emphasis on Hamiltonian quantum computing, including quantum annealing and the gate-model.Solution method: Open system dynamics is simulated in HOQST by numerically solving the Redfield master equation [1], thepolaron-transformed Redfield equation (PTRE) [2], the adiabatic master equation (AME) [3], the coarse-grained master equation(CGME) [4], the universal Lindblad equation (ULE) [5], or the stochastic Schrödinger equation with spin-fluctuators [6]. In allcases both time-dependent and time-independent system Hamiltonians are supported. ODE solvers currently used are Tsitouras5/4 Runge-Kutta [7], TR-BDF2 [8], adaptive Exponential-Rosenbrock [9] and Linear Exponential. A large collection of additionalODE solvers (not tested) is available through DifferentialEquations.jl [10].

1. Introduction

The theory of open quantum system has been an important subfield of quantum physics during the past decadeswith a rich collection of well established method [11–16]. Since perfect isolation of quantum systems is impossible,any quantum mechanical system must be treated as an open system in practice. The theory of open quantum systemthus plays a major role in various applications of quantum physics, e.g., quantum optics [17–19], quantum control [20],and quantum computing (QC) [21]. It becomes even more relevant in the context of Hamiltonian quantum computing(HQC), broadly defined as ‘analog’ QC performed via continuously and smoothly driven Hamiltonians, as opposed

∗Corresponding author.E-mail address: huochen@usc.edu

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to ‘discrete’ gate-model QC, where Hamiltonians are driven discontinuously. Well-known example of HQC includeadiabatic quantum computing (AQC) [22] and quantum annealing (QA) [23] (see, e.g., Refs. [24, 25] for reviews), aswell as holonomic QC [26, 27]. For example, in QA the Hamiltonian needs to move continuously from the the initialdriver Hamiltonian to the final problem Hamiltonian, and therefore, unlike idealized gate-model quantum computerswhose description often involves effective noise channels (completely positive maps), practical quantum annealers arebetter described by noise models derived directly from first principles [28–32]. As the entire field of QC is now inthe noisy intermediate-scale quantum (NISQ) era [33], an efficient and evolving framework of open quantum systemsimulation is essential for advancing our understanding of noise in quantum devices, as well for helping in the searchfor more effective error suppression and correction techniques [34]. Moreover, the distinction between analog anddiscrete models of QC is to some degree arbitrary, since in reality even gate-model QC involves continuous drivingdue to the finite bandwidth of signal generators and controllers. We thus view the gate-model of QC as part of HQCfor the purposes of this work.

At present, there is an increasing number of software tools being developed for open system simulations. Animportant example of open-source software in this area is QuTiP [35, 36]. It is one of the first packages in the fieldto adopt the modern software engineering paradigm, and is actively maintained on Github since its release, with newfeatures and enhancements being continuously added. However, since QuTiP is designed to be as general as possible,it lacks several tools and the computational performance required to address the new challenges we are now facing inthe field of HQC.

Inspired by this unique challenge, and by the success of QuTiP, we present here a complementary and alternativeopen system simulation framework, which we call “Hamiltonian Open Quantum System Toolkit" (HOQST). As thename suggests, the goal of HOQST is not to cover the entire field of open quantum system simulation but to focuson Hamiltonian QC, while retaining the flexibility to simulate systems subject to arbitrary time-dependent Hamil-tonians. This focus gives us the ability to adopt domain-specific design choices and optimizations. The resultingimplementation distinguishes itself from other available software by offering the following advantages:

• HOQST is written in the Julia programming language [37], which is designed for high performance computing.

• HOQST is built upon the excellent ordinary differential equations (ODE) package DifferentialEquations.jl[10]; thus HOQST also benefits from progress in the field of ODE solvers.

• Focusing solely on Hamiltonian QC, HOQST features several newly developed open quantum system toolssuch as recently published master equations.

• HOQST includes tools that work beyond the weak coupling limit.

• HOQST provides a naive interface for HPC clusters.

• HOQST provides native GPU support with the CUDA.jl package.

HOQST is developed following the Julia design philosophy [38]. We would like it to be as user-friendly as possiblewithout compromising performance. Although there is room for optimization, the first release of HOQST featuresreliable and efficient implementations of several key master equations (MEs) adopted in the HQC field, together witha highly modularized framework suitable for future development. Since the HOQST project started as an attempt tobuild a tool to simulate quantum annealing, it displays a certain bias towards QA. However, we reemphasize that it isbroadly applicable to open quantum systems evolving subject to any time-dependent Hamiltonian.

This paper is organized as follows. In Sec. 2 we briefly review the open system models and corresponding masterequations supported by this package. In Sec. 3 we introduce several numerical techniques to speed up computations.In Sec. 4 we introduce the capabilities of HOQST and provide a comparison with other quantum simulators. In Sec. 5we provide examples to illustrate typical workflows of HOQST. We conclude in Sec. 6, and provide various additionaltechnical details in the Appendix.

2. Methods

Throughout this work we consider a quantum mechanical system S coupled to a bath B. The total Hamiltonian isassumed to have the following form

H = HS + HI + HB , (1)

2

where HS and HB denote, respectively, the free system and bath Hamiltonians. HI is the system-bath interaction,which is often written as

HI =∑α

gαAα ⊗ Bα , (2)

where Aα and Bα are dimensionless Hermitian operators acting on the system and the bath, respectively, and excludeboth IS and IB (the identity operators on the system and the bath, respectively). The parameters gα are sometimesabsorbed into Bα but are kept explicit in HOQST, are have units of energy. In addition, we assume for simplicity thefactorized initial condition ρ(0) = ρS(0) ⊗ ρB for the joint system-bath state at the initial time t = 0, where ρB is aGibbs state at inverse temperature β

ρB =e−βHB

Tr[e−βHB

] , (3)

though we note that factorization is not necessary for a valid description of open system dynamics [39, 40]. We workin units such that kB = 1, so that β has units of inverse energy, or time (since we also set ~ = 1).

Before proceeding, we transform the original Hamiltonian Eq. (1) into a rotating frame defined by U(t), i.e.

H(t) = U†(t)HU(t) . (4)

The specific form of the unitary operator U(t) will lead to different master equations and will be discussed in detailin subsequent sections. The goals of the rotation are to remove the pure-bath Hamiltonian HB and to identify termsthat remains small in different system bath coupling regimes. As long as U(t) acts non-trivially on the bath, we mayassume without loss of generality we that the rotating frame Hamiltonian H has the following form:

H = HS + HI , (5)

where HS acts only the system and HI acts jointly on the system and the bath. The Liouville von Neumann equationin this rotating frame is

∂

∂tρ(t) = −i[H(t), ρ(t)] ≡ L(t)ρ(t) , (6)

where L(t) denotes the Liouvillian superoperator. We work in units of ~ = 1 throughout. Once again we can alwayswrite

HI =∑α

gαAα ⊗ Bα (7)

where Aα and Bα are, respectively, system and bath operators (excluding identity). However, it is important to note thatAα and Bα do not necessarily correspond to U†(t)AαU(t) and U†(t)BαU(t) in Eq. (2) because U(t) may not preservethe tensor product structure (i.e., we allow for U†A ⊗ BU , U†AU ⊗ U†BU).

Let ⟨X⟩

= Tr(XρB

)(8)

denote the expectation value of any rotating frame bath operator X with respect to ρB. Then the two-point correlationfunction is

Cαβ(t1, t2) = gαgβ⟨Bα(t1)Bβ(t2)

⟩. (9)

If the correlation function is time-translation-invariant

Cαβ(t1, t2) = Cαβ(t1 − t2, 0) ≡ Cαβ(τ) , τ ≡ t1 − t2 , (10)

then the noise spectrum of the bath can be properly defined by taking the Fourier transform

γαβ(ω) =

∫ ∞

−∞

Cαβ(τ)eiωτ dω . (11)

The widely used Ohmic bath case is

γOhmicαβ (ω) = 2πηgαgβ

ωe−|ω|/ωc

1 − e−βω. (12)

HOQST provides a built-in Ohmic bath object. It also provides flexible APIs to define a CustomBath object byspecifying either the correlation function or the noise spectrum. However, to avoid the added computational cost ofthe Fourier transform and the numerical instability associated with large frequencies, HOQST does not convert thesequantities into each other, so the user is responsible for supplying the correct function depending on the solver used.

3

2.1. Timescales

Following [4], we define the two timescales

1τS B

=

∫ ∞

0|C(τ)|dτ , τB =

∫ t f

0 τ|C(τ)|dτ∫ ∞0 |C(τ)|dτ

(13)

Here t f is the total evolution time, used as a cutoff which can often be taken as ∞. The quantity τS B is the fastestsystem decoherence timescale, or timescale over which the system density matrix ρS changes due to the coupling tothe bath, in the interaction picture. The quantity τB is the characteristic timescale of the decay of C(τ). Note that theexpression for τB becomes an identity if we choose |C(τ)| ∝ e−τ/τB and take the limit t f → ∞.

As shown in Ref. [4], τS B and τB are the only two parameters relevant for determining the range of applicabilityof the various master equations discussed here (with the ULE case being no exception, but discussed separatelyin Ref. [5]). For convenience we collect the corresponding error bounds here, before discussing the various MEs.Namely, the error bound of the Redfield master equation is

‖ρtrue(t) − ρR(t)‖1 ≤ O(τB

τS Be12t/τS B

)ln

(τS B

τB

), (14)

where ρtrue(t) denotes the true (approximation-free) state.The error bound of the Davies-Lindblad master equation is

‖ρtrue(t) − ρD(t)‖1 ≤ O((τB

τS B+

1√τS BδE

)e12t/τS B

), (15)

where δE =mini, j|Ei − E j| is the level spacing, with Ei the eigenenergies of the system Hamiltonian HS. This originalversion of this ME [41] does not directly allow for time-dependent driving, and we shall consider an adiabatic variantthat does, the AME [3]. The same error bound should apply in this case, since the difference is only in that theLindblad operators are rotated in the AME case with the (adiabatically) changing eigenstates of HS, and the normsused to arrive at Eq. (15) are invariant under this unitary transformation.

The error bound of the coarse-grained master equation is

‖ρtrue(t) − ρC(t)‖1 ≤ O(√

τB

τS Be6t/τS B

). (16)

The error of the PTRE master equation can be separated into two parts. The first part comes from truncating theexpansion to 2nd order. It can be bounded using the same expression as in Eq. (14), with timescales defined by thepolaron frame correlation K(τ):

1τS B

= ∆2m

∫ ∞

0|K(τ)|dτ , τB =

∫ t f

0 τ|K(τ)|dτ∫ ∞0 |K(τ)|dτ

. (17)

A detailed discussion of the above quantities is presented in Appendix B and Appendix C, and as far as we know theerror bounds we derive here for the PTRE are new. We mention here that if the system-bath coupling strength gα inEq. (2) is sent to infinity, both 1/τS B and τB/τS B go to 0. Thus the PTRE works in the strong coupling regime. Thesecond part of the error is caused by ignoring the 1st and 2nd order inhomogeneous terms, which themselves are dueto the polaron transformation breaking the factorized initial condition. We do not have a bound on this error yet, butnumerical studies suggest it is small when ρS(0) is diagonal [42].

These bounds assume a Gaussian bath. For a non-Gaussian bath extra timescales relating to higher-order correla-tion functions generally appear, and the error bounds will contain additional terms.

4

2.2. Cumulant expansion

The cumulant expansion is a technique originally designed for the perturbation expansion of stochastic differentialequations [43, 44]. This technique can be generalized to the open quantum system setting and allows a systematicdescription of the reduced system dynamics [13, 45]. By applying this technique in different rotating frames, masterequations with different ranges of applicability can be derived [2, 13, 32]. Defining the projection operator P

Pρ = trB{ρ} ⊗ ρB ≡ ρS ⊗ ρB , (18)

the formal cumulant expansion of Eq. (6) is

∂

∂tPρ(t) =

∑n

Kn(t)Pρ(t) , (19)

where the nth order generator Kn(t) is

Kn(t) =

∫ t

0dt1

∫ t1

0dt2 · · ·

∫ tn−2

0dtn−1

⟨L(t)L(t1) · · · L(tn−1)

⟩oc, (20)

and the quantities⟨L(t)L(t1) · · · L(tn−1)

⟩oc

are known as ordered cumulants [13, 45]. In HOQST, we consider onlythe first and second order generators, which are given by:⟨

L(t)⟩

oc= PL(t)P ,

⟨L(t)L(t1)

⟩oc

= PL(t)L(t1)P . (21)

Incorporating higher order cumulants generally leads to more accurate results [13].

2.3. Redfield equation

The oldest and one of the most well-known MEs in this category is the Redfield equation [1] (also known asTCL2 [13], where TCLn stands for time-convolutionless at level n, arising from an expansion up to an includingKn(t)), which directly follows from Eq. (19) after choosing the rotation U(t) to be

U(t) = US(t) ⊗ UB(t) , US(t) = T+ exp{−i

∫ t

0HS(τ) dτ

}, UB(t) = exp{−iHBt} , (22)

where T+ denotes the forward time-ordering operator. After rotating back to the Schrödinger picture, one of the mostcommon forms of the Redfield equation is

ρS(t) = −i[HS(t), ρS(t)

]−

∑α

[Aα(t),Λα(t)ρS(t)

]+ h.c. (23)

where

Λα(t) =∑β

∫ t

0Cαβ(t − τ)US(t, τ)Aβ(τ)U†S(t, τ) dτ . (24)

This is the form used in HOQST. The error bound for the Redfield ME is given in Eq. (14). We note that the currentrelease of HOQST supports correlated baths for the Redfield and adiabatic master equation solvers. However, forsimplicity, we henceforth focus on uncorrelated baths where Cαβ(t) = δαβCα(t).

The most significant drawback of the Redfield equation is the fact that it does not generate a completely-positiveevolution, and in particular can result in unphysical negative states (density matrices with negative eigenvalues).Though formal fixes for this problem have been proposed [46, 47], to address the issue in HOQST an optional positiv-ity check routine is implemented at the code level, which can stop the solver if the density matrix become negative. Inaddition, three variants of Redfield equation that guarantee positivity, namely the adiabatic master equation (AME) [3],the coarse-grained master equation (CGME) [4, 48], and the universal Lindblad equation (ULE) [5], are included inHOQST. We detail these MEs next.

5

2.4. Coarse-grained master equation (CGME)

The CGME can be obtained from Eq. (23) by first time-averaging the Redfield part, i.e., shifting t 7→ t + t1 andapplying 1

Ta

∫ Ta/2Ta/2

dt1. One then neglects a part of the integral to regain complete positivity. The result is [4]:

ρ = −i[HS + HLS, ρ

]+

∑α

1Ta

∫ −Ta/2

Ta/2dt1

∫ −Ta/2

Ta/2dt2 Cα(t2 − t1)

[Aα(t + t1)ρSA†α(t + t1) −

12{Aα(t + t2)Aα(t + t1), ρS}

],

(25)where Aα(t + t1) = U†(t + t1, t)Aα(t)U(t + t1, t) and the Lamb shift is given by

HLS =i

2Ta

∫ −Ta/2

Ta/2dt1

∫ −Ta/2

Ta/2dt2 sgn(t1 − t2)Cα(t2 − t1)A(t + t2)A(t + t1) . (26)

The quantity Ta is the coarse-graining time, a phenomenological parameter that can be manually specified or automati-cally chosen based on the bath correlation function [48]. HOQST uses a multidimensional ‘h-adaptive’ algorithm [49]based on [50] to perform the 2-dimensional integration. The error bound for the CGME is given in Eq. (16).

2.5. Universal Lindblad equation (ULE)

The ULE [5] is a Lindblad-form master equation that shares the same error bound as the Redfield equation, i.e.,Eq. (14). The formal form of the ULE is identical to the Lindblad equation:

ρS(t) = −i[HS(t) + HLS(t), ρS(t)

]+

∑α

[Lα(t)ρL†α(t) −

12

{L†α(t)Lα(t), ρ

}], (27)

where the time-dependent Lindblad operators are

Lα(t) =

∫ ∞

−∞

gα(t − τ)US(t, τ)Aα(τ)U†(t, τ) dτ , (28)

and the Lamb shift is

HLS(t) =∑α

12i

∫ ∞

−∞

ds ds′ U(t, s)Aα(s)gα(s − t)U(s, s′)gα(t − s′)Aα(s′)U†(t, s′) sgn(s − s′) . (29)

In the above expression, gα(t) is called the jump correlation and is the inverse Fourier transform of the square root ofthe noise spectrum

gα(t) =1

2π

∫ ∞

−∞

√γα(ω)e−iωt dω . (30)

The integration limits of Eqs. (28) and (29) are problematic in practice because the unitary US(t) does not go beyond[0, t f ]. In numerical implementation, we replace the integral limit with

∫ t f

0 . This is a good approximation when g(t)decays much faster than t f . This is the form of ULE used in HOQST.

2.6. Adiabatic master equation (AME)

To derive the AME, we replace US(t − τ) in Eq. (24) with the ‘ideal adiabatic evolution’ and apply the standardMarkov assumption and the rotating wave approximation (RWA). The resulting equation is

ρS(t) = −i[HS(t) + HLS(t), ρS(t)

]+

∑αβ

∑ω

γαβ(ω)[Lω,β(t)ρS(t)L†ω,α(t) −

12

{L†ω,α(t)Lω,β(t), ρS(t)

}]. (31)

The AME is in Davies form [41] and the Lindblad operators are defined by

Lω,α(t) =∑

εb−εa=ω

〈ψa|Aα|ψb〉 |ψa〉〈ψb| , (32)

6

where εa is the instantaneous energy of the a’th level of the system Hamiltonian, i.e., HS(t) |ψa(t)〉 = εa(t) |ψa(t)〉.Finally, the Lamb shift term is

HLS(t) =∑αβ

∑ω

L†ω,α(t)Lω,β(t)S αβ(ω) , (33)

where

S αβ(ω) =1

2π

∫ +∞

−∞

γαβ(ω′)P(1

ω − ω′) dω′ , (34)

with P denoting the Cauchy principal value. The error bound is given in Eq. (15).If the RWA is not applied, the resulting equation is called the one-sided AME:

ρS(t) = −i[HS(t), ρS(t)

]+

∑αβ

∑ω

Γαβ(ω)[Lω,β(t)ρS(t), Aα

]+ h.c. , (35)

where

Γαβ(ω) =

∫ ∞

0Cαβ(t)eiωt dt =

12γαβ(ω) + iS αβ(ω) . (36)

These two forms of the AME behave differently when the energy gaps are small because the RWA breaks down in suchregions [4]. More importantly, like the Redfield equation, the one-sided AME does not generate a completely-positiveevolution. The code-level positivity check routine works with this version of the AME as well.

2.7. Classical 1/ f noise

HOQST includes the ability to model 1/ f noise, which is an important and dominant source of decoherence inmost solid-state quantum NISQ platforms [6], in particular those based on superconducting qubits [51–54]. Fullyquantum treatments of 1/ f noise have been proposed [55, 56], including for quantum annealing [32]. In HOQST weadopt the simpler approach of modeling 1/ f noise as classical stochastic noise generated by a summation of telegraphprocesses, which has proved to be a good approximation to the fully quantum version [6]. Specifically, we provide aquantum-trajectory simulation of the following stochastic Schrödinger equation∣∣∣Φ⟩

= −i(HS +

∑α

δα(t)Aα

)|Φ〉 , (37)

where each δα(t) is a sum of telegraph processes

δα(t) =

N∑i=1

Ti(t) , (38)

where Ti(t) switches randomly between ±bi with rate γi. As shown in [6], in the limit of N → ∞ and bi → b,if the γi’s are log-uniformly distributed in the interval [γmin, γmax] (with γmax � γmin), the noise spectrum of δα(t)approaches a 1/ f spectrum within the same interval. Empirically, we find that a good approximation can be achievedwith relatively small N.

2.8. Hybrid model

The most significant drawback of a purely classical noise model is that if its steady state is unique then it is themaximally mixed state. To see this, we first realize that each trajectory of Eq. (37) generates a unitary acting on thespace S(HS ) of density matrices

Uk(t)ρS = Uk(t)ρSU†k (t) . (39)

Averaging over the trajectories over a distribution p(k) creates a unital (identity preserving) map from S(HS ) intoitself

U(t)ρS(0) =

∫p(k)Uk(t)ρS dk . (40)

7

If the steady state ρ∞ is unique then we can define it as

limt→∞U(t)ρS(0) = ρ∞ ∀ρS(0) . (41)

By unitality it would then follow that ρ∞ = I, since we can choose ρS(0) = I.However, this is not what is observed in real devices, e.g., in experiments with superconducting flux [30] or

transmon [57] qubits. To account for this, HOQST includes a hybrid classical-quantum noise model:

ρ = −i

HS +∑α

δα(t)Aα, ρ

+L(ρ) , (42)

where δα(t) is the same random process as in Eq. (38), and L is the superoperator generated by the cumulant expan-sion (20). At present, HOQST supports the combination of 1/ f noise with both the Redfield and adiabatic masterequations.

2.9. Polaron transformIf the bath operators in Eq. (2) are bosonic

Bα =∑

k

λk(b†α,k + bα,k) , HB =

∑α,k

ωα,kb†α,kbα,k , (43)

we can choose the joint system-bath unitary U(t) in Eq. (22) as [2]

Up(t) = exp

−i∑α,k

Adα

gαλk

iωα,k(b†α,k − bα,k)

UB(t) , (44)

where Adα is the diagonal component of Aα in the interaction Hamiltonian (2) and UB(t) is given in Eq. (22)1. The

corresponding second order ME (23) is known as the polaron-transformed Redfield equation (PTRE) [2] or thenoninteracting-blip approximation (NIBA) [15, 58]. The PTRE has a different range of applicability than the pre-vious MEs we have discussed. Whereas the latter apply under weak-coupling conditions, the transformation definedin Eq. (44) leads to a complementary range of applicability under strong-coupling. This particular form of Eq. (44)does not preserve the factored initial state, so that inhomogeneous terms are present after the transformation. How-ever, if ρS(0) is diagonal then numerical studies of the effects of the inhomogeneous terms suggest that they can beignored [42].

In addition, the PTRE can be extended beyond the spin-boson model by choosing a different form of the jointsystem-bath unitary (44), as [32, 59]2

Up(t) = UB(t)T+ exp

−i∑α

Adαgα

∫ t

0Bα(τ) dτ

. (45)

The two transformations in Eq. (44) and (45) lead to MEs with identical structure but slightly different expressions(see Appendix A for details). Whether those differences make any physical significance is an interesting topic forfurther study. In this paper we choose to work with Eq. (44).

Because the general form of the PTRE is unwieldy, we present its form for a standard quantum annealing model

H(t) = HS(t) + HI + HB , HI =∑

i

giσzi ⊗ Bi (46a)

HS(t) = a(t)Hdriver + b(t)Hprob , (46b)

1We use λ instead of g in Bα to distinguish it from the expansion parameter in Eq. (7).2Note that in references [32, 59], different lower integration limits, 0 or −∞, are chosen for the rotation (45). Similar results are derived despite

this technical difference.

8

where a(t) and b(t) are the annealing schedules, and Hdriver and Hprob are the standard driver and problem Hamiltonians,respectively:

Hdriver = −∑

i

σxi , Hprob =

∑i

hiσzi +

∑i< j

Ji jσziσ

zj , (47)

where the Pauli matrix σx acting on qubit i is denoted by σxi , etc. The transformed Hamiltonian is

H(t) = a(t)[∑

i

σ+i ⊗ ξ

+i (t) + σ−i ⊗ ξ

−i (t)

]+ b(t)Hprob , (48)

where

ξ±i (t) = U†B(t) exp

±∑k

2giλk

ωk(b†i,k − bi,k)

UB(t) . (49)

The Redfield equation corresponding to Eq. (48) is

∂

∂tρS(t) = −i

HS(t) + a(t)∑

i

κiσx, ρS(t)

−∑i,α

[σαi ,Λ

αi (t)ρS(t)

]+ h.c. , (50)

where

Λαi (t) = a(t)

∑β

∫ t

0a(τ)Kαβ

i (t, τ)US(t, τ)σβi U†S(t, τ) dτ (51a)

Kαβi (t, τ) =

⟨ξαi (t)ξβi (τ)

⟩(51b)

κi =⟨ξ±i (t)

⟩(51c)

and

US(t, τ) = T+ exp{−i

∫ t

τ

HS(τ′) dτ′}, HS(t) = b(t)Hprob . (52)

Here Kαβi (t, τ) is the two-point correlation function in the polaron frame [akin to the correlation function defined in

Eq. (9)], and κi corresponds to the first order cumulant generator in Eq. (20) and is also known as the reorganizationenergy; it contributes a Lamb-shift-like term in Eq. (50). It is also worth mentioning that the polaron transforma-tion (44) can be done partially, which means that in Eqs. (44) and (45), α can be summed over a subset of system-bathcoupling terms.

To solve this form of the PTRE in HOQST, the user can define a new correlation function Cαβi (t, τ) = a(t)a(τ)Kαβ(t, τ)

and use the Redfield solver. An alternative approach is to make the Markov approximation in Eq. (51a)∫ t

0a(τ) · · · dτ→ a(t)

∫ ∞

0· · · dτ (53)

and write Eq. (50) in Davies form [60] (see Appendix D for more details). This leads to the same expression as theAME [Eq. (31)], but with different Lindblad operators:

Lω,αi (t) = a(t)∑

εb−εa=ω

〈ψa|σαi |ψb〉 |ψa〉〈ψb| , (54)

where now |ψa〉 is the energy eigenstate of the Hamiltonian HS(t), and the noise spectrum is

γαβi (ω) =

∫ ∞

−∞

Kαβi (t)eiωt dt . (55)

Then the AME solver can be used to solve this Lindblad-form PTRE.

9

3. Numerical techniques

3.1. Redfield backward integrationTo solve the Redfield or Redfield-like master equation (24), one needs to integrate the unitary US backward in

time at each ODE step. Such integrations are computationally expensive for long evolution times and become thebottleneck of the solver. To improve the efficiency of the solver, we introduce an additional parameter Ta as the lowerintegration limit:

Λα(t) =∑β

∫ t

Ta

Cαβ(t − τ)US(t, τ)Aβ(τ)U†S(t, τ) dτ . (56)

To justify this, note first that∥∥∥∥∥∥∫ Ta

0Cαβ(t − τ)US(t, τ)Aβ(τ)U†S(t, τ) dτ

∥∥∥∥∥∥ ≤∫ t

t−Ta

∣∣∣C(τ′)∣∣∣ dτ′ , (57)

where ‖·‖ is any unitarily invariant norm. To obtain this inequality, we perform a change of variable t − τ → τ′

and make use of the fact that the operator Aβ(t) can always be normalized by absorbing a constant factor into thecorresponding bath operator Bβ. Second, note that in most applications the bath correlation function C(τ′) decaysfast compared with the total evolution time. As a result, the r.h.s. of Eq. (57) is small for sufficiently large t. Theneglected part, i.e., the integral over [0,Ta], can thus be safely ignored so long as the r.h.s. of Eq. (57) is below theerror tolerance of the numerical integration algorithm.

The same technique can also be applied to the ULE. The integration limits in Eqs. (28) and (29) can be localizedaround t, i.e. replaced by

∫ t+Ta

t−Ta. However, choosing an appropriate Ta is a process of trial and error. The user needs to

determine its value in a case-by-case manner.

3.2. Precomputing the Lamb shiftHOQST adopts numerical techniques proposed in [61] to calculate Cauchy principle value in the Lamb shift (34).

Instead of evaluating it at each ODE step, to speed up the computations all the ME solvers support precomputing theLamb shift on a predefined grid and use interpolation to fill up the values between the grid points.

3.3. Adiabatic frameFor a typical annealing process, the total annealing time is usually much larger than the inverse energy scale of the

problem

t f �1

mins∈0,1[max(A(s), B(s))

] . (58)

Informally, the frequency of the oscillation between the real and imaginary part of the off-diagonal elements of ρSin the neighborhood of s is positively proportional to both A(s) and B(s). As a consequence, directly solving thedynamics in the Schrödinger picture is challenging because the algorithm needs to deal with the fast oscillationsinduced by the Hamiltonian, thus impacting the step size. HOQST includes an optional pre-processing step to rotatethe Hamiltonian into the adiabatic frame [62]. If the evolution is in the adiabatic limit, the off-diagonal elements ofthe density matrix in this frame should approximately vanish. The fast oscillation is absent and a large step size can betaken by the ODE solver. This technique provides advantages if the user wishes to repeatedly solve the same problemwith different parameters.

We briefly summarize the adiabatic frame transformation, following Ref. [63]. For general multi-qubit annealing,the system Hamiltonian given in Eqs. (46b) and (47) can be formally diagonalized and written as

HS(s) =∑

n

En(s) |n〉〈n| , (59)

where {|n(s)〉} is the instantaneous energy eigenbasis [eigenvectors of HS(s)], and s = t/t f is the dimensionless instan-taneous time. We assume that HS(s) is real for all s. The system density matrix can be written in the instantaneousenergy eigenbasis:

ρ(s) =∑nm

ρnm |n〉〈m| . (60)

10

Solvers Descriptionsolve_schrodinger Schrödinger equationsolve_von_neumann Von Neumann equationsolve_unitary Closed-system unitarysolve_redfield Redfield equationsolve_lindblad Lindblad equationsolve_ame Adiabatic master equationsolve_cgme Coarse-grained master equationsolve_ule Universal Lindblad equation

Table 1: The common solver APIs. The PTRE can be solved using solve_redfield and solve_ame by defining the corresponding polaronframe Hamiltonian and bath.

We call the associated matrix ρ =[ρnm

]the density matrix in the adiabatic frame. It can shown [63] that

ρnm = −it f (En − Em)ρnm − i

−i∑n′,n

⟨n∣∣∣n′⟩ ρn′m + i

∑m′,m

ρnm′⟨m′

∣∣∣m⟩ , (61)

where the dot denotes differentiation with respect to s. I.e., ρ obeys the von Neumann equation ˙ρ = −i[H, ρ

], with the

effective Hamiltonian

H =

t f E0 −i

⟨0∣∣∣1⟩ . . .

i⟨0∣∣∣1⟩ t f E1 . . ....

. . .

. (62)

3.4. Quantum trajectories method

HOQST implements a quantum-trajectory solver for the AME based on Ref. [31]. Using the native distributedmemory parallel computing interface of both Julia and DifferentialEquations.jl, the quantum-trajectorysimulations can take advantage of HPC clusters with minimum changes in the code. In addition, classical 1/ f noisecan be infused into the AME trajectory solver to generate the hybrid dynamics described in Eq. (42).

4. Capabilities of HOQST and comparison with other quantum simulators

HOQST provides two sets of APIs for direct and trajectory solvers. The APIs for direct solvers, which are listedin Table 1, adopt the naming convention solve_name, where name is the type of equation to be solved. The APIsfor the quantum-trajectory solver is built around a single function build_ensembles. Calling this function buildsan EnsembleProblem object that can be distributed to remote workers for parallel simulation [64]. An example ofa complete workflow is given in Sec. 5.

Recent developments in the field of QC have led to an explosion of quantum software platforms, such as [65–68]. A review of various major platforms is given in [69]. Because some of these platforms include the capabilityto simulate noisy quantum circuits, we briefly compare their respective noise models and solver types in Table 2.Furthermore, for packages that support arbitrary time-dependent Hamiltonian and rely on MEs as solvers, we list theircompatible MEs in Table 3.

3The frequency form is supported via the one-sided AME.

11

Table 2: Comparison chart for noise models and solver typesHOQST QuTiP [35, 36] Qiskit [65] Qiskit Pulse [66]

QC model t-dependent Hamiltonian t-dependent Hamiltonian circuit t-dependent HamiltonianNoise model system-bath coupling system-bath coupling Kraus map constant Lindblad operatorSolver type ME ME noisy gates ME

pyQuil [67] ProjectQ [68]QC model circuit circuitNoise model Kraus map Stochastic noiseSolver type Noisy gates N/A

Table 3: Comparison chart for ME supportME HOQST QuTiP Qiskit Pulseconstant Lindblad equationRedfield equation time/frequency form3 frequency form ×

AME (t-dependent Lindblad) × ×

CGME × ×

ULE × ×

PTRE × ×

Floquet-Markov formalisms [70] × ×

Stochastic Schrodinger spin-fluctuator non-Hermitian effective Hamiltonian ×

5. Examples

HOQST has a large collection of tutorials located at a dedicated Github repo [71]. Tables 4 and 5 list theintroductory-level and advanced tutorials, respectively.

Notebook Description01-closed_system Introductory tutorial for solving closed system dynamics02-lindblad_equation Time-independent Lindblad equation03-single_qubit_ame Introductory open system simulation tutorial based on [29]04-polaron_transformed_redfield Polaron transformed Redfield equation [Eq. (50)]05-CGME_ULE Coarse-grained ME and universal Lindblad equation [Eqs. (25) and (27)]06-spin_fluctuators Classical 1/ f noise simulation [Eq. (37)]

Table 4: List of introductory-level tutorials

Table 5: List of advanced tutorials

Notebook Descriptionhamiltonian/01-custom_eigen Using user-defined eigendecomposition routineredfield/01-non_positivity_redfield An example of non-positivity in the Redfield equationredfield/02-redfield_multi_axis_noise Solving the Redfield equation with multi-axis noiseadvanced/01-ame_spin_fluctuators Adiabatic master equation with spin-fluctuators [Eq. (42)]advanced/02-3_qubit_entanglement_witness 3-qubit entanglement witness experiment

As an illustrative yet non-trivial example, we next discuss the simulation of a three-qubit quantum annealing

12

entanglement witness experiment [72, 73]. This example will serve to illustrate the typical workflow of HOQST,which includes:

1. Define the system Hamiltonian and initial state.2. Define the coupling operators and bath objects.3. (Optional) Combine coupling and bath objects into InteractionSet if there are multiple system-bath cou-

plings.4. Combine these objects into an Annealing4 object.5. (Optional) Construct EnsembleProblem from Annealing object.6. Call the solver.7. Handle the solution.

5.1. Entanglement witness experiment modelingThe entanglement witness experiment was proposed to provide evidence of entanglement in a D-Wave quantum

annealing device [72]. An open system analysis of these experiments was performed using the AME in Ref. [73]. Thecrux of the experiment is actually a form of tunneling spectroscopy [74], where the goal is to find the energy gaps ofthe Hamiltonian −a(s)

∑i σ

ix + b(s)HIsing. This is done by observing the location of a peak in the tunneling rate as

measured using a probe qubit.The Hamiltonian of the 3-qubit-version of the experiment is

HS(τ) = −a(s(τ))2∑

i=1

σxi − a(sp(τ))σx

p + b(s(τ))HIsing , (63)

where a(s) and b(s) are the annealing schedules, and s(τ) and sp(τ) are functions of the dimensionless time τ =

t/t f . The Hamiltonian consists of two system qubits coupled to an ancilla system qubit, as shown in Fig. 1. Theaforementioned location of the tunneling rate peak can be controlled by varying hP, and this information can be usedto extract the energy gaps as a function of s.

We refer interested readers to Refs. [72, 73] for more information. The annealing schedules and annealing pa-

1

h1 = −J1P

2

h2 = 0

p

−hP

J1P JS

Figure 1: The Ising Hamiltonian HIsing in Eq. (63) of the 3-qubit entanglement witness experiment. The goal is to demonstrate entanglementbetween qubits 1 and 2, by probing the ancilla qubit (denoted p). In our simulations, J1P and JS are fixed at J1P = −1.8, JS = −2.5.

rameters are illustrated in Fig. 2. The particular chip used in the experiments is described in Ref. [72]. To extractthe tunneling rate, we first perform the simulation with the initial ‘all-one’ state |ψ(0)〉 = |1〉⊗3 for different hp andt2 = τ2t f values. The population of the all-one state at the end of anneal is then obtained as a function of hp and

τ2: P|1〉(hp, t2) =∣∣∣∣⟨ψ(t f )

∣∣∣ψ(0)⟩∣∣∣∣2. Lastly, we fit P|1〉(hp, t2) to the function aebt2 + cedt2 , from which the rate Γ can be

estimated [75]

Γ(hp) = −∂P|1〉∂t2

∣∣∣∣∣t2=0

= −ab − cd . (64)

For the open system model, we follow the strategy used in Ref. [73]. We assume the qubits are coupled toindependent baths

H(t) = HS(t) +

2∑i=1

giσzi ⊗ Bi + gpσ

pz ⊗ Bp + HB , (65)

4It is called Annealing for historical reasons. Evolution is also supported.

13

0.00 0.25 0.50 0.75 1.000

1

2

3

4

5

(a)

0.00 0.25 0.50 0.75 1.00

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(b)

Figure 2: (a) Annealing schedules. (b) Annealing parameters as functions of dimensionless time. There are three stages in the experiment: (1) firstevolve s(τ) from 1 to s∗ for τ ∈ [0, τ1] and then evolve sp(τ) from 1 to s∗p for τ ∈ [τ1, 2τ1]; (2) pause for a time τ2; (3) reverse the first stage. In oursimulation, we choose s∗ = 0.339, τ1 ∗ t f = 10µs following [73] and s∗p = 0.612 such that 2A(s∗p) ≈ 1MHz [72]. The value of τ2 is varied to obtainthe tunneling rate.

but the bath coupling to the probe qubit gp is much stronger than the coupling to the two system qubits g1 = g2 = gs.In addition, we assume the bath is Ohmic [Eq. (12)] with coupling strength ηg2

s/~2 = 1.2732×10−4, cut-off frequencyfc = 4GHz, and temperature T = 12.5mK. We run the simulation with different models of Bp:

1. Ohmic bath with interaction strength gp = 10gs: Sec. 5.2.2. Hybrid Ohmic bath whose coupling strength to the Ohmic component is gp = 10gs and varying macroscopic

resonant tunneling (MRT) width: Sec. 5.3.

The following pseudo-code block demonstrates the workings of an AME simulation in HOQST, following theworkflow above. The exact codes are hosted in the tutorial repo [71].

jp = -1.8; js = -2.5; hp = 0.5;# the natural unit of time in HOQST is (ns)τ1 = 10000; τ2 = 5000; tf = 4*τ1 + τ2;# construct each term in the HamiltonianH_list = [-σx ⊗ σi ⊗ σi, -σi ⊗ σx ⊗ σi - σi ⊗ σi ⊗ σx,

local_field_term([hp, -jp, 0], [1, 2, 3], 3) + two_local_term([jp, js], [[1, 2], [2, 3]], 3)]

# we assume here s and sp are already defined according to figure 2f_list = [(t)->A(sp(t)), (t)->A(s(t)), (t)->B(s(t))]# define the system HamiltonianH = DenseHamiltonian(f_list, H_list)# define the initial stateu0 = PauliVec[3][2] ⊗ PauliVec[3][2] ⊗ PauliVec[3][2]# define the system-bath couplingcoupling_ohmic = ConstantCouplings(["10ZII", "IZI", "IIZ"], unit = ~)bath_ohmic = Ohmic(1.2732e-4, 4, 12.5)# combine the components into an Annealing objectannealing = Annealing(H, u0, coupling = coupling, bath = bath)# call the solversol = solve_ame(annealing, tf, alg = Tsit5(), ω_hint = range(-15,15,length=200),

d_discontinuities = [τ1, 2*τ1, 2*τ1+τ2, 3*τ1+τ2],saveat = range(0,tf,length=100), maxiters = 1e8, reltol = 1e-5

)

The solution sol is returned as an ODESolution type, the details of which are listed in Ref. [76].

5.2. AME with an Ohmic bathSimulations using two different flavors of the AME, the one-sided AME (35) and the Lindblad form AME (31),

are shown in Fig. 3. The results demonstrate that these two AME flavors only differ significantly near the small gapregion.

14

5.3. PTRE

To use the PTRE for the entanglement witness problem, we perform the polaron transformation (44) only on theprobe qubit and leave the system qubits unchanged. After following through the same procedure as detailed in Sec. 2.9and Appendix D, the following ME can be derived:

∂

∂tρS(t) = −i

[HS(t) + HLS(t), ρS(t)

]+LA

[ρS(t)

]+LP

[ρS(t)

], (66)

where

HS(t) = −a(t)2∑

i=1

σxi + b(t)HIsing , (67)

and the Liouville operators LA and LP corresponds to the AME part and PTRE part of this equation, respectively:

LA(ρ) =

2∑i=1

∑ω

γ(ω)[Lω,i(t)ρL†ω,i(t) −

12

{L†ω,i(t)Lω,i(t), ρ

}](68)

LP(ρ) =∑

α∈{+,−}

∑ω

γP(ω)[Lω,αp (t)ρLω,α†p (t) −

12

{Lω,α†p (t)Lω,αp (t), ρ

}], (69)

where the Lindblad operators are defined in Eq. (32) and Eq. (54) respectively. The function γ(ω) is the standardOhmic spectrum and γP(ω) is the polaron frame spectrum with a hybrid Ohmic form [32, 59] discussed in AppendixE. We provide the explicit form of γP(ω) here

γP(ω) =

∫K(t)eiωt dt =

∫dx2π

GL(ω − x)GH(x) dx , (70)

where

GL(ω) =

√π

2W2 exp[−

(ω − 4εL)2

8W2

], (71)

andGH(ω) =

4γ(ω)ω2 + 4γ(0)2 . (72)

GL(ω) is the contribution of low frequency component, characterized by the MRT width W. Because W and εL areconnected through the fluctuation-dissipation theorem we have W2 = 2εLT ; thus, hybridizing low frequency noisewith an Ohmic bath introduces one additional parameter.

5.4. Results

The tunneling rates obtained via different ME simulations are compared with the experimental results [72] inFig. 3. In comparison with the AME, the PTRE exhibits a larger Gaussian line-width broadening and closer agreementwith the experimental data. There are two additional observations, which will be the subject of future studies:

1. If we increase W while fixing T , the curve is stretched to the right. This the result of the fluctuation-dissipationtheorem, where εL scales quadratically with W. Such a shift can be compensated by increasing the temperaturetogether with W [see the black dashed curve in Fig. (3)]. To better match the experimental curve, we need ahigher T than the reported value in Ref. [72].

2. There is still a mismatch between the theoretical and experimental amplitudes of the tunneling rate curves,whose physical origin is currently unclear.

15

Figure 3: Tunneling rates obtained via different MEs compared with the experimental results (extracted from [72] using WebPlotDigitizer [77]).The bath coupled to the system qubits is Ohmic with parameters: ηg2

s/~2 = 1.2732×10−4, fc = 4GHz and T = 12.5mK. The corresponding modelsand parameters of the bath coupled to the probe qubit used for different simulations are (1) AME: Ohmic with gp = 10gs. (2) PTRE: hybrid-Ohmicwith gp = 10gs and various W, T values. The cut-off frequency fc is the same across different models.

6. Conclusions

We presented a software package called Hamiltonian Open Quantum System Toolkit (HOQST). It is user-friendlyand written in Julia. It supports various master equations with a wide joint range of applicability, as well as stochasticHamiltonians to model 1/ f noise. We briefly reviewed the theories behind these master equations and illustrated howto use HOQST to simulate the open system dynamics in a 3-qubit entanglement witness experiment. We also derivednew error bounds for the polaron-transformed Redfield equation.

We expect HOQST to be useful for researchers working in the field of open quantum systems, dealing with systemsgoverned by time-dependent Hamiltonians. HOQST provides both basic and advanced numerical simulation tools inthis area, which can be applied to simulate superconducting qubits of all types, trapped ions, NV centers, siliconquantum dot qubits, etc. Future releases of HOQST will expand both the suite of open system models and range ofquantum control and computation models it supports.

Acknowledgments

The authors are grateful to Jenia Mozgunov, Tameem Albash, Ka Wa Yip and Vinay Tripathi for useful discus-sions and feedback. The authors also thanks Grace Chen for the HOQST logo design. This research is based uponwork (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Re-search Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. ArmyResearch Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authorsand should not be interpreted as necessarily representing the official policies or endorsements, either expressed orimplied, of the ODNI, IARPA, DARPA, ARO, or the U.S. Government. The U.S. Government is authorized to re-produce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Theauthors acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern Californiafor providing computing resources that have contributed to the research results reported within this publication. URL:https://carc.usc.edu.

16

Appendix A. The polaron transformation

Appendix A.1. Two different formulationsWe illustrate two different formulations of the polaron transformation via the simplest Hamiltonian

H = HS ⊗ IB + HSB + IS ⊗ HB

≡ (εσz + ∆σx) ⊗ IB + gσz ⊗∑

k

λk(b†k + bk) + IS ⊗

∑k

ωkb†kbk

. (A.1)

The standard polaron transformation rotates this Hamiltonian via the unitary [2]

Up = exp

−iσz

∑k

gλk

iωk(b†k − bk)

(A.2)

which leads to the interaction picture Hamiltonian

H = εσz + ∆ξ+σ+ + ∆ξ−σ− + HB , (A.3)

where

ξ± = exp

±2i∑

k

gλk

iωk(b†k − bk)

. (A.4)

The only quantities that matter in the 2nd order TCL formalism (20) are the average and two-point correlation functionof ξ±(t) given by

〈ξ±(t)〉 = TrB{U†

B(t)ξ±UB(t)ρB} = TrB{ξ±ρB} (A.5a)⟨ξα(t)ξβ(0)

⟩= TrB{U

†

B(t)ξαUB(t)ξβρB} , (A.5b)

where α, β ∈ {+,−} and UB(t) = exp{−iHBt}. To arrive at the second equality in Eq. (A.5a) we used [ρB,UB] = 0,which is true for ρB in a Gibbs state, as in Eq. (3).

An alternative approach [32] is to first rotate the Hamiltonian (A.1) w.r.t. HB and then rotate it again by

Ud(t) = T+ exp{−iσz

∫ t

0B(τ) dτ

}, (A.6)

whereB(τ) = U†B(τ)g

∑k

λk(b†k + bk)UB(τ) . (A.7)

It is worth noting that in this formalism the bath does not need to be bosonic. Let us define

Ξ±(t) = T+ exp{±i

∫ t

0B(τ) dτ

}. (A.8)

Then the effective Hamiltonian in this two-fold interaction picture is

H = εσz + ∆ζ+(t)σ+ + ∆ζ−(t)σ− , (A.9)

whereζ±(t) = Ξ

†∓(t)Ξ±(t) . (A.10)

Again, the 2nd order ME only depends on

〈ζ±(t)〉 = TrB{Ξ†∓(t)Ξ±(t)ρB} (A.11)⟨

ζα(t1)ζβ(t2)⟩

= TrB{Ξ†

α(t1)Ξα(t1)Ξ†β(t2)Ξβ(t2)ρB} , (A.12)

where α means the opposite symbol of α in {+,−}. Appendices D.4 and D.5 in [32] give detailed calculations forEq. (A.11) and (A.12).

17

Appendix A.2. Correlation function in polaron frameWe now discuss Eqs. (A.5a, A.5b, A.11, A.12) for the spin-boson model. We first introduce the bath spectral

function [78]J(ω) =

∑k

λ2kδ(ω − ωk) , (A.13)

which is usually taken into the continuous limit. The standard result [78, 79] for Eqs. (A.5a) and (A.5b) are

κ = 〈ξ±(t)〉 = exp

−2g2∑

k

λ2k

ω2k

coth(βωk/2)

= exp{−2g2

∫ ∞

0

J(ω)ω2 coth(βω/2) dω

}, (A.14)

and

〈ξα(t)ξα(0)〉 = κ2(e−φ(t) − 1) (A.15a)⟨ξα(t)ξβ(0)

⟩= κ2(eφ(t) + 1) , (A.15b)

where

φ(t) = 4g2∫ ∞

0dω

J(ω)ω2

(coth

(βω

2

)cos(ωt) − i sin(ωt)

). (A.16)

For spectral functions which lead to κ → 0 and φ(t) → ∞ (e.g., an Ohmic bath), κ2eφ(t) converges to a finite number.Thus the standard result for an Ohmic bath is

〈ξ+(t)ξ+(0)〉 = 〈ξ−(t)ξ−(0)〉 = 0 (A.17a)〈ξ+(t)ξ−(0)〉 = 〈ξ−(t)ξ+(0)〉 = exp(−4Q2(t) − 4iQ1(t)) , (A.17b)

where Q2(t) and Q1(t) [78] are

Q1(t) ≡ g2∫ ∞

0

J(ω)ω2 sin(ωt) dω (A.18a)

Q2(t) ≡ g2∫ ∞

0

J(ω)(1 − cos(ωt))ω2 coth(β~ω/2) dω . (A.18b)

By substituting the Ohmic spectral function J(ω) = ηωeω/ωc , the explicit form of Q1 and Q2 can be calculated. Herewe take another approach by noticing that

− Q2(t) − iQ1(t) =

∫ t

0

∫ 0

−∞

C(t1, t2) dt1 dt2 , (A.19)

where C(t1, t2) is the bath correlation function defined in Eq. (9) that can also be expressed in terms of the spectralfunction

C(t1, t2) = g2∫ ∞

0dω J(ω) coth(βω/2) cos(ω(t1 − t2)) − iJ(ω) sin(ω(t1 − t2)) . (A.20)

Thus Eq. (A.17b) can be rewritten as

〈ξ+(t)ξ−(0)〉 = 〈ξ−(t)ξ+(0)〉 = exp{

4∫ ∞

−∞

γ(ω)e−iωt − 1ω2 dω

}, (A.21)

where γ(ω) is the spectral density defined in Eq. (11).For Eq. (A.11) and (A.12), we follow Ref. [32] and write down the the result here for comparison. The 1st order

average is〈ζ±(t)〉 = exp{−4Q2(t) − 4iQ1(t)} , (A.22)

which is supposed to decrease exponentially fast for t > 0. The two-point correlation functions are

〈ζ+(t)ζ+(0)〉 = 〈ζ−(t)ζ−(0)〉 = 0 (A.23a)

〈ζ+(t1)ζ−(t2)〉 = 〈ζ−(t1)ζ+(t2)〉 = exp{− 4Q2(t) − 4iQ1(t) + 4i

[C(t1) −C(t2)

]}. (A.23b)

The difference between Eqs. (A.17b) and (A.23b) are easily noticeable. The physical origins and significance of thisdifference is a subject for future studies.

18

Appendix B. Error analysis

We present an error analysis based on the standard form of the polaron transformation (A.2). Our strategy is todivide the error into separate parts and use Lemma 1 in Ref. [4] to estimate the total error of the master equation.Besides those already presented in [4], there are two additional error terms in the PTRE, the physical origin of whichare the inhomogeneous terms resulting from the breakdown of the factorized initial condition by Up:

U†pρS(0) ⊗ ρBUp , ρS(0) ⊗ ρB . (B.1)

In addition to the projector defined in Eq. (18), we define its complementary Q ≡ I − P. The standard projectionoperator technique [13] leads to

∂

∂tPρ(t) =

∑n

Kn(t)Pρ(t) +∑

n

In(t)Qρ(0) , (B.2)

where the rightmost expression is the contribution of the inhomogeneous terms. Ref. [80] calculates In(t) up ton = 4 and shows they are identical to those for Kn(t) except that the final P in each term is replaced by a Q. Herewe conjecture that this is also true for n > 4, and then use the same error bound as in Ref. [4] (called the Bornapproximation error there), to bound the error of truncating to the order of I2:

E(∞)I =

∥∥∥∥∥∥∥∞∑

n=1

In(t)Qρ(0) −2∑

n=1

In(t)Qρ(0)

∥∥∥∥∥∥∥1

. (B.3)

To simplify the notation, we define ρtrue(t) as the solution of Eq. (B.2) when n → ∞. The PTRE error bound can bewritten as the sum of three terms:

‖ρtrue(t) − ρPTRE(t)‖1 ≤ EBM + E(∞)I + E

(2)I , (B.4)

where EBM is the error bound presented in Ref. [4]:5

EBM ≤ O(e

12tτS B

τB

τS B

)lnτS B

τB, (B.5)

E(∞)I is given in Eq. (B.3) and E(2)

I =∥∥∥∑2

n=1 In(t)Qρ(0)∥∥∥

1. Here we argue that E(∞)I can be bounded by the the same

expression given in (B.5). Because Qρ(0) = ρ(0) − Pρ(0), the error (B.3) can be bounded by the triangle inequality

E(∞)I ≤

∥∥∥∥∥∥∥∞∑

n=1

In(t)ρ(0) −2∑

n=1

In(t)ρ(0)

∥∥∥∥∥∥∥1

+

∥∥∥∥∥∥∥∞∑

n=1

In(t)Pρ(0) −2∑

n=1

In(t)Pρ(0)

∥∥∥∥∥∥∥1

, (B.6)

where the second term automatically satisfies the bound given in Eq. (B.5). To bound the first term, we first write outρ(0) explicitly for the case of a single system qubit and HI = gσz ⊗

∑k λk(b†k + bk):

ρ(0) = U†pρS(0) ⊗ ρBUp (B.7a)

=(eiD |0〉〈0| + e−iD |1〉〈1|

)ρS(0) ⊗ ρB

(e−iD |0〉〈0| + eiD |1〉〈1|

)(B.7b)

= ρ00S eiDρBe−iD + ρ11

S e−iDρBeiD + ρ10S e−iDρBe−iD + ρ01

S eiDρBeiD , (B.7c)

where Up was defined in Eq. (44), and

D =∑

k

gλk

iωk(b†k − bk) , ρ

i jS = 〈i|ρS(0)| j〉 |i〉〈 j| . (B.8)

5The error bound in Ref. [4] includes the error due to changing∫ t→

∫ ∞ in the Redfield equation. We employ Eq. (B.5) despite not makingthis approximation, because the upper bound remains valid without it.

19

Using our conjecture, the formal expansion of In(t) is the same as Kn(t) with the average operation in the rightmostterm being replaced by

〈·〉mn = TrB

{· ρmn

B

}, (B.9)

where ρmnB = eniDρBemiD and m, n ∈ {−1,+1}. To illustrate this point, let us consider one term in the 4th order

expansion I4(t)ρ(0): ∫ t

0

∫ t1

0

∫ t2

0TrB

[L(t)L (t3) ρB

]TrB

[L (t1) L (t2) ρ(0)

]dt1 dt2 dt3 . (B.10)

After substituting Eq. (B.7c) into the above expression and expanding the commutators, one term in the summation is

F++ = ∆4∫ t

0

∫ t1

0

∫ t2

0dt1 dt2 dt3 〈ξ+(t)ξ−(t3)〉 〈ξ+(t1)ξ−(t2)〉++ σ

+σ−σ+σ−ρ01S , (B.11)

where we use +,− in the subscript instead of +1,−1 to simplify the notation. Thus, the next step is to calculate theaverage and two point correlation function under the new “average” operator (B.9):

〈ξ±(t)〉mn = TrB{eniDU†B(t)ξ±UB(t)emiDρB} (B.12a)⟨ξα(t1)ξβ(t2)

⟩mn

= TrB{eniDU†B(t1)ξαUB(t1)U†B(t2)ξβUB(t2)emiDρB} . (B.12b)

Eq. (B.12) can be explicitly carried out using the following identities:

ξ±(t) ≡ U†B(t)ξ±UB(t) = exp

±2iU†B(t)∑

k

gλk

iωk

(b†k − bk

)UB(t)

= exp

±2i∑

k

gλk

iωk

(b†keiωk t − bke−iωk t

) (B.13a)[b†keiωk t − bke−iωk t, b†k′ − bk′

]= 2i sin(ωkt)δkk′ , (B.13b)

and the Baker-Campbell-Hausdorff (BCH) formula

eXeY = eX+Y+ 12 [X,Y]+... . (B.14)

After some algebraic manipulations, we get

〈ξ±(t)〉mn = e±4in

∑k

g2λ2k

ω2k

sin(ωk t) ⟨ξ±(t)ei(n+m)D

⟩=

f ±n (t) 〈ξ±(t)〉 m , nf ±n (t) 〈ξ±(t)ξl(0)〉 m = n, l = sgn(n)

, (B.15)

where f αn (t) = exp{4iαn

∑k

g2λ2k

ω2k

sin(ωkt)}. For simplicity, we consider only the Ohmic bath case, for which 〈ξ±(t)〉 = 0

and 〈ξ±(t)ξl(0)〉 decay exponentially for t > 0. As a result, we can ignore the first order term 〈ξ±(t)〉mn ≈ 0. The two-point correlation function is

⟨ξα(t1)ξβ(t2)

⟩mn

= f αn (t1) f βn (t2)⟨ξα(t1)ξβ(t2)ei(n+m)D

⟩=

f αn (t1) f βn (t2)⟨ξα(t1)ξβ(t2)

⟩m , n

f αn (t1) f βn (t2)⟨ξα(t1)ξβ(t2)ξl(0)

⟩m = n, l = sgn(n)

.

(B.16)where the second line equals 0 because 〈ξ±(t)〉 = 0 and the noise is Gaussian. The above result can be generalized tothe multi-point correlation function

⟨ξα1 (t1)ξα2 (t2) · · · ξαk (tk)

⟩mn =

f α1n (t1) f α2

n (t2) · · · f αkn (tk)

⟨ξα1 (t1)ξα2 (t2) · · · ξαk (tk)

⟩m , n

f α1n (t1) f α2

n (t2) · · · f αkn (tk)

⟨ξα1 (t1)ξα2 (t2) · · · ξαk (tk)ξl(0)

⟩m = n, l = sgn(n)

. (B.17)

20

For the m , n case, the correlation function under 〈·〉mn is the same as the correlation function under 〈·〉 up to anadditional phase factor. For the m = n case, the right hand side of Eq. (B.17) can be decomposed into two-pointcorrelation functions by Wick’s theorem. After the decomposition, each term would appear inside the integral as∫ t1

0dt2

∫ t2

0dt3 · · ·

∫ tk−1

0dtk

⟨ξα1 (tα1 )ξα2 (tα2 )

⟩· · ·

⟨ξαs (ts)ξl(0)

⟩. (B.18)

Again, because the particular pair⟨ξαs (ts)ξl(0)

⟩decreases exponentially for ts > 0 and thus has approximately zero

overlap with other two point correlation functions under the integral, we ignore all these terms. Finally, we can sumup every term over the indices m and n. For example, for terms that have the same form as Eq. (B.11), the summationis

F++ + F+− + F−+ + F−− = ∆4∫ t

0

∫ t1

0

∫ t2

0dt1 dt2 dt3 〈ξ+(t)ξ−(t3)〉 〈ξ+(t1)ξ−(t2)〉σ+σ−σ+σ−o , (B.19)

where o = ρ00S f +− (t1) f −− (t2) + ρ11

S f ++ (t1) f −+ (t2). Eq. (B.19) can be bounded by

‖F++ + F+− + F−+ + F−−‖1 ≤ ∆4∫ t

0

∫ t1

0

∫ t2

0dt1 dt2 dt3 |〈ξ+(t)ξ−(t3)〉||〈ξ+(t1)ξ−(t2)〉| , (B.20)

where we make use of the fact |o| ≤ |ρ00S | + |ρ

11S | = 1. Formally applying the same technique to every term after

expanding In(t)ρ(0), allows us to show that the first part of Eq. (B.6) also satisfies the error bound given in Eq. (B.5).6

In conclusion, the error due to truncation of the inhomogeneous parts to second order has the same scaling as theerror due to truncation of the homogeneous parts [Eq. (14)]. As a result, the total truncation error can be combinedusing a single big-O notation:

‖ρtrue(t) − ρPTRE(t)‖1 ≤ O(τB

τS Be12t/τS B

)ln

(τS B

τB

)+ E

(2)I , (B.21)

We discuss the timescales in the polaron frame in Appendix C. Before proceeding, we remark that:

1. The same analysis also applies to the multi-qubit PTRE described in Eq. (50) by replacing ρi jS with ρi j

k

ρi jk = ZkρS(0)Zk , (B.22)

where k is the index for different qubits.2. The analysis is also applicable for time-dependent ∆(t). In this case, the corresponding two-point correlation

function is defined as Cαβ(t1, t2) = ∆(t1)∆(t2)⟨ξα(t1)ξβ(t2)

⟩, which can be upper bounded by∣∣∣Cαβ(t1, t2)

∣∣∣ ≤ ∆2m

∣∣∣∣⟨ξα(t1)ξβ(t2)⟩∣∣∣∣ , (B.23)

where ∆m = maxt∈[0,t f ] ∆(t). This introduces a constant ∆m in the definition of the timescales.3. The error bound we analyzed in this section does not include the error due to ignoring the first and second order

inhomogeneous terms E(2)I (which is usually the case in the standard PTRE treatment). Numerical studies show

that they can be ignored if ρS(0) is diagonal [42]. More rigorous error bounding is an interesting topic for futurestudy.

Appendix C. Time scales

The time scales given in Eq. (13) can be computed directly for the polaron frame two-point correlation func-tion (A.17b). Because the error bound in Eq. (14) scales with 1/τS B and τB/τS B, we explicitly write out those two

6In this case we do not have the Markov approximation error and the error due to changing the integration limit. But the bound can still berelaxed to the form of Eq. (B.5).

21

quantities

1τS B

= ∆2m

∫ ∞

0|K(τ)| dτ =

∫ ∞

0exp

{−4g2

∫ ∞

0

J(ω)(1 − cos(ωτ))ω2 coth(β~ω/2) dω

}dτ (C.1)

τB

τS B= ∆2

m

∫ t f

0τ|K(τ)| dτ =

∫ ∞

0τ exp

{−4g2

∫ ∞

0

J(ω)(1 − cos(ωτ))ω2 coth(β~ω/2) dω

}dτ , (C.2)

where ∆m = maxt∈[0,t f ] ∆(t). We can see that, when the system-bath coupling strength is sent to infinity, both of thesequantities go to zero: 1/τS B → 0 and τB/τS B → 0 as g→ ∞. Thus, the PTRE works in the strong coupling regime.

Appendix D. Lindblad form of the PTRE via the adiabatic approximation

In this section, we derive an adiabatic form of the PTRE from Eq. (50):

∂

∂tρS(t) = −i

HS(t) + a(t)∑

i

κiσx, ρS(t)

−∑i

[σαi ,Λ

αi (t)ρS(t)

]+ h.c. , (D.1)

where

Λαi (t) = a(t)

∑β

∫ t

0a(τ)Kαβ

i (t − τ)US(t, τ)σβi U†S(t, τ) dτ (D.2a)

Kαβi (t − τ) =

⟨ξαi (t − τ)ξβi (0)

⟩(D.2b)

κi =⟨ξ±i (t)

⟩, (D.2c)

and

US(t, τ) = T+ exp{−i

∫ t

τ

HS(τ′) dτ′}. (D.3)

Three approximations need to be applied to obtain the final result:

Markov approximationThe first step is to move the function a(t) in Eq. (D.2a) outside the integral∫ t

0a(τ) · · · dτ→ a(t)

∫ t

0· · · dτ , (D.4)

then perform a change of variable t − τ→ τ. Eq. (D.2a) becomes:

Λαi (t) = a2(t)

∑β

∫ t

0Kαβ

i (τ)US(t, t − τ)σβi U†S(t, t − τ) dτ . (D.5)

Finally, the integration limit is taken to infinity:∫ t

0 →∫ ∞

0 .

Adiabatic approximationA detailed description of the adiabatic approximation is given in Ref. [3]. The core idea is to rewrite the unitary

in Eq. (D.5) asUS(t, t − τ) = US(t, 0)U†S(t − τ, 0) , (D.6)

and then replace US with an appropriate adiabatic evolution

US(t − τ, 0) ≈ eiτHS(t)UadS (t, 0) (D.7a)

US(t, 0) ≈ UadS (t, 0) , (D.7b)

22

where UadS (t, 0) is the ideal adiabatic evolution

UadS

(t, t′

)=

∑a

|εa(t)〉⟨εa

(t′)∣∣∣ e−iµa(t,t′) , (D.8)

with a phase

µa(t, t′

)=

∫ t

t′dτ

[εa(τ) − φa(τ)

]. (D.9)

In the above expressions, |εa(t)〉 is the instantaneous eigenstate of HS(t) and φa(t) = i 〈εa(t) | εa(t)〉 is the Berryconnection. After this procedure, Eq. (D.5) becomes:

Λαi (t) = a2(t)

∑β,a,b

Γαβi (ωba)Lab

iβ (t) , (D.10)

where ωba = εb − εa, andLab

iβ (t) = 〈εa(t)|σβi |εb(t)〉 |εa(t)〉〈εb(t)| . (D.11)

Γαβi is the one-sided Fourier transform of the correlation function

Γαβi (ω) =

∫ ∞

0dt Kαβ

i (t)eiωt . (D.12)

Substituting Eq. (D.10) into Eq. (D.1), we have the one-sided adiabatic PTRE.

Rotating wave approximation(RWA)To perform the RWA, we first need to move the one-sided adiabatic PTRE into the interaction picture with respect

to HS(t) via the approximated unitary in Eq. (D.7b). The non-Hamiltonian part of Eq. (D.1) is transformed into

− U†S(t)∑

i

[σαi ,Λ

αi (t)ρS(t)

]US(t) + h.c. ≈

∑a,b,a′,b′

e−i[µba(t,0)+µb′a′ (t,0)]a2(t)∑

i

Γαβi (ω)

{〈εa(t)|σβi |εb(t)〉Πab(0) ≈ρS(t)Πa′b′ (0) 〈εa′ (t)|σαi |εb′ (t)〉

− 〈εa′ (t)|σαi |εb′ (t)〉 〈εa(t)|σβi |εb(t)〉Πab(0)Πa′b′ (0) ≈ρS(t)}

+ h.c. , (D.13a)

where Πab(0) is a shorthand notation for |εa(0)〉〈εb(0)|. The RWA amounts to keeping only terms where either a′ = b,b′ = a or a = b, a′ = b′. After the fast-oscillating terms are ignored, we move back to the polaron frame and makeuse of Eq. (36) to derive the Lindblad-form PTRE:

˙ρS(t) = −i

HS(t) + HLS(t) + a(t)∑

i

κiσx, ρS(t)

+∑i,α,β

∑ω

γαβi (ω)

[Lω,βi (t)ρS(t)Lω,α†i (t) −

12

{Lω,α†i (t)Lω,βi (t), ρS(t)

}],

(D.14)where the new Lindblad operators are given by

Lω,αi (t) = a(t)∑

εb−εa=ω

〈εa(t)|σαi |εb(t)〉 |εa(t)〉〈εb(t)| . (D.15)

The notation α again means the opposite symbol of α in {+,−}. Since σ±i are Hermitian conjugates of each other:

L−ω,αi (t) =[Lω.αi (t)

]†. (D.16)

Finally, the Lamb shift term isHLS(t) =

∑i,α,β

∑ω

Lω,α†i Lω,βi (t)S αβi (ω) . (D.17)

Because the techniques used here are the same as those in [3, 4], the error bounds in these references are still valid.

23

Appendix E. Hybrid noise

The two point correlation function in the polaron frame [Eq. (A.21)] can be extended to a hybrid noise model [59].The core assumption here is that the bath spectral density can be separated into low frequency and high frequencyparts:

γ(ω) = γL(ω) + γH(ω) . (E.1)

The integral inside the exponent of Eq. (A.21) can also be separated into

fL(t) =

∫ ∞

−∞

dωγL(ω)e−iωt − 1ω2 , fH(t) =

∫ ∞

−∞

dωγH(ω)e−iωt − 1ω2 . (E.2)

If γL(ω) is concentrated on low frequencies, a 2nd order Taylor expansion of e−iωt is justified. As a result, the firstterm in Eq. (E.2) can be written as

fL(t) = −12

W2t2 − iεLt (E.3)

whereW2 =

∫γL(ω) dω εL = P

∫γL(ω)ω

dω , (E.4)

and P stands for the Cauchy principal value. W and εL, usually known as the MRT linewidth and reorganizationenergy, respectively, are experimentally measurable quantities that are connected through the fluctuation-dissipationtheorem [59]: W2 = 2εLT .

Finally, the two point correlation function under this hybrid noise model is

K(t) = 〈ξ+(t)ξ−(0)〉 = 〈ξ−(t)ξ+(0)〉 = e−4iεLt−2W2t2exp

{4∫

dωγH(ω)e−iωt − 1ω2

}. (E.5)

The corresponding spectral density of K(t) is

γK(ω) =

∫K(t)eiωt dt =

∫dx2π

GL(ω − x)GH(x) dx , (E.6)

where GL(ω) and GH(ω) are the Fourier transforms of exp{4 fL(t)} and exp{4 fH(t)}, respectively. The low frequencycomponent is Gaussian:

GL(ω) =

∫dω e−4iεLt−2W2t2

eiωt =

√π

2W2 exp[−

(ω − 4εL)2

8W2

], (E.7)

while the high frequency part can be approximated as a single Lorentzian [32]:

GH(ω) =4γH(ω)

ω2 + 4γH(0)2 . (E.8)

Finally, it should be mentioned that the fluctuation-dissipation theorem guarantees that the Kubo-Martin-Schwinger(KMS) condition be satisfied for GL(ω). So if γH(ω) satisfies KMS condition, the spectral density in the polaron framealso satisfies the KMS condition

γK(|ω|) = eβ|ω|γK(−|ω|) . (E.9)

References

[1] A. G. Redfield, The theory of relaxation processes, in: J. S. Waugh (Ed.), Advances in Magnetic and Optical Resonance, Vol. 1, AcademicPress, 1965, pp. 1–32.URL http://www.sciencedirect.com/science/article/pii/B9781483231143500076

[2] D. Xu, J. Cao, Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformationapproach, Frontiers of Physics 11 (4) (2016) 110308. doi:10.1007/s11467-016-0540-2.URL https://doi.org/10.1007/s11467-016-0540-2

24

[3] T. Albash, S. Boixo, D. A. Lidar, P. Zanardi, Quantum adiabatic markovian master equations, New J. of Phys. 14 (12) (2012) 123016.URL http://iopscience.iop.org/article/10.1088/1367-2630/14/12/123016/meta

[4] E. Mozgunov, D. Lidar, Completely positive master equation for arbitrary driving and small level spacing, Quantum 4 (2020) 227. doi:10.22331/q-2020-02-06-227.URL https://quantum-journal.org/papers/q-2020-02-06-227/

[5] F. Nathan, M. S. Rudner, Universal Lindblad equation for open quantum systems, arXiv:2004.01469 [cond-mat, physics:quant-ph]ArXiv:2004.01469 (Apr. 2020).URL http://arxiv.org/abs/2004.01469

[6] E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/ f noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86 (2014)361–418.URL http://link.aps.org/doi/10.1103/RevModPhys.86.361

[7] C. Tsitouras, Runge–Kutta pairs of order 5(4) satisfying only the first column simplifying assumption, Computers & Mathematics withApplications 62 (2) (2011) 770–775. doi:10.1016/j.camwa.2011.06.002.URL http://www.sciencedirect.com/science/article/pii/S0898122111004706

[8] R. Bank, W. Coughran, W. Fichtner, E. Grosse, D. Rose, R. Smith, Transient Simulation of Silicon Devices and Circuits, IEEE Transactions onComputer-Aided Design of Integrated Circuits and Systems 4 (4) (1985) 436–451, conference Name: IEEE Transactions on Computer-AidedDesign of Integrated Circuits and Systems. doi:10.1109/TCAD.1985.1270142.

[9] M. Tokman, J. Loffeld, P. Tranquilli, New Adaptive Exponential Propagation Iterative Methods of Runge–Kutta Type, SIAM Journal on Sci-entific Computing 34 (5) (2012) A2650–A2669, publisher: Society for Industrial and Applied Mathematics. doi:10.1137/110849961.URL https://epubs.siam.org/doi/abs/10.1137/110849961

[10] C. Rackauckas, Q. Nie, DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia,Journal of Open Research Software 5 (1) (May 2017). doi:10.5334/jors.151.URL http://openresearchsoftware.metajnl.com/articles/10.5334/jors.151/

[11] K. Blum, Density Matrix Theory and Applications, Plenum Press, New York, 1981.[12] R. Alicki and M. Fannes, Quantum Dynamical Systems, Oxford University Press, 2001.[13] H.-P. Breuer, F. Petruccione, The theory of open quantum systems, Oxford University Press, 2002.[14] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Springer Science & Business Media, 2007.[15] U. Weiss, Quantum dissipative systems, Vol. 13, World scientific, 2012.[16] D. A. Lidar, Lecture notes on the theory of open quantum systems, arXiv preprint arXiv:1902.00967 (2019).

URL https://arxiv.org/abs/1902.00967[17] H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Springer, Berlin, 1999.[18] C.W. Gardiner and P. Zoller, Quantum Noise, Vol. 56 of Springer Series in Synergetics, Springer, Berlin, 2000.[19] A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics 63 (2) (2014) 77–149, publisher: Taylor &

Francis.[20] H. Wiseman, G. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.[21] M. A. Nielsen, I. Chuang, Quantum computation and quantum information, American Association of Physics Teachers, 2002.[22] E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Quantum computation by adiabatic evolution, arXiv preprint quant-ph/0001106 (2000).

URL http://arxiv.org/abs/quant-ph/0001106[23] T. Kadowaki, H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E 58 (5) (1998) 5355.

URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.58.5355[24] T. Albash, D. A. Lidar, Adiabatic quantum computation, Reviews of Modern Physics 90 (1) (2018) 015002. doi:10.1103/

RevModPhys.90.015002.URL https://link.aps.org/doi/10.1103/RevModPhys.90.015002

[25] P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, W. D. Oliver, Perspectives of quantum annealing: Methods and implementations,Reports on Progress in Physics (2020).URL https://iopscience.iop.org/article/10.1088/1361-6633/ab85b8

[26] P. Zanardi, M. Rasetti, Holonomic quantum computation, Physics Letters A 264 (2–3) (1999) 94–99.URL http://www.sciencedirect.com/science/article/pii/S0375960199008038

[27] L. M. Duan, J. I. Cirac, P. Zoller, Geometric manipulation of trapped ions for quantum computation, Science 292 (5522) (2001) 1695–1697.URL http://www.sciencemag.org/content/292/5522/1695.abstract

[28] A. M. Childs, E. Farhi, J. Preskill, Robustness of adiabatic quantum computation, Phys. Rev. A 65 (1) (2001) 012322.URL http://journals.aps.org/pra/abstract/10.1103/PhysRevA.65.012322

[29] T. Albash, D. A. Lidar, Decoherence in adiabatic quantum computation, Physical Review A 91 (6) (2015) 062320. doi:10.1103/PhysRevA.91.062320.URL https://link.aps.org/doi/10.1103/PhysRevA.91.062320

[30] M. H. Amin, Searching for quantum speedup in quasistatic quantum annealers, Physical Review A 92 (5) (2015) 052323. doi:10.1103/PhysRevA.92.052323.URL https://link.aps.org/doi/10.1103/PhysRevA.92.052323

[31] K. W. Yip, T. Albash, D. A. Lidar, Quantum trajectories for time-dependent adiabatic master equations, Physical Review A 97 (2) (2018)022116. doi:10.1103/PhysRevA.97.022116.URL https://link.aps.org/doi/10.1103/PhysRevA.97.022116

[32] A. Y. Smirnov, M. H. Amin, Theory of open quantum dynamics with hybrid noise, New Journal of Physics 20 (10) (2018) 103037. doi:10.1088/1367-2630/aae79c.URL https://doi.org/10.1088%2F1367-2630%2Faae79c

[33] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2 (2018) 79, arXiv: 1801.00862. doi:10.22331/

25

q-2018-08-06-79.URL http://arxiv.org/abs/1801.00862

[34] D. A. Lidar, T. A. Brun, Quantum error correction, Cambridge University Press, 2013.[35] J. R. Johansson, P. D. Nation, F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Computer

Physics Communications 183 (8) (2012) 1760–1772. doi:10.1016/j.cpc.2012.02.021.URL http://www.sciencedirect.com/science/article/pii/S0010465512000835

[36] J. R. Johansson, P. D. Nation, F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Computer PhysicsCommunications 184 (4) (2013) 1234–1240. doi:10.1016/j.cpc.2012.11.019.URL http://www.sciencedirect.com/science/article/pii/S0010465512003955

[37] J. Bezanson, A. Edelman, S. Karpinski, V. Shah, Julia: A Fresh Approach to Numerical Computing, SIAM Review 59 (1) (2017) 65–98.doi:10.1137/141000671.URL https://epubs.siam.org/doi/10.1137/141000671

[38] J. Bezanson, S. Karpinski, V. B. Shah, A. Edelman, The Julia Language, library Catalog: julialang.org.URL https://julialang.org/blog/2012/02/why-we-created-julia/

[39] C. A. Rodríguez-Rosario, K. Modi, A.-M. Kuah, A. Shaji, E. C. G. Sudarshan, Completely positive maps and classical correlations, J. ofPhys. A 41 (20) (2008) 205301.URL http://stacks.iop.org/1751-8121/41/i=20/a=205301

[40] J. M. Dominy, D. A. Lidar, Beyond complete positivity, Quant. Inf. Proc. 15 (4) (2016) 1349.URL http://link.springer.com/article/10.1007%2Fs11128-015-1228-1

[41] E. B. Davies, Markovian master equations, Comm. Math. Phys. 39 (2) (1974) 91–110.URL http://dx.doi.org/10.1007/BF01608389

[42] S. Jang, Theory of coherent resonance energy transfer for coherent initial condition, The Journal of Chemical Physics 131 (16) (2009) 164101.doi:10.1063/1.3247899.URL https://aip.scitation.org/doi/10.1063/1.3247899

[43] N. G. Van Kampen, A cumulant expansion for stochastic linear differential equations. II, Physica 74 (2) (1974) 239–247. doi:10.1016/0031-8914(74)90122-0.URL http://www.sciencedirect.com/science/article/pii/0031891474901220

[44] N. G. Van Kampen, A cumulant expansion for stochastic linear differential equations. I, Physica 74 (2) (1974) 215–238. doi:10.1016/0031-8914(74)90121-9.URL http://www.sciencedirect.com/science/article/pii/0031891474901219

[45] R. Kubo, Generalized cumulant expansion method, Journal of the Physical Society of Japan 17 (7) (1962) 1100–1120.URL http://journals.jps.jp/doi/abs/10.1143/JPSJ.17.1100

[46] P. Gaspard, M. Nagaoka, Slippage of initial conditions for the redfield master equation, Journal of Chemical Physics 111 (13) (1999) 5668–5675. doi:10.1063/1.479867.URL https://doi.org/10.1063/1.479867

[47] R. S. Whitney, Staying positive: going beyond lindblad with perturbative master equations, Journal of Physics A: Mathematical and Theoret-ical 41 (17) (2008) 175304.URL http://stacks.iop.org/1751-8121/41/i=17/a=175304

[48] C. Majenz, T. Albash, H.-P. Breuer, D. A. Lidar, Coarse graining can beat the rotating-wave approximation in quantum Markovian masterequations, Physical Review A 88 (1) (2013) 012103. doi:10.1103/PhysRevA.88.012103.URL https://link.aps.org/doi/10.1103/PhysRevA.88.012103

[49] JuliaMath/HCubature.jl.URL https://github.com/JuliaMath/HCubature.jl

[50] A. C. Genz, A. A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangularregion, Journal of Computational and Applied Mathematics 6 (4) (1980) 295–302. doi:10.1016/0771-050X(80)90039-X.URL http://www.sciencedirect.com/science/article/pii/0771050X8090039X

[51] E. Paladino, L. Faoro, G. Falci, R. Fazio, Decoherence and 1/ f noise in josephson qubits, Physical Review Letters 88 (22) (2002) 228304–.URL https://link.aps.org/doi/10.1103/PhysRevLett.88.228304

[52] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L.Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, W. D. Oliver, The flux qubit revisited to enhance coherence and reproducibility, NatureCommunications 7 (2016) 12964 EP –.URL http://dx.doi.org/10.1038/ncomms12964

[53] L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, V. E. Manucharyan, High-Coherence Fluxonium Qubit, Physical Review X9 (4) (2019) 041041, publisher: American Physical Society. doi:10.1103/PhysRevX.9.041041.URL https://link.aps.org/doi/10.1103/PhysRevX.9.041041

[54] C. M. Quintana, Y. Chen, D. Sank, A. G. Petukhov, T. C. White, D. Kafri, B. Chiaro, A. Megrant, R. Barends, B. Campbell, Z. Chen,A. Dunsworth, A. G. Fowler, R. Graff, E. Jeffrey, J. Kelly, E. Lucero, J. Y. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, P. Roushan,A. Shabani, V. N. Smelyanskiy, A. Vainsencher, J. Wenner, H. Neven, J. M. Martinis, Observation of classical-quantum crossover of 1/ f fluxnoise and its paramagnetic temperature dependence, Physical Review Letters 118 (5) (2017) 057702–. doi:10.1103/PhysRevLett.118.057702.URL https://link.aps.org/doi/10.1103/PhysRevLett.118.057702

[55] R. Bauernschmitt, Y. V. Nazarov, Detailed balance in single-charge traps, Physical Review B 47 (15) (1993) 9997–10000. doi:10.1103/PhysRevB.47.9997.URL https://link.aps.org/doi/10.1103/PhysRevB.47.9997

[56] C.-Y. Hsieh, J. Cao, A unified stochastic formulation of dissipative quantum dynamics. I. Generalized hierarchical equations, The Journal of

26

Chemical Physics 148 (1) (2018) 014103. doi:10.1063/1.5018725.URL https://aip.scitation.org/doi/10.1063/1.5018725

[57] B. Pokharel, N. Anand, B. Fortman, D. A. Lidar, Demonstration of Fidelity Improvement Using Dynamical Decoupling with SuperconductingQubits, Physical Review Letters 121 (22) (2018) 220502, publisher: American Physical Society. doi:10.1103/PhysRevLett.121.220502.URL https://link.aps.org/doi/10.1103/PhysRevLett.121.220502

[58] H. Dekker, Noninteracting-blip approximation for a two-level system coupled to a heat bath, Physical Review A 35 (3) (1987) 1436–1437.doi:10.1103/PhysRevA.35.1436.URL https://link.aps.org/doi/10.1103/PhysRevA.35.1436

[59] M. H. S. Amin, D. V. Averin, Macroscopic Resonant Tunneling in the Presence of Low Frequency Noise, Physical Review Letters 100 (19)(2008) 197001. doi:10.1103/PhysRevLett.100.197001.URL https://link.aps.org/doi/10.1103/PhysRevLett.100.197001

[60] E. B. Davies, Markovian master equations, Communications in Mathematical Physics 39 (2) (1974) 91–110. doi:10.1007/BF01608389.URL https://doi.org/10.1007/BF01608389

[61] P. Keller, I. Wróbel, Computing Cauchy principal value integrals using a standard adaptive quadrature, Journal of Computational and AppliedMathematics 294 (2016) 323–341. doi:10.1016/j.cam.2015.08.021.URL http://www.sciencedirect.com/science/article/pii/S0377042715004422

[62] S. Klarsfeld, J. A. Oteo, Magnus approximation in the adiabatic picture, Physical Review A 45 (5) (1992) 3329–3332. doi:10.1103/PhysRevA.45.3329.URL https://link.aps.org/doi/10.1103/PhysRevA.45.3329

[63] H. Chen, D. A. Lidar, Why and When Pausing is Beneficial in Quantum Annealing, Physical Review Applied 14 (1) (2020) 014100, publisher:American Physical Society. doi:10.1103/PhysRevApplied.14.014100.URL https://link.aps.org/doi/10.1103/PhysRevApplied.14.014100

[64] Parallel Ensemble Simulations · DifferentialEquations.jl.URL https://diffeq.sciml.ai/stable/features/ensemble/

[65] H. Abraham, AduOffei, R. Agarwal, I. Y. Akhalwaya, G. Aleksandrowicz, T. Alexander, M. Amy, E. Arbel, Arijit02, A. Asfaw, A. Avkhadiev,C. Azaustre, AzizNgoueya, A. Banerjee, A. Bansal, P. Barkoutsos, G. Barron, G. S. Barron, L. Bello, Y. Ben-Haim, D. Bevenius, A. Bhobe,L. S. Bishop, C. Blank, S. Bolos, S. Bosch, Brandon, S. Bravyi, Bryce-Fuller, D. Bucher, A. Burov, F. Cabrera, P. Calpin, L. Capelluto, J. Car-ballo, G. Carrascal, A. Chen, C.-F. Chen, E. Chen, J. C. Chen, R. Chen, J. M. Chow, S. Churchill, C. Claus, C. Clauss, R. Cocking, F. Correa,A. J. Cross, A. W. Cross, S. Cross, J. Cruz-Benito, C. Culver, A. D. Córcoles-Gonzales, S. Dague, T. E. Dandachi, M. Daniels, M. Dartiailh,DavideFrr, A. R. Davila, A. Dekusar, D. Ding, J. Doi, E. Drechsler, Drew, E. Dumitrescu, K. Dumon, I. Duran, K. EL-Safty, E. Eastman,G. Eberle, P. Eendebak, D. Egger, M. Everitt, P. M. Fernández, A. H. Ferrera, R. Fouilland, FranckChevallier, A. Frisch, A. Fuhrer, B. Fuller,M. GEORGE, J. Gacon, B. G. Gago, C. Gambella, J. M. Gambetta, A. Gammanpila, L. Garcia, T. Garg, S. Garion, A. Gilliam, A. Giridharan,J. Gomez-Mosquera, S. de la Puente González, J. Gorzinski, I. Gould, D. Greenberg, D. Grinko, W. Guan, J. A. Gunnels, M. Haglund,I. Haide, I. Hamamura, O. C. Hamido, F. Harkins, V. Havlicek, J. Hellmers, Ł. Herok, S. Hillmich, H. Horii, C. Howington, S. Hu, W. Hu,J. Huang, R. Huisman, H. Imai, T. Imamichi, K. Ishizaki, R. Iten, T. Itoko, JamesSeaward, A. Javadi, A. Javadi-Abhari, Jessica, M. Jivrajani,K. Johns, S. Johnstun, Jonathan-Shoemaker, V. K, T. Kachmann, N. Kanazawa, Kang-Bae, A. Karazeev, P. Kassebaum, J. Kelso, S. King,Knabberjoe, Y. Kobayashi, A. Kovyrshin, R. Krishnakumar, V. Krishnan, K. Krsulich, P. Kumkar, G. Kus, R. LaRose, E. Lacal, R. Lambert,J. Lapeyre, J. Latone, S. Lawrence, C. Lee, G. Li, D. Liu, P. Liu, Y. Maeng, K. Majmudar, A. Malyshev, J. Manela, J. Marecek, M. Mar-ques, D. Maslov, D. Mathews, A. Matsuo, D. T. McClure, C. McGarry, D. McKay, D. McPherson, S. Meesala, T. Metcalfe, M. Mevissen,A. Meyer, A. Mezzacapo, R. Midha, Z. Minev, A. Mitchell, N. Moll, J. Montanez, M. D. Mooring, R. Morales, N. Moran, M. Motta, MrF,P. Murali, J. Müggenburg, D. Nadlinger, K. Nakanishi, G. Nannicini, P. Nation, E. Navarro, Y. Naveh, S. W. Neagle, P. Neuweiler, J. Nicander,P. Niroula, H. Norlen, NuoWenLei, L. J. O’Riordan, O. Ogunbayo, P. Ollitrault, R. Otaolea, S. Oud, D. Padilha, H. Paik, S. Pal, Y. Pang,S. Perriello, A. Phan, F. Piro, M. Pistoia, C. Piveteau, P. Pocreau, A. Pozas-iKerstjens, V. Prutyanov, D. Puzzuoli, J. Pérez, Quintiii, R. I.Rahman, A. Raja, N. Ramagiri, A. Rao, R. Raymond, R. M.-C. Redondo, M. Reuter, J. Rice, M. L. Rocca, D. M. Rodríguez, RohithKarur,M. Rossmannek, M. Ryu, T. SAPV, SamFerracin, M. Sandberg, H. Sandesara, R. Sapra, H. Sargsyan, A. Sarkar, N. Sathaye, B. Schmitt,C. Schnabel, Z. Schoenfeld, T. L. Scholten, E. Schoute, J. Schwarm, I. F. Sertage, K. Setia, N. Shammah, Y. Shi, A. Silva, A. Simonetto,N. Singstock, Y. Siraichi, I. Sitdikov, S. Sivarajah, M. B. Sletfjerding, J. A. Smolin, M. Soeken, I. O. Sokolov, I. Sokolov, SooluThomas,Starfish, D. Steenken, M. Stypulkoski, S. Sun, K. J. Sung, H. Takahashi, T. Takawale, I. Tavernelli, C. Taylor, P. Taylour, S. Thomas, M. Tillet,M. Tod, M. Tomasik, E. de la Torre, K. Trabing, M. Treinish, TrishaPe, D. Tulsi, W. Turner, Y. Vaknin, C. R. Valcarce, F. Varchon, A. C.Vazquez, V. Villar, D. Vogt-Lee, C. Vuillot, J. Weaver, J. Weidenfeller, R. Wieczorek, J. A. Wildstrom, E. Winston, J. J. Woehr, S. Woerner,R. Woo, C. J. Wood, R. Wood, S. Wood, S. Wood, J. Wootton, D. Yeralin, D. Yonge-Mallo, R. Young, J. Yu, C. Zachow, L. Zdanski, H. Zhang,C. Zoufal, Zoufalc, a kapila, a matsuo, bcamorrison, brandhsn, nick bronn, chlorophyll zz, dekel.meirom, dekelmeirom, dekool, dime10,drholmie, dtrenev, ehchen, elfrocampeador, faisaldebouni, fanizzamarco, gabrieleagl, gadial, galeinston, georgios ts, gruu, hhorii, hykavitha,jagunther, jliu45, jscott2, kanejess, klinvill, krutik2966, kurarrr, lerongil, ma5x, merav aharoni, michelle4654, ordmoj, sagar pahwa, rmo-yard, saswati qiskit, scottkelso, sethmerkel, strickroman, sumitpuri, tigerjack, toural, tsura crisaldo, vvilpas, welien, willhbang, yang.luh,yotamvakninibm, M. Cepulkovskis, Qiskit: An open-source framework for quantum computing. doi:10.5281/zenodo.2562110.

[66] T. Alexander, N. Kanazawa, D. J. Egger, L. Capelluto, C. J. Wood, A. Javadi-Abhari, D. McKay, Qiskit Pulse: Programming QuantumComputers Through the Cloud with Pulses, arXiv:2004.06755 [quant-ph]ArXiv: 2004.06755 (Apr. 2020).URL http://arxiv.org/abs/2004.06755

[67] R. S. Smith, M. J. Curtis, W. J. Zeng, A Practical Quantum Instruction Set Architecture (Aug. 2016).URL https://arxiv.org/abs/1608.03355v2

[68] D. S. Steiger, T. Häner, M. Troyer, ProjectQ: an open source software framework for quantum computing, Quantum 2 (2018) 49, publisher:Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. doi:10.22331/q-2018-01-31-49.

27

URL https://quantum-journal.org/papers/q-2018-01-31-49/[69] R. LaRose, Overview and Comparison of Gate Level Quantum Software Platforms, Quantum 3 (2019) 130. doi:10.22331/

q-2019-03-25-130.URL https://quantum-journal.org/papers/q-2019-03-25-130/

[70] M. Grifoni, P. Hänggi, Driven Quantum Tunneling, 1998.[71] USCqserver/HOQSTTutorials.jl.

URL https://github.com/USCqserver/HOQSTTutorials.jl[72] T. Lanting, A. Przybysz, A. Smirnov, F. Spedalieri, M. Amin, A. Berkley, R. Harris, F. Altomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud,

J. Hilton, E. Hoskinson, M. Johnson, E. Ladizinsky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M. Thom, E. Tolkacheva,S. Uchaikin, A. Wilson, G. Rose, Entanglement in a Quantum Annealing Processor, Physical Review X 4 (2) (2014) 021041. doi:10.1103/PhysRevX.4.021041.URL https://link.aps.org/doi/10.1103/PhysRevX.4.021041

[73] T. Albash, I. Hen, F. M. Spedalieri, D. A. Lidar, Reexamination of the evidence for entanglement in a quantum annealer, Physical Review A92 (6) (2015) 062328. doi:10.1103/PhysRevA.92.062328.URL https://link.aps.org/doi/10.1103/PhysRevA.92.062328

[74] A. J. Berkley, A. J. Przybysz, T. Lanting, R. Harris, N. Dickson, F. Altomare, M. H. Amin, P. Bunyk, C. Enderud, E. Hoskinson, M. W.Johnson, E. Ladizinsky, R. Neufeld, C. Rich, A. Y. Smirnov, E. Tolkacheva, S. Uchaikin, A. B. Wilson, Tunneling spectroscopy using a probequbit, Phys. Rev. B 87 (2) (2013) 020502–.URL http://link.aps.org/doi/10.1103/PhysRevB.87.020502

[75] T. Albash, private communication.[76] Solution Handling · DifferentialEquations.jl.

URL https://diffeq.sciml.ai/stable/basics/solution/[77] A. Rohatgi, Webplotdigitizer: Version 4.3.

URL https://automeris.io/WebPlotDigitizer[78] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. Fisher, A. Garg, W. Zwerger, Dynamics of the dissipative two-state system, Reviews of

Modern Physics 59 (1) (1987) 1.URL http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.59.1

[79] C. K. Lee, J. Moix, J. Cao, Accuracy of second order perturbation theory in the polaron and variational polaron frames, The Journal ofChemical Physics 136 (20) (2012) 204120. doi:10.1063/1.4722336.URL https://aip.scitation.org/doi/abs/10.1063/1.4722336

[80] T. M. Chang, J. L. Skinner, Non-Markovian population and phase relaxation and absorption lineshape for a two-level system stronglycoupled to a harmonic quantum bath, Physica A: Statistical Mechanics and its Applications 193 (3) (1993) 483–539. doi:10.1016/0378-4371(93)90489-Q.URL http://www.sciencedirect.com/science/article/pii/037843719390489Q

28