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Succinct Description and Efficient Simulation of Non-Markovian

Open Quantum Systems

Xiantao Li* and Chunhao Wang†

*Department of Mathematics, Pennsylvania State University†Department of Computer Science and Engineering, Pennsylvania State University

Email: xiantao.li,[email protected]

Abstract

Non-Markovian open quantum systems are the most general dynamics when the quantumsystem is coupled with a bath environment. The quantum dynamics arising from many impor-tant applications are non-Markovian. Although for special cases such as Hamiltonian evolutionand Lindblad evolution, quantum simulation algorithms have been extensively studied, efficientquantum simulations of the dynamics of non-Markovian open quantum systems remain under-explored. The most immediate obstacle for studying such systems is the lack of a universalsuccinct description of their dynamics. In this work, we fulfill the gap of studying such dynam-ics by 1) providing a succinct representation of the dynamics of non-Markovian open quantumsystems with quantifiable error, and 2) developing an efficient quantum algorithm for simulatingsuch dynamics with cost O(t polylog(t/ǫ)) for evolution time t and precision ǫ. Our derivationof the succinct representation is based on stochastic Schrödinger equations, which could lead tonew alternatives to deal with other types of open quantum systems.

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http://arxiv.org/abs/2111.03240v1

1 Introduction

As the size of many modern-day electronic devices is continuously being reduced, quantum me-chanical properties start to become dominant. Many novel designs have emerged, e.g., quantumwires, quantum dots, and molecular transistors, to take advantage of these properties. What iscommon in these applications is that the quantum properties are vital. However, these quantumdynamics do not evolve in isolation. Known as open quantum systems [Bre02], they are alwaysinteracting with their environment, which due to the large dimension, can not be included explicitlyin the computation. Another challenge comes from the fact that the continuous interactions cangive rise to non-Markovian behavior, for which classical descriptions break down. The implicationof non-Markovian dynamics to the quantum properties have been analyzed through many modelexamples [BLPV16, PZA+19, dVA17, SP16, SH04, AM21, MEL20]. More importantly, there areimportant experimental observations of non-Markovian dynamics [GTP+15, MALH+11], and suchbehavior has a strong impact on the electronic properties of molecular devices. Furthermore, it hasbe discovered that non-Markovianity can enhance quantum entanglement [TER+09].

Generally speaking, the dynamics of a quantum system interacting with a bath environment canbe described by the von Neumann equation1

i∂tρ = [Htot, ρ], ρ(0) = ρS(0)⊗ ρB , (1)

where the coupled Hamiltonian is given by,

Htot = HS ⊗ IB + IS ⊗HB + λ

M∑

α=1

Sα ⊗Bα. (2)

Here, λ is the coupling parameter, and M refers to the number of interaction terms. To considerthe problem in the context of quantum simulations, we let 2n be the dimension of the system (S),i.e., HS ∈ C

2n×2n is acting on n qubits. The simplest example is the spin Boson model, where HS

and Sα can be chosen to be Pauli matrices. A notable example is the dynamics of qubits coupled toa boson bath [WC13]. There are also many other important examples, e.g., the Anderson-Holsteinmodel [CL17a] whereHS also includes the kinetic and potential energy terms. In electronic transportproblems, the Hamiltonian is often expressed in terms of molecular or atomic orbitals [GN05].

When λ = 0, the system is completely decoupled from the environment, and the dynamics re-duces to Hamiltonian evolution. In recent decades, the problem of simulating Hamiltonian evolutionhas been extensively studied. A subset of notable works includes [Llo96, BCG14, BCC+15, BCK15,BCC+17, LC17, LC19, CGJ19]. This line of research has led to optimal Hamiltonian simulationalgorithms [LC17, GSLW19].

However, the progress of developing fast simulation algorithms is much slower in the regime ofλ 6= 0, i.e., open quantum systems. In particular, when 0 < λ ≪ 1 and the dynamics of the bathoccurs at a much faster rate than the system, the dynamics is referred to as Markovian open quantumsystems. Intuitively, if an open quantum system is Markovian, the bath Hamiltonian is fast enoughto “forget” the disturbance caused by the system-bath interaction, and therefore information onlyflows from system to bath with no transfer of information back to the system. In 2011, Kliesch,Barthel, Gogolin, Kastoryano, and Eisert [KBG+11] gave the first quantum algorithm for simulatingMarkovian open quantum systems with cost O(t2/ǫ) for evolution time t and precision ǫ. In 2017,

1The von Neumann equation is a generalization of the Schrödinger equation to the context of density matrices.

2

Childs and Li [CL17b] improved the cost to O(t1.5/√ǫ), and Cleve and Wang [CW17] further improve

the cost to O(t polylog(t/ǫ)), which is nearly optimal. The difficulty of simulating Markovian openquantum systems lies in the observation that the advanced algorithmic techniques for Hamiltoniansimulation cannot be directly used for open quantum systems because of the presence of decoherence.In fact, Cleve and Wang [CW17] have shown that it is impossible to achieve linear dependence onevolution time by a direct reductionist approach.

In sharp contrast to quantum Markovian processes, for which a complete description that onlyinvolves operators on the system has been developed by Lindblad [Lin76] (i.e., the Lindblad equa-tion for general positive and trace-preserving maps), there is no universal form for the quantummaster equation for non-Markovian dynamics. In the regime when the rate of the bath dynamicsis comparable to the rate of the system dynamics, the system becomes non-Markovian. Looselyspeaking, the non-Markovian dynamics can be interpreted as the backflow of information for theenvironment to the open quantum system, and it can be more precisely characterized using variousmetrics [BLP09, VMP+11, LFS12]. For such systems, many approaches have been developed todescribe such dynamics [CK10, dVA17, DS97, GN99, IT05, KMCWM16, MT99, MCR16, MCR17,PMCKM19, PRRF+18, Sha04, SG03, SKK+18, Str96, SDG99, SES14, Tan06, Li21, Tan06]. Theearliest approach dates back to the projection formalism of Nakajima and Zwanzig [Nak58, Zwa60].At the level of density matrices, the non-Markovian property is reflected in a memory integralthat involves the bath correlation function, which can be approximated by a linear combination ofLorentzian terms [RE14]. This representation enables an embedding procedure, where the memoryintegral can be replaced by the dynamics of additional density-matrices [MT99, Li21]. Although aunified framework is not present, in most such methods, the density matrix of the system is embed-ded in an extended density matrix, for which the dynamics is Markovian. The coefficients in theextended dynamics are connected to the spectral properties of the bath. Such a quantum masterequation is often referred to as the generalized quantum master equation (GQME).

1.1 Main results

In this work, we 1) provide a succinct GQME representation of the dynamics of non-Markovianquantum systems, and 2) develop an efficient quantum algorithm for simulating the dynamics ofopen quantum systems in this new representation. It is worth noting that we only consider openquantum systems with coupling parameter λ ≪ 1. This is because, when the coupling betweenthe system and the bath is strong, a formulation of the dynamics will inevitably require moreinformation from the bath Hamiltonian HB and the bath part Bα of the interaction terms. Thiswill make the simulation problem intractable even for quantum computers due to the enormous sizeof the bath.

For the GQME, we choose to work with one that can be unravelled [Bre02]. Namely, it corre-sponds to a stochastic Schrödinger equation and the state-matrix is automatically positive semidefi-nite. Let K be the number of Lorentzian terms in representing the bath correlation function [RE14].We consider an embedding of the system state ρS into an O(n logK)-qubit system (with dimensionK2n). Let Γ denote the (unnormalized) state density matrix of this larger system. We propose thefollowing quantum master equation to describe the dynamics of Γ:

∂tΓ = −i(HΓ− ΓH†) +K∑

k=1

VkΓV†k , (3)

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where H (in general non-Hermitian) is defined in Eq. (62) which involves HS, Sα, and the bathcorrelation function (assumed to be part of input), and Vk is defined in Eq. (60) which involves thebath correlation function. The initial state Γ(0) of this GQME is

Γ(0) := |0〉〈0| ⊗ ρS(0) +

K∑

k=1

(|4k − 2〉〈4k − 2|+ |4k〉〈4k| |4k〉〈4k|)⊗ ρS(0). (4)

The system state is embedded in Γ in the sense that the upper-left block of Γ(t) — the unnormalizedstate of the larger system at time t — approximates the exact system state ρS(t) at time t witherror up to O

(λ3)

(see Theorem 10). This error is referred to as the model error as it characterizesthe accuracy of modelling the actual dynamics determined by Eq. (1) as a succinct GQME. Ourmodel error of O

(λ3)

improves the model error O(λ3/α3

)of the Lindblad equation with α ≪ 1

representing the time scale separation between the system and the bath, which (surprisingly) hadnot been precisely characterized until recently [CL17a]. A specific example of the GQME (Eq. (3))can be found at the end of Section 3.2.

Input model and problem definition The computational problem we consider is to simulatethe dynamics generated by Eq. (3), which consists of an initial state preparation problem and atarget state approximation problem. More formally, we formulate the problem as follows.

Problem 1 (Simulating non-Markovian open quantum systems). Consider the dynamics definedby Eq. (3). For any initial state ρS(0) of the system, evolution time t, and precision parameter ǫ,we need to

1. prepare the initial state Γ(0) as in Eq. (4), and

2. produce a quantum state ρS(t) for the system so that the trace-distance between this state andthe upper-left block of Γ(t), which is ρS(t), is at most ǫ, where Γ(t) is the (unnormalized) stateof evolving Eq. (3) for time t with initial state Γ(0).

To solve this simulation problem, we need efficient descriptions of the operators in Eq. (3). Themost straightforward input model is to assume that we are given these efficient descriptions directly.In the literature of simulating Markovian open quantum systems [KBG+11, CL17b, CW17], someefficient access to Lindblad operators was assumed. However, this straightforward input model isoften not physically feasible: in many cases, we only have low-level information about the system,such as the system Hamiltonian and the system part of interaction Hamiltonians, while high-levelinformation such as descriptions of the operators in Eq. (3) is not readily available. In this work,we consider a low-level input model, i.e., we only assume information that arises in Eq. (2), whichis more convenient in real-world applications. In particular, when we consider Hamiltonians suchas HS and Sα, we use a widely-used input model that has been recently introduced by Low andChuang [LC19], and Chakraborty, Gilyén, and Jeffery [CGJ19] — the block-encodings of Hamilto-nians. Roughly speaking, a block-encoding with normalizing factor α of a matrix A is a unitary Uwhose upper-left block is A/α. This input model is general enough to include almost all efficientrepresentations that arises in physics applications, including linear combinations of tensor productsof Paulis, sparse-access oracles, and local Hamiltonians.

Completely excluding the information of the bath Hamiltonian and the bath part of the inter-action Hamiltonians seems unattainable because the dynamics is no longer Markovian and the bath

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is no longer memoryless. On the other hand, we cannot afford modeling the entire bath, as its sizetypically exceeds the capacities of quantum computers. The influence of the bath is usually reflectedin the bath correlation function [Bre02]. In this work, we consider a typical representation of thecorrelation function, expressed as,

Cα,β(t) :=K∑

k=1

θ2k 〈α|Qk〉〈Qk|β〉 exp(−id∗kt), (5)

where |Qk〉 ∈ CM , θk ∈ R, and dk ∈ C. Note that for all t, Cα,β is an M ×M matrix — a size the is

tractable for classical computers. In practice, Eq. (5) is often obtained by a least squares approach[RE14]. For example, the power spectrum (The Fourier transform) of the matrix C(t) is related tothe spectral density of the bath that can be further related to optical properties. We consider thepoles dk within a cut-off frequency dmax, i.e., |dk| ≤ dmax.

Hence, in our quantum algorithm, we assume we are given the following quantities, as an efficientdescription of Eq. (3):

1. a block-encoding UHSof HS;

2. a block-encoding USα of Sα for each α ∈ [M ];

3. real number θk, vector |Qk〉 (all entries), and complex number dk for k ∈ [K] as in Eq. (5),and an upper bound dmax on |dk|.

Main algorithmic result Our main algorithmic contribution is a quantum algorithm that solvesProblem 1. We informally state this result as follows.

Theorem 2 (Informal version of Theorem 18). Suppose that we are given a block-encoding UHS

of HS, block-encodings USα of Sα (for α ∈ [M ]), θk, |Qk〉 (all entries), and dk for k ∈ [K] as inEq. (5). Then there exists a quantum algorithm that solves Problem 1 using

O (t polylog(t/ǫ)poly(α,M,K, dmax)), (6)

queries to UHS, and USa and additional 1- and 2-qubit gates, where α is the normalizing factor of

the block-encodings, M is the number of interaction terms in Eq. (1), K is the number of Lorentzianterms in Eq. (5), and dmax is an upper bound of |dk|.

Our algorithm follows the high-level idea of [CW17], but we have generalized their techniquesto work with block-encoded inputs. The building block of our algorithm is an implementation ofcompletely positive maps given block-encodings of the Kraus operators (see Lemma 6 for more de-tails). Suppose the normalizing factors of the block-encodings of the Kraus operators are α1, . . . , αm,then our construction gives the success probability parameter 1/

∑mj=1 α

2j , while a straightforward

construction using Stinespring dilation yields a worse success probability parameter 1/(∑m

j=1 αj)2,

which does not permit the desired dependence on t and ǫ.Since the dynamics Eq. (3) generates is completely positive, we consider an infinitesimal ap-

proximation map that approximates the dynamics for a small enough evolution time. From thelow-level input model, we can construct the block-encodings of the Kraus operators of this infinites-imal approximation map, and it can be implemented using our building block (Lemma 6) with high

5

success probability parameter. We repeat this construction until the success probability parame-ter becomes a constant and we have obtained a normalized version of Γ(t) for t proportional to aconstant. Now, to extract the upper-left block of the resulting (normalized) density matrix, we useoblivious amplitude amplification for isometries ([CW17] and Lemma 17) to achieve this with anextra factor

√K, as the trace of Γ(t) is upper bounded by O(K) for all t (see Corollary 12).

1.2 Summary of contributions

We highlight our contributions as follows:

1. We derive a succinct representation of non-Markovian dynamics, where the density matrixfrom the GQME is proved to be consistent with that from the full quantum dynamics upto O

(λ3). A notable feature of our new succinct representation is that the positivity is

guaranteed, which is the key requirement for designing quantum simulation algorithms.

2. We develop a quantum algorithm based on this representation. The cost of our algorithmscales linearly in t, poly-logarithmically in ǫ, and polynomially in M and K. To the best of ourknowledge, this algorithm is the first to achieve linear dependence on t and poly-logarithmicdependence on ǫ for simulating non-Markovian open quantum systems. In addition, ouralgorithm works with low-level input models, which are readily available in many real-worldapplications.

3. Other technical contributions: We have shown that the GQMEs can be unravelled (see [Bre02])into stochastic Schrödinger equations, which provides another potential alternative to obtainthe density matrix. In addition, we prove that the extended density matrix from the GQMEis bounded over the time scale O

(λ−1

).

1.3 Related work

In the context of modeling open quantum systems [Bre02], the hierarchical equations of motion(HEOM) approach [Tan06, Tan20] also yields an extended dynamics for the density matrix, butwithout using the weak coupling assumption. Rather, the truncation of the hierarchy of densitymatrix equations is determined by assuming a frequency separation. The positivity property hasbeen verified numerically, while a theoretical analysis still remains open.

It has been shown in [CK10] that non-Markovian dynamics can either be written in a non-localform, i.e., with memory, or in a time-local form, with time-dependent generators. Sweke, Sanz,Sinayskiy, Petruccione, and Solano [SSS+16] considered such time-local quantum master equations,and constructed quantum algorithms. Their algorithms rely on Trotter splittings, by decomposingthe entire generator into local operators.

1.4 Open questions

Modeling open quantum systems outside the weak coupling regime is still an outstanding challenge.The HEOM approach [Tan20] relies on a frequency cut-off to achieve a Markovian embedding.Quantifying the error associated with such an approximation, and ensuring the positivity are twoof the remaining issues.

6

Another interesting scenario is when the open quantum system is subject to an external potential.Quantum optimal control is one important example. Deriving a GQME in the presence of a time-dependent external field while still maintaining the control properties is still an open problem tothe best of our knowledge.

The cost of our quantum algorithm is O(t polylog(t/ǫ)). Is there a faster quantum algorithmthat achieves an additive cost, i.e., O(t + polylog(1/ǫ))? This additive cost is the lower boundfor Hamiltonian simulation [BACS07, BCK15, BCC+17], and it is hence a lower bound for thenon-Markovian simulation problem. The optimal Hamiltonian simulation was achieved by quantumsignal processing due to Low and Chuang [LC17], which has been generalized to quantum singularvalue transformation by Gilyén, Su, Low, and Wiebe [GSLW19]. Unfortunately, these techniquesdo not immediately generalize to open quantum systems, as it is not clear what the correspondenceof singular values and eigenvalues should be for superoperators.

2 Preliminaries

2.1 Notation

In this paper, we use the ket-notation to denote a vector only when it is normalized. For a vector v,we use ‖v‖ to denote its Euclidean norm. For a matrix M , we use ‖M‖ to denote its spectral norm

and use ‖M‖1 to denote its trace norm, i.e., ‖M‖1 = tr(√

M †M). The identity operator acting

on a Hilbert space of dimension N is denoted by IN , i.e., I2n is the identity operator acting on nqubits. When the context is clear, we drop the subscript and simply use I. We use calligraphicfont, such as K, L, and M to denote superoperators, which maps matrices to matrices. We considersuperoperators of the form

M : CN×N → CM×M (7)

and use T(CN ,CM ) to denote the set of all such superoperators. In particular, we use I to denotethe identity map, which maps every matrix to itself. Whenever necessary, we use the subscript Nof IN ∈ T(CN ,CN ) to denote the dimension of matrices it acts on. For example, I2n is acting onn-qubit operators. The induced trace norm of a superoperator M ∈ T(CN ,CM ), denoted by ‖M‖1,is defined as

‖M‖1 = max‖M(A)‖1 : A ∈ CN×N , ‖A‖1 ≤ 1. (8)

The diamond norm of a superoperator M ∈ T(CN ,CM ), denoted by ‖M‖⋄ is defined as

‖M‖⋄ := ‖M⊗ IN‖1. (9)

For a positive integer m, we use [m] to denote the set 1, 2, . . . ,m.

2.2 The block-encoding method

We use the notion of block-encoding as an efficient description of input operators. To have access toan operator A, we assume we have access to a unitary U whose upper-left block encodes A in thesense that

U =

(A/α ·· ·

), (10)

7

which implies that A = α(〈0| ⊗ I)U |0〉 ⊗ I). More precisely, we have the following definition.

Definition 3. Let A be an n-qubit operator. For a positive real number α > 0 and natural numberm, we say that an (n+m)-qubit unitary U is an (α,m, ǫ)-block-encoding of A if

‖A− α(〈0| ⊗ I2n)U |0〉 ⊗ I2n)‖ ≤ ǫ. (11)

The following lemma shows how to construct a block-encoding for sparse matrices.

Lemma 4 ([GSLW19, Lemma 48]). Let A ∈ C2n×2n be an n-qubit operator with at most s nonzero

entries in each row and column. Suppose A is specified by the following sparse-access oracles:

OA : |i〉 |j〉 |0〉 7→ |i〉 |j〉 |A(i, j)〉 , and (12)

OS : |i〉 |k〉 |0〉 7→ |i〉 |j〉 |ri,k〉 , (13)

where ri,k is the k-th nonzero entry of the i-th row of A. Suppose |Ai,j | ≤ 1 for i ∈ [m] and j ∈ [n].Then for all ǫ ∈ (0, 1), an (s, n + 3, ǫ)-block-encoding of A can be implemented using O(1) queriesto OA and OS, along with O(n+ polylog(1/ǫ)) 1- and 2-qubit gates.

A linear combination of block-encodings can be constructed using [GSLW19, Lemma 52]. Here,we slightly generalize their construction to achieve better performance when the normalizing factorsof each block-encoding are different.

Lemma 5. Suppose A :=∑m

j=1 yjAj ∈ C2n×2n , where Aj ∈ C

2n×2n and yj > 0 for all j ∈1, . . . m. Let Uj be an (αj , a, ǫ)-block-encoding of Aj, and B be a unitary acting on b qubits (with

m ≤ 2b − 1) such that B |0〉 =∑2b−1

j=0

√αjyj/s |j〉, where s =

∑mj=1 yjαj. Then a (

∑j yjαj , a +

b,∑

j yjαjǫ)-block-encoding of∑m

j=1 yjAj can be implemented with a single use of∑m−1

j=0 |j〉〈j| ⊗Uj + ((I −∑m−1

j=0 |j〉〈j|)⊗ IC2a ⊗ IC2n ) plus twice the cost for implementing B.

Proof. The proof is similar to that of [GSLW19, Lemma 52]. The difference is that, instead of

preparing the state∑m

j=1

√yj√

∑mj=1 yj

|j〉, we use the state preparation gate B here. First note that

∥∥(〈0|⊗a ⊗ I2n)Uj(|0〉⊗a ⊗ I2n)−Aj/αj

∥∥ ≤ ǫ. (14)

Let W =∑m−1

j=0 |j〉〈j| ⊗Uj +((I−∑m−1j=0 |j〉〈j|)⊗ I2a ⊗ I2n) and define W = (B†⊗ I2a ⊗ I2n)W (B⊗

I2a ⊗ I2n). We have∥∥∥∥∥∥

m∑

j=1

yjAj − s(〈0|⊗b ⊗ 〈0|⊗a ⊗ I2n

)W(|0〉⊗b ⊗ |0〉⊗a ⊗ I2n

)∥∥∥∥∥∥

(15)

=

∥∥∥∥∥∥

m∑

j=1

yjAj −m∑

j=1

αjyj(〈0|⊗a ⊗ I2n)Uj(|0〉⊗a ⊗ I2n)

)∥∥∥∥∥∥

(16)

≤m∑

j=1

yjαj

∥∥Aj/αj −(〈0|⊗a ⊗ I2n)Uj(|0〉⊗a ⊗ I2n)

)∥∥ (17)

≤m∑

j=1

yjαjǫ. (18)

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2.3 Stochastic processes

In the context of open quantum system, the dynamics of the wave function usually involves randomnoises [BD12], which induces a random process. For a random variable X ∈ C

d, we use E[X] todenote its mean, and E[XX†] for the covariance matrix, which is Hermitian and positive semidefinite.A stochastic process X(t) : R → C

d is said to be stationary in the wide sense [Ris84] if the meanis a constant for all time, i.e., E[X(t)] =const., and the time correlation has the property thatE[X(t)X(t′)†

]= E

[X(t− t′)X(0)†

], ∀t, t′ ∈ R.

One way to generate a stationary stochastic process is through a Itô stochastic differentialequations [Øks03]:

X(t) = f(X(t)) + σ(X(t))W (t). (19)

Here σ(X) : Cn → Cn×m, and W (t) : R → C

m is the m-dimensional, complex-valued Brownianmotion. X := d

dtX indicates the time derivative.

A simple example is a scalar linear SDE,

ix = −ax+ σw. (20)

The solution forms a stationary process, if γ2k = 2Imα. The corresponding process is known as theOrnstein–Uhlenbeck (OU) process [Ris84].

The Itô Lemma [Øks03] states that a twice continuously differential function g(X(t)), withgradient and hessian denoted respectively by ∇g and ∇2g, satisfies the equation,

d

dtg = (f · ∇g + 1

2tr(σT∇2gσ

)+∇gTσW (t). (21)

This formula is a convenient route to derive equations governing the statistics of X [KP13].

2.4 Technical tool for implementing completely positive maps

In this subsection, we provide the technical primitives for developing the simulation algorithm. Thefollowing lemma generalizes the technique of linear combination of unitaries for completely positivemaps [CW17] to the context of block-encoding. This tool might be of independent interest as well.

Lemma 6. Let A1, . . . , Am ∈ C2n be the Kraus operators of a completely positive map, and let

U1, . . . , Um ∈ C2n+n′

be their corresponding (sj, n′, ǫ)-block-encodings, i.e.,

‖Aj − sj(〈0| ⊗ I)Uj |0〉 ⊗ I)‖ ≤ ǫ, for all 1 ≤ j ≤ m. (22)

Let |µ〉 := 1√

∑mj=1 s

2j

∑m−1j=0 sj |j〉. Then (

∑mj=1 |j〉〈j| ⊗ Uj) |µ〉 |0〉 ⊗ I implements this completely

positive map in the sense that

∥∥∥∥∥∥I ⊗ 〈0| ⊗ I

m∑

j=1

|j〉〈j| ⊗ Uj

|µ〉 |0〉 |ψ〉 − 1√∑m

j=1 s2j

m∑

j=1

|j〉Aj |ψ〉

∥∥∥∥∥∥≤ mǫ√∑m

j=1 s2j

(23)

for all |ψ〉.

9

Proof. It is easy to verify that

m∑

j=1

|j〉〈j| ⊗ Uj

|µ〉 |0〉 |ψ〉 = 1√∑m

j=1 s2j

∑

j

sj |j〉Uj(|0〉 |ψ〉). (24)

Then we have∥∥∥∥∥∥(I ⊗ 〈0| ⊗ I)

m∑

j=1

|j〉〈j| ⊗ Uj

|µ〉 |0〉 |ψ〉 − 1√∑m

j=1 s2j

∑

j

|j〉Aj |ψ〉

∥∥∥∥∥∥(25)

=

∥∥∥∥∥∥(I ⊗ 〈0| ⊗ I)

1√∑mj=1 s

2j

∑

j

sj |j〉Uj(|0〉 |ψ〉)−1√∑mj=1 s

2j

∑

j

|j〉Aj |ψ〉

∥∥∥∥∥∥(26)

=1√∑mj=1 s

2j

∥∥∥∥∥∥

∑

j

sj |j〉 (〈0| ⊗ I)Uj(|0〉 ⊗ I) |ψ〉 −∑

j

|j〉Aj |ψ〉

∥∥∥∥∥∥(27)

≤ mǫ√∑mj=1 s

2j

. (28)

3 Non-Markovian Quantum Master Equation

In this section, we present the derivation of the GQME [Li21]. Non-Markovian dynamics hasbeen extensively studied in the context of open quantum systems [Bre02]. The starting point forconsidering an open quantum system is a quantum dynamics that couples a quantum system and abath environment,

i∂tρ = [Htot, ρ], ρ(0) = ρS(0)⊗ ρB , (29)

where the coupled Hamiltonian is given by,

Htot = HS ⊗ IB + IS ⊗HB + λM∑

α=1

Sα ⊗Bα. (30)

Here HS ∈ C2n×2n is acting on n qubits, and M refers to the number of interaction terms.

One typical assumption is that ρB corresponds to a canonical ensemble, i.e.,

ρB =exp(−βHB)

Z, Z := tr (exp(−βHB)) , (31)

with β being the inverse temperature. When λ = 0, the system and bath will completely decouple,and they will evolve according to their own Hamiltonian dynamics. Without loss of generality, wecan assume that tr(ρBBα) = 0, α = 1, 2, · · · ,M, which can be ensured by properly shifting of theoperators [GN99]. This helps to eliminate O(λ) terms in the asymptotic expansion [GN99].

10

To arrive at a quantum master equation that embodies non-Markovian properties, we start withthe non-Markovian stochastic Schrödinger equation (NMSSE), which was derived in [GN99] fromthe wave function representation of Eq. (29),

i∂tψ = HSψ − iλ2M∑

α,β=1

∫ t

0Cα,β(τ)S

†αe

−iHSτSβψ(t− τ)dτ + λM∑

β=1

ηβ(t)Sβψ(t). (32)

Here i =√−1 and λ is the coupling parameter with 0 < λ ≪ 1. The matrix C(t) : R → C

M×M

with elements Cα,β(t) corresponds to the correlation among the bath variances Bα1≤α≤M .Each noise term ηα(t) : R → C is Gaussian with mean zero and correlation given by,

E[η∗α(t)ηβ(t′)] = Cα,β(t− t′), 1 ≤ α, β ≤M. (33)

The stationarity of the process also implied that C(t) = C(−t)†. Thus, it suffices to consider thecorrelation function for t ≥ 0. This relation between a dissipation kernel and the time correlation ofthe noise is well known in non-equilibrium statistical physics, and it is often labeled as the secondfluctuation-dissipation theorem [Kub66].

3.1 The Markovian regime: The Lindblad equation

The dynamics will reduce to the Markovian regime when the dynamics of the bath occurs at a muchfaster rate, especially when,

Cα,β(t− t′) ≈ Aα,βδ(t− t′). (34)

This occurs when the time scale of the bath is much faster than that of the system. In this case,the SSE in Eq. (32) can be simplified to the stochastic Schrödinger equation,


α,β=1

S†αAα,βSβψ(t) + λ

M∑

β=1

ηβ(t)Sβψ(t). (35)

The matrix A still includes the correlation among the bath variables. This can be treated byusing a change of basis. For instance, due to the positive definite property we can express A = RΣR†,where Σ is diagonal matrix with non-negative diagonals given by θ2j , and R is unitary. For instance,this can be obtained using the matrix singular values decomposition. Since R is unitary, the columnsform a set of orthonormal basis in C

M :

IM =

M∑

j=1

|Rj〉〈Rj| . (36)

As a result, the matrix A can be written as,

A =M∑

j=1

θ2j |Rj〉〈Rj | . (37)

With this spectral decomposition, one can generate the noise ηα that satisfies the relations inEqs. (33) and (34). More specifically, we define, for each 1 ≤ β ≤M,

ηβ(t) =

M∑

j=1

〈Rj|β〉 θjwj(t), (38)

11

where wj(t), j = 1, 2, · · · ,M are independent complex-valued white noise. Namely,

E[w∗j (t)wk(t

′)] = δj,kδ(t − t′).

Next we show how this can simplify the NMSSE in Eq. (32). We first define the operators,

Vj =

M∑

β=1

θj 〈Rj|β〉Sβ, (39)

with which we can rewrite Eq. (35) as follows,


j

V †j Vjψ(t) + λ

M∑

j=1

wj(t)Vjψ(t). (40)

The stochastic equation in Eq. (40) is linear, and for such SDEs, the second moment,

ρ(t) = E [|ψ(t)〉〈ψ(t)|] ,

follows a closed-form equation. Specifically, by using the Itô’s Lemma, followed by taking theexpectation, one arrives at,

i∂tρ = [HS , ρ]− λ2M∑

j=1

(V †j Vjρ+ ρV †

j Vj − 2V †j ρVj

). (41)

This corresponds to the Lindblad equation [Lin76], which is often referred to as the quantum masterequation (QME). In the next section, we will go beyond the assumption in Eq. (34) and considermore general bath correlation functions.

3.2 The generalized quantum master equation (GQME)

As a building block to approximate general stationary Gaussian processes, we consider the Orn-stein–Uhlenbeck type of processes [Ris84], expressed as the solution of the following linear SDEs,

iζk = −dkζk + γkwj(t), (42)

for k = 1, 2, . . . ,K, with K ≥M . Here γk ≥ 0 and dk is a complex number with positive imaginarypart. In addition, each of these independent OU processes has an initial variance θ2k,

E[ζk(0)†ζk(0)] = θ2k.

If we pick γk, such that γ2k = 2Im(dk), then ζk(t) is a stationary Gaussian process with correlation,

E[ζ†j (t)ζk(t′)] = θ2j δj,k exp

(− id∗j (t− t′)

). (43)

We now construct an ansatz for approximating the bath correlation function. The case whenM = 1 has been thoroughly investigated in [RE14]. As a time correlation function, C(t) can beexpressed in terms of its power spectrum, denoted here by C(ω), as a Fourier integral,

C(t) =

∫ ∞

−∞C(ω)e−iωtdω. (44)

12

Using a contour integral in the upper half plane, the Fourier integral can be reduced to a summationover the poles,

C(t) =∑

j

C(d∗k)e−id∗

kt.

Meanwhile, the power spectrum C(ω) is Hermitian and positive semidefinite, and it can be diago-nalized using the transformation:

C(ω) =∑

j

|Qj(ω)〉 θj(ω) 〈Qj| (ω).

Combining the above two equations leads to the following ansatz:

Cα,β(t) ≈ Cα,β(t) :=

K∑

k=1

θ2k 〈α|Qk〉〈Qk|β〉 exp(− id∗kt

). (45)

To ensure the approximation accuracy, in practice, the ansatz in Eq. (45) is often obtained by aleast squares approach. In addition, we will consider the poles dk within a cut-off frequency dmax,i.e., |dk| ≤ dmax.

Similar to the previous section, we approximate the noise η(t) according to Eq. (38), but withthe white noise wk(t) replaced by the colored noise (OU process) ζk(t),

ηβ(t) =

K∑

k=1

〈Qk|β〉 θkζk(t). (46)

In light of Eq. (43), the approximation in Eq. (46) corresponds to an approximation of the timecorrelation function in Eq. (45).

For the case of a single interaction term, where M = 1 and the matrix C(t) is corresponding toa scalar function, this reduces to the standard approach using a sum of exponentials [RE14]. ButEq. (45) provides a more general scheme to handle multiple interaction terms.

Next we show how this can simplify the NMSSE in Eq. (32) when the bath correlation functionsare expressed as Eq. (45). We first extended the definition in Eq. (39) (notice that the index j goesup to K, which can be greater than M),

Tj =M∑

β=1

θj 〈Qj |β〉Sβ, (47)

This simplifies the NMSSE in Eq. (32) to,

i∂tψ = HSψ − iλ2K∑

k=1

∫ t

0T †ke

−i(HS+d∗k)τTkψ(t− τ)dτ + λ

K∑

j=1

ζj(t)Tjψ(t). (48)

Here the multiplication by d∗k represents the operator d∗kIS , where IS is the identity matrix.Eq. (48) still contains memory. But compared to the original NMSSE in Eq. (32), the correlation

C(t) has been broken down to exponential functions, which can be combined with the unitary

operator e−itHS . In addition, the noise is expressed as the OU process ζj’s, which can be treatedusing Itô calculus.

13

Next, we demonstrate how to embed the dynamics in Eq. (48) into an extended, but Markovian,dynamics. The simple observation that motivated the Markovian embedding is that a convolutionintegral in time can be represented as the solution of a differential equation.

f(t) =

∫ t

0exp(−a(t− τ))g(τ)dτ =⇒ f ′ = −af + g, f(0) = 0. (49)

Here f will be regarded as an auxiliary variable, introduced to reduce the memory integral. Com-pared to computing the integral at every time step using direct quadrature formulas, it is muchmore efficient to solve the differential equation.

We now show that we can use the same idea to define auxiliary orbitals. Specifically, by insertingEq. (45) in the SSE in Eq. (32), and by letting, for k = 1, 2, . . . ,K,

χIk(t) =

∫ t

0e−i(HS+d∗

k)τTkψ(t− τ)dτ,

we arrive at the equation,i∂tχ

Ik = (HS + d∗k)χ

Ik + iσλTkψ(t). (50)

This simplifies Eq. (48) to,


j=1

T †j χ

Ij + λ

K∑

j=1

Tjψ(t)ζj(t). (51)

To incorporate the noise term, we define,

χIIk(t) = iψ(t)ζk(t). (52)

This reduces Eq. (51) to,


j=1

T †j χ

Ij − iλ

K∑

j=1

TjχIIj (t). (53)

It remains to derive a closed-from equation for Eq. (52). Using the Itô’s formula, we obtain,

i∂tχIIk = (HS − dk)χ

IIk + iγkψ(t)wk + λζk(t)T

†kχ

Ik + λζk(t)Tkχ

IIk. (54)

When the coupling parameter λ is sufficiently small, one can add or drop O(λ) terms, which willcontributed to an O(λ2) error when substituted into Eq. (53). If such an error is acceptable, bycollecting equations, we have extended Schrödinger equations,

i∂tψ = HSψ − iλ

K∑

k=1

T †kχ

Ik − iλ

∑

k

TkχIIk,

i∂tχIk = (HS + d∗k)χ

Ik + iλTkψ(t),


IIk + iλT †

kψ(t) + iγkψ(t)wk.k = 1, 2, . . . ,K.

(55)

Rather than dropping O(λ) terms in Eq. (54), one can continue such a procedure and incorporatethe high order terms. Specifically, noticing the similarity between the O(λ) terms in Eq. (54) withEq. (52), we define,

χIIIk = iζk(t)χ

Ik, χIV

k = iζk(t)χIIk. (56)

14

By repeating the above procedure, one can derive similar equations for these auxiliary wave func-tions. These embedding steps yield the following extended Schrödinger equations (ESE),

i∂tψ = HSψ − iλK∑

k=1

T †kχ

Ik − iλ

∑

k

TkχIIk,

i∂tχIk = (HS + d∗k)χ

Ik + iλTkψ(t),


IIk − iλT †

kχIIIk − iλTkχ

IVk + iγkψ(t)wk(t),

i∂tχIIIk = iλTkχ

IIk + (HS − dk − d∗k)χ

IIIk + iγkχ

Ikwk(t),

i∂tχIVk = iλT †

kχIIk + (HS − 2dk)χ

IVk + 2iγkχ

IIkwk(t),

k = 1, 2, . . . ,K.

(57)

We have dropped O(λ) terms in the last two equations, which will contributed to an O(λ3) errorwhen substituted into Eq. (53). Namely, the accuracy is the same as the NMSSE.

From the definitions of the auxiliary functions, we can deduce their initial conditions,

χIk = 0, χII

k = iζk(0)ψ(0), χIIIk = 0, χIII

k = −ζk(0)2ψ(0).

To derive the corresponding GQME of the stochastic model in Eq. (55), we first write it as asystem of SDEs,

i∂tΨ = HΨ+

K∑

k=1

VkΨwk. (58)

Here the function Ψ includes the wave function ψ and the auxiliary wave functions χIk, χ

IIk, χ

IIIk , χ

IVk Kk=1.

One can arrange the wave functions as follows,

Ψ =

ψχI1

χII1

χIII1

χIV1

χI2

χII2

χIII2

χIV2...

, V1 =

0 0 0 0 0 0 . . .0 0 0 0 0 0 . . .√2ν1 0 0 0 0 0 . . .0

√2ν1 0 0 0 0 . . .

0 0 2√2ν1 0 0 0 . . .

0 0 0 0 0 0 . . .0 0 0 0 0 0 . . .0 0 0 0 0 0 . . .0 0 0 0 0 0 . . ....

......

......

.... . .

⊗ IS . (59)

Similarly,

V2 =

0 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 . . .√2ν2 0 0 0 0 0 0 0 . . .0 0 0 0 0

√2ν2 0 0 . . .

0 0 0 0 0 0 2√2ν2 0 . . .

......

......

......

......

. . .

⊗ IS .

15

Clearly, Vk’s are sparse and low rank. Consider the standard basis in R4K+1, here abbreviated

simply into |0〉 , |1〉 , . . . , |4K〉. Then we have,

Vk =√2νk

(|4k − 2〉〈0|+ |4k − 1〉〈4k − 3|+ 2 |4k〉〈4k − 2|

)⊗ IS , k = 1, 2, . . . ,K. (60)

Recall that νk = Imdk, and they are nonnegative.The operator H in Eq. (58) can also be written in a block form,

Hs −iλT†1

−iλT1 0 0 −iλT†2

−iλT2 0 0 · · ·iλT1 Hs + d∗

10 0 0 0 0 0 0 · · ·

0 0 Hs − d1 −iλT†1

−iλT1 0 0 0 0 · · ·0 0 iλT1 Hs + d∗

1− d1 0 0 0 0 0 · · ·

0 0 iλT†1

0 Hs − 2d1 0 0 0 0 · · ·iλT2 0 0 0 0 Hs + d∗

20 0 0 · · ·

0 0 0 0 0 0 Hs − d2 −iλT†2

−iλT2 · · ·0 0 0 0 0 0 iλT2 Hs + d∗

1− d1 0 · · ·

0 0 0 0 0 0 iλT2 0 Hs − 2d1 · · ·...

......

......

......

......

. . .

. (61)

The GQME introduced an auxiliary space that mimics the effect of the quantum bath. RecallIS ∈ C

2n×2n is the identity operator. Similarly, we let IA ∈ C(4K+1)×(4K+1) be the identify operator

in an auxiliary space labelled by A. Then the Hamiltonian in Eq. (61) can be expressed in thermsof tensor products,

H = IA ⊗HS +HA ⊗ IS + iλ

K∑

k=1

Dk ⊗ Tk + iλ

K∑

k=1

Ek ⊗ T †k . (62)

Here HA ∈ C(4K+1)×(4K+1) is a diagonal matrix:

HA = diag 0, d∗1,−d1, d∗1 − d1,−2d1, d∗2,−d2, d∗2 − d2,−2d2, . . . .

In addition, the matrices Dk, Ek ∈ C(4K+1)×(4K+1) are given by

Dk = |4k − 3〉〈0| − |0〉〈4k − 2|+ |4k − 1〉〈4k − 2| − |4k − 2〉〈4k − 1| ,Ek =− |0〉〈4k − 3|+ |4k〉〈4k − 2| − |4k − 2〉〈4k| .

(63)

Assuming that the dimension of the original wave function |ψ〉 ∈ Cn, the dimension of Ψ is (4K+1)n.

Hence, the dimension of Γ is [4K + 1)n]× [(4K + 1)n].Altogether, the density matrix associated with the combined wave functions Ψ in Eq. (59) is

defined as the entry-wise expectation,

Γα,β(t) = E[Ψα(t)Ψβ(t)∗]. (64)

The Itô’s formula in Eq. (21) implies the following quantum master equation,

∂tΓ = K(Γ) := −i(HΓ− ΓH†) +K∑

k=1

VkΓV†k . (65)

16

The noise has been averaged out by the expectation. In fact, it has been shown in [KP13] that forany linear SDEs, the first and second moments satisfy close-form equations.

We can deduce the initial condition of Γ from the definitions of the auxiliary wave functions.Since ζk is Gaussian with mean zero and variance 1, we have

E[⟨χIk

∣∣χIk

⟩] = 0, E[

⟨χIIk

∣∣χIIk

⟩] = ρs(0), E[

⟨χIIIk

∣∣χIIIk

⟩] = 0, E[

⟨χIVk

∣∣χIVk

⟩] = 3ρs(0).

Therefore, the initial density matrix Γ is a block-diagonal matrix,

Γ(0) = diagρS(0), 0, ρS (0), 0, 3ρS(0), 0, ρS (0), 0, 3ρS(0), · · ·

. (66)

We notice that the trace of Γ(0) is 4K + 1.One illustration example is a two-qubit model coupled to a common bosonic bath [WC13],

where HEOM type of equations were derived. Specifically, the system Hamiltonian is written asHs = ω0

2 (σIz + σII

z ), where I and II label the two qubits and ω0 is the Zeeman energy. In addition,in the coupling term, S = σI

x + σIIx. Thus M = 1. Furthermore, the study in [WC13] considered the

bath correlation function C(t) = λγ2 e

−(γ+iω0)t. In light of Eq. (45), we have that K = 1, θ1 =√

λγ2 ,

|Qk〉 = 1, and d1 = ω0 + iγ. From Eq. (47), we also have T1 = θ1S. In this case, the matrix Γis a 5 × 5 block matrix with total dimension being 20. Meanwhile, since the HEOM approach in[WC13] does not make a weak coupling assumption, many more equations need to be introduced inthe extended system until the resulting density matrix converges.

3.3 Properties of the GQME

We first provide some basic estimates on the extended density matrix Γ. We begin by writing theHamiltonian in Eq. (61) as,

H = H0 + λH1. (67)

Here we made the observation that H0 is block diagonal. On the other hand, H1 only containsnonzero blocks in the first row and the first column. To refer to the block entries of the densitymatrix Γ, we write it in the following block form,

Γ =

Γ0,0 Γ0,1 Γ0,2 · · ·Γ1,0 Γ1,1 Γ1,2 · · ·

......

.... . .

Γ4K,0 Γ4K,1 Γ4K,2 · · ·

. (68)

In particular, ρS is embedded into Γ as the first block: ρS = Γ0,0.For such block matrices, we will used the following induced norm,

‖Γ‖∞ := max0≤j≤4K

∑

0≤i≤4K

‖Γi,j‖. (69)

Namely, for each entry, we use the spectral norm. But among the blocks, we use the ∞-norm.One can verify that this norm still has the submultiplicative property. We choose this norm merelybecause we will estimate the bound of each block, and the formula in Eq. (69) can easily connectsuch estimates to the bound of the entire matrix. In principle, since matrix norms are continuouswith respect to the entries, one can also use other norms among the blocks.

17

Lemma 7. The exponential operator U(t) := exp(− itH0

)is bounded for all time:

‖U(t)‖ ≤ 1,∀t ∈ R.

Here we used the spectral norm. This can be seen from the fact that H0 is block diagonal, Hs

is Hermitian, and dk has non-negative imaginary parts.By separating the O(λ) term in Eq. (67), we can write Eq. (65) in a perturbative form,

∂tΓ = −i(H0Γ− ΓH†0) +

K∑

k=1

VkΓV†k − iλ(H1Γ− ΓH†

1). (70)

In particular, we let Γ(0) be the solution of the “unperturbed” equation,

∂tΓ = −i(H0Γ− ΓH†0) +

K∑

k=1

VkΓV†k . (71)

Let L0 be the corresponding operator on the right hand side, and we express the solution of Eq. (71)as,

Γ(0)(t) = exp(tL0)Γ(0)(0). (72)

The following Lemma provides a bound for the perturbation term in Eq. (70).

Lemma 8. Let Λ = max1≤k≤K ‖Tk‖1, and let

Ξ = H1Γ− ΓH†1, (73)

be the commutator for any Hermitian matrix Γ. Then the trace of the first diagonal block of Ξ isbounded by, ∣∣tr

(Ξ0,0

)∣∣ ≤ 4Λ∑

k>0

‖Γ0,k‖. (74)

For the remaining diagonal blocks, one has,

∑

k>0

∣∣tr(Ξk,k

)∣∣ ≤ 4Λ∑

0<j<k≤4K

‖Γj,k‖ . (75)

These estimates can be obtained by direct calculations. For instance, with direct calculations,one can show that the first diagonal is given by,

Ξ0,0 =K∑

k=1

(Γ0,4k−3Tk − T †

kΓ4k−3,0 + Γ0,4k−2T†k − TkΓ4k−2,0

). (76)

Thus the bound Eq. (74) follows from the triangle inequalities, together with the von Neumann’strace inequality. The important observation is that the trace of the perturbation term in Eq. (70)is only controlled by the norms of the off-diagonal blocks of Γ.

We now show that the “unperturbed term” in Eq. (70), i.e., the solution of the GQME in Eq. (65)when λ = 0, has bounded solutions.

18

Lemma 9. Assume that the imaginary parts of dk are non-negative, i.e., νk ≥ 0 for all k. Assumeλ = 0. The solution of the GQME in Eq. (65) is denoted by Γ(0)(t) (also see Eq. (71)). Then thefollowing statements hold.

1. The first diagonal block is given by,

ρ(t) = US(t)ρS(0)US(t)†, US(t) := exp(−itHs).

2. If Γ(0)(0) is block diagonal, then Γ(0)(t) remains block diagonal.

3. If Γ(0)(0) is given by Eq. (66), then, the trace of Γ(0)(t) is

tr(Γ(0)(t)

)= 2K + 1 + 2

∑

k

e−4νkt. (77)

4. The solution Γ(0)(t) of Eq. (71) is bounded for general initial conditions. Namely, the exists aconstant c, independent of t, such that,

‖Γ(0)(t)‖∞ ≤ c‖Γ(0)(0)‖∞, ∀t ∈ R+.

We included the proof in the Appendix.

The GQME in Eq. (65) is considered as a route to obtain an approximation to the density matrixρS(t) from Eq. (29). Assuming that the representation of the bath correlation function in Eq. (45)is exact, we can show that the error associated with the approximation of ρS(t) by the GQME inEq. (65) is O

(λ3).

Theorem 10. Let ρS(t) be the density matrix from the full quantum model in Eq. (29) with bathcorrelation given by Eq. (45). In addition, let ρS(t) be the first diagonal block of the density matrixΓ(t) from the GQME in Eq. (65). Then,

ρS(t) = ρS(t) +O(λ3). (78)

This asymptotic error is consistent with that of the NMSSE in Eq. (32).

In the computation, we work with the GQME in Eq. (65), and the next theorem provides somebounds on the solution Γ(t).

Theorem 11. For any t > 0, the density matrix Γ(t) from the GQME in Eq. (65) is positivesemidefinite: Γ(t) ≥ 0, and it has the following properties.

(i) The norm of the density matrix follows the bound,

‖Γ(t)‖∞ ≤ ‖Γ(0)‖∞ exp(2λC‖H1‖t). (79)

The constant C is the same as that in the Lemma 4.

(ii) The norms of the off-diagonal blocks of Γ(t) is of order λ.

(iii) For any initial condition Γ(0), not necessarily positive, the trace of Γ(t) is bounded as,

∣∣tr(Γ(t)

)∣∣ ≤ 3K∣∣tr(Γ1,1(0)

)∣∣+∑

k>1

∣∣tr(Γk,k(0)

)∣∣ . (80)

19

Corollary 12. Fix T > 0. The trace of Γ(t) for t ∈ [0, T ] is bounded as follows,

tr (Γ(t)) = 2K + 1 + 2

K∑

k=1

e−4νkt +O(λ2). (81)

More importantly, the trace of ρS(t), as the first block diagonal of Γ(t), is bounded by,

tr (ρS(t)) = tr (Γ0,0(t)) = 1 +O(λ3). (82)

4 Quantum algorithm for simulating generalized quantum master

equations

Before presenting the quantum algorithm for this problem, we first prove some useful technicalresults that will be used in our quantum algorithm.

Lemma 13 (Initial state preparation). Given a copy of any initial state |ψ〉 of the system, anormalized version of Γ0 as defined in Eq. (66) can be prepared using O(K) 1- and 2-qubit gates.

Proof. Let K ′ ≤ 2K + 1 be the smallest integer larger than K such that (4K ′ + 1) is some powerof 2. We initialize a log(4K ′ + 1)-qubit state which is a normalized version of

|0〉+K∑

k=1

(|4k − 2〉+ |4k〉

)(83)

This can be implemented using O(K) 1- and 2-qubit gates. Append an additional register that isinitialized to |0〉 and use CNOT gates to “copy” the basis states to the second register. We have thenormalized version of

|0〉|0〉+K∑

k=1

(|4k − 2〉|4k − 2〉+ |4k〉|4k〉

). (84)

Then, simply appending the original state ρS of the system and tracing out the second register, weobtain the normalized version of Γ0.

Constructing block-encodings In the next two lemmas, we show how to obtain block-encodingsof H and Vj for j ∈ [K] using block-encodings of HS and Sβ for β ∈ [M ].

Lemma 14. Given access to an (α, a, ǫ)-block-encoding UHSof HS, an (α, a, ǫ)-block-encoding USβ

for each β ∈ [M ], a (1+δ(α+dmax+2αλ√M), a+O(logM+logK), (1+δ(α+dmax+2αλ

√M))ǫ)-

block-encoding of I − iδH, where H is defined in Eq. (62), can be implemented with one use to eachof UHS

and USβtogether with O(K) 1- and 2-qubit gates.

Proof. Recall the components of H as defined in Eq. (62). We first show that it is easy to construct ablock-encoding of the tensor product of two block-encodings. Let UA be an (α, a, ǫ)-block-encodingof A and UB be an (β, b, ǫ)-encoding of B, then it is straightforward to see that the unitary UA⊗UB

together with O(a + b) swap gates is an (αβ, a + b, ǫ)-block-encoding of A ⊗ B. As a result, an(α, a, ǫ)-block-encoding of IA⊗HS can be implemented. SinceHA is diagonal, it is easy to implement

20

a (dmax,O(logK), 0)-block-encoding of HA. Hence, a (dmax,O(logK), 0)-block-encoding of HA⊗IScan be implemented.

Given an (α, a, ǫ)-block-encoding USβof Sβ, we use Lemma 5 to implement a (

√Mα, a +

O(logM),√Mαǫ)-block-encoding UTj

of Tj . Since Dk is O(1)-sparse, we use Lemma 4 to imple-ment an (O(1),O(log(K)), ǫ)-block-encoding UDk

of Dk. Using the observation on tensor productsof block-encodings, we can implement a (O(

√Mα), a +O(logM + logK),

√Mαǫ)-block-encoding

of Dk ⊗ Tk can be implemented. Ek ⊗ T †k can be implemented similarly.

Now, by Lemma 5, we can implement a (1+ δ(α+ dmax +2αλ√M), a+O(logM + logK), (1+

δ(α + dmax + 2αλ√M))ǫ)-block-encoding of I − iδH can be implemented with one use to each of

UHSand USβ

. The additional 1- and 2-qubit gates are used for the state preparation in Lemma 5and this cost if O(K)

Lemma 15. Given dk’s as in Eq. (45), a (dmax,O(log(K)), ǫ)-block-encoding for Vk for k ∈ [K]can be constructed using O(logK + polylog(1/ǫ)) 1- and 2-qubit gates.

Proof. Each Vk is 1-sparse, and it is straightforward to implement the sparse-access oracles specifiedin Eqs. (12) and (13). Then we use Lemma 4 to implement a (1,O(log(K)), ǫ)-block-encoding forVk/dmax, which implies a (dmax,O(log(K)), ǫ)-block-encoding for Vk.

Infinitesimal approximation by completely-positive maps An important step of our quan-tum algorithm is to use the following superoperator to approximate eKδ when δ is small.

Mδ(X) = A0XA†0 +

K∑

k=1

AkXA†k, (85)

where

A0 = I − iδH, and Ak =√δVk for all k ∈ [K]. (86)

We use the following lemma, which is proved in Appendix E to bound the error of this approx-imation:

Lemma 16. Let Mδ be a superoperator defined in Eq. (85), and let K be as defined in Eq. (65).Define Λ := ‖H‖+∑K

k=1 ‖Vk‖2. Then it holds that

∥∥∥Mδ − eKδ∥∥∥⋄≤ 5(δΛ)2. (87)

Oblivious amplitude amplification for isometries In our algorithm, we need to apply ampli-tude amplification to boost the success probability. However, in our context, the underlying operatoris an isometry instead of a unitary (i.e., part of the input is restricted to be some special state).We use oblivious amplitude amplification for isometries, which was first introduced in [CW17] toachieve this. In [CW17], only a special case where the initial success probability is exactly 1/4 wasconsidered. Here, we give a more general version.

Lemma 17. For any a, b ∈ N+, let |0〉 := |0〉⊗a and |µ〉 := |µ〉⊗b for an arbitrary state |µ〉. Forany n-qubit state |ψ〉, define |ψ〉 := |0〉 |µ〉 |ψ〉. Let the target state |φ〉 be defined as

|φ〉 := |0〉 |φ〉 , (88)

21

Proof of the main theorem Now, we have all the tools for proving the quantum algorithm forsimulating Problem 1.

Theorem 18. Suppose we are given a block-encoding UHSof HS, a block-encodings USα of Sα (for

α ∈ [M ]), θk, |Qk〉 (all entries), and dk for k ∈ [K] as in Eq. (5). There exists a quantum algorithmthat solves Problem 1. Let τ = t(α

√M +Kd2max). This quantum algorithm uses

O(τ√K

log(τ/ǫ)

log log(τ/ǫ)

), (104)

queries to UHSand USa , and

O(τ√K log2(τ/ǫ)

log log(τ/ǫ)+ τK2.5

)(105)

additional 1- and 2-qubit gates.

Proof. We use Lemma 13 to prepare a normalized version of Γ0 as in Eq. (66).To simulate the dynamics, first note that the solution to Eq. (65) is eKt. We use the superoperator

Mδ defined in Eqs. (85) and (86) to approximate eδK for small δ. By Lemma 16, we know that theapproximation error is at most 5(δΛ)2, where Λ = ‖H‖+∑K

k=1 ‖Vk‖2.

To use Lemma 6 to implement Mδ, we first need to implement the block-encoding of Aj forj ∈ 0, . . . ,K − 1. Using Lemma 14, a (1 + δ(α + dmax + 2αλ

√M), a + O(logM + logK), (1 +

δ(α+dmax+2αλ√M))ǫ)-block-encoding of A0 can be implemented. Also, a (

√δdmax,O(log(K)), ǫ)-

block-encoding of Aj can be implemented for each j ∈ [K] using O(logK + polylog(1/ǫ)) 1- and2-qubit gates by Lemma 4. Now we use Lemma 6 to obtain the unnormalized state

∑mj=1 |j〉Aj |ψ〉

with the “success probability” parameter

1

(1 + δ(α + dmax + 2αλ√M))2 +

∑Kj=1 δd

2max

(106)

=1

(1 + δ(2α + 2dmax + 4αλ√M +Kd2max) + δ2(α+ dmax + 2αλ

√M)2

(107)

=1− δ(2α + 2dmax + 4αλ√M +Kd2max) +O(δ2(α+ dmax + 2αλ

√M)2). (108)

Setting the stepsize,

δ = Θ

(1

r(2α+ 2dmax + 4αλ√M +Kd2max)

), (109)

and repeating the above procedure r times, this success probability parameter becomes O(1).For the approximation error, note that

∥∥∥Mδ − eδK∥∥∥⋄≤ 5δ2Λ2 = O

(Λ2

r2(2α + 2dmax + 4αλ√M +Kd2max)

2

)≤ O

(1

r2

). (110)

We have

∥∥Mrδ − etK

∥∥⋄ ≤ O

(1

r

), (111)

23

for evolution t = rδ = Θ(

12α+2dmax+4αλ

√M+Kd2max

). The parameter r can be chosen larger enough

so that this error is at most ǫ.Further, conditioned on we have obtained an approximation Γt of Γt, we can extract an approx-

imation of |ψ〉〈ψ| by measuring the first log(4K + 1) qubits of Γt and post-select the outcome being0. According to Corollary 12, this probability of the outcome being 0 is Ω(1/(2K + 1)). Now, thesuccess probability is Ω(1/K). It follows from Lemma 17 that using O(

√K) iterations of oblivious

amplitude amplification for isometries, we can obtain the desired state.

Now, the total evolution time we have simulated so far is rδ = Θ(

12α+2dmax+4αλ

√M+Kd2max

),

and the total cost in terms of queries to UHSand USβ

is O(r). In the following, we show how toachieve poly-logarithmic cost.

Note that when applying Lemma 5 to construct a block-encoding of A0 = I− iδH and applyingLemma 6 to implement the superoperator specified by Kraus operators A0, A1, . . . , AK , the firstregister (containing O(logK) qubits) is used for the |µ〉 state in Lemma 6, and the second register(containing O(logK) qubits) is used for state B |0〉 in Lemma 5. The coefficients of the state inthe two registers are concentrated to |0〉|0〉, which corresponds to I (nothing to implement). Morespecifically, recall that the parameter s0 in Lemma 6 is the block-encoding normalization factor forI − iδH, which is

∑2K+2j=1 yiαj by Lemma 14. As a result, the amplitude for |0〉 |0〉 is

s0√∑Kj=0 s

2j

· 1√∑2K+2j=1 yjαj

=

√√√√∑2K+2

j=1 yjαj∑K

j=0 s2j

(112)

=

√1 + δ(α+ dmax + 2αλ

√M)

(1 + δ(α + dmax + 2αλ√M))2 +Kdmaxδ

(113)

=

√1 + δ(α + dmax + 2αλ

√M)

1 + 2δ(α + dmax + 2αλ√M +Kdmax/2) + Θ(δ2(α+ dmax + αλ

√M +Kdmax)2)

(114)

=

√1− δ(α + dmax + 2αλ

√M +Kdmax/2) + Θ(δ2(α+ dmax + αλ

√M +Kdmax)2). (115)

Therefore, the probability that the first two register are not measured 0, 0 is proportional to2

δ(α + dmax + 2αλ√M +Kdmax/2) + Θ(δ2(α+ dmax + αλ

√M +Kdmax)

2) = O(1

r

)(116)

We apply the techniques used in [CW17] (first introduced in [BCC+14]) to bypass the state prepa-ration for |µ〉 in Lemma 5 and for B |0〉 in Lemma 5. Instead, we use a state with Hamming weightat most

h = O(

log(1/ǫ)

log log(1/ǫ)

), (117)

to approximate the state after the state preparation procedure while causing error at most ǫ. Fol-lowing similar analysis as in [CW17], the number of 1- and 2-qubit gates for implementing this

2Note that we use the term “proportional to” because the actual probability is normalized according to the trace

ratio of the first block and the whole matrix of Γ, which incurs a factor O(1/K).

24

compressed encoding procedure is O(log(1/ǫ)h+K2). Now, the number of queries to UHSand USβ

becomes

O(√

K log(1/ǫ)

log log(1/ǫ)

). (118)

For arbitrary simulation time t, we repeat this O(t(2α+2dmax+4αλ√M+Kd2max)) = O(t(α

√M+

Kd2max)) times where each segment has the error parameter ǫ/(t(α√M +Kd2max)), which has total

cost

O(τ√K

log(τ/ǫ)

log log(τ/ǫ)

), (119)

where τ = t(α√M +Kd2max).

The additional 1- and 2-qubit gates are used in the compressed encoding procedure and thereflections in the oblivious amplitude amplification for isometries. This is bounded by

O(τ√K(log(τ/ǫ)h+K2

))= O

(τ√K log2(τ/ǫ)

log log(τ/ǫ)+ τK2.5

). (120)

Acknowledgement

XL’s research is supported by the National Science Foundation Grants DMS-2111221. CW thanksYudong Cao and Peter D. Johnson for helpful discussions on the HEOM approach for modelingnon-Markovian open quantum systems.

A The proof of Lemma 9

Proof. Our proof will mainly target the statement (4). The rest of the lemma will become selfevident throughout the proof. In light of the structure of the GQME in Eq. (65), it is enough toconsider the case K = 1. In this case, Γ can be viewed as a 5× 5 block matrix. We first write V1 ina block matrix form,

V1 =

(0 0R1 0

),

where R1 is a 3× 3 block matrix with diagonals√2ν1IS ,

√2ν1IS and 2

√2ν1IS . With direct calcu-

lations, we can show that the last term in Eq. (65) can be written as,

V1ΓV†1 =

(0 0

0 R1Γ0:2,0:2R†1

).

Here Γ0:2,0:2 refers to the first 3× 3 sub-matrix of Γ.

We first look at the scenario when Γ(0) is block diagonal. The zero blocks in V1ΓV†1 , along

with the observation that H0 is block diagonal, imply that the off-diagonals do not change. For thediagonal blocks of Γ(t), we first have,

Γ0,0(t) =US(t)ρS(0)US(t)†,

Γ1,1(t) = exp(−it(−d1 + d∗1))US(t)Γ1,1(0)US(t)†.

(121)

25

As a result, these two blocks have norms given respectively by,

‖Γ0,0(t)‖ = ‖Γ0,0(0)‖, ‖Γ1,1(t)‖ = ‖Γ1,1(0)‖e−2ν1t.

The next three block diagonals will pick up non-homogeneous terms. For instance, we have,

∂tΓ2,2 = −i[Hs − d1,Γ2,2] + 2ν1Γ0,0.

Using the variation-of-constant formula, we have,

Γ2,2(t) = e−2ν1tUS(t)Γ2,2(0)US(t)† + 2ν1

∫ t

0e−2ν1τUS(τ)Γ0,0(t− τ)US(τ)

†dτ. (122)

Also by noticing that ‖Γ1,1(t)‖ is constant in time, the diagonal block Γ3,3 can be boundeddirectly as,

‖Γ2,2(t)‖ ≤ ‖Γ2,2(0)‖ exp(−2ν1t) + ‖Γ0,0(0)‖(1 − e−2ν1t). (123)

The right hand side remains bounded for all time. Similarly, the next diagonal block can be expressedas,

Γ3,3(t) = exp(−4ν1t)US(t)Γ3,3(0)US(t)† + 2ν1

∫ t

0exp(−4ν1t)US(τ)Γ1,1(t− τ)US(τ)

†dτ. (124)

Notice that ‖Γ1,1(t)‖ is proportional to exp(−2ν1t). Essentially, what leads to the boundedness ofthe solution is the fact that there is no secular term, implying that ‖Γ3,3(t)‖ follows a similar boundas ‖Γ2,2(t)‖2. The estimate of ‖Γ4,4(t)‖ follows the same steps.

We now turn to the off-diagonal blocks. By direct calculations, we have, for j = 0, 1, k = 1, 2, 3, 4,and k > j

∂tΓj,k = −i(HSΓj,k − Γj,k(Hs + d1)

),

which yields,

Γj,k(t) = exp(−id∗1t)US(t)Γj,k(0)U†S(t) =⇒ ‖Γj,k(t)‖ = ‖Γj,k(0)‖ exp(−ν1t). (125)

For the remaining off-diagonal entries, we will check Γ2,3 as an example. It follows the equation,

∂tΓ2,3 = −i((Hs − d1)Γ2,3 − Γ2,3(Hs + d1 + d∗1)

)+ 2ν1Γ0,1.

This implies that,

‖Γ3,4(t)‖ ≤ ‖Γ3,4(0)‖e−3ν1t + ‖Γ1,2(0)‖2e−2ν1t(1− e−ν1t).

We also see from these calculations that these off-diagonal blocks will become zero if the initialmatrix Γ(0) is block diagonal.

By examining the block entries of Γ(t), we have shown the boundedness of the solution statedin Lemma 9 for all time.

26

B The proof of Theorem 11

Proof. In Lemma 9, we have proved a bound for the case when λ = 0. Denote the solution byΓ0(t), and the solution operator by exp(tL0). Therefore, the GQME in Eq. (65) can be written ina perturbation form,

∂tΓ = L0Γ− iλ(H1Γ− ΓH†1). (126)

The solution can be recast in an integral form,

Γ(t) = Γ0(t)− iλ

∫ t

0exp ((t− τ)L0) (H1Γ− ΓH†

1)dτ. (127)

From Lemma 8, the super-operator exp(tL0) is bounded. Therefore, we have,

‖Γ(t)− Γ0(t)‖ ≤ 2λC‖H1‖∫ t

0‖Γ(t)‖dt.

As a result, the bound can be obtained by directly using the Gronwall’s inequality.Since Γ0(t) is block diagonal, the above inequality also shows that the off-diagonal blocks of Γ(t)

is of order λ. Finally, the bounds for the trace can be verified from Eqs. (121), (122) and (124) inthe proof of Lemma 9.

C The proof of Corollary 12

Proof. From Eq. (126), we may take the trace.

tr(Γ(t))− tr(Γ(0)(t)

)= −iλ

∫ t

0tr(Σ(t, τ))dτ, (128)

where,Σ(t, τ) = exp(tL0)Ξ,

with Ξ from Eq. (73). By the definition of exp(tL0), Σ(t, τ) is the solution of the equation,

∂tΣ = L0Σ, Σ(τ, τ) = (H1Γ− ΓH†1).

Using the property in Eq. (80) of the super-operator exp(tL0) in Theorem 11, we obtain the bound,

∣∣tr(Σ(t, τ)

)∣∣ ≤ 3K∣∣tr(Σ0,0(τ, τ)

)∣∣+∑

k>0

∣∣tr(Σk,k(τ, τ)

)∣∣ .

We now invoke the estimate in Lemma 8. The trace of each diagonal block of Σ(τ, τ) is boundedby the off-diagonal blocks of Γ(t), which is of order λ. Namely, there exists a constant, such that,

∣∣tr(Σ(τ, τ)

)∣∣ ≤ Cλ.

Collecting terms, we have, ∣∣∣tr(Γ(t))− tr(Γ(0)(t)

)∣∣∣ ≤ Ctλ2.

27

Alternatively, one can start with Eq. (126), and apply the formula again to replace the Γ(t) inthe integral:

Γ(t)− Γ(0)(t) =− iλ

∫ t

0exp ((t− τ)L0) (H1Γ

(0)(τ)− Γ(0)(τ)H†1)dτ

− λ2∫ t

0

∫ τ

0exp ((t− τ)L0) (H1Σ(τ, s)− Σ(τ, s)H†

1)dsdτ +O(λ3).

Here Σ(τ, s) = exp ((τ − s)L0) Ξ0, with

Ξ(0) = H1Γ0(s)− Γ0(s)H†1. (129)

For the O(λ) term, we notice that Γ(0) is block diagonal, and so in light of Lemma 8, the matrixΞ(0) has zero trace. This implies that,

tr(Γ(t))− Γ(0)(t)

)= O(λ2).

For the O(λ2) term, from Eq. (76), we have, the trace of the first block of H1Σ(τ, s)−Σ(τ, s)H†1

is given by,K∑

k=1

tr((Σ0,4k−3 − Σ4k−2,0)Tk

)+

K∑

k=1

tr(T †k (Σ0,4k−2 − Σ4k−3,0)

). (130)

Meanwhile, from the proof of Lemma 9, we see that the superoperator does not change the

off-diagonal blocks in the first row and column. Thus, Σ0,j(τ, s) = Σ0,j(s, s) = Ξ(0)0,j(s).

With direct matrix multiplications, we find that,

Ξ4k−3,0 = −TkΓ4k−3,4k−3 − Γ0,0T†k , Ξ4k−2,0 = TkΓ4k−2,4k−2.

From the proof of Lemma 9, we also have Γ4k−3,4k−3(t) = 0. Combining these steps, we find that,the trace in Eq. (130) is zero. Therefore, we have

tr(ρS(t)) = 1 +O(λ3).

D The Proof of Theorem 10

Proof. We will prove the asymptotic bound using an expansion of the Eq. (29). More specifically,we write the total density matrix in terms of powers of λ,

ρ(t) = ρ(0)(t) + λρ(1)(t) + λ2ρ(2)(t) +O(λ3). (131)

By taking a partial trace over the bath space, we obtain a similar expansion for ρS :

ρS(t) = ρ(0)S (t) + λρ

(1)S (t) + λ2ρ

(2)S (t) +O(λ3). (132)

28

By inserting Eq. (131) into Eq. (29) and separate terms of different order, we arrive at

i∂tρ(0) = [HS ⊗ IB + IS ⊗HB , ρ

(0)], ρ(0)(0) = ρ(0),

i∂tρ(1) = [HS ⊗ IB + IS ⊗HB , ρ

(1)] +M∑

α=1

[Sα ⊗Bα, ρ(0)], ρ(1)(0) = 0,

i∂tρ(2) = [HS ⊗ IB + IS ⊗HB , ρ

(2)] +

M∑

α=1

[Sα ⊗Bα, ρ(1)], ρ(2)(0) = 0.

(133)

Within this expansion, the dynamics of ρ(0) contains no coupling. Let

U(t) = US(t)⊗ UB(t), US = exp (−itHS) , UB(t) = exp (−itHB) ,

be the unitary operators. Then we have,

ρ(0)(t) = U(t)ρ(0)(0)U(t)†. (134)

Since all the operators on the right hand side are in tensor product forms, ρ(0)(t) remains a tensorproduct:

ρ(0)(t) = ρ(0)S (t)⊗ ρB. (135)

Here ρ(0)S (t) = US(t)ρS(0)US(t)

†. Meanwhile, the matrix ρB stays because it commutes with HB.The term ρ(1) can be expressed using the variation-of-constant formula,

ρ(1)(t) =− i∑

α

∫ t

0U(t− t′)Sα ⊗Bαρ

(0)(t′)U(t− t′)†dt′

+ i∑

α

∫ t

0U(t− t′)ρ(0)(t′)Sα ⊗BαU(t− t′)†dt′.

In light of Eq. (135), we can make the same observation that ρ(1)(t) consists of terms that are tensorproducts. By following standard notations [Bre02], i.e.,

Sα(t) = US(t)†SαUS(t), Bα(t) = UB(t)

†BαUB(t), (136)

we can simplify ρ(1)(t) as follows,

ρ(1)(t) =− i∑

α

∫ t

0Sα(t

′ − t)ρ(0)S (t)⊗Bα(t

′ − t)ρBdt′

+ i∑

α

∫ t

0ρ(0)S (t)Sα(t

′ − t)⊗Bα(t′ − t)ρBdt

′.

Since ρB commutes with HB, it commutes with UB . Therefore,

tr(Bα(t

′ − t)ρB)= tr(BαρB) = 0,

which shows thattrB

(ρ(1)(t)

)= 0.

29

Therefore, ρ(1)S (t) = 0. The correction to ρS comes from ρ

(s)S (t), which is similarly expressed as,

ρ(2)(t) =− i∑

α

∫ t

0U(t− t′)Sα ⊗Bαρ

(1)(t′)U(t− t′)†dt′

+ i∑

α

∫ t

0U(t− t′)ρ(1)(t′)Sα ⊗BαU(t− t′)†dt′,

=−∑

α

∑

β

∫ t

0

∫ t′

0Sα(t

′ − t)Sβ(t′′ − t)ρ

(0)S (t)⊗Bα(t

′ − t)Bβ(t′′ − t)ρBdt

′′dt′,

+∑

α

∑

β

∫ t

0

∫ t′

0Sβ(t

′′ − t)ρ(0)S (t)Sα(t

′ − t)⊗Bβ(t′′ − t)ρBBα(t

′ − t)dt′′dt′,

+∑

α

∑

β

∫ t

0

∫ t′

0Sα(t

′ − t)ρ(0)S (t)Sβ(t

′′ − t)⊗Bα(t′ − t)ρBBβ(t

′′ − t)dt′′dt′,

−∑

α

∑

β

∫ t

0

∫ t′

0ρ(0)S (t)Sβ(t

′′ − t)Sα(t′ − t)⊗ ρBBβ(t

′′ − t)Bα(t′ − t)dt′′dt′.

Invoking the bath correlation function,

Cα,β(t) = tr(Bα(t)BβρB), (137)

we arrive at an expansion of ρS(t) up to O(λ2),

ρS(t) = ρ(0)S (t)− λ2

∑

α

∑

β

∫ t

0

∫ t′

0Sα(t

′ − t)Sβ(t′′ − t)ρ

(0)S (t)Cα,β(t

′ − t′′)dt′′dt′

+ λ2∑

α

∑

β

∫ t

0

∫ t′

0Sβ(t

′′ − t)ρ(0)S (t)Sα(t

′ − t)Cα,β(t′ − t′′)dt′′dt′

+ λ2∑

α

∑

β

∫ t

0

∫ t′

0Sα(t

′ − t)ρ(0)S (t)Sβ(t

′′ − t)Cβ,α(t′ − t′′)∗dt′′dt′

− λ2∑

α

∑

β

∫ t

0

∫ t′

0ρ(0)S (t)Sβ(t

′′ − t)Sα(t′ − t)Cα,β(t

′ − t′′)∗dt′′dt′

+O(λ3).

(138)

Here we have used the property of the bath correlation function: Cα,β(t) = Cβ,α(−t)∗. Now we

30

incorporate the function form of the bath correlation function in Eq. (45). We find that,

ρS(t) = ρ(0)S (t)− λ2

∑

k

∫ t

0

∫ t′

0Tk(t

′ − t)Tk(t′′ − t)ρ

(0)S (t)e−i(t′−t′′)dktdt′′dt′

+ λ2∑

k

∫ t

0

∫ t′

0Tk(t

′′ − t)ρ(0)S (t)Tk(t

′ − t)e−i(t′−t′′)dktdt′′dt′

+ λ2∑

k

∫ t

0

∫ t′

0Tk(t

′ − t)ρ(0)S (t)Tk(t

′′ − t)ei(t′−t′′)dktdt′′dt′

− λ2∑

k

∫ t

0

∫ t′

0ρ(0)S (t)Tk(t

′′ − t)Tk(t′ − t)ei(t

′−t′′)dktdt′′dt′

+O(λ3).

(139)

Now we show that the GQME in Eq. (65) has an asymptotic expansion that is consistent withEq. (138). Expanding Γ as,

Γ(t) = Γ(0)(t) + λΓ(1)(t) + λ2Γ(2)(t) +O(λ3),

and substituting it into Eq. (65), one gets,

Γ(0)(t) = exp (tL0) Γ(0),

Γ(1)(t) = −i∫ t

0exp

((t− t′)L0

)[H1,Γ

(0)(t′)]dt′,

Γ(2)(t) = −i∫ t

0exp

((t− t′)L0

)[H1,Γ

(1)(t′)]dt′.

(140)

The leading term Γ(0)(t) has been shown to be a block diagonal matrix in the previous section. The

first diagonal block is precisely ρ(0)S (t), which is consistent with the O (1) term in Eq. (138). To

examine Γ(1)(t), we first notice that the commutator in the integral has the following structure,

Ξ(0)(t′) = [H1,Γ(0)(t′)] = i

0 Ξ(0)0,1(t

′) Ξ(0)0,2(t

′) 0 0 · · ·Ξ(0)1,0(t

′) 0 0 0 0 · · ·Ξ(0)2,1(t

′) 0 0 Ξ(0)3,4(t

′) Ξ(0)3,5(t

′) · · ·0 0 Ξ

(0)4,3(t

′) 0 0 · · ·0 0 Ξ

(0)5,3(t

′) 0 0 · · ·...

......

......

. . .

. (141)

Here we highlighted the leading 5 × 5 submatrix and the zero blocks within it. This is enough forthe purpose of the proof.

By inspecting the solutions that correspond to exp (tL0), we find that the first diagonal blockof∫ t

0 exp ((t− t′)L0) Ξ(0)(t′) is zero. Therefore, Γ(1) has no contribution to the density matrix ρS ,

i.e., there is no O (λ) term. This is consistent with Eq. (138).

31

To proceed further, we have to identify the nonzero blocks in Eq. (141). With direct calculations,we have,

Ξ(0)0,4k−3(t) =− Γ

(0)1,1(t)Tj = −ρ(0)S (t)Tj ,

Ξ(0)0,4k−2(t) =TkΓ

(0)4k−2,4k−2(t).

From Eq. (140), we can extract the equation,

i∂tΓ(1)0,4k−3 = HSΓ

(1)0,4k−3 − Γ

(1)0,4k−3(HS + dk).

Combining the two equations above, we obtain,

Γ(1)0,4k−3(t

′) =

∫ t′

0US(t

′ − t′′)ρ(0)S (t′′)TkUS(t

′ − t′′)†ei(t′−t′′)dkdt′′

=

∫ t′

0ρ(0)S (t′)Tk(t

′′ − t′)ei(t′−t′′)dkdt′′.

Again using the solution properties associated with the superoperator exp (tL0), we have thatthe first block of Γ(1) is given by,

− i

∫ t

0US(t− t′)Ξ(1)(t′)US(t− t′)†dt′

=∑

k

∫ t

0US(t− t′)

([Γ0,4k−3(t

′), Tk] + [Γ0,4k−2(t′), T †

k ])US(t− t′)†dt′.

Similar to Eq. (139), we have also obtained four terms after expanding the commutators. Let usexamine the first integral term,

∑

k

∫ t

0

∫ t′

0US(t− t′)ρ

(0)S (t′)Tk(t

′′ − t′)Tkei(t′−t′′)dkUS(t− t′)†)dt′′

=∑

k

∫ t

0

∫ t′

0ρ(0)S (t′)Tk(t

′′ − t)Tk(t′ − t)ei(t

′−t′′)dkdt′′dt′.

This is the same as the first last integral in Eq. (139). The rest of the integrals can be similarlyverified.

E The proof of Lemma 16

Proof. We use an intermediate superoperator I + δK. We denote by CN the Hilbert space K is

acting on. Let CN ′be a Hilbert space of arbitrary dimension N ′. For any operator Q on C

N ⊗CN ′

with ‖Q‖1 = 1, we have

‖(Mδ ⊗ IN ′ − (IN + δK) ⊗ IN ′)(Q)‖1 =

∥∥∥∥∥∥

m∑

j=0

(Aj ⊗ I)Q(Aj ⊗ I)† − (Q+ δ(K ⊗ IN ′)(Q)

∥∥∥∥∥∥1

(142)

=∥∥∥δ2(H ⊗ I)Q(H† ⊗ I)

∥∥∥1

(143)

≤ ‖H ⊗ I‖2 (144)

≤ (δΛ)2. (145)

32

Now, we have

‖(Mδ − (IN + δK)‖⋄ ≤ (δΛ)2. (146)

To bound the distance between I + δK and eδK, we assume 0 ≤ δ‖K‖⋄ ≤ 1. Consider any Xsuch that ‖X‖1 ≤ 1, we have

∥∥∥(eδK − (I + δK))(X)∥∥∥1=

∥∥∥∥∥

∞∑

s=2

δs

s!Ks(X)

∥∥∥∥∥1

≤∞∑

s=2

δs

s!‖Ks(X)‖1 (147)

≤∞∑

s=2

δs

s!‖K(X)‖s1 ≤ (δ‖K(X)‖1)2 ≤ (δ‖K‖1)2, (148)

where the penultimate inequality follows from the fact that ez − (1 + z) ≤ z2 when 0 ≤ z ≤ 1.Now, we extend this bound to the diamond norm. Note that, for two Hilbert spaces C

N andCN ′

,

(eδK − (IN + δK)) ⊗ IN ′ = eδ(K⊗IN′ ) − (IN×N ′ + δ(K ⊗ IN ′)). (149)

When N = N ′, we have that ‖K ⊗ IN ′‖1 = ‖K‖⋄. This implies that

∥∥∥(eδK − (IN ′ + δK)∥∥∥⋄=∥∥∥(eδK − (IN + δK)

)⊗ IN ′

∥∥∥1

(150)

=∥∥∥eδ(K⊗IN′ ) − (IN×N ′ + δ(K ⊗ IN ′))

∥∥∥1

(151)

≤ (δ‖K ⊗ IN ′‖1)2 (152)

≤ (δ‖K‖⋄)2. (153)

To see the relationship between ‖L‖⋄ and ‖K‖1, first observe that ‖K‖1 ≤ 2Λ. Then using thefact the ‖M ⊗ I‖ = ‖M‖ for all M , t follows that ‖K‖⋄ ≤ 2Λ. Together with Eq. (153), we have

∥∥∥(eδK − (IN ′ + δK)∥∥∥⋄≤ (2δΛ)2. (154)

This lemma follows from Eqs. (146) and (154).

33

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succinct description and eﬃcient simulation of non

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