Relationship between costs for quantum error mitigation and non-Markovian measures
TThe relationship between costs for quantum error mitigation and non-Markovianmeasures
Hideaki Hakoshima, Yuichiro Matsuzaki, ∗ and Suguru Endo † Research Center for Emerging Computing Technologies,National Institute of Advanced Industrial Science and Technology (AIST),1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan. NTT Secure Platform Laboratories, NTT Corporation, Musashino 180-8585, Japan
Quantum error mitigation (QEM) has been proposed as an alternative method of quantum errorcorrection (QEC) to compensate errors in quantum systems without qubit overhead. While Marko-vian gate errors on digital quantum computers are mainly considered previously, it is indispensableto discuss a relationship between QEM and non-Markovian errors because non-Markovian noiseeffects inevitably exist in most of the solid state systems. In this work, we investigate the QEM fornon-Markovian noise, and show that there is a clear relationship between costs for QEM and non-Markovian measures. We exemplify several non-Markovian noise models to bridge a gap betweenour theoretical framework and concrete physical systems. This discovery may help designing betterQEM strategies for realistic quantum devices with non-Markovian environments.
I. INTRODUCTION
It is now widely accepted that quantum computingwill enable us to solve classically intractable tasks suchas Shor’s algorithm for prime factorization [1], quan-tum simulation for quantum many-body systems [2], andHHL (Harrow-Hassidim-Lloyd) algorithm for solving lin-ear equations [3]. However, the effect of decoherence doesimpose an inevitable impact on the reliability and effi-ciency of quantum computation, and hence suppressingphysical errors is crucial to obtain reliable results [2, 4–7]. Although fault-tolerant quantum computing based onquantum error correction can resolve that difficulty, it isnot likely to happen for a while because of requiring thelarge number of physical qubits per single logical qubit.Quantum error mitigation (QEM) methods have beenproposed to mitigate errors in digital quantum comput-ing, which is compatible with near-term quantum com-puters with the restricted number of qubits and gateoperations as it does not rely on encoding required infault-tolerant quantum computing [8–12]. For example,probabilistic error cancellation can perfectly cancel theeffect of noise if the complete description of the noisemodel is given [9, 10]. We apply recovery quantum op-erations E R to invert noise processes of gates N G suchthat E R = N − G . Since inverse channel of noisy processis generally unphysical channel, we need to realize thisby applying single-qubit operations with a classical post-processing of measurement outcomes. Also, the repeti-tion of quantum circuits need to be C times greater toachieve the same accuracy as before QEM, where C is anoverhead factor determined by the noise model and oper-ations used in the QEM procedure. We call this overheadfactor as QEM costs throughout this paper. We can writeQEM costs of the error mitigation of the quantum circuit ∗ [email protected] † [email protected] as C = (cid:81) N g k =1 c k where c k is a QEM cost of k -th noisygate and N g is the number of gates. Therefore, in or-der to suppress costs of QEM, we need to investigate theproperty of c k and optimize it.Recently, Sun et al. [13] proposed a general quantumerror mitigation scheme that can also be applied to con-tinuous quantum systems such as analog quantum sim-ulators. The continuous time evolution of a quantumsystem is described by dρ N ( t ) dt = − i [ H ( t ) , ρ N ( t )] + L (cid:2) ρ N ( t ) (cid:3) , (1)where H is the Hamiltonian, ρ N ( t ) denotes a densitymatrix under noisy dynamics, L (cid:2) ρ N ( t ) (cid:3) is the super-operator describing the effect of the environment, whichshould be mitigated. Even when L consists of local Lind-blad operators, the effects of them easily propagate to theentire system, resulting in highly correlated noise. Notethat Eq. (1) can be rewritten as ρ N ( t + δt ) = E N ( ρ N ( t ))where E N denotes a superoperator of noisy dynamics fora small time interval δt . Thus, similarly to probabilis-tic cancellation, denoting E I as the ideal process, we canapply the recovery channel E R such that E R E N = E I us-ing additional single-qubit operations and classical post-processing of measurement results. In Ref. [13], thestochastic QEM method was introduced to implementrecovery operations in the limit of δt → E R ( t ) can be described as c ( t ) ≈ c (cid:48) ( t ) δt . Therefore,a QEM cost from t = 0 to t = T can be described as C ( T ) = exp[ (cid:82) T dtc (cid:48) ( t )].So far, QEM for Markovian noise was mainly consid-ered and non-Markovian noise was not investigated well.The development of the QEM for non-Markov noise ispractically important, because non-Markovian noise isrelevant in most of the solid state systems such as super-conducting qubits, nitrogen vacancy centers in diamond,and spin qubits in quantum dots [14–19].In addition to the practical motivation for non-Markovian noise, the concept of quantum non- a r X i v : . [ qu a n t - ph ] S e p Markovianity has been extensively studied from a fun-damental interest of the characterization and the quan-tification of the backflow of the information from an en-vironment [20, 21].
There are some definitions of non-Markovianity such as semigroup definition [22], divisi-bility definition [23, 24], and BLP (proposed by Breuer,Laine and Piilo) definition [25] (Ref. [20] discusses a hier-archical relation between these definitions). Throughoutthis paper, we adopt divisible maps as Markovian pro-cesses and we will show that precise definition in latersection.Moreover, there are many applications to utilize non-Markovianity in a positive way for quantum informa-tion processing, including quantum Zeno effects [26, 27],dynamical decoupling [28], Loschmidt echo and criti-cality [29], continuous-variable quantum key distribu-tion [30], time-invariant discord [31], quantum chaos [32],quantum resource theory [33], and quantum metrol-ogy [34, 35]. These motivate the researchers to inves-tigate the properties of non-Markovianity.In this paper, we investigate QEM costs for the case ofnon-Markovian noise. The stochastic QEM can be nat-urally applied to time-dependent non-Markovian noiseto fully compensate for physical errors. We show thatQEM costs reduce in the non-Markovian region. Wealso find a clear relationship between costs of QEM andpreviously reported non-Markovian measures, decay ratemeasure [36] and RHP (Rivas, Huelga, and Plenio) mea-sure [24, 36]. We calculate QEM costs for two exper-imental setups showing non-Markovianity as examples:The one is a controllable open quantum system, whichconsists of a long-lived qubit coupled with a short-livedqubit. This system has been realized in the NMR (nu-clear magnetic resonance) experiments [37, 38]. Theother example is a qubit dispersively coupled with a dis-sipative resonator [39–41]. This discovery may illuminatehow to construct efficient QEM procedures.The rest of this paper is organized as follows. In Sec. II,we review the stochastic QEM proposed in Ref. [13]. InSec. III, we review the definition and the measure of non-Markovianity. In Sec. IV, we discuss the relation betweenQEM costs and the measure of non-Markovianity, andstudy QEM costs for specific models. Finally, we sum-marize and discuss our results in Sec. V.
II. STOCHASTIC QUANTUM ERRORMITIGATION
In this section, we review the stochastic QEM [13].Suppose that the dynamics of the system of interest canbe described by Eq. (1). Here, we assume that the localnoise and the coupling to the environment is sufficientlyweak and the continuous dynamics of the system can bedescribed by the time-dependent Lindblad master equa-tion. Now we express the evolution of the state from t to t + δt as ρ ( t + δt ) = E N ( t )( ρ ( t )) and ρ ( t + δt ) = E I ( t )( ρ ( t )),corresponding to the noisy and the ideal process, respec- tively. The ideal process represents the unitary dynam-ics without any noisy operators in Eq. (1). We hope toemulate the ideal evolution E I by mitigating errors ofthe process E N . When the evolution is affected by localnoise operators, i.e., L can be decomposed as a linearcombination of local noise operators, by using a recoveryoperation E Q ( t ), we can efficiently find a decomposition: E I ( t ) = E Q ( t ) E N ( t ) (2) E Q ( t ) = (cid:88) i µ i R i = c ( t ) (cid:88) i sgn( µ i ) p i R i . (3)Here, c ( t ) = (cid:80) i | µ i | , p i = | µ i | /c ( t ), and {R i } is a setof polynomial number of physical operations applied forQEM. Each R i is a tensor product of single-qubit oper-ations. For a given decomposition, the ideal process U T from t = 0 to t = T can be decomposed as U T ≈ N d − (cid:89) n =0 E I ( nδt )= C ( T ) (cid:88) (cid:126)i p (cid:126)i s (cid:126)i N d − (cid:89) n =0 R i n E N ( nδt ) + O ( T δt ) , (4)where N d = T /δt , (cid:126)i = ( i , i , ..., i N d ), p (cid:126)i = (cid:81) N d − n =0 p i n , s (cid:126)i = (cid:81) N d − n =0 sgn( µ i n ) and a QEM cost can be describedas C ( T ) = (cid:81) N d − n =0 c ( nδt ). Suppose that the initial statefor the quantum circuit is ρ in , we have ρ I ( T ) = C ( T ) (cid:88) (cid:126)i p (cid:126)i s (cid:126)i ρ (cid:126)i + O ( T δt ) , (5)where ρ I ( T ) is the density operator after the ideal process ρ I ( T ) = U T ( ρ in ) and ρ (cid:126)i = ( (cid:81) N d − n =0 R i n E N ( nδt ))( ρ in ).When measuring a observable M , since the expectationvalue for the state ρ equals to (cid:104) M (cid:105) ρ = Tr[ ρM ], we obtain (cid:104) M (cid:105) ρ I = C ( T ) (cid:88) (cid:126)i p (cid:126)i s (cid:126)i (cid:104) M (cid:105) ρ (cid:126)i + O ( T δt ) . (6)We can obtain the (cid:104) M (cid:105) ρ I of the Eq. (6) from the actualexperiment as follows: Firstly, we generate the recoveryoperation R i with a probability p i with a time interval δt until time T , and measure the observable M , and werecord the measurement outcome after multiplying thefactor of s (cid:126)i . Secondly, we repeat the same procedure toreduce the statistical uncertainty. Finally, we estimatethe value of C ( T ) (cid:80) (cid:126)i p (cid:126)i s (cid:126)i (cid:104) M (cid:105) ρ (cid:126)i from the measurementresults, and this approximates the error-free expectationvalue.Since E N ≈ E I for a small δt and the recovery operationbecomes an identity operation in almost all the cases, wecan use the Monte Carlo method to stochastically real-ize continuous recovery operations R i corresponding to δt → +0 to eliminate a discretization error O ( T δt ). Thisprocedure is similar to the one employed in the simula-tion of stochastic Schr¨odinger equation. Refer to Ref. [13]for details. Let c ( t ) = 1 + c (cid:48) ( t ) δt , the QEM cost becomes C ( T ) = lim δt → +0 T/δt − (cid:89) n =0 (1 + c (cid:48) ( nδt ) δt )= exp (cid:18) (cid:90) T dtc (cid:48) ( t ) (cid:19) . (7) III. PROPERTIES OF NON-MARKOVIANITY
In this section, we review typical properties of non-Markovianity [20, 21].
A. Definition of non-Markovianity
There are some definitions of Markovianity in quan-tum dynamics, but throughout this paper, we adopt adefinition introduced in [23, 24]. Here, a Markovian mapis defined as a CP-divisible map : a dynamical map E ( t, from 0 to t is CP-divisible if the map E ( t,s ) (0 ≤ s ≤ t )defined by E ( t,s ) = E ( t, E − s, (8)is completely positive for all time s . Otherwise, dynam-ical maps are non-Markovian. Particularly when the in-verse of dynamical maps E ( t, exists, even though thedynamical maps are non-Markovian, the equations of thedynamics can be written in the canonical form of thetime-local master equation [36, 42] dρ ( t ) dt = − i [ H ( t ) , ρ ( t )] + (cid:88) k γ k ( t ) (cid:104) L k ( t ) ρ ( t ) L † k ( t ) − (cid:110) L † k ( t ) L k ( t ) , ρ ( t ) (cid:111)(cid:105) , (9)where γ k ( t ) is a decay rate and L k ( t ) is a time-dependent decoherence operator satisfying Tr[ L k ( t )] = 0and Tr[ L † k ( t ) L k ( t )] = 1. Here, { A, B } = AB + BA de-note anticommutator. It is worth mentioning that theEq. (9) has a similar form to the Lindblad Markovianmaster equation, except that the decay rate γ k ( t ) canbe negative in some time interval. In fact, if and onlyif all the decay rates γ k ( t ) are non-negative for all thetime t , the dynamical maps are CP-divisible [36]. Inother words, the sign of γ k ( t ) characterizes whether thedynamical maps are Markovian or non-Markovian. B. Measure of non-Markovianity
There are several non-Markovian measures proposed inprevious studies (for example, see review papers [20, 21]).In this paper, to quantify non-Markovianity, we adopt the decay rate measure [36]: F ( t (cid:48) , t ) = (cid:88) k (cid:90) t (cid:48) t ds | γ k ( s ) | − γ k ( s )2 . (10)Since Markovian dynamical maps give F ( t, t (cid:48) ) = 0,this measure can be interpreted as the total amountof non-Markovianity. Moreover, it is shown that thismeasure is equivalent to the RHP (Rivas, Huelga, andPlenio) measure [24, 36], which quantifies the degreeof non-completeness of the map E ( t,s ) based on Choi-Jamio(cid:32)lkowski isomorphism [43, 44]. IV. RELATION BETWEEN QEM COSTS ANDTHE MEASURE FOR NON-MARKOVIANITY
We derive QEM costs for non-Markovian dynamics anddiscuss the direct relation between costs and the mea-sure of non-Markovianity. Note that, since the time-localmaster equation Eq. (9) is derived from a given Hamil-tonian, modifications of the Hamiltonian for applying re-covery operations could affect the form of time-local mas-ter equation. However, to derive a relation between QEMcosts and non-Markovian measures, we assume that re-covery operations do not change the equation Eq. (9).
A. General form of QEM costs
Here, we derive the general form of QEM costs forthe time-local quantum master equation Eq. (9). Thekey idea is to represent the decoherence operators L k ( t )using the process matrix form [45] L k ( t ) = d − (cid:88) i =1 d Tr[ L k ( t ) G i ] G i , (11)where the operators G i satisfy the conditions G = I ⊗ N , G i = G † i , Tr[ G i G j ] = d × δ ij , and ( G i ) = I ⊗ N , and d = 2 N is the dimension of the state vector of N qubits.An example of { G i } i is a set of Pauli products, i.e., G i ∈{ I, X, Y, Z } ⊗ N . Then, Eq. (9) can be rewritten as ddt ρ N ( t ) = d − (cid:88) i,j =0 M ij ( t ) G i ρ N ( t ) G j , (12)where M ( t ) is an d × d Hermitian matrix defined by M ij ( t ) = d (cid:80) k γ k ( t )Tr[ L k ( t ) G i ]Tr[ L † k ( t ) G j ] ( i, j ≥ − d (cid:80) k γ k ( t )Tr[ L † k ( t ) L k ( t ) G i ] ( i ≥ , j = 0) − d (cid:80) k γ k ( t )Tr[ L † k ( t ) L k ( t ) G j ] ( i = 0 , j ≥ − (cid:80) k γ k ( t ) ( i = j = 0)and M ( t ) can be diagonalized using an unitary matrix uM ij ( t ) = d − (cid:88) l =0 u il ( t ) q l ( t ) u ∗ jl ( t ) . (13)Therefore, we obtain ddt ρ N ( t ) = d − (cid:88) l =0 q l ( t ) B l ( t ) ρ N ( t ) B † l ( t ) , (14)where B l ( t ) is an operator B l ( t ) = (cid:80) d − i =0 u il ( t ) G i .Using Eq. (14), we can derive QEM costs. By choosingthe recovery operation at time t E Q ( t ) = c ( t ) p ( t ) I + (cid:88) l ≥ sgn( − q l ( t )) p l ( t ) B l ( t ) , (15)where c ( t ) = 1+( − q ( t )+ (cid:80) l ≥ | q l ( t ) | ) δt , p l ( t ) = | q l ( t ) | δt ( l ≥ p ( t ) = 1 − (cid:80) l ≥ p l ( t ), and B l ( t ) ρ = B l ( t ) ρB † l ( t ).Hence, the general form of QEM costs is give by C ( T ) = exp (cid:90) T − q ( t ) + (cid:88) l ≥ | q l ( t ) | dt . (16) B. The effect of non-Markovianity on QEM costs
We assume L † k ( t ) L k ( t ) = L k ( t ) L † k ( t ) = I ⊗ N for all k in Eq. (9), such as Pauli products. In this case, thematrix M can be easily diagonalized because M i = 0and M i = 0 for all i ( i ≥
1) and the unitary matrix u is determined by u ik = d Tr[ L k ( t ) G i ] ( i, k ≥ u i = u l = 0 ( i, k ≥ u = 1, and the eigenvalues of M are q k = γ k ( t ) ( k ≥
1) and q = − (cid:80) k ≥ γ k ( t ). We canderive QEM costs as C ( T ) = exp (cid:34)(cid:88) k (cid:90) T (cid:16) | γ k ( t ) | + γ k ( t ) (cid:17) dt (cid:35) . (17)Here, we define the quantity D ( t (cid:48) , t ) = (cid:80) k (cid:82) t (cid:48) t ds | γ k ( s ) | ,which is equivalent to the QEM costs e D ( t (cid:48) ,t ) for theMarkovian case with decay rates | γ k ( t ) | . By using D ( t (cid:48) , t )and F ( t (cid:48) , t ), we can rewrite the QEM costs as C ( T ) = exp (cid:104) (cid:16) D ( T, − F ( T, (cid:17)(cid:105) . (18)From this equation, we can understand that as theamount of non-Markovianity in Eq. (10) increases, QEMcosts are reduced. More specifically, in a time region with γ k ( t ) < k , QEM costs do not increase at all. C. Study of specific models
Here, we study QEM costs for specific models. Al-though the implementation of recovery operations couldchange the form of time local master equation, we discussthe case it is invariant, i.e., recovery operations commute with both the system Hamiltonian and the noise oper-ators. In this case, since we can perform the recoveryoperations at the end of the dynamics, the time localmaster equation is not affected by recovery processes.We consider a two-qubit system where a long-livedqubit is coupled with a short-lived qubit. (Another ex-ample of a qubit dispersively coupled with a dissipativeresonator is illustrated in Appendix A.) Importantly, thissystem has been realized with nuclear magnetic reso-nance, and the non-Markovian noise has been controlledby implementation of the pulse [37, 38]. The equation ofthe two-qubit model is given by dρ (1+2) N ( t ) dt = i (cid:20) ρ (1+2) N ( t ) , J Z ⊗ Z (cid:21) + L (cid:2) ρ (1+2) N ( t ) (cid:3) , L (cid:2) ρ (cid:3) = (cid:88) k =1 γ k (cid:16) L k ρL † k − { L † k L k , ρ } (cid:17) , (19)where ρ (1+2) N denotes the density operator of the two-qubit system, J denotes a coupling strength between thequbits, γ = 2 γs denotes a thermalization rate associatedwith a Lindblad operator L = I ⊗ σ + , γ = 2 γ (1 − s )denotes an energy relaxation rate associated with a Lind-blad operator L = I ⊗ σ − , σ + = ( X + iY ) / σ − =( X − iY ) /
2) denotes a raising (lowering) operator, s de-notes a control parameter determined by the environmen-tal temperature (0 ≤ s ≤ / /J denotes a time scale of theexchange of the information between the first qubit andthe second qubit, while 1 /γ denotes the decoherence timefor the second qubit. In the regime of J/γ > ρ (1+2) N (0) = | + (cid:105) (cid:104) + |⊗ ρ Gibbs , where | + (cid:105) = ( | (cid:105) + | (cid:105) ) / | (cid:105) and | (cid:105) , and ρ Gibbs = s | (cid:105) (cid:104) | + (1 − s ) | (cid:105) (cid:104) | is the Gibbs state correspondingto the Lindbladian in Eq. (19), In this case, the equationof its dynamics is given by dρ (1) N ( t ) dt = i (cid:20) ρ (1) N ( t ) , S ( t )2 Z (cid:21) + γ ( t )2 (cid:16) Zρ (1) N ( t ) Z − ρ (1) N ( t ) (cid:17) , (20)where ρ (1) N ( t ) is the reduced density operator of the firstqubit and Z is a Pauli Z matrix. In this case, the decayrate γ ( t ) and S ( t ) are given by γ ( t ) − iS ( t ) = − f ( t ) ddt f ( t ) , (21) tC t t FIG. 1. (Color online) The decay rate γ ( t ) in Eq. (21) andQEM costs C ( t ) in Eq. (22) at time t . We choose the param-eters as J = 2 π ×
215 rad s − , γ = 6 . s = 0 .
3. The vertical lines repre-sent the solutions satisfying γ ( t ) = 0. where f ( t ) = iJ (2 s − e λ + t − e λ − t λ + − λ − − λ − e λ + t − λ + e λ − t λ + − λ − and λ ± are the two solutions of an equation λ + γλ + (2 iJγ (1 − s ) + J ) / C ( T ) canbe derived as C ( T ) = exp (cid:32)(cid:90) T dt | γ ( t ) | + γ ( t )2 (cid:33) . (22)Fig. 1 shows the numerical results of γ ( t ) and C ( t )at time t with experimental parameters in Ref. [37]. InFig. 1, γ ( t ) becomes negative in some time interval andtherefore the dynamics is actually non-Markovian. More-over, the area where γ ( t ) ≥ C ( t ), as shown in Fig. 1. In other words,QEM costs C ( t ) do not increase at all in the region sat-isfying γ ( t ) <
0, and the area of its region is equivalentto the non-Markovian measure in Eq. (10).
V. DISCUSSIONS
In this work, we discuss a relationship between theQEM and non-Markov measures. Non-Markovianity ischaracterized by a negative decay rate of the dissipator.Interestingly, QEM costs do not increase at all when theall decay rates are negative. This demonstrates that non-Markovianity can contributes to reduce cost of the QEM.We show specific physical systems as examples that sup-port our theoretical analysis. We focus on the case wherethe decoherence operators can be described by a set oforthogonal operators such as Pauli operators, and leavemore general cases for a future work. Our work helpsunderstanding properties of QEM and may lead to so-phisticated construction of QEM for realistic quantumsystems with non-Markovian noise.
ACKNOWLEDGMENTS
This work was supported by Leading Initiative forExcellent Young Researchers MEXT Japan and JSTpresto (Grant No. JPMJPR1919) Japan. This pa-per is partly based on results obtained from a project,JPNP16007, commissioned by the New Energy and In-dustrial Technology Development Organization (NEDO),Japan. This work was supported by MEXT QuantumLeap Flagship Program (MEXT Q- LEAP) (Grant No.JPMXS0120319794, JPMXS0118068682) and JST ER-ATO (Grant No. JPMJER1601).
Appendix A: A qubit dispersively coupled with adissipative resonator
Here, we also consider a qubit-resonator system wherea qubit is dispersively coupled with a lossy resonator.The qubit is affected by a dephasing induced from theinteraction with the resonator, and this dynamics hasbeen studied in Refs. [39–41]. When the frequency of thequbit is significantly detuned from that of the resonator,the Hamiltonian is given by H q + r = χZa † a, (A1)where χ/π is the dispersive frequency shift of the qubitper photon and a † ( a ) is the creation (annihilation) op-erator of the photon in the resonator. For simplicity,we assume that we are in a rotating frame, and we onlyconsider the interaction Hamiltonian in Eq. (A1). Thedynamics of the system can be described by the Lindbladmaster equation dρ ( q + r ) N ( t ) dt = i (cid:104) ρ ( q + r ) N ( t ) , H q + r (cid:105) + κ (cid:16) aρ ( q + r ) N ( t ) a † − { a † a, ρ ( q + r ) N ( t ) } (cid:17) , (A2)where κ is a decay rate. We set the initial stateto | + (cid:105) ⊗ | α (cid:105) , where | α (cid:105) is a coherent state for α ∈ C . This equation can be easily solved as ρ ( q + r ) N ( t ) = (cid:80) i,j =0 c ij ( t ) | i (cid:105) (cid:104) j | ⊗ | α i ( t ) (cid:105) (cid:104) α j ( t ) | , where c ( t ) = c ( t ) = and c ( t ) = c ( t ) ∗ and c ( t ) = c (0) (cid:104) α ( t ) | α ( t ) (cid:105) exp (cid:20) −| α | − e (2 iχ − κ ) t − iκ/ χ (cid:21) . (A3)Here, α ( t ) = e ( iχ − κ/ t α and α ( t ) = e ( − iχ − κ/ t α .The reduced density operator of the qubit is given by ρ ( q ) N ( t ) = (cid:80) i,j =0 c (cid:48) ij ( t ) | i (cid:105) (cid:104) j | , where c (cid:48) ( t ) = c (cid:48) ( t ) = and c (cid:48) ( t ) = c (cid:48) ( t ) ∗ and c (cid:48) ( t ) = c ( t ) × e (2 iχ − κ ) t = e −| α | ( x + iy ) . From this, the time-local master equationcan be derived as dρ ( q ) N ( t ) dt = i (cid:20) ρ ( q ) N ( t ) , S ( t )2 Z (cid:21) + γ ( t )2 (cid:16) Zρ ( q ) N ( t ) Z − ρ ( q ) N ( t ) (cid:17) , (A4)where S ( t ) = | α | dxdt and the decay rate of the qubit γ ( t ) = | α | dydt are given by S ( t ) = | α | e − κt ( κ (1 − cos 2 χt ) − χ sin 2 χt )+ | α | e − κt κ/ χ ) (cid:18) κ cos 2 χt + (cid:18) χ − κ χ (cid:19) sin 2 χt (cid:19) , (A5) γ ( t ) = | α | κe − κt κ/ χ ) (cid:32) κχ cos 2 χt + (cid:32) − (cid:18) κ χ (cid:19) (cid:33) sin 2 χt (cid:33) . (A6)From Eq. (A6), γ ( t ) behaves as a damped oscillation withthe time constant κ and the angular frequency 2 χ . QEMcosts C ( T ) are the same form as that in Section IV Cand therefore C ( T ) can decrease in the case of κ < χ . | α = χ / κ = | α = / χ / κ = κ t - - γ ( κ t ) FIG. 2. (Color online) The decay rate γ ( κt ) in Eq. (A6) at κt . We choose the parameters as | α | = 1 , χ/κ = 3 (red line)and | α | = 1 / , χ/κ = 12 (blue line). | α = χ / κ = | α = / χ / κ = κ t1.051.101.15C ( κ t ) FIG. 3. (Color online) QEM costs C ( κt ) at κt . All the pa-rameters are the same as those in Fig. 2. In Figs. 2 and 3, we show the numerical results of γ ( κt )and C ( κt ). As is the same in Section IV C, the dynam-ics is actually non-Markovian because of some negativeregions of γ ( κt ) in Fig. 2, and QEM costs C ( κt ) do notincrease at all for those regions. [1] M. A. Nielsen and I. Chuang, “Quantum computationand quantum information,” (2002).[2] I. M. Georgescu, S. Ashhab, and F. Nori, Reviews ofModern Physics , 153 (2014).[3] A. W. Harrow, A. Hassidim, and S. Lloyd, Physical re-view letters , 150502 (2009).[4] P. Hauke, F. M. Cucchietti, L. Tagliacozzo, I. Deutsch,and M. Lewenstein, Reports on Progress in Physics ,082401 (2012).[5] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara,K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S.Tsai, and W. D. Oliver, Nature Physics , 565 (2011).[6] J.-M. Reiner, F. Wilhelm-Mauch, G. Sch¨on, andM. Marthaler, Quantum Science and Technology ,035005 (2019).[7] R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Simon,and D. I. Schuster, Nature , 51 (2019).[8] J. Preskill, Quantum , 79 (2018). [9] K. Temme, S. Bravyi, and J. M. Gambetta, Physicalreview letters , 180509 (2017).[10] S. Endo, S. C. Benjamin, and Y. Li, Physical Review X , 031027 (2018).[11] Y. Li and S. C. Benjamin, Physical Review X , 021050(2017).[12] S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin,and X. Yuan, Reviews of Modern Physics , 015003(2020).[13] J. Sun, X. Yuan, T. Tsunoda, V. Vedral, S. C. Bejamin,and S. Endo, arXiv preprint arXiv:2001.04891 (2020).[14] F. Yoshihara, K. Harrabi, A. Niskanen, Y. Nakamura,and J. S. Tsai, Physical review letters , 167001 (2006).[15] K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba,H. Takayanagi, F. Deppe, and A. Shnirman, Physicalreview letters , 047004 (2007).[16] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, andR. L. Walsworth, Nature communications , 1 (2013). [17] G. De Lange, Z. Wang, D. Riste, V. Dobrovitski, andR. Hanson, Science , 60 (2010).[18] E. Kawakami, P. Scarlino, D. R. Ward, F. Braakman,D. Savage, M. Lagally, M. Friesen, S. N. Coppersmith,M. A. Eriksson, and L. Vandersypen, Nature nanotech-nology , 666 (2014).[19] T. Watson, S. Philips, E. Kawakami, D. Ward, P. Scar-lino, M. Veldhorst, D. Savage, M. Lagally, M. Friesen,S. Coppersmith, et al. , Nature , 633 (2018).[20] A. Rivas, S. F. Huelga, and M. B. Plenio, Reports onProgress in Physics , 094001 (2014).[21] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini,Reviews of Modern Physics , 021002 (2016).[22] H.-P. Breuer, F. Petruccione, et al. , The theory of openquantum systems (Oxford University Press on Demand,2002).[23] M. M. Wolf and J. I. Cirac, Communications in Mathe-matical Physics , 147 (2008).[24] ´A. Rivas, S. F. Huelga, and M. B. Plenio, Physical reviewletters , 050403 (2010).[25] H.-P. Breuer, E.-M. Laine, and J. Piilo, Physical reviewletters , 210401 (2009).[26] B. Misra and E. G. Sudarshan, Journal of MathematicalPhysics , 756 (1977).[27] W. M. Itano, D. J. Heinzen, J. Bollinger, andD. Wineland, Physical Review A , 2295 (1990).[28] L. Viola and S. Lloyd, Physical Review A , 2733(1998).[29] P. Haikka, J. Goold, S. McEndoo, F. Plastina, andS. Maniscalco, Physical Review A , 060101 (2012). [30] R. Vasile, S. Olivares, M. A. Paris, and S. Maniscalco,Physical Review A , 042321 (2011).[31] P. Haikka, T. Johnson, and S. Maniscalco, Physical Re-view A , 010103 (2013).[32] M. ˇZnidariˇc, C. Pineda, and I. Garcia-Mata, Physicalreview letters , 080404 (2011).[33] E. Wakakuwa, arXiv preprint arXiv:1709.07248 (2017).[34] Y. Matsuzaki, S. C. Benjamin, and J. Fitzsimons, Phys-ical Review A , 012103 (2011).[35] A. W. Chin, S. F. Huelga, and M. B. Plenio, Physicalreview letters , 233601 (2012).[36] M. J. Hall, J. D. Cresser, L. Li, and E. Andersson, Phys-ical Review A , 042120 (2014).[37] L. Binho, Y. Matsuzaki, M. Matsuzaki, Y. Kondo, et al. ,New Journal of Physics , 093008 (2019).[38] S. Kukita, Y. Kondo, and M. Nakahara, arXiv preprintarXiv:2007.11382 (2020).[39] J. Govenius, Y. Matsuzaki, I. Savenko, andM. M¨ott¨onen, Physical Review A , 042305 (2015).[40] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Physical Review A , 062320 (2004).[41] P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. Har-mans, D. P. DiVincenzo, and J. Mooij, Physical reviewletters , 257002 (2005).[42] I. De Vega and D. Alonso, Reviews of Modern Physics , 015001 (2017).[43] M.-D. Choi, Linear algebra and its applications , 285(1975).[44] A. Jamio(cid:32)lkowski, Reports on Mathematical Physics3