Quantum error mitigation (QEM) has been proposed as an alternative method of quantum error correction to compensate errors in quantum systems without qubit overhead. While Markovian gate errors on digital quantum computers have been mainly considered previously, it is indispensable to discuss a relationship between QEM and non-Markovian errors because non-Markovian noise effects inevitably exist in most of the solid-state systems. In this work, we investigate the QEM for non-Markovian noise, and show that there is a clear relationship between costs for QEM and non-Markovian measures. As examples, we show several non-Markovian noise models to bridge a gap between our theoretical framework and concrete physical systems. This discovery may help in designing better QEM strategies for realistic quantum devices with non-Markovian environments.
In this study, we explore the non-Markovian cost function for quantum error mitigation (QEM) and the representation of two-qubit operators using Dirac Gamma matrices, central to the structure of relativistic quantum mechanics. The primary focus of quantum computing research, particularly with noisy intermediate-scale quantum (NISQ) devices, is on reducing errors and decoherence for practical application. While much of the existing research concentrates on Markovian noise sources, the study of non-Markovian sources is crucial given their inevitable presence in most solid-state quantum computing…
The relation between the thermodynamic entropy production and non-Markovian evolutions is matter of current research. Here, we study the behavior of the stochastic entropy production in open quantum systems undergoing unital non-Markovian dynamics. In particular, for the family of Pauli channels we show that in some specific time intervals both the average entropy production and the variance can decrease, provided that the quantum dynamics fails to be P-divisible. Although the dynamics of the system is overall irreversible, our result may be interpreted as a transient tendency towards reversib…
We consider Zero Noise Extrapolation (ZNE) as an error mitigation strategy in quantum metrology. It is shown that noise expansion can be systematically performed over sufficiently short time scales for general Markovian noise models described by the time homogeneous Lindblad master equation. This suggests that ZNE can be an effective, resource efficient error mitigation alternative when strategies employing full quantum error correcting codes are unavailable. The ZNE method is then applied quantum phase estimation in a Mach-Zehnder interferometer subject to photon losses. Numerical simulations…
In the scale-up of quantum computers, the framework underpinning fault-tolerance generally relies on the strong assumption that environmental noise affecting qubit logic is uncorrelated (Markovian). However, as physical devices progress well into the complex multi-qubit regime, attention is turning to understanding the appearance and mitigation of correlated -- or non-Markovian -- noise, which poses a serious challenge to the progression of quantum technology. This error type has previously remained elusive to characterisation techniques. Here, we develop a framework for characterising non-Mar…
We construct a non-Markovian canonical dynamical map that accounts for systems correlated with the environment. The physical meaning of not completely positive maps is studied to obtain a theory of non-Markovian quantum dynamics. The relationship between inverse maps and correlations with the environment is established. A generalized non-Markovian master equation is derived from the canonical dynamical map that goes beyond the Kossakowski-Lindblad Markovian master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory.
Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the v…
Fermi's golden rule applies to a situation in which a single quantum state $|ψ\rangle$ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of $|ψ\rangle$. Here we show that the coupling of a quantum system to the standard model of a thermal environment, a bath of harmonic oscillators, can be decomposed into a "cascade" made up of the quasi-continuum coupling structures of Fermi's golden rule. This clarifies the connection between the physics of the golden rule and that of a thermal bath, and provides a non-rigorous but physica…
It is vital to minimise the impact of errors for near-future quantum devices that will lack the resources for full fault tolerance. Two quantum error mitigation (QEM) techniques have been introduced recently, namely error extrapolation [Li2017,Temme2017] and quasi-probability decomposition [Temme2017]. To enable practical implementation of these ideas, here we account for the inevitable imperfections in the experimentalist's knowledge of the error model itself. We describe a protocol for systematically measuring the effect of errors so as to design efficient QEM circuits. We find that the effe…
We provide an experimental study of the relationship between the action of different classical noises on the dephasing dynamics of a two-level system and the non-Markovianity of the quantum dynamics. The two-level system is encoded in the photonic polarization degrees of freedom and the action of the noise is obtained via a spatial light modulator, thus allowing for an easy engineering of different random environments. The quantum non-Markovianity of the dynamics driven by classical Markovian and non-Markovian noise, both Gaussian and non-Gaussian, is studied by means of the trace distance. Ou…
We study the competing effects of simultaneous Markovian and non-Markovian decoherence mechanisms acting on a single spin. We show the existence of a threshold in the relative strength of such mechanisms above which the spin dynamics becomes fully Markovian, as revealed by the use of several non-Markovianity measures. We identify a measure-dependent nested structure of such thresholds, hinting at a causality relationship amongst the various non-Markovianity witnesses used in our analysis. Our considerations are then used to argue the unavoidably non-Markovian evolution of a single-electron qua…
We show the connection between a witness that detects dynamical maps with initial system-environment correlations and a witness that detects non-Markovian open quantum systems. Our analysis is based on studying the role that state preparation plays in witnessing violations of contractivity of open quantum system dynamics. Contractivity is a property of some quantum processes where the trace distance of density matrices decrease with time. From this, we show how a witness of initial-correlations is an upper bound to a witness of non-Markovianity. We discuss how this relationship shows further c…
Non-Markovian Noise Suppression Simplified through Channel Representation
Tsinghua University（清华大学）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Technology Innovation Institute（技术创新研究所）；A*STAR Quantum Innovation Centre (Q.InC)（A*STAR量子创新中心）；Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）；Institute of High Performance Computing (IHPC)（高性能计算研究所）；University of Oxford（牛津大学）；Imperial College London（帝国理工学院）；Quantum Motion
A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment i…
By employing the full counting statistics formalism, we characterize the first moment of energy that is exchanged during a generally non-Markovian evolution in non-driven continuous variables systems. In particular, we focus on the evaluation of the energy flowing back from the environment into the open quantum system. We apply these results to the quantum Brownian motion, where these quantities are calculated both analytically, under the weak coupling assumption, and numerically also in the strong coupling regime. Finally, we characterize the non-Markovianity of the reduced dynamics through a…
In this work, we investigate the non-Markovianity, quantum Fisher information (QFI) and quantum coherence of a qubit in a nonequilibrium environment and have obtained the expressions of QFI and quantum coherence as well as their relationship. We have also discussed in detail the influences of the different noise parameters on these quantum sffects. The results show that the suitable parameters of the nonequilibrium environment can retard the QFI and quantum coherence in both Markovian and non-Markovian regions. In addition, the smaller memory effects and the larger the jumping rate, the greate…
We investigate the relationship between non-Markovianity and the effectiveness of a dynamical decoupling protocol for qubits undergoing pure dephasing. We consider an exact model in which dephasing arises due to a bosonic environment with a spectral density of the Ohmic class. This is parametrised by an Ohmicity parameter by changing which we can model both Markovian and non-Markovian environments. Interestingly, we find that engineering a non-Markovian environment is detrimental to the efficiency of the dynamical decoupling scheme, leading to a worse coherence preservation. We show that each …
Reducing measurement errors in multi-qubit quantum devices is critical for performing any quantum algorithm. Here we show how to mitigate measurement errors by a classical post-processing of the measured outcomes. Our techniques apply to any experiment where measurement outcomes are used for computing expected values of observables. Two error mitigation schemes are presented based on tensor product and correlated Markovian noise models. Error rates parameterizing these noise models can be extracted from the measurement calibration data using a simple formula. Error mitigation is achieved by ap…
The degree of non-Markovianity of quantum processes has been characterized in several different ways in the recent literature. However, the relationship between the non-Markovian behavior and the flow of information between the system and the environment through an entropic measure has not been yet established. We propose an entanglement-based measure of non-Markovianity by employing the concept of assisted knowledge, where the environment E, acquires information about a system S, by means of its measurement apparatus A. The assisted knowledge, based on the accessible information in terms of v…
We explore the behavior in time of the energy exchange between a system of interest and its environment, together with its relationship to the non-Markovianity of the system dynamics. In order to evaluate the energy exchange we rely on the full counting statistics formalism, which we use to evaluate the first moment of its probability distribution. We focus in particular on the energy backflow from environment to system, to which we associate a suitable condition and quantifier, which enables us to draw a connection with a recently introduced notion of non-Markovianity based on information bac…
In this paper, we investigate the relationship between the quantum speedup, nonMarkovianity and formation of a system-environment bound state. Previous results show a monotonic relation between these three such that providing stronger bound states can lead to a higher degree of non-Markovianity, and hence to greater speed of quantum evolution. By studying dynamics of a dissipative two-level system or a V-type three-level system in the presence of similar and additional systems, we reveal that the quantum speedup is exclusively related to the formation of the system-environment bound state, whi…
To explain aspects of the quantum-to-classical transition, quantum Darwinism explores the fact that, due to interactions between a quantum open system and its surrounding environment, information about the system can be spread redundantly to the environment. Here we recall that there are in the literature two distinct and non-equivalent ways to make this statement precise and quantitative. We first point out the difference with some simple but illustrative examples. We then consider a model where Darwinism can be seen from both perspectives. Moreover, the non-Markovianity of our model can be v…
We explore signatures of the non-Markovianity in the time-resolved energy transfer processes for quantum open systems. Focusing on typical systems such as the exact solvable damped Jaynes-Cummings model and the general spin-boson model, we establish quantitative links between the time-resolved energy current and the symmetric logarithmic derivative quantum Fisher information (SLD-QFI) flow, one of measures quantifying the non-Markovianity, within the framework of non-Markovian master equations in time-local forms. From the relationships, we find in the damped Jaynes-Cummings model that the SLD…
It is widely spread in the literature that non-Markovianity (NM) may be regarded as a resource in quantum mechanics. However, it is still unclear how and when this alleged resource may be exploited. Here, we study the relationship between NM and quantum optimal control under the objective of generating entanglement within M non-interacting subsystems, each one coupled to the same non-Markovian environment. Thus, we design a variety of entangling protocols that are only achievable due to the existence of the environment. We show that NM plays a crucial role in all the entangling protocols consi…
We introduce a generalized approach to characterize the non-Markovianity of quantum dynamical maps via breakdown of monotonicity of thermodynamic functions. By adopting an entropy-based formulation of quantum thermodynamics, we use the relationship between heat and entropy to propose a measure of non-Markovianity based on the heat flow for single-qubit quantum evolutions. This measure can be applied for unital dynamical maps that do not invert the sign of the internal energy. Under certain conditions, it can also be extended for other thermodynamic functions, such as internal energy and work f…
As discovered by P. W. Anderson, excitations do not propagate freely in a disordered lattice, but, due to destructive interference, they localise. As a consequence when an atom interacts with a disordered lattice one indeed observes, a non-trivial excitation exchange between atom and lattice. Such non-trivial atomic dynamics will in general be characterised also by a non-trivial quantum information backflow, a clear signature of non-Markovian dynamics. To investigate the above scenario we consider a quantum emitter, or atom, weakly coupled to a uniform coupled-cavity array (CCA). If initially …
The quantum speed limit sets a bound on the minimum time required for a quantum system to evolve between two states. For open quantum systems this quantity depends on the dynamical map describing the time evolution in presence of the environment, on the evolution time τ , and on the initial state of the system. We consider a general single qubit open dynamics and show that there is no simple relationship between memory effects and the tightness of the quantum speed limit bound. We prove that only for specific classes of dynamical evolutions and initial states, there exists a link between non-M…
Quantum computing has the potential to revolutionize computing for certain classes of problems with exponential scaling, and yet this potential is accompanied by significant sensitivity to noise, requiring sophisticated error correction and mitigation strategies. Here we simulate the relaxations of stationary states at different frequencies on several quantum computers to obtain unique spectroscopic fingerprints of their noise. Response functions generated from the data reveal a clear signature of non-Markovian dynamics, demonstrating that each of the quantum computers acts as a non-Markovian …
On the non-Markovian quantum stochastic network dynamics
The Chinese University of Hong Kong（香港中文大学）；The Hong Kong Polytechnic University（香港理工大学）；The Hong Kong Polytechnic University Shenzhen Research Institute（香港理工大学深圳研究院）
Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higher-dimensional systems of the relation $T_2 \leq 2 T_1$ for qubit decoherence and relaxation times. It als…
We study the relationship between (non-)Markovian evolutions, established correlations, and the entropy production rate. We consider a system qubit in contact with a thermal bath and in addition the system is strongly coupled to an ancillary qubit. We examine the steady state properties finding that the coupling leads to effective temperatures emerging in the composite system, and show that this is related to the creation of correlations between the qubits. By establishing the conditions under which the system reaches thermal equilibrium with the bath despite undergoing a non-Markovian evoluti…
In this study, we explore optimal environments for efficient EET between end sites in a number of dimer and trimer model pigment-protein molecular aggregates when the EET dynamics is modeled by the HEOM-method. For these optimal environmental parameters, we further quantify the non-Markovianity by the BLP-measure to elucidate its possible connection to efficient EET. We also quantify coherence in the pigment systems by means of the measure l1-norm of coherence to analyze its interplay with environmental effects when EET efficiency is maximal. Our aim is to investigate possible environmental de…
Motivated by recent numerous works on the interplay among various measures of quantum correlations, we aim to investigate the relationship between the violation of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and geometric measure of quantum discord for two-qubit systems. Exact lower and upper bounds of Bell violation versus geometric discord are obtained for a specific and significant class of states, Bell diagonal states, and the respective states which suffice those bounds are also characterized. The dynamical behavior of these two quantifiers is carefully analyzed in the presence of d…
We investigate the behavior in time of the energy current between a quantum spin chain and its surrounding non-Markovian, finite temperature baths, together with its relationship to the coherence dynamics of the system. To be specific, both the system and the baths are assumed to be initially in thermal equilibrium at temperature $T_s$ and $T_b$, respectively. This model plays a fundamental role for the study of quantum system evolution towards thermal equilibrium in an open system. The non-Markovian quantum state diffusion (NMQSD) equation approach is used to calculate the dynamics of the spi…
