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Benchmarking Non-Markovianity Measures in Digital Quantum Simulation of Open Quantum Systems

S. M. Yousuf Iqbal Tomal

Published Jul 21, 2026• 0 citations
Publisher

Abstract

Abstract Digital quantum simulation of open quantum systems has produced a range of methods for reproducing non-Markovian dynamics, including pseudomode embeddings, the hierarchical equations of motion, and collision models. These methods are validated against populations, coherences, or spectra. They are not validated against the quantities that define memory itself, and the behavior of those quantities under the pressures of a real simulation is largely unexamined. We study three such pressures. First, a single-qubit mode encoding misrepresents the Breuer-Laine-Piilo (BLP) measure with an error whose sign flips between regimes: it fabricates memory at strong coupling, by about a factor of 2.6 over the converged value, and it erases memory at moderate temperature. The Rivas-Huelga-Plenio (RHP) measure corroborates the temperature-driven destruction, while at strong coupling it is numerically unstable and does not track the BLP result. We derive a cross-validated rule for the required Fock dimension that collapses both regimes onto the occupied number extent, d req ≈ ⌈⟨n⟩+c σ n ⌉+1, and we show that the constant must be calibrated on super-Poissonian statistics. Second, using the exact noisy measure as computable ground truth, we show that the raw finite-shot estimate of the BLP measure can nearly double the true value under realistic noise. We provide a debiased estimator that is validated near-unbiased in that regime, together with a Markovian null-floor control that ports to hardware. Third, temporally correlated device noise is conflated with physical memory by the measure. A resource-fair comparison finds that a pseudomode and an ancilla-train collision model reach parity at equal qubit cost for a single-Lorentzian bath. Every result is simulator-based, validated against an exact reference, and reproducible from recorded data. We make no claim of quantum advantage.

DOI
10.21203/rs.3.rs-10406502/v1

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Abstract Digital quantum simulation of open quantum systems has produced a range of methods for reproducing non-Markovian dynamics, including pseudomode embeddings, the hierarchical equations of motion, and collision models. These methods are validated against populations, coherences, or spectra. They are not validated against the quantities that define memory itself, and the behavior of those quantities under the pressures of a real simulation is largely unexamined. We study three such pressures. First, a single-qubit mode encoding misrepresents the Breuer-Laine-Piilo (BLP) measure with an error whose sign flips between regimes: it fabricates memory at strong coupling, by about a factor of 2.6 over the converged value, and it erases memory at moderate temperature. The Rivas-Huelga-Plenio (RHP) measure corroborates the temperature-driven destruction, while at strong coupling it is numerically unstable and does not track the BLP result. We derive a cross-validated rule for the required Fock dimension that collapses both regimes onto the occupied number extent, d req ≈ ⌈⟨n⟩+c σ n ⌉+1, and we show that the constant must be calibrated on super-Poissonian statistics. Second, using the exact noisy measure as computable ground truth, we show that the raw finite-shot estimate of the BLP measure can nearly double the true value under realistic noise. We provide a debiased estimator that is validated near-unbiased in that regime, together with a Markovian null-floor control that ports to hardware. Third, temporally correlated device noise is conflated with physical memory by the measure. A resource-fair comparison finds that a pseudomode and an ancilla-train collision model reach parity at equal qubit cost for a single-Lorentzian bath. Every result is simulator-based, validated against an exact reference, and reproducible from recorded data. We make no claim of quantum advantage.
Abstract We explore in a rigorous manner the intuitive connection between the non-Markovianity of the evolution of an open quantum system and the performance of the system as a quantum memory. Using the paradigmatic case of a two-level open quantum system coupled to a bosonic bath, we compute the recovery fidelity, which measures the best possible performance of the system to store a qubit of information. We deduce that this quantity is connected, but not uniquely determined, by the non-Markovianity, for which we adopt the Breuer–Laine–Piilo measure proposed in Breuer et al (2009 Phys. Rev. Lett . 103 210401). We illustrate our findings with explicit calculations for the case of a structured environment.
Simulating open quantum systems on digital quantum computers typically relies on the use of auxiliary qubits, resulting in overheads of noisy multi-qubit gates, that severely limit execution on near-term hardware. In this work, we explore the simulation of non-Markovian dynamics as well as memory channels leveraging the method of trajectory mixing, valid for mixed unitary channels. This allows to drastically reduce circuit depth, trading entangling gates for a statistical mixture of independent, pure state trajectories. Using a realistic noise model calibrated to modern quantum processors, we show the benefits of this approach, yielding higher state fidelity and better preservation of quantum correlations. This shows the possibility of simulating long-time non-Markovian evolutions with low noise and limited resources.
The quantification and characterization of non-Markovian dynamics in quantum systems is an essential endeavor both for the theory of open quantum systems and for a deeper understanding of the effects of non-Markovian noise on quantum technologies. Here, we introduce the robustness of non-Markovianity, an operationally-motivated, optimization-free measure that quantifies the minimum amount of Markovian noise that can be mixed with a non-Markovian evolution before it becomes Markovian. We show that this quantity is a bonafide non-Markovianity measure, since it is faithful, convex, and monotonic under composition with Markovian channels. A two-fold operational interpretation of this measure is provided, with the robustness measure quantifying an advantage in both a state discrimination and a channel discrimination task. Furthermore, we provide a closed-form analytical expression for this measure and show that, quite remarkably, the robustness measure is exactly equal to half the Rivas-Huelga-Plenio (RHP) measure [Phys. Rev. Lett. \textbf{105}, 050403 (2010)]. As a result, we provide a direct operational meaning to the RHP measure while endowing the robustness measure with the physical characterizations of the RHP measure.
Different measures have been presented to depict the deviation of quantum time evolution in open systems from Markovian processes. We demonstrate that the measure proposed by Breuer, Laine, and Piilo [Phys. Rev. Lett. 103, 210401 (2009)] and the two measures proposed by Rivas, Huelga, and Plenio [Phys. Rev. Lett. 105, 050403 (2010)] have exactly the same non-Markovian time-evolution intervals and thus are really equivalent to each other when they are applied to open two-level systems coupled to environments via the Jaynes-Cummings or dephasing models. This equivalence implies that the three measures, in different ways, capture the intrinsic character of the non-Markovianity of quantum evolutional processes. We also show that the maximization in the definition of the first measure can be actually removed for the considered models without influencing the sensibility of the measure to detect non-Markovianity.
We study non-Markovian effects present in a driven qubit coupled to a finite environment using a recently proposed model developed in the context of calorimetric measurements of open quantum systems. To quantify the degree of non-Markovianity we use the Breuer-Laine-Piilo (BLP) measure [H.-P. Breuer, Phys. Rev. Lett. 103, 210401 (2009)]. We show that information backflow only occurs in the case of driving, in which case we investigate the dependence of memory effects on the environment size, driving amplitude, and coupling to the environment. We show that the degree of non-Markovianity strongly depends on the ratio between the driving amplitude and the coupling strength. We also show that the degree of non-Markovianity does not decrease monotonically as a function of the environment size.
We present a quantum computing framework for simulating open-quantum-system approaches based on Markovian and non-Markovian dynamics, which is relevant to heavy-ion collisions. To simulate the non-Markovian evolution on quantum computers, we introduce an auxiliary two-level pseudomode that carries the memory forward and couples to both the subsystem and the residual Markovian bath. We explicitly show that tracing out the pseudomode reproduces the non-Markovian evolution with the exact memory kernel. Moreover, in the relevant time scale hierarchy, the quantum circuit construction of the pseudomode smoothly converges to the Markovian limit. For a given bath memory kernel, our results demonstrate the feasibility of quantum simulations of both Markovian and non-Markovian dynamics, establishing a framework for future studies of hard probes such as jets, heavy quarks, and quarkonia in heavy-ion collisions.