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arXiv:2609.13456

Reusable selected-time channels for quantum simulation of non-Markovian dynamics

Yusui Chen

quant-phSubmitted Sep 11, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

Non-Markovian quantum-state diffusion (QSD) provides a microscopic trajectory description of open-system dynamics, but trajectories associated with a chosen initial state are not directly reusable as quantum channels. We develop a constructive framework that converts the QSD propagator into an input-independent Kraus channel for a fixed bath preparation and a selected evolution time. The construction incorporates reservoir memory through a finite-mode approximation of the bath covariance and yields a deterministic channel representation that can be compressed for approximate implementation with a system-ancilla circuit. For the qubit systems considered here, the construction closes exactly in vacuum and extends systematically to finite temperatures. Two-qubit examples with structured reservoirs validate the vacuum channel construction, illustrate its finite-temperature extension for a chosen initial state, and show that the complexity of the reservoir description can differ substantially from the ancilla space required for quantum realization. The framework therefore establishes a direct bridge from microscopic reservoir memory to reusable non-Markovian quantum simulation and provides a foundation for optimized hardware implementations and extensions to larger systems and multitime dynamics.

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Non-Markovian quantum-state diffusion (QSD) provides a microscopic trajectory description of open-system dynamics, but trajectories associated with a chosen initial state are not directly reusable as quantum channels. We develop a constructive framework that converts the QSD propagator into an input-independent Kraus channel for a fixed bath preparation and a selected evolution time. The construction incorporates reservoir memory through a finite-mode approximation of the bath covariance and yields a deterministic channel representation that can be compressed for approximate implementation with a system-ancilla circuit. For the qubit systems considered here, the construction closes exactly in vacuum and extends systematically to finite temperatures. Two-qubit examples with structured reservoirs validate the vacuum channel construction, illustrate its finite-temperature extension for a chosen initial state, and show that the complexity of the reservoir description can differ substantially from the ancilla space required for quantum realization. The framework therefore establishes a direct bridge from microscopic reservoir memory to reusable non-Markovian quantum simulation and provides a foundation for optimized hardware implementations and extensions to larger systems and multitime dynamics.
Digital quantum simulation on quantum systems require algorithms that can be implemented using finite quantum resources. Recent studies have demonstrated digital quantum simulation of open quantum systems on Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we develop quantum circuits for optimal simulation of Markovian and Non-Markovian open quantum systems. The circuits use ancilla qubits to simulate the environment, and memory effects are induced by storing information about the system on extra qubits. We simulate the amplitude damping channel and dephasing channel as examples of the framework and infer (Non-)Markovianity from the (non-)monotonic behaviour of the dynamics. Further, we develop a method to optimize simulations by decomposing complex open quantum dynamics into smaller parts, that can be simulated using a small number of qubits. We show that this optimization reduces quantum space complexity from O(l)O(l)O(l) to O(1)O(1)O(1) for simulating the environment.
The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.
We present an error-controlled framework for the quantum simulation of non-Markovian open-system dynamics in Gaussian environments. For a system coupled linearly to a Gaussian environment with zero mean, the bath's influence on the reduced dynamics is completely determined by its two-time correlation kernel. This allows us to replace the microscopic environment with an equivalent bath that has the same kernel. On a finite simulation interval, a rank-NNN approximation of the kernel yields an effective bath of NNN bosonic modes with time-dependent couplings to the system. To encode these modes in qubits, we first truncate each mode to its lowest ddd occupation levels. When stronger couplings lead to large occupations, we introduce coupling dilution, which replaces each mode with several more weakly coupled copies while preserving the bath correlation kernel. This distributes excitations across more modes and trades additional modes for smaller local dimensions. We derive explicit reduced-state error bounds and establish polynomial circuit width and depth in simulation time and inverse accuracy under stated spectral-decay and coupling-regularity assumptions, provided elementary unitaries admit polynomial-cost implementations.
A long-standing open problem in the non-Markovian quantum state diffusion (QSD) approach to open quantum systems is to establish the non-Markovian QSD equations for multiple-qubit systems. In this paper, we settle this important question by explicitly constructing a set of exact time-local QSD equations for NNN-qubit systems. Our exact time-local (convolutionless) QSD equations have paved the way towards simulating quantum dynamics of many-body open systems interacting with a common bosonic environment. The applicability of this multiple-qubit stochastic equation is exemplified by numerically solving several quantum open many-body systems concerning quantum coherence dynamics and dynamical control.
Non-Markovian dynamics is studied for two interacting qubits strongly coupled to a dissipative bosonic environment. We derive a non-Markovian quantum-state-diffusion (QSD) equation for the coupled two-qubit system without any approximations, and in particular, without the Markov approximation. As an application and illustration of our derived time-local QSD equation, we investigate the temporal behavior of quantum coherence dynamics. In particular, we find a strongly non-Markovian regime where entanglement generation is significantly modulated by the environmental memory. Additionally, we study residual entanglement in the steady state by analyzing the steady-state solution of the QSD equation. Finally, we discuss an approximate QSD equation.
In this work, we present a quantum algorithm for simulating non-Markovian quantum dynamics based on the Feynman-Vernon influence functional formalism, which provides an exact description of the dynamics of a quantum system linearly coupled to a harmonic bath. In condensed-phase environments, the bath correlation functions typically decay over time, allowing them to be truncated beyond a finite time scale and enabling an iterative propagation scheme. We implement the finite-memory propagator using a fast-forwarding algorithm that achieves constant circuit depth with respect to the number of propagation steps, while the nonunitary matrices are realized on quantum circuits using unitary dilation. We further exploit the structure of the unitary dilation to show that the additional circuit complexity introduced by the dilation scales polynomially with the number of dilation ancilla qubits, while the dominant computational cost arises from the implementation of the underlying 1-dilation. The algorithm is compared with classical benchmark calculations for the spin-boson model and shows close agreement. Together, these developments establish a quantum-circuit framework for fast forwarding non-Markovian dynamics and identify the principal computational bottlenecks that must be overcome to achieve quantum advantage.