Ancilla-train quantum algorithm for simulating non-Markovian open quantum systems

TL;DR

The algorithm can reproduce the true dynamics of such problems at arbitrary accuracy and, for a broad range of problems, only adds a minor resource cost relative to Trotterized time evolution; the cost is low-degree polynomial in the inverse target accuracy.

Abstract

We present a quantum algorithm for simulating open quantum systems coupled to Gaussian environments valid for any configuration and coupling strength. The algorithm is applicable to problems with strongly coupled, or non-Markovian, environments, problems with multiple environments out of mutual equilibrium, and problems with time-dependent Hamiltonians. We show that the algorithm can reproduce the true dynamics of such problems at arbitrary accuracy and, for a broad range of problems, only adds a minor resource cost relative to Trotterized time evolution; the cost is low-degree polynomial in the inverse target accuracy. The algorithm is based on the insight that any Gaussian environment can be represented as a train of ancillary qubits that sequentially interact with the system through a time-local coupling, given by the convolution square root of the bath correlation function; this is a secondary result of our work. Our results open up new applications of quantum computers for efficient simulation of non-equilibrium and open quantum systems.

Figures (5)

• Figure 1: Ancilla-train algorithm (ATA): The ATA simulates the evolution of any open quantum system with Gaussian environments to arbitrary accuracy via Trotterized evolution. Each iteration---here the $m$th---propagate the system through a time step $\Delta t$ via two gate sequences, $U_m$ and $V_m$, with $U_m$ the time-evolution generated by the system Hamiltonian, while $V_m$ connects one or more system observables to a register of ancilla qubits (red, purple, and blue boxes; black lines indicate couplings). After the iteration, a subset of ancillas are carried over to the next [$(m+1)$th] iteration (purple); the remainder are discarded (red), and an identical number of new ancillas are (re)introduced in the $|0\rangle$ state (blue). With a suitable system--ancilla coupling structure obtained from the jump correlator of the bath ULE, $g(t)$, we show that the ATA can reproduce the true dynamics of the system to arbitrary accuracy with a finite ancilla register.
• Figure 2: Regimes of applicability of the ATA, in relation to Quantum Gibbs Samplers (QGS) chen2023efficientLinLin2024GSpreplloyd2025quantumthermalstatepreparationhahn2025Matthies_2024, and non-Markovian collision models based on systematic, nonsecular, Lindblad equations (SL algorithms) mozgunov_completely_2020ULECICCARELLO20221davidovic_completely_2020; see Introduction and Sec. \ref{['sec:qgs']} for details.
• Figure 3: Ancilla-train representation of classical colored noise: Environments with even power spectral densities are equivalent to classical noise signals acting on the system breuer_theory_2007. Here our results imply that any classical noise signal with Gaussian correlations---or colored noise signals---can be generated by summing a sequence of identical pulses with random signs (red/white dots) and spacing $\Delta \xi$, and taking the limit $\Delta \xi \to 0$. Here the is pulse given by $g(t)\sqrt{\Delta \xi}$brockwell_introduction_2016.
• Figure 4: Pseudocode for Ancilla-train algorithm. Input parameters denote the initial system state $\rho(t_0)$, bath PSD $\Tilde{J}(\omega)$, system Hamiltonian $H_{\rm S}$, system coupling operator $S$, system--bath coupling $\gamma$, total simulation time $T$, and (relative) target accuracy $\varepsilon$. $U_m$, given in \ref{['eq: Trotter U']}, denotes a Trotter interaction generated by the system Hamiltonian and $V_m$, given in \ref{['eq: Trotter V 1']} and \ref{['eq: Trotter V']}, denotes a Trotter iteration in the time-evolution generated by the ancilla(s)-system coupling. We write $\lvert \Psi \rangle \sim \rho$ to indicate that the expected value of $\lvert \Psi \rangle \langle \Psi \rvert$ is $\rho$.
• Figure 5: Overview of approximations leading from the physical system to the ATA. Orange boxes indicate the sub-appendices where the approximations are introduced and bounded.