Faster Quantum Simulation Of Markovian Open Quantum Systems Via Randomisation

TL;DR

The paper tackles the challenge of simulating Markovian open quantum system dynamics on quantum computers by extending randomisation techniques to the GKSL framework. It introduces two non-probabilistic randomised methods—first-order and second-order randomised Trotter-Suzuki formulas—and a QDRIFT-inspired channel, each with rigorous diamond-norm error bounds that avoid the mixing lemma used in Hamiltonian simulation. Two implementation paradigms, Classical Sampling (CS) and Quantum Forking (QF), are developed to realize these protocols, with detailed gate-complexity analyses showing favorable scaling in the number of GKSL terms $M$ compared to deterministic TS formulas. The work also discusses trade-offs: CS implementations can be efficient for large $M$, while QF offers a fully quantum path at the cost of linear or factorial dependencies in $M$ for certain formulas, and QDRIFTCS provides an $M$-independent gate cost at the expense of longer-time scaling. Together, these results establish randomisation as a practical route to faster, more scalable simulations of open quantum system dynamics on near- and mid-term quantum hardware, with clear directions for future improvements (e.g., higher-order randomised schemes and non-Markovian extensions).

Abstract

When simulating the dynamics of open quantum systems with quantum computers, it is essential to accurately approximate the system's behaviour while preserving the physicality of its evolution. Traditionally, for Markovian open quantum systems, this has been achieved using first and second-order Trotter-Suzuki product formulas or probabilistic algorithms. In this work, we introduce novel non-probabilistic algorithms for simulating Markovian open quantum systems using randomisation. Our methods, including first and second-order randomised Trotter-Suzuki formulas and the QDRIFT channel, not only maintain the physicality of the system's evolution but also enhance the scalability and precision of quantum simulations. We derive error bounds and step count limits for these techniques, bypassing the need for the mixing lemma typically employed in Hamiltonian simulation proofs. We also present two implementation approaches for these randomised algorithms: classical sampling and quantum forking, demonstrating their gate complexity advantages over deterministic Trotter-Suzuki product formulas. This work is the first to apply randomisation techniques to the simulation of open quantum systems, highlighting their potential to enable faster and more accurate simulations.

Figures (8)

• Figure 1: The key ideas behind the digital quantum simulation of an OQS are illustrated. The state $\rho(0)$ is the initial state of the system, $T_{t}$ is the quantum channel describing the systems evolution and $\rho(t)$ is the state after evolving for some time $t$. The quantum circuit implements the second order deterministic TS product formula $S_{2}^{(det)}$ which produces an output state $\tilde{\rho}(t)$ which approximates the state $\rho(t)$ up to a precision $\epsilon/2$.
• Figure 2: Quantum circuit showing the implementation of $S_{1}^{(ran)}(\tau)^{N}$ using CS. The circuit shows how the gate set $G_{1}^{(ran)}$ is constructed and applied to an initial state $\rho(0)$ with the output state $\tilde{\rho}(t)$. The double wires are used to show that the quantum channels depend on information from a classical computer.
• Figure 3: Quantum circuit showing the implementation of $S_{2}^{(ran)}(\tau)^{N}$ by CS.
• Figure 4: Quantum circuit depicting the implementation of the QDRIFT channel using CS. Here each $j_{k}$ is sampled from the discrete distribution $p_{k}$ as defined in equation (\ref{['eq5-2']}).
• Figure 5: Quantum circuit for implementing $S_{1}^{(ran)}$ using QF. The state $\rho_{\phi}$ can be any arbitrary state that has the same dimensions as the state $\rho(0)$. The controlled-SWAP gates correspond to controlled-SWAP channels. The measurement operation with discard means that we should measure the register then discard the outcome which is equivalent to partially tracing out that register.

Theorems & Definitions (30)

• Lemma 2.1
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Lemma 4.1
• proof
• Lemma 4.2
• proof