Faster Quantum Simulation Of Markovian Open Quantum Systems Via Randomisation

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.

💡 Research Summary

The paper addresses the challenge of efficiently simulating Markovian open quantum systems (OQS) on quantum computers while preserving the completely positive trace‑preserving (CPTP) nature of the dynamics. Traditional approaches rely on first‑ and second‑order Trotter‑Suzuki (TS) product formulas, which guarantee CPTP maps but suffer from poor scaling with the number of Lindblad terms (M) and become impractical at higher orders because the recursive construction can produce non‑CPTP maps. Probabilistic algorithms have also been proposed, but they introduce a non‑zero failure probability, which is undesirable for many applications.

The authors introduce two non‑probabilistic, randomisation‑based algorithms that improve gate‑complexity scaling with respect to M while maintaining physicality:

1. Randomised Trotter‑Suzuki (RTS) formulas – Both first‑order (S⁽¹⁾_ran) and second‑order (S⁽²⁾_ran) versions are constructed by randomly selecting a Lindblad term according to a probability distribution proportional to its norm and applying the corresponding elementary channel for a short time step Δt. Repeating this process N = t/Δt times yields an overall channel that approximates the exact evolution. Crucially, the authors derive diamond‑norm error bounds without invoking the mixing lemma (which only applies to Hamiltonian simulation). The bounds are:

   • First‑order RTS (classical sampling or quantum forking): ‖T_t – (S⁽¹⁾ran)^N‖⋄ = O(t^{3/2} M^{5/2} / √ε).
   • Second‑order RTS (classical sampling only, because quantum forking would require O(M!) controlled‑SWAPs): ‖T_t – (S⁽²⁾ran)^N‖⋄ = O(t^{3/2} M^{2} / √ε).

   Compared with deterministic TS, whose complexities scale as O(t² M³/ε) for first order and O(t^{3/2} M^{5/2}/√ε) for second order, the randomised formulas achieve markedly better dependence on M, especially the second‑order RTS which attains a quadratic M‑dependence.

2. QDRIFT channel for OQS – Inspired by the QDRIFT algorithm for Hamiltonian simulation, the authors sample Lindblad terms directly according to probabilities p_k = ‖L_k‖/Γ, where Γ = Σ_k ‖L_k‖. Each sampled term is exponentiated for a time step τ = t Γ Ω / N, where Ω is a weighted average norm. The total number of steps required to achieve error ε is N = O((t Γ Ω)² / ε). Importantly, the gate complexity of the QDRIFT channel is independent of M when implemented via classical sampling, making it ideal for systems with a very large number of terms (e.g., 2‑D dissipative Jaynes‑Cummings lattices or boundary‑driven Heisenberg models). When implemented via quantum forking, the complexity becomes linear in M (O((t Γ Ω)² M/ε)) due to the need for 2(M−1) controlled‑SWAP channels, which is still better than any deterministic TS approach.

Implementation strategies are explored in depth:

• Classical Sampling (CS) – A classical processor draws samples from the prescribed probability distribution, constructs a deterministic gate sequence, and feeds it to the quantum processor. This approach is straightforward but incurs additional classical preprocessing time.
• Quantum Forking (QF) – The quantum circuit itself performs the sampling using ancilla qubits and controlled‑SWAP operations, eliminating the need for a classical sampler. The authors show that for the first‑order RTS and QDRIFT, QF incurs no extra gate overhead compared with CS. However, the second‑order RTS would require O(M!) controlled‑SWAPs, rendering QF impractical for that case.

A comprehensive gate‑complexity table (Table 1) summarizes the scaling of each method under both CS and QF implementations. The randomised approaches uniformly improve the dependence on M relative to deterministic TS. The first‑order RTS matches the second‑order deterministic TS scaling, while the second‑order RTS offers the best asymptotic scaling (quadratic in M). QDRIFT provides M‑independent complexity for short‑time simulations, and its QF variant adds only a linear M factor.

Practical considerations are discussed. The authors note that while the parameter Γ (or Ω) may hide an implicit M‑dependence, it can usually be computed classically beforehand, so it does not affect the asymptotic gate count. They also emphasize that the quadratic scaling in simulation time t limits QDRIFT to short‑time regimes; for longer simulations, the RTS formulas become more advantageous. The need for ancilla qubits and controlled‑SWAP gates in QF is highlighted as a hardware constraint on near‑term devices.

Related work is positioned: concurrent papers have explored randomisation for OQS (e.g., a qDRIFT‑type approach with ensembles of generators, and a second‑order product formula combined with randomised compiling). This work distinguishes itself by providing a direct, general extension of both randomised TS and QDRIFT to arbitrary GKSL generators, and by deriving rigorous error bounds that do not rely on the mixing lemma.

Conclusion – The authors demonstrate that randomisation, previously successful for closed‑system Hamiltonian simulation, can be effectively transferred to the open‑system domain. Their algorithms retain CPTP guarantees, achieve superior gate‑complexity scaling with the number of Lindblad terms, and offer flexible implementation pathways (CS vs. QF). This opens the door to faster, more accurate quantum simulations of large, dissipative many‑body systems, and sets the stage for future work on higher‑order randomised formulas, hybrid error‑mitigation techniques, and experimental validation on emerging quantum hardware.