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arXiv:2509.10425 · Quantum

Quantum simulation algorithms based on quantum trajectories

Evan Borras, Milad Marvian

Published Apr 13, 2026• 1 citation
arXivHTMLPDF

Abstract

Quantum simulation has emerged as a key application of quantum computing, with significant progress made in algorithms for simulating both closed and open quantum systems. The simulation of open quantum systems, particularly those governed by the Lindblad master equation, has received attention recently with the current state-of-the-art algorithms having an input model query complexity of O ( T p o l y l o g ( T / ϵ ) ) , where T and ϵ are the desired time and precision of the simulation respectively. For the Hamiltonian simulation problem it has been show that the optimal Hamiltonian query complexity is O ( T + log ⁡ ( 1 / ϵ ) ) , which is additive in the two parameters, but for Lindbladian simulation this question remains open. In this work we show that the additive query complexity to a Lindbladian's jump operators is reachable for the simulation of a large class of Lindbladians by constructing a novel quantum algorithm based on quantum trajectories.

DOI
10.22331/q-2026-04-13-2063

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Quantum simulation has emerged as a key application of quantum computing, with significant progress made in algorithms for simulating both closed and open quantum systems. The simulation of open quantum systems, particularly those governed by the Lindblad master equation, has received attention recently with the current state-of-the-art algorithms having an input model query complexity of O ( T p o l y l o g ( T / ϵ ) ) , where T and ϵ are the desired time and precision of the simulation respectively. For the Hamiltonian simulation problem it has been show that the optimal Hamiltonian query complexity is O ( T + log ⁡ ( 1 / ϵ ) ) , which is additive in the two parameters, but for Lindbladian simulation this question remains open. In this work we show that the additive query complexity to a Lindbladian's jump operators is reachable for the simulation of a large class of Lindbladians by constructing a novel quantum algorithm based on quantum trajectories.
We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian HHH and a projected unitary encoding of the stacked jump operator B=∑k=1m∣k⟩⊗LkB=\sum_{k=1}^m \lvert k\rangle\otimes L_kB=∑k=1m​∣k⟩⊗Lk​, with normalization factors αHα_HαH​ and αBα_BαB​, respectively. For evolution time ttt, set τ=(αH+αB2)tτ=(α_H+α_B^2)tτ=(αH​+αB2​)t. The algorithm approximates the evolution channel to diamond-norm error ε\varepsilonε using O ⁣(τ+log⁡(1/ε)log⁡ ⁣(e+log⁡(1/ε)/τ))O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)O(τ+log(e+log(1/ε)/τ)log(1/ε)​) oracle queries, matching the query lower bound for Hamiltonian simulation. The number of additional one- and two-qubit gates is linear in the query complexity up to polylogarithmic factors. The query- and gate-complexity bounds extend to Lipschitz-continuous time-dependent Lindbladians under coherent time-indexed oracle access. Our construction uses a one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst. We bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels. The gate implementation combines a compressed Kraus-label representation, which stores only the positions and values of the nonzero labels, with the rotation factorization of Chen et al.
Simulating the dynamics of open quantum systems is a crucial task in quantum computing, offering wide-ranging applications but remaining computationally challenging. In this paper, we propose two quantum algorithms for simulating the dynamics of open quantum systems governed by Lindbladians. We introduce a new approximation channel for short-time evolution, inspired by the quantum trajectory method, which underpins the efficiency of our algorithms. The first algorithm achieves a gate complexity independent of the number of jump operators, m , marking a significant improvement in efficiency. The second algorithm achieves near-optimal dependence on the evolution time t and precision ϵ and introduces only an additional O ~ ( m ) factor, which strictly improves upon state-of-the-art gate-based quantum algorithm that has an O ~ ( m 2 ) factor. The improvement stems from the integration of the new approximation channel with a novel structured linear combination of unitaries method. In both our algorithms, the reduction of dependence on m significantly enhances the efficiency of simulating practical dissipative processes characterized by a large number of jump operators.
For the problem of simulating Lindblad evolution for time ttt to precision εεε, Hamiltonian simulation provides an additive query lower bound, informally, Ω(t+polylog(1/ε))Ω(t + \mathrm{polylog}(1/ε))Ω(t+polylog(1/ε)). However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally, O(t polylog(1/ε))\mathcal{O}(t\,\mathrm{polylog}(1/ε))O(tpolylog(1/ε)), in query complexity. It has remained open whether this multiplicative dependence is necessary. In this paper, we close the gap in query complexity by giving an algorithm with optimal additive dependence on evolution time and precision in the block-encoding model. Our approach uses the transducer framework to reduce the query cost of composing first-order approximations to the evolution channel, together with linear combinations of reuse circuits of different lengths to suppress catalyst-removal error. We further achieve nearly optimal gate complexity in evolution time and precision through history compression and an efficient implementation of the query-free part of the transducer using operation reordering and linear combinations of unitaries, while preserving the optimal query complexity.
Abstract The Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on repeated interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian e t L within error ε scales in most physical scenarios as ν ∈ O ( t 2 ‖ L ‖ 1 → 1 2 / ϵ ) . This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative qubitization approach and Trotter–Suzuki formulas, and specifically show that for iterative qubitization the number of operations needed to simulate the dynamics (for a fixed value of ν) scales as O ( α 0 t + ν log ⁡ ( 1 / ϵ ) / log ⁡ log ⁡ ( 1 / ϵ ) ) in the limit where α 0 (the coefficient 1-norm for the system and bath Hamiltonians) asymptotically dominates over the corresponding factor for the interaction Hamiltonian, which is often the case in weak coupling. This scaling would appear to be optimal if the complexity of ν is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.
We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an nnn-qubit system for time ttt within precision εεε. If the Lindbladian consists of poly(n)\mathrm{poly}(n)poly(n) operators that can each be expressed as a linear combination of poly(n)\mathrm{poly}(n)poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/ε)poly(n))O(t\, \mathrm{polylog}(t/ε)\mathrm{poly}(n))O(tpolylog(t/ε)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of O(t2/ε)O(t^2/ε)O(t2/ε) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.
The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.