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Physical Review Letters

Lévy-Khintchine Structure Enables Fast-Forwardable Lindbladian Simulation

Anonymous, Zhengfeng Ji, Chenghua Liu

Published Aug 18, 2026• 0 citations
Publisher

Abstract

Simulation of open quantum systems is an area of active research in quantum algorithms. In this Letter, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. By applying the Lévy-Khintchine representation theorem, we clarify when and how dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with Gaussian, compound Poisson, and symmetric stable distributions and their algorithmic implications. These give Lindbladian families whose expected Hamiltonian simulation time ranges from Θ(sqrt[t]) for Gaussian dephasing to Θ(t) for compound-Poisson jumps, and Θ(t^{1/α}) for symmetric α-stable laws with 1<α≤2. Moreover, we prove that these time complexities are asymptotically optimal for the classes studied here using an information-theoretic approach, establishing fundamental lower bounds on fast-forwarding quantum simulation algorithms for the first time to the best of our knowledge.

DOI
10.1103/qy4b-gzlq

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•Jul 31, 2026•0 citations
Simulation of open quantum systems is an area of active research in quantum algorithms. In this Letter, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. By applying the Lévy-Khintchine representation theorem, we clarify when and how dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with Gaussian, compound Poisson, and symmetric stable distributions and their algorithmic implications. These give Lindbladian families whose expected Hamiltonian simulation time ranges from Θ(sqrt[t]) for Gaussian dephasing to Θ(t) for compound-Poisson jumps, and Θ(t^{1/α}) for symmetric α-stable laws with 1<α≤2. Moreover, we prove that these time complexities are asymptotically optimal for the classes studied here using an information-theoretic approach, establishing fundamental lower bounds on fast-forwarding quantum simulation algorithms for the first time to the best of our knowledge.
Simulation of open quantum systems is an area of active research in quantum algorithms. In this work, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. By applying the Lévy-Khintchine representation theorem, we clarify when and how a dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with Gaussian, compound Poisson and symmetric stable distributions and their algorithmic implications. These give wide classes of Lindbladians that can be simulated in Θ(t1/α)Θ(t^{1/α})Θ(t1/α) Hamiltonian simulation time without any extra ancilla or other quantum gates for 1≤α≤21\le α\le 21≤α≤2. Moreover, we prove that these time complexities are asymptotically optimal using an information theoretic approach, which, to the best of our knowledge, is the first result of lower bounds on fast-forwarding simulation algorithms.
Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no–fast-forwarding theorem for simulating sparse Lindbladians in black-box models.
The fundamental difference between closed and open quantum dynamics lies in their environmental interaction: closed systems are perfectly isolated and evolve reversibly under unitary Hamiltonian dynamics, whereas open systems continuously couple to an external bath, resulting in irreversible dissipation and information loss. In this work, we show internal Hamiltonian dynamics can be "faked`` via external pure dissipation, i.e., Lindbladians without a coherent Hamiltonian part. More concretely, we show that, in a GKSL representation with zero explicit Hamiltonian term but nontraceless jump operators, bounded-norm dissipative generators can approximate Hamiltonian dynamics within εεε error in diamond norm using O(t2/ε)\mathcal{O}(t^2/ε)O(t2/ε) evolution time. We further prove that for time-independent dynamics this O(t2/ε)\mathcal{O}(t^2/ε)O(t2/ε) scaling is in the worst case, necessary and optimal from a geometric perspective, which captures the fundamental decoherence cost for catching up with the speed of Hamiltonian dynamics. Our construction leads to various implications, including the BQP-completeness of purely dissipative dynamics even before reaching approximate equilibrium, a Zeno-adjacent state-independent freezing effect, the no super-quadratic fast-forwarding theorem of a class of purely dissipative dynamics, and reducing Lindbladian simulation cost via gauge changing.
Simulating the dynamics of dissipative quantum many-body systems governed by local Lindbladians is a fundamental task in quantum computation. While the near-optimal gate count has been achieved for Hamiltonian simulation, comparable results for Lindbladian simulation remain elusive. In this work, we develop quantum algorithms for lattice Lindbladians by exploiting their locality. First, we consider sparsely dissipative systems, in which the dissipation is sparsely located, including boundary-driven systems. We establish a near-optimal quantum algorithm for their dynamics with gate count O(Nt polylog(Nt/ε))O(Nt \, \mathrm{polylog}(Nt/\varepsilon))O(Ntpolylog(Nt/ε)), where NNN is the system size, ttt is the evolution time, and ε\varepsilonε is the allowable error. We then consider generic lattice Lindbladians with finite-range interactions and dissipation, and develop an algorithm for simulating time-evolved observables with gate count O((Nt)4/3 polylog(Nt/ε))O((Nt)^{4/3} \, \mathrm{polylog}(Nt/\varepsilon))O((Nt)4/3polylog(Nt/ε)) per sample with the sampling complexity Θ(ε−2)Θ(\varepsilon^{-2})Θ(ε−2). The gate count has the smallest known dependence on the system size among algorithms retaining polylogarithmic dependence on 1/ε1/\varepsilon1/ε. Our algorithms are based on two techniques that exploit locality: patching and merging. Patching decomposes dissipative dynamics into dynamics on subsystems with exponentially small error, generalizing a key idea underlying the Haah-Hastings-Kothari-Low algorithm for near-optimal Hamiltonian simulation. Merging absorbs reversed dissipative dynamics into other parts of the evolution, substantially reducing the overhead associated with quasi-probabilistic sampling. These results demonstrate that locality can be fully exploited to achieve fast quantum simulation of dissipative many-body dynamics, opening the way to applications such as predicting nonequilibrium phenomena and preparing desirable quantum states.
Quantum phase estimation (QPE) and Lindbladian dynamics are both foundational in quantum information science and central to quantum algorithm design. In this work, we bridge these two concepts: certain simple Lindbladian processes can be adapted to perform QPE-type tasks. However, unlike QPE, which achieves Heisenberg-limit scaling, these Lindbladian evolutions are restricted to standard quantum limit complexity. This indicates that, different from Hamiltonian dynamics, the natural dissipative evolution speed of such Lindbladians does not saturate the fundamental quantum limit, thereby suggesting the potential for quadratic fast-forwarding. We confirm this by presenting a quantum algorithm that simulates these Lindbladians for time ttt within an error ε\varepsilonε using O(tlog⁡(ε−1))\mathcal{O}\left(\sqrt{t\log(\varepsilon^{-1})}\right)O(tlog(ε−1)​) cost, whose mechanism is fundamentally different from the fast-forwarding examples of Hamiltonian dynamics. As a bonus, this fast-forwarded simulation naturally serves as a new Heisenberg-limit QPE algorithm. Therefore, our work explicitly bridges the standard quantum limit-Heisenberg limit transition to the fast-forwarding of dissipative dynamics. We also adopt our fast-forwarding algorithm for efficient Gibbs state preparation and demonstrate the counter-intuitive implication: the allowance of a quadratically accelerated decoherence effect under arbitrary Pauli noise.
Abstract The fundamental difference between closed and open quantum dynamics lies in their environmental interaction: closed systems are perfectly isolated and evolve reversibly under unitary Hamiltonian dynamics, whereas open systems continuously couple to an external bath, resulting in irreversible dissipation and information loss. In this work, we show internal Hamiltonian dynamics can be "faked`` via external pure dissipation, i.e., Lindbladians without a coherent Hamiltonian part. More concretely, we show that, in a GKSL representation with zero explicit Hamiltonian, bounded-norm dissipative generators can approximate Hamiltonian dynamics within ϵ\epsilonϵ error in diamond norm using O(t2/ϵ)\mathcal{O}(t^2/\epsilon)O(t2/ϵ) evolution time. We further prove that for time-independent dynamics this O(t2/ϵ)\mathcal{O}(t^2/\epsilon)O(t2/ϵ) scaling is in the worst case, necessary and optimal from a geometric perspective, which captures the fundamental decoherence cost for catching up with the speed of Hamiltonian dynamics. For practical considerations, we establish an autonomous QDRIFT protocol and show how to exponentially improve the dependence on ϵ\epsilonϵ for expectation value estimation via extrapolations. Our construction leads to various implications, including the BQP-completeness of purely dissipative dynamics even before reaching approximate equilibrium, a Zeno-adjacent state-independent freezing effect, the no super-quadratic fast-forwarding theorem of a class of purely dissipative dynamics, and reducing Lindbladian simulation cost via gauge changing.