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Hamiltonian dynamics from pure dissipation

Daniel Stilck França, Zhong-Xia Shang

Published Jul 31, 2026• 0 citations
Publisher

Abstract

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 $\epsilon$ error in diamond norm using $\mathcal{O}(t^2/\epsilon)$ evolution time. We further prove that for time-independent dynamics this $\mathcal{O}(t^2/\epsilon)$ 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 $\epsilon$ 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.

DOI
10.21203/rs.3.rs-10130455/v1

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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.
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 open quantum systems reveals how environmental coupling shapes relaxation, excitation transport, and the dynamics of quantum correlations. On quantum hardware, dissipative channels add operations and might seem to increase cost. However, we show that a broad class of Pauli noise, including depolarization, can ease quantum and classical simulation by exponentially suppressing high-weight components of Heisenberg-evolved observables. For bounded-degree Lindblad dynamics, this permits system-size-independent Trotter steps when estimating local observables. Combining this compression with Richardson extrapolation, we provide classical and quantum error bounds with respect to the 2-norm of the propagated observable. These give high-probability expectation-value guarantees for random input states or random Hamiltonians acting on a fixed state and control (out-of-time-order) correlators. Concretely, we show that expectation values of local observables at time ttt can be estimated to accuracy ε\varepsilonε using circuits of maximum depth O ⁣([1+(t/γ)3/2]log⁡2(1/ε))O\!\left(\left[1+\left(t/γ\right)^{3/2}\right]\log^2(1/\varepsilon)\right)O([1+(t/γ)3/2]log2(1/ε)), independent of the system size nnn. Classically, we show that sparse Pauli propagation runs in time polynomial in nnn, ttt and 1/ε1/\varepsilon1/ε for every fixed γ>0γ>0γ>0, with polynomial degree O(1/γ)O(1/γ)O(1/γ) in the weak dissipation limit. The gap between our quantum and classical upper bounds leaves room for a substantial polynomial quantum advantage as dissipation weakens, an observation further supported by our numerical results.
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.
Experimental implementations of Hamiltonian dynamics are often affected by dissipative noise arising from interactions with the environment. This raises the question of whether one can detect the presence or absence of such dissipation using only access to the observed time evolution of the system. We consider the following decision problem: given black-box access to the time-evolution channels etLe^{t\mathcal{L}}etL generated by an unknown time-independent Lindbladian L\mathcal{L}L, determine whether the dynamics are purely Hamiltonian or contain dissipation of magnitude at least εεε in normalized Frobenius norm. We give a randomized procedure that solves this task using total evolution time O(ε−1)\mathcal{O}(ε^{-1})O(ε−1), which is information-theoretically optimal. This guarantee holds under the assumptions that the Lindblad generator has bounded strength and its dissipative part is of constant locality with bounded degree. Our work provides a practical method for detecting dissipative noise in experimentally implemented quantum dynamics.
Understanding the structure and stability of open quantum dynamics is increasingly important for both fundamental studies of nonequilibrium quantum systems and the development of quantum simulation algorithms. In this work, we introduce a structural framework for Markovian open quantum systems that characterizes Lindbladian generators in terms of two scalar quantities: the dissipative strength and the non-normality. We show that normal generators admit an exact decoupling between dissipative and norm-preserving dynamics, leading to purely exponential behavior governed by the dissipative scale. In contrast, non-normality is an intrinsically dissipative feature: it vanishes in the absence of dissipation but is not implied by it. Moreover, it is structurally constrained by the interplay between the Hermitian and anti-Hermitian components of the generator. For generic Markovian open quantum systems, we identify parametric regimes controlled by a dimensionless ratio between non-normality and dissipative strength, governing the onset of transient amplification. These structural features have direct implications for quantum simulation. While Hamiltonian and normal dissipative dynamics exhibit stable evolution with standard scaling behavior, non-normal generators can induce transient growth that amplifies numerical errors and increases simulation cost. Our results provide a unified generator-level perspective on irreversibility, stability, and quantum simulation of open quantum systems.
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.