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arxiv: 2509.09517 · v2 · pith:YJA7S6SKnew · submitted 2025-09-11 · 🪐 quant-ph

Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties

Zhong-Xia Shang , Dong An , Changpeng Shao This is my paper

Pith reviewed 2026-05-25 07:32 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Lindbladian simulationfast-forwardingGibbs state estimationdissipative quantum dynamicsunitary jump operatorscoherence conditionsquantum algorithms
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The pith

Lindbladian simulation with unitary jumps achieves additive query complexity and exponential fast-forwarding for block-diagonal Pauli structures

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a quantum algorithm for simulating purely dissipative Lindbladians with unitary jump operators that achieves additive query complexity O(t + log(ε^{-1})). When the jump operators have block-diagonal Pauli structure, the algorithm is modified to achieve exponential fast-forwarding with circuit depth O(log(t + log(ε^{-1}))), preserving query complexity through parallel access. These techniques are applied to estimate Gibbs state properties of the form ⟨ψ1|e^{-β(H+I)}|ψ2⟩ up to additive error ε, yielding exponential complexity improvement O(2^{-n/2} ε^{-1} log β) for input states satisfying specific coherence conditions such as ⟨0|^{⊗n} e^{-β(H+I)} |+⟩^{⊗n}. The method also covers applications including ground state overlap testing and amplitude estimation, with the degree of improvement depending on the coherence resource in the input states.

Core claim

For Lindbladians with unitary jump operators the simulation algorithm attains additive query complexity O(t + log(ε^{-1})); when the operators possess block-diagonal Pauli structure this extends to exponential fast-forwarding with logarithmic circuit depth, and the same techniques produce an exponential reduction in complexity for estimating certain Gibbs-state matrix elements whenever the input states obey the stated coherence conditions.

What carries the argument

Quantum algorithm for Lindbladian simulation with unitary jump operators, modified for block-diagonal Pauli structure to enable fast-forwarding via parallel access and then applied to coherent Gibbs-state estimation

If this is right

  • Query complexity stays additive O(t + log(ε^{-1})) even after the circuit-depth reduction via parallel access.
  • Exponential circuit-depth improvement applies directly to estimating Gibbs-state matrix elements under the coherence conditions.
  • The same fast-forwarding yields better resource scaling for ground-state overlap testing and amplitude estimation.
  • The level of improvement varies continuously with the amount of coherence present in the input states |ψ1⟩ and |ψ2⟩.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exponential fast-forwarding may extend to other open-system models that admit analogous block structures in their jump operators.
  • Coherence in the input states functions as a quantifiable resource that trades off against the inverse-temperature parameter β in thermal estimation tasks.
  • Small-system numerical simulations could directly test the predicted logarithmic depth scaling for concrete Pauli-structured Lindbladians.
  • The approach suggests possible speed-ups in related tasks such as preparing approximate thermal states when coherence is available.

Load-bearing premise

The jump operators must be unitary and, for exponential fast-forwarding, must have block-diagonal Pauli structure; the input states must satisfy the coherence conditions such as the given overlap between all-zero and all-plus states after the thermal factor.

What would settle it

Run the algorithm on a small number of qubits with block-diagonal Pauli jump operators and verify whether the circuit depth required for fixed error scales logarithmically rather than linearly with simulation time t.

read the original abstract

Fast-forwarding refers to the ability to simulate a system of time $t$ using significantly fewer than $t$ queries or circuit depth. While various Hamiltonian systems are known to circumvent the no fast-forwarding theorem, analogous results for dissipative dynamics, governed by Lindbladians, remain largely unexplored. We first present a quantum algorithm for simulating purely dissipative Lindbladians with unitary jump operators, achieving additive query complexity $\mathcal{O}\left(t + \log(\varepsilon^{-1})\right)$ up to error~$\varepsilon$, improving previous algorithms. When the jump operators have certain structures (i.e., block-diagonal Paulis), the algorithm can be modified to achieve exponential fast-forwarding, attaining circuit depth $\mathcal{O}\left(\log\left(t + \log(\varepsilon^{-1})\right)\right)$, while preserving query complexity via parallel access. Using these fast-forwarding techniques, we develop a quantum algorithm for estimating Gibbs state properties of the form $\langle \psi_1 | e^{-\beta(H + I)} | \psi_2 \rangle$, up to additive error $\epsilon$, with $H$ the Hamiltonian and $\beta$ the inverse temperature. For input states exhibiting certain coherence conditions -- e.g.,~$\langle 0|^{\otimes n} e^{-\beta(H + I)} |+\rangle^{\otimes n}$ -- our method achieves exponential improvement in complexity (measured by circuit depth), $\mathcal{O} (2^{-n/2} \epsilon^{-1} \log \beta ),$ compared to the quantum singular value transformation-based approach, with complexity $\tilde{\mathcal{O}} (\epsilon^{-1} \sqrt{\beta} )$. We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general $| \psi_1 \rangle$ and $| \psi_2 \rangle$, we also show how the level of improvement is changed with the coherence resource in $| \psi_1 \rangle$ and $| \psi_2 \rangle$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims quantum algorithms for simulating purely dissipative Lindbladians with unitary jump operators that achieve additive query complexity O(t + log(ε^{-1})), with a modification for block-diagonal Pauli jump operators yielding exponential fast-forwarding to circuit depth O(log(t + log(ε^{-1}))) while preserving query complexity via parallel access. These techniques are applied to estimate Gibbs-state matrix elements ⟨ψ₁|e^{-β(H+I)}|ψ₂⟩ to additive error ε, achieving complexity O(2^{-n/2} ε^{-1} log β) under specific coherence conditions on the input states (e.g., overlap with |0⟩^{⊗n} and |+⟩^{⊗n}), compared to Õ(ε^{-1} √β) for QSVT-based methods; applications to ground-state overlap testing and amplitude estimation are indicated, with a general discussion of coherence resources.

Significance. If the algorithmic constructions and analyses hold, the work would constitute a meaningful advance in quantum algorithms for open-system simulation by establishing fast-forwarding results for structured Lindbladians and coherence-dependent exponential improvements for Gibbs-property estimation. The explicit restrictions to unitary jumps and coherence conditions are appropriately foregrounded.

major comments (3)
  1. [Abstract / Introduction] Abstract and introduction: the central complexity claims (additive O(t + log(ε^{-1})) query complexity, exponential fast-forwarding to O(log(t + log(ε^{-1}))) depth, and O(2^{-n/2} ε^{-1} log β) Gibbs estimation) are stated without any derivation, error analysis, or proof sketch, which is load-bearing for assessing correctness of the fast-forwarding and amplification results.
  2. [Lindbladian simulation section] The section presenting the Lindbladian simulation algorithm: no explicit construction, Trotterization or product-formula analysis, or query-complexity accounting is supplied to support the claimed additive (rather than multiplicative) dependence on t.
  3. [Gibbs estimation section] The section on Gibbs-state estimation: the exponential improvement is conditioned on specific coherence overlaps (e.g., ⟨0|^{⊗n} e^{-β(H+I)} |+⟩^{⊗n}), yet no quantitative relation between the overlap magnitude and the stated complexity is derived or bounded.
minor comments (2)
  1. Notation: the distinction between query complexity and circuit depth should be made uniform across all stated bounds; the parallel-access model for preserving query complexity under exponential fast-forwarding requires a brief definition.
  2. The comparison to QSVT is given only asymptotically; a short table contrasting the two approaches under the stated coherence assumptions would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the potential significance of the fast-forwarding and coherence-dependent results. We address each major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract / Introduction] Abstract and introduction: the central complexity claims (additive O(t + log(ε^{-1})) query complexity, exponential fast-forwarding to O(log(t + log(ε^{-1}))) depth, and O(2^{-n/2} ε^{-1} log β) Gibbs estimation) are stated without any derivation, error analysis, or proof sketch, which is load-bearing for assessing correctness of the fast-forwarding and amplification results.

    Authors: We agree that the abstract and introduction would benefit from additional context on the derivations. In the revised version we will expand the introduction with concise proof sketches that outline the key steps establishing the additive query complexity, the parallel-access construction for exponential depth reduction under block-diagonal Pauli jumps, and the coherence-dependent scaling for the Gibbs estimator. Full technical details and error analyses will continue to reside in the dedicated sections. revision: yes

  2. Referee: [Lindbladian simulation section] The section presenting the Lindbladian simulation algorithm: no explicit construction, Trotterization or product-formula analysis, or query-complexity accounting is supplied to support the claimed additive (rather than multiplicative) dependence on t.

    Authors: The manuscript contains a high-level algorithmic description, but we concur that an explicit construction and rigorous accounting are required for verification. We will revise the Lindbladian simulation section to supply the concrete circuit construction for unitary-jump Lindbladians, specify the product formula employed, and provide a complete query-complexity analysis demonstrating the additive O(t + log(ε^{-1})) scaling together with the associated error bounds. revision: yes

  3. Referee: [Gibbs estimation section] The section on Gibbs-state estimation: the exponential improvement is conditioned on specific coherence overlaps (e.g., ⟨0|^{⊗n} e^{-β(H+I)} |+⟩^{⊗n}), yet no quantitative relation between the overlap magnitude and the stated complexity is derived or bounded.

    Authors: We will augment the Gibbs estimation section with an explicit derivation that relates the magnitude of the coherence overlaps (including the example ⟨0|^{⊗n} e^{-β(H+I)} |+⟩^{⊗n} and generalizations) to the achieved complexity. This will include quantitative bounds showing how the overlap controls the effective circuit depth and query cost, together with the corresponding scaling for general input states. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are explicit algorithmic constructions

full rationale

The paper presents quantum algorithms for Lindbladian simulation and Gibbs state property estimation conditioned on unitary jump operators (with block-diagonal Pauli structure for exponential fast-forwarding) and specific input coherence conditions. These improvements are derived from explicit constructions (additive query complexity O(t + log(1/ε)) and circuit depth O(log(t + log(1/ε))), with exponential gains for coherent states like ⟨0|⊗n e^{-β(H+I)} |+⟩⊗n). No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claims remain independent of the target results and are externally falsifiable via the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the claims rest on standard quantum-computing and Lindblad-equation assumptions with no visible free parameters or new postulated entities.

axioms (2)
  • domain assumption Lindblad master equation with unitary jump operators governs the dynamics
    The paper assumes standard dissipative evolution under the Lindblad equation with unitary jumps.
  • standard math Quantum circuit model with query and depth complexity measures
    Relies on the standard framework of quantum query complexity and circuit depth.

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Forward citations

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