arXiv:2609.17490

Query-optimal quantum simulation of Lindblad evolution

Chunhao Wang, Christopher Ye

quant-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations• Revised Sep 25, 2026 (v2)

Abstract

For the problem of simulating Lindblad evolution for time $t$ to precision $ε$, Hamiltonian simulation provides an additive query lower bound, informally, $Ω(t + \mathrm{polylog}(1/ε))$. However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally, $\mathcal{O}(t\,\mathrm{polylog}(1/ε))$, 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.

Comments: 43 pages, no figures. Improved gate complexity

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Query-Optimal and Gate-Efficient Lindbladian Simulation

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Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

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The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

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arXiv:2609.17490

Query-optimal quantum simulation of Lindblad evolution

Chunhao Wang, Christopher Ye

quant-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations• Revised Sep 25, 2026 (v2)

arXiv HTML PDF

Abstract

For the problem of simulating Lindblad evolution for time

t

to precision

ε

, Hamiltonian simulation provides an additive query lower bound, informally,

Ω(t + \mathrm{polylog}(1/ε))

. However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally,

\mathcal{O}(t\,\mathrm{polylog}(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.

Comments: 43 pages, no figures. Improved gate complexity

Related papers

Query-Optimal and Gate-Efficient Lindbladian Simulation

Boyang Chen, Minbo Gao, Xinzhao Wang +1 more

We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian $H$ and a projected unitary encoding of the stacked jump operator $B=\sum_{k=1}^m \lvert k\rangle\otimes L_k$ , with normalization factors $α_H$ and $α_B$ , respectively. For evolution time $t$ , set $τ=(α_H+α_B^2)t$ . The algorithm approximates the evolution channel to diamond-norm error $\varepsilon$ using $O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)$ 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.We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian $H$ and a projected unitary encoding of the stacked jump operator $B=\sum_{k=1}^m \lvert k\rangle\otimes L_k$, with normalization factors $α_H$ and $α_B$, respectively. For evolution time $t$, set $τ=(α_H+α_B^2)t$. The algorithm approximates the evolution channel to diamond-norm error $\varepsilon$ using $O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)$ 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.

quant-pharXiv:2609.18757Sep 17, 20260 citations

Causal Query Compression for Lindblad Dynamics: Optimal Queries and Nearly Linear Local Simulation

Jacob Kitchen

We construct a causal query compiler for time-dependent Lindblad dynamics and establish query and gate bounds under distinct input models. With coherent block encodings of the Hamiltonian and individual jump operators, a Lipschitz generator can be simulated to diamond-norm error $\varepsilon$ using $O(1+τ+\log(1/\varepsilon)/\log(e+\log(1/\varepsilon)/τ))$ controlled queries, where $τ=T(α_H+\sum_μα_μ^2)$ . This bound is independent of the time-grid size and is worst-case optimal for $τ\ge1$ . Weak active entry and clean return yield a factorial history tail; combining two reuse lengths removes the auxiliary input. A three-call polar correction supplies the Lindblad cells. The compiler also characterizes the minimum oracle action of bounded adaptive Markovian protocols by quantum query complexity and the general adversary bound, up to constant factors. For finite-range dynamics on $n$ lattice sites, efficient coherent evaluation of the local matrices and fixed local parameters give $O(n(T+1)\operatorname{polylog}X)$ elementary gates, where $X=\max\{e,n(T+1)(1+K_t+B)/\varepsilon\}$ , for piecewise Hölder generators with fixed exponent, local variation bound $K_t$ , and at most $B$ breakpoints per term. The jumps need not commute. This gate bound includes evaluation, arithmetic, selection, routing, and environment storage. Its proof combines a spatial decomposition on a common bath, compilation on occupied inputs, and routing on compressed records. Reusing bath storage between short segments gives depth $(T+1)\operatorname{polylog}X$ and space $n\operatorname{polylog}X$ .We construct a causal query compiler for time-dependent Lindblad dynamics and establish query and gate bounds under distinct input models. With coherent block encodings of the Hamiltonian and individual jump operators, a Lipschitz generator can be simulated to diamond-norm error $\varepsilon$ using $O(1+τ+\log(1/\varepsilon)/\log(e+\log(1/\varepsilon)/τ))$ controlled queries, where $τ=T(α_H+\sum_μα_μ^2)$. This bound is independent of the time-grid size and is worst-case optimal for $τ\ge1$. Weak active entry and clean return yield a factorial history tail; combining two reuse lengths removes the auxiliary input. A three-call polar correction supplies the Lindblad cells. The compiler also characterizes the minimum oracle action of bounded adaptive Markovian protocols by quantum query complexity and the general adversary bound, up to constant factors. For finite-range dynamics on $n$ lattice sites, efficient coherent evaluation of the local matrices and fixed local parameters give $O(n(T+1)\operatorname{polylog}X)$ elementary gates, where $X=\max\{e,n(T+1)(1+K_t+B)/\varepsilon\}$, for piecewise Hölder generators with fixed exponent, local variation bound $K_t$, and at most $B$ breakpoints per term. The jumps need not commute. This gate bound includes evaluation, arithmetic, selection, routing, and environment storage. Its proof combines a spatial decomposition on a common bath, compilation on occupied inputs, and routing on compressed records. Reusing bath storage between short segments gives depth $(T+1)\operatorname{polylog}X$ and space $n\operatorname{polylog}X$.

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Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

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The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right)$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$ . However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right]$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$ . The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

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Chenhao Zhao, Yinan Li, Dong An

We develop a quantum algorithm for slow analytic Hamiltonians $\widetilde H(t)=H(t/T)$ with $\|H(s)\|\leqα$ that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of $H(s)$ , together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity $\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)$ and additional gate complexity $\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)$ , assuming coherent access to $H'(s)$ and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.We develop a quantum algorithm for slow analytic Hamiltonians $\widetilde H(t)=H(t/T)$ with $\|H(s)\|\leqα$ that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of $H(s)$, together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity $\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)$ and additional gate complexity $\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)$, assuming coherent access to $H'(s)$ and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.

quant-pharXiv:2608.17653Aug 19, 20260 citations

Optimal Lower Bounds for Hamiltonian Simulation

Alexander Zlokapa, Richard R. Allen, Aram W. Harrow

For Hamiltonian $H = \sum_j h_j$ , we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms $\|h_j\|$ , time $t$ , and trace-distance error $ε$ . The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case $\|h_j\|$ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in $1/ε$ , contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.For Hamiltonian $H = \sum_j h_j$, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms $\|h_j\|$, time $t$, and trace-distance error $ε$. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case $\|h_j\|$ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in $1/ε$, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.

quant-pharXiv:2607.19852Jul 23, 20260 citations

For the problem of simulating Lindblad evolution for time

t

to precision

ε

, Hamiltonian simulation provides an additive query lower bound, informally,

Ω(t + \mathrm{polylog}(1/ε))

. However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally,

\mathcal{O}(t\,\mathrm{polylog}(1/ε))

We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian

H

and a projected unitary encoding of the stacked jump operator

B=\sum_{k=1}^m \lvert k\rangle\otimes L_k

, with normalization factors

α_H

and

α_B

, respectively. For evolution time

t

, set

τ=(α_H+α_B^2)t

. The algorithm approximates the evolution channel to diamond-norm error

\varepsilon

using

O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)

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.

\varepsilon

using

O(1+τ+\log(1/\varepsilon)/\log(e+\log(1/\varepsilon)/τ))

controlled queries, where

τ=T(α_H+\sum_μα_μ^2)

. This bound is independent of the time-grid size and is worst-case optimal for

τ\ge1

. Weak active entry and clean return yield a factorial history tail; combining two reuse lengths removes the auxiliary input. A three-call polar correction supplies the Lindblad cells. The compiler also characterizes the minimum oracle action of bounded adaptive Markovian protocols by quantum query complexity and the general adversary bound, up to constant factors. For finite-range dynamics on

n

lattice sites, efficient coherent evaluation of the local matrices and fixed local parameters give

O(n(T+1)\operatorname{polylog}X)

elementary gates, where

X=\max\{e,n(T+1)(1+K_t+B)/\varepsilon\}

, for piecewise Hölder generators with fixed exponent, local variation bound

K_t

, and at most

B

breakpoints per term. The jumps need not commute. This gate bound includes evaluation, arithmetic, selection, routing, and environment storage. Its proof combines a spatial decomposition on a common bath, compilation on occupied inputs, and routing on compressed records. Reusing bath storage between short segments gives depth

(T+1)\operatorname{polylog}X

and space

n\operatorname{polylog}X

We give a query-optimal algorithm for simulating a general

n

-qubit time-dependent Hamiltonian

H(t)

[0,T]

, assuming that

H

is Lipschitz continuous and

\|H(t)\|\leqα

. In the standard $\mathrm{HAM\mbox{-}T}$ access model, the algorithm approximates the time-ordered propagator

U_H(T)

to error

\varepsilon

using

O\left( αT+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(αT))} \right)

$\mathrm{HAM\mbox{-}T}$ queries. This matches the known query lower bound for time-independent Hamiltonians, showing that time dependence incurs no asymptotic query overhead. Our method first constructs a one-query transducer that, given an auxiliary state, implements an approximation to

U_H(T)

and returns the state unchanged. A weighted combination of circuits that apply the transducer different numbers of times makes the error caused by omitting this state decay factorially, yielding the stated optimal precision dependence. For time-independent Hamiltonians, the same method also gives a query-optimal alternative to qubitization.

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses

q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right)

queries to $\mathrm{HAM\mbox{-}T}$ within

\varepsilon

error for a Lipschitz-continuous time-dependent Hamiltonian

H(t)

[0,T]

satisfying

\left\lVert H(t)\right\rVert\leqα

. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses

O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right]

one- and two-qubit gates, where

a

is the number of block-encoding ancilla qubits and

β

is the Lipschitz constant of

H

. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

We develop a quantum algorithm for slow analytic Hamiltonians

\widetilde H(t)=H(t/T)

with

\|H(s)\|\leqα

that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of

H(s)

, together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity

\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)

and additional gate complexity

\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)

, assuming coherent access to

H'(s)

and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.

For Hamiltonian

H = \sum_j h_j

, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms

\|h_j\|

, time

t

, and trace-distance error

ε

. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case

\|h_j\|

to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in

1/ε

, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.