arXiv:2608.17653

Quantum simulation of slow analytic time-dependent Hamiltonians

Chenhao Zhao, Yinan Li, Dong An

quant-phmath.NASubmitted Aug 18, 2026• Announced Aug 19, 2026• 0 citations

Abstract

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.

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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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Hamiltonian Simulation by Qubitization

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arXiv:1610.06546QuantumJul 12, 2019854 citations

arXiv:2608.17653

Quantum simulation of slow analytic time-dependent Hamiltonians

Chenhao Zhao, Yinan Li, Dong An

quant-phmath.NASubmitted Aug 18, 2026• Announced Aug 19, 2026• 0 citations

arXiv HTML PDF

Abstract

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.

Related papers

Time-Dependent Hamiltonian Simulation with Optimal Query Complexity

Boyang Chen, Minbo Gao, Xinzhao Wang +1 more

We give a query-optimal algorithm for simulating a general $n$ -qubit time-dependent Hamiltonian $H(t)$ on $[0,T]$ , 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.We give a query-optimal algorithm for simulating a general $n$-qubit time-dependent Hamiltonian $H(t)$ on $[0,T]$, 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.

quant-pharXiv:2608.06094Aug 7, 20260 citations

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

Boyang Chen, Minbo Gao, Zhengfeng Ji +3 more

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.

quant-pharXiv:2608.30629Sep 1, 20260 citations

Root-sparsity scaling for Hamiltonian simulation, differential equations and linear-system solvers

Dominic W. Berry, Abhishek Rawat, Sophia Simon +1 more

We give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an $s$ -sparse Hermitian matrix with row and column Euclidean norms bounded by $η$ , evolution for time $t$ to error $ε$ uses $\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(1/ε)}+\log(1/ε))$ queries, where $τ=η|t|$ . The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in $\sqrt{s}τ$ and $\log(1/ε)$ . For dissipative linear differential equations we obtain $\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))$ queries, where $τ$ is the evolution time multiplied by a norm bound, $\bar g$ bounds the ratio of combined input and source norms to the final-solution norm, and $C$ is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give $\mathcal{O}(κ\sqrt{s}\log(1/ε))$ queries when $\|A\|\le1$ and $\|A^{-1}\|\leκ$ , and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.We give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an $s$-sparse Hermitian matrix with row and column Euclidean norms bounded by $η$, evolution for time $t$ to error $ε$ uses $\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(1/ε)}+\log(1/ε))$ queries, where $τ=η|t|$. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in $\sqrt{s}τ$ and $\log(1/ε)$. For dissipative linear differential equations we obtain $\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))$ queries, where $τ$ is the evolution time multiplied by a norm bound, $\bar g$ bounds the ratio of combined input and source norms to the final-solution norm, and $C$ is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give $\mathcal{O}(κ\sqrt{s}\log(1/ε))$ queries when $\|A\|\le1$ and $\|A^{-1}\|\leκ$, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.

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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

Query-optimal quantum simulation of Lindblad evolution

Chunhao Wang, Christopher Ye

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.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.

quant-pharXiv:2609.17490Sep 16, 20260 citations

Hamiltonian Simulation by Qubitization

Guang Hao Low, Isaac L. Chuang

We present the problem of approximating the time-evolution operator e−iH^t to error ϵ , where the Hamiltonian H^=(⟨G|⊗I^)U^(|G⟩⊗I^) is the projection of a unitary oracle U^ onto the state |G⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log⁡(1/ϵ)) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d -sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^ in an invariant SU(2) subspace. A large class of operator functions of H^ can then be computed with optimal query complexity, of which e−iH^t is a special case.We present the problem of approximating the time-evolution operator e−iH^t to error ϵ , where the Hamiltonian H^=(⟨G|⊗I^)U^(|G⟩⊗I^) is the projection of a unitary oracle U^ onto the state |G⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log⁡(1/ϵ)) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d -sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^ in an invariant SU(2) subspace. A large class of operator functions of H^ can then be computed with optimal query complexity, of which e−iH^t is a special case.

arXiv:1610.06546QuantumJul 12, 2019854 citations

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)

\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)

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 give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an

s

-sparse Hermitian matrix with row and column Euclidean norms bounded by

η

, evolution for time

t

to error

ε

uses

\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(1/ε)}+\log(1/ε))

queries, where

τ=η|t|

. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in

\sqrt{s}τ

and

\log(1/ε)

. For dissipative linear differential equations we obtain

\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))

queries, where

τ

is the evolution time multiplied by a norm bound,

\bar g

bounds the ratio of combined input and source norms to the final-solution norm, and

C

is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give

\mathcal{O}(κ\sqrt{s}\log(1/ε))

queries when

\|A\|\le1

and

\|A^{-1}\|\leκ

, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.

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.

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.