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arXiv:2607.19852

Optimal Lower Bounds for Hamiltonian Simulation

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

quant-phSubmitted Jul 22, 2026• Announced Jul 23, 2026• 0 citations
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Abstract

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.

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For Hamiltonian H=∑jhjH = \sum_j h_jH=∑j​hj​, 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 ∥hj∥\|h_j\|∥hj​∥, time ttt, and trace-distance error εεε. 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 ∥hj∥\|h_j\|∥hj​∥ 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/ε1/ε1/ε, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.
Recent work has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian HHH for a given simulation problem into subsets AAA and BBB such that H=A+BH=A+BH=A+B, where the terms in AAA are simulated with a Trotter-Suzuki channel and the BBB terms are randomly sampled via the QDrift algorithm. Here we show that this approach holds in imaginary time, making it a candidate classical algorithm for quantum Monte-Carlo calculations. We upper-bound the induced Schatten-1→11 \to 11→1 norm on both imaginary-time QDrift and Composite channels. Another recent result demonstrated that simulations of Hamiltonians containing geometrically-local interactions for systems defined on finite lattices can be improved by decomposing HHH into subsets that contain only terms supported on that subset of the lattice using a Lieb-Robinson argument. Here, we provide a quantum algorithm by unifying this result with the composite approach into ``local composite channels" and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form e−iHjte^{-iH_j t}e−iHj​t and e−Hjβe^{-H_j β}e−Hj​β to meet a certain error tolerance εεε. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a ≈20\approx 20≈20 fold speedup that occurs for a simulation of Jellium.
We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian HHH and a projected unitary encoding of the stacked jump operator B=∑k=1m∣k⟩⊗LkB=\sum_{k=1}^m \lvert k\rangle\otimes L_kB=∑k=1m​∣k⟩⊗Lk​, with normalization factors αHα_HαH​ and αBα_BαB​, respectively. For evolution time ttt, set τ=(αH+αB2)tτ=(α_H+α_B^2)tτ=(αH​+αB2​)t. The algorithm approximates the evolution channel to diamond-norm error ε\varepsilonε using O ⁣(τ+log⁡(1/ε)log⁡ ⁣(e+log⁡(1/ε)/τ))O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)O(τ+log(e+log(1/ε)/τ)log(1/ε)​) 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.
Randomized product formulas such as qDrift offer a resource-efficient alternative to deterministic Trotter--Suzuki decompositions for Hamiltonian simulation, removing their polynomial dependence on the number of Hamiltonian terms. qDrift, however, is intrinsically limited to first order in the evolution time, so its query complexity remains linear in the inverse of the target accuracy, 1/ε1/ε1/ε. We introduce Pathwise Random Hamiltonian Simulation (PRHS), which extends qDrift to arbitrary order by subdividing each time step into MMM correlated slices, each evolving under a term sampled from a quasi-probability distribution that we construct in closed form and prove unique, with a bias decaying factorially in MMM. Optimizing jointly over the number of slices MMM and the number of independent blocks NNN interpolates between the standard qDrift protocol at long times and a high-precision regime where the query cost grows slower than any power of 1/ε1/ε1/ε, without requiring ancillary qubits. Numerical simulations of five molecular Hamiltonians confirm this advantage, with PRHS achieving accuracies two to four orders of magnitude beyond qDrift at equal query cost.
We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on nnn qubits for time TTT up to error ϵ\epsilonϵ using O(nTpolylog(nT/ϵ)){\mathcal O}( nT {polylog} (nT/\epsilon))O(nTpolylog(nT/ϵ)) gates with depth O(Tpolylog(nT/ϵ)){\mathcal O}(T { polylog} (nT/\epsilon))O(Tpolylog(nT/ϵ)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nTnTnT and polylogarithmic in 1/ϵ1/\epsilon1/ϵ. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires Ω~(nT){\widetilde{\Omega}}(nT)Ω(nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
Hamiltonian simulation is a central application of quantum computing. Randomized Hamiltonian simulation approximates the target dynamics by sampling quantum circuits and often allows simpler circuit implementations. We develop a framework for randomized Hamiltonian simulation in which Hamiltonian terms are sampled with arbitrary probabilities and the corresponding short-time evolutions are implemented sequentially. The leading channel error relative to ideal time evolution is governed by the variance of the sampled generators. Minimizing the variance-based upper bound on the worst-case error recovers qDRIFT, a leading randomized Hamiltonian simulation algorithm that samples terms in proportion to their operator norms. For typical input states, however, this sampling distribution need not be optimal. Minimizing Haar-averaged error bounds instead yields sampling probabilities proportional to the Hilbert-Schmidt norms of the terms. This choice can provide smaller average-error bounds than conventional qDRIFT. For Hamiltonians expressed as sums of Pauli strings, we combine this framework with commuting Pauli grouping to obtain smaller error bounds than conventional qDRIFT at fixed RzR_zRz​ depth. Numerical benchmarks for the Sachdev-Ye-Kitaev model show error-bound improvement factors consistent with linear scaling in the number of qubits. For the 108-qubit FeMoco Hamiltonian, the error bounds are reduced by a factor of about 5.65. These results establish average-error optimization as a practical design principle for randomized Hamiltonian simulation.
For the problem of simulating Lindblad evolution for time ttt to precision εεε, Hamiltonian simulation provides an additive query lower bound, informally, Ω(t+polylog(1/ε))Ω(t + \mathrm{polylog}(1/ε))Ω(t+polylog(1/ε)). However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally, O(t polylog(1/ε))\mathcal{O}(t\,\mathrm{polylog}(1/ε))O(tpolylog(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.