Randomized product formulas beyond optimal deterministic scaling

• URL: http://arxiv.org/abs/2608.07720v1
• Date: Fri, 07 Aug 2026 19:13:09 GMT
• Title: Randomized product formulas beyond optimal deterministic scaling
• Abstract summary: Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems.<n>We introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, $H=A+αB$, where $α$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal O(α^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $Ω(α)$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+αB_\ell$ for $B = \sum_{\ell}B_\ell$, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves $\mathcal O(α^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $Ω(α^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.

Related papers

• Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas [23.101202672272027]
  We develop a high-order nested-commutator algorithm that preserves the main advantages of product formulas.<n>The resulting time dependence matches that of a product formula of order $2K+1$.
  arXiv  Detail & Related papers  (2026-07-13T17:42:25Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Practical framework for simulating permutation-equivariant quantum circuits [0.0]
  We introduce a practical algorithm for $S_n$-equivariant circuits under the assumption that the gate generators are at most $k$-local, with $kin O(1)$.<n>The resulting method runs in $O(n+1)$ time for constant depth, where $$ is the matrix multiplication exponent, significantly lowering the degree compared to existing techniques.
  arXiv  Detail & Related papers  (2026-03-13T15:18:18Z)
• Improving time dynamics simulation by sampling the error unitary [0.0]
  We introduce an algorithm to improve the error scaling of product formulas by randomly sampling the generator of their exact error unitary.<n>Our approach takes an arbitrary product formula of time $t$, $S_k(t)$ with error $O(tk+1)$ and produces a formula with expected error scaling as $O(t2k+2)$ with respect to the exact dynamics.<n>For a given fixed error $epsilon$ and total evolution time $T$ this leads to an improved gate complexity of $N=O(T/epsilon)frac1
  arXiv  Detail & Related papers  (2025-08-24T22:31:15Z)
• Trotter error and gate complexity of the SYK and sparse SYK models [0.8192907805418583]
  We study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie-Trotter- Suzuki formulas.<n>We find near-optimal gate complexities for simulating these models using Lie-Trotter- Suzuki formulas.
  arXiv  Detail & Related papers  (2025-02-25T18:16:56Z)
• Provable Scaling Laws for the Test-Time Compute of Large Language Models [84.00141420901038]
  We propose two algorithms that enjoy provable scaling laws for the test-time compute of large language models.<n>One is a two-stage knockout-style algorithm, where each candidate is evaluated by its average win rate against multiple opponents.<n>The other is a two-stage league-style algorithm, where each candidate is evaluated by its average win rate against multiple opponents.
  arXiv  Detail & Related papers  (2024-11-29T05:29:47Z)
• Improved Time-independent Hamiltonian Simulation [0.0]
  We describe a simple method for simulating time-independent Hamiltonian $H$ that could be decomposed as $H = sum_i=1m H_i$. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate $H_i$ in an alternative way.
  arXiv  Detail & Related papers  (2024-10-20T02:49:14Z)
• Unbiased random circuit compiler for time-dependent Hamiltonian simulation [8.694056486825318]
  Time-dependent Hamiltonian simulation is a critical task in quantum computing. We develop an unbiased random compiler for TDHS. We perform numerical simulations for a spin model under the interaction picture and the adiabatic ground state preparation for molecular systems.
  arXiv  Detail & Related papers  (2022-12-19T13:40:05Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Minimax Optimal Quantization of Linear Models: Information-Theoretic Limits and Efficient Algorithms [59.724977092582535]
  We consider the problem of quantizing a linear model learned from measurements. We derive an information-theoretic lower bound for the minimax risk under this setting. We show that our method and upper-bounds can be extended for two-layer ReLU neural networks.
  arXiv  Detail & Related papers  (2022-02-23T02:39:04Z)
• Average-case Speedup for Product Formulas [69.68937033275746]
  Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states. Our results open doors to the study of quantum algorithms in the average case.
  arXiv  Detail & Related papers  (2021-11-09T18:49:48Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.