Hamiltonian simulation with random inputs

• URL: http://arxiv.org/abs/2111.04773v1
• Date: Mon, 8 Nov 2021 19:08:42 GMT
• Title: Hamiltonian simulation with random inputs
• Authors: Qi Zhao, You Zhou, Alexander F. Shaw, Tongyang Li, and Andrew M. Childs
• Abstract summary: Theory of average-case performance of Hamiltonian simulation with random initial states. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models.
• Score: 74.82351543483588
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and $k$-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with $n$ spins, the error is quadratically reduced from $\mathcal O(n)$ in the worst case to $\mathcal O(\sqrt{n})$ on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.

Related papers

• Principal Trotter Observation Error with Truncated Commutators [0.0]
  Hamiltonian simulation is one of the most promising applications of quantum computers. In this work, we consider the simulation error under a fixed observable. For highly commuting observables, the simulation error indicated by this upper bound can be significantly compressed.
  arXiv  Detail & Related papers  (2024-08-07T16:44:21Z)
• Scaling Laws in Linear Regression: Compute, Parameters, and Data [86.48154162485712]
  We study the theory of scaling laws in an infinite dimensional linear regression setup. We show that the reducible part of the test error is $Theta(-(a-1) + N-(a-1)/a)$. Our theory is consistent with the empirical neural scaling laws and verified by numerical simulation.
  arXiv  Detail & Related papers  (2024-06-12T17:53:29Z)
• Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation [0.0]
  We show that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. We improve the additive observable error bounds to uniform-in-$h$ observable error bounds. This is, to our knowledge, the first uniform observable error bound for semiclassical Schr"odinger equation.
  arXiv  Detail & Related papers  (2022-08-16T21:34:49Z)
• The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion [84.66087478797475]
  Quantum error mitigation (QEM) is a class of promising techniques for reducing the computational error of variational quantum algorithms. We consider a practical channel inversion strategy based on Monte Carlo sampling, which introduces additional computational error. We show that when the computational error is small compared to the dynamic range of the error-free results, it scales with the square root of the number of gates.
  arXiv  Detail & Related papers  (2022-01-20T00:05:01Z)
• 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)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Time-dependent unbounded Hamiltonian simulation with vector norm scaling [2.973326951020451]
  The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all.
  arXiv  Detail & Related papers  (2020-12-24T05:01:58Z)
• Hamiltonian Simulation Algorithms for Near-Term Quantum Hardware [6.445605125467574]
  We develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model. We analyse the impact of these techniques under the standard error model. We derive analytic circuit identities for efficiently synthesising multi-qubit evolutions from two-qubit interactions.
  arXiv  Detail & Related papers  (2020-03-15T18:22:02Z)
• 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.