Quadratic Speed-up in Infinite Variance Quantum Monte Carlo

• URL: http://arxiv.org/abs/2401.07497v2
• Date: Thu, 7 Mar 2024 01:46:13 GMT
• Title: Quadratic Speed-up in Infinite Variance Quantum Monte Carlo
• Authors: Jose Blanchet, Mario Szegedy, Guanyang Wang
• Abstract summary: We give an extension of Montanaro's arXiv/archive:1504.06987 quantum Monte Carlo method. It is tailored for computing expected values of random variables that exhibit infinite variance.
• Score: 1.2891210250935148
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this study, we give an extension of Montanaro's arXiv/archive:1504.06987 quantum Monte Carlo method, tailored for computing expected values of random variables that exhibit infinite variance. This addresses a challenge in analyzing heavy-tailed distributions, which are commonly encountered in various scientific and engineering fields. Our quantum algorithm efficiently estimates means for variables with a finite $(1+\delta)^{\text{th}}$ moment, where $\delta$ lies between 0 and 1. It provides a quadratic speedup over the classical Monte Carlo method in both the accuracy parameter $\epsilon$ and the specified moment of the distribution. We establish both classical and quantum lower bounds, showcasing the near-optimal efficiency of our algorithm among quantum methods. Our work focuses not on creating new algorithms, but on analyzing the execution of existing algorithms with available additional information about the random variable. Additionally, we categorize these scenarios and demonstrate a hierarchy in the types of supplementary information that can be provided.

Related papers

• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Quantum Algorithms for the Pathwise Lasso [1.8058773918890538]
  We present a novel quantum high-dimensional linear regression algorithm based on the classical LARS (Least Angle Regression) pathwise algorithm. Our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. We prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm is robust to errors.
  arXiv  Detail & Related papers  (2023-12-21T18:57:54Z)
• Generalized Hybrid Search and Applications to Blockchain and Hash Function Security [50.16790546184646]
  We first examine the hardness of solving various search problems by hybrid quantum-classical strategies. We then construct a hybrid quantum-classical search algorithm and analyze its success probability.
  arXiv  Detail & Related papers  (2023-11-07T04:59:02Z)
• Quantum Adversarial Learning in Emulation of Monte-Carlo Methods for Max-cut Approximation: QAOA is not optimal [0.0]
  We apply notions of emulation to Variational Quantum Annealing and the Quantum Approximate Algorithm Optimization (QAOA) Our Variational Quantum Annealing schedule is based on a novel parameterization that can be optimized in a similar gradient-free way as QAOA, using the same physical ingredients. In order to compare the performance of ans"atze types, we have developed statistical notions of Monte-Carlo methods.
  arXiv  Detail & Related papers  (2022-11-24T19:02:50Z)
• Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants [8.453228628258778]
  We develop quantum algorithms for sampling logconcave distributions and for estimating their normalizing constants. We exploit quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/epsilon1-o(1)$ quantum lower bound for estimating normalizing constants.
  arXiv  Detail & Related papers  (2022-10-12T19:10:43Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• 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)
• Quantum algorithm for stochastic optimal stopping problems with applications in finance [60.54699116238087]
  The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in optimal stopping theory. We propose a quantum LSM based on quantum access to a process, on quantum circuits for computing the optimal stopping times, and on quantum techniques for Monte Carlo.
  arXiv  Detail & Related papers  (2021-11-30T12:21:41Z)
• Quantum algorithms for approximate function loading [0.0]
  We introduce two approximate quantum-state preparation methods for the NISQ era inspired by the Grover-Rudolph algorithm. A variational algorithm capable of loading functions beyond the aforementioned smoothness conditions is proposed.
  arXiv  Detail & Related papers  (2021-11-15T17:36:13Z)
• Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance [1.128265591164748]
  We study quantum algorithms for differential equations (SDEs) We provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance.
  arXiv  Detail & Related papers  (2020-12-11T12:34:55Z)

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.