量子贝叶斯推理：一种探索（英）

Quantum Bayesianinference: an exploration by Jon Frost, Carlos Madeira, Yash Rastogi and HaraldUhlig Monetary and Economic Department April2026 JEL classification: C11, C20, C30, C50, C60 Keywords:Quantum computing,Bayesian estimator,Bayesian inference, Markov chain Monte Carlo (MCMC) BISWorking Papers are written by members of the Monetary and EconomicDepartment of the Bank for International Settlements, and from time to time by othereconomists, and are published by the Bank. The papers are on subjects of topical This publication is available on the BIS website (www.bis.org). ©Bank for International Settlements 2026. All rights reserved. Brief excerpts may be ISSN 1020-0959 (print)ISSN 1682-7678 (online) Quantum Bayesian Inference: An Exploration Jon Frost, Carlos Madeira, Yash Rastogi and Harald Uhlig∗ First draft: July 21, 2025This revision: February 20, 2026 Abstract This paper introduces a framework for performing Bayesian infer-ence using quantum computation. It presents a proof-of-concept quan-tum algorithm that performs posterior sampling. We provide an acces-sible introduction to quantum computation for economists and a prac-tical demonstration of quantum-based posterior sampling for Bayesianestimation.Our key contribution is the preparation of a quantum Keywords: Quantum computing; Bayesian estimator; Bayesianinference; Markov chain Monte Carlo (MCMC) algorithms; Gibbs JEL codes: C11; C20; C30; C50; C60 1Introduction This paper explores the potential of quantum computation as a framework forBayesian inference. Quantum bits (qubits) are probabilistic objects, makingquantum computation conceptually aligned with Bayesian inference, where belief updates are driven by probabilistic rules.This conceptual parallelmotivates our key idea:encoding a discretized posterior distribution over In practice, quantum computation is a field largely driven forward byengineers and computer scientists, while statisticians and econometriciansstudy Bayesian inference.Bridging this disciplinary divide requires trans-lating core ideas across fields. To that end, we provide an accessible intro-duction to quantum computation tailored to economists, alongside a concisereview of Bayesian inference. We then present a simple quantum workflow Despite the early excitement surrounding quantum computation, the fieldremains in its infancy when it comes to solving computationally intense prob-lems at scale. Progress has been constrained by significant engineering chal- due to the current limitations in quantum hardware.1For example, numberfactorization using Shor’s algorithm so far has been limited to small numbers,since quantum computing remains noisy and only operates at intermediatescale (Mart´ın-L´opez et al.2012 [16]).2 Only recently has a quantum com- Some critics have questioned whether quantum advantage will ever beachieved at scale due to the computational requirements for noise-reductionas exercises scale up (Kalai 2020 [13]).Even optimistic assessments sug- Despite these diverging views, however, some quantum computing devel-opers believe that significant advances will be made for the study of molec-ular reactions, materials research and maybe for some managerial processessuch as logistics optimization (Brooks 2023 [4]). Some teams are exploringnewer quantum computing methods and hardware, including the combina- (Acharya et al.2024 [1], Auer et al.2024 [2], McMahon et al.2024 [17],Ghysels et al.2025 [9] and Ghysels and Morgan 2025 [10]).Recent workhas also shown advances in terms of applying quantum annealers to dynamicprogramming problems, achieving order-of-magnitude speedups relative tobenchmarks in solving real business cycle models (Fernandez and Hull 2023 Nonetheless, now is an opportune moment to explore the theoretical ca-pabilities of quantum computing, particularly in domains like Bayesian in-ference where probabilistic reasoning is central.Our aim is to contribute 2A brief introduction to quantum comput- The purpose of this section is to provide a brief introduction to quantumcomputing, including widely used concepts, notations, and ideas.We fo- The basis for all classical computation is a bit, which can take one of thetwo values 0 or 1. In quantum computation, the Dirac or bra-ket notation is often used. This entails writing the value 0 as|0⟩and the value 1 as|1⟩or, al-ternatively, as the two-dimensional vectorsh10i′andh01i′. As a firstpass, it might be helpful to think of a qubit as assigning a probabilitypto|0⟩and the probability 1−pto|1⟩, with either outcome realized with these prob-abilities, once the qubit is observed or read out.3 Matters are slightly morecomplicated, however. The probabilitiespand 1−pare actually parameter-ized by two complex-valued numbersαandβwithp=|α|2and 1−p=|β|2.This, in turn, leads to parameterizing a qubit withHopf coordinates[ν, ϕ, δ]′withν, ϕ, δ∈[0,2π] andα=eiδcos(ν/2), β=ei(δ+ϕ)sin(ν/2). Theglobal phaseδ∈[0,2π] has no physically observable consequen