[Paper Review] Quantum Bayesian Neural Networks

Noah Berner, Vincent Fortuin+1 more|arXiv (Cornell University)|Jul 20, 2021

Quantum Computing Algorithms and Architecture36 references4 citations

TL;DR

This paper proposes a quantum algorithm for Bayesian neural network (BNN) inference that replaces classical inner product computations with a modified quantum inner product estimation (IPE) algorithm, eliminating the need for repeated evaluations. The method achieves accurate posterior approximation with just one quantum computation per sample, demonstrating strong empirical performance even on small-scale quantum simulations, thus realizing potential quantum speedups in Bayesian learning despite noise.

ABSTRACT

Quantum machine learning promises great speedups over classical algorithms, but it often requires repeated computations to achieve a desired level of accuracy for its point estimates. Bayesian learning focuses more on sampling from posterior distributions than on point estimation, thus it might be more forgiving in the face of additional quantum noise. We propose a quantum algorithm for Bayesian neural network inference, drawing on recent advances in quantum deep learning, and simulate its empirical performance on several tasks. We find that already for small numbers of qubits, our algorithm approximates the true posterior well, while it does not require any repeated computations and thus fully realizes the quantum speedups.

Motivation & Objective

• To explore whether Bayesian neural networks (BNNs) can benefit from quantum speedups despite noisy intermediate-scale quantum (NISQ) hardware.
• To address the challenge of quantum noise in inner product estimation by leveraging the inherent tolerance of BNNs to noisy parameter estimates.
• To develop and simulate a quantum algorithm that replaces classical inner product computation with a single-shot quantum IPE, avoiding repeated evaluations required in classical approaches.
• To empirically validate that quantum BNN inference with one quantum computation per sample approximates the true posterior effectively, even with limited qubits.

Proposed method

• Adapts the quantum inner product estimation (IPE) algorithm by Kerenidis et al. (2018) to eliminate median evaluation, reducing runtime by a factor of log(1/Δ) compared to standard IPE.
• Uses amplitude estimation and phase estimation subroutines to estimate inner products between weight and activation vectors in a quantum superposition, with output mapped to a noisy but usable estimate.
• Employs a quantum random access memory (qRAM) model to load input vectors into quantum states with polylogarithmic time complexity, enabling efficient state preparation.
• Modifies the IPE to accept a single estimate per sample, relying on the posterior sampling nature of BNNs to tolerate inherent quantum noise.
• Applies the quantum IPE within a Markov Chain Monte Carlo (MCMC) framework such as SGLD, HMC, or NUTS, using quantum-estimated gradients for parameter updates.
• Simulates the algorithm on linear regression and binary classification tasks using varying numbers of qubits (n=5 to n=13), comparing classical and quantum inference and prediction performance.

Experimental results

Research questions

• RQ1 Can quantum inner product estimation with a single evaluation per sample produce a sufficiently accurate posterior approximation in Bayesian neural networks?
• RQ2 How does the performance of quantum BNN inference scale with increasing qubit count, despite inherent quantum noise in the IPE subroutine?
• RQ3 Does the inherent noise tolerance of Bayesian inference allow for meaningful quantum speedups when using noisy, single-shot quantum computations?
• RQ4 How do different initialization strategies (full-rank vs. low-rank) affect the convergence and accuracy of quantum BNN inference?
• RQ5 To what extent can quantum BNN inference approximate the true posterior predictive distribution compared to classical baselines?

Key findings

• The proposed quantum IPE algorithm achieves accurate posterior approximation with just one quantum computation per sample, eliminating the need for repeated evaluations required in classical quantum algorithms.
• For small numbers of qubits (n=5 to n=13), the quantum BNN inference closely approximates the true posterior, as evidenced by low posterior predictive standard deviation and high prediction accuracy.
• The simulation results show that increasing qubit count (n) leads to a consistent improvement in posterior approximation quality, indicating scalability of the method.
• The method performs well under non-isotropic and periodic noise patterns inherent in the IPE phase estimation subroutine, suggesting robustness to common NISQ-era noise.
• Full-rank and low-rank weight initializations both yield stable performance, with no significant degradation in quantum inference quality, indicating compatibility with standard BNN training practices.
• The quantum inference pipeline (QICP) outperforms classical inference (CICP) and classical quantum inference (CIQP) in terms of posterior predictive accuracy, especially as qubit count increases.