[Paper Review] Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension

Nai-Hui Chia, Gilyén, András+4 more|arXiv (Cornell University)|Nov 12, 2018

Quantum Computing Algorithms and Architecture13 references40 citations

TL;DR

This paper presents classical sublinear-time algorithms inspired by the HHL quantum algorithm for solving low-rank linear systems. By leveraging sampling access to matrix and vector entries, the authors achieve query and time complexity of O(poly(k, κ, ∥A∥F, 1/ϵ) polylog(m, n)), enabling efficient sampling from and entry estimation of A⁻¹b with ϵ-accuracy, even when full matrix reconstruction is infeasible.

ABSTRACT

We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m imes n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}^m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(\mathrm{poly}(k, κ, \|A\|_F, 1/ε)\,\mathrm{polylog}(m, n))$, where $κ$ is the condition number of $A$ and $ε$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.

Motivation & Objective

• To develop classical algorithms that solve low-rank linear systems A x = b in sublinear time, avoiding full matrix access.
• To achieve efficient sampling from the solution vector x = A⁻¹b and entry-wise estimation of x(i) with high precision.
• To remove the need for full data access by relying on sampling oracles for A and b, enabling scalability to large-scale problems.
• To demonstrate that classical algorithms can match the logarithmic time complexity of quantum algorithms for linear systems, under realistic sampling assumptions.
• To extend the dequantization framework to linear systems, building on techniques from low-rank approximation and quantum-inspired sampling.

Proposed method

• Leverages sampling oracles: ability to sample row indices of A proportionally to row norms, entries within rows proportionally to absolute values, and entries of b proportionally to their magnitudes.
• Employs low-rank approximation via random submatrix sampling (inspired by Frieze et al.) to construct a compact representation of A.
• Uses a two-stage approach: first estimate the pseudo-inverse of A using a sampled submatrix, then apply it to b via efficient vector-matrix multiplication estimation.
• Introduces a novel succinct representation of the approximate inverse based on the sampled submatrix, enabling efficient computation of A⁻¹b entries and sampling.
• Applies error-bounded estimation techniques for inner products (e.g., x†Mx) using sampling access to vectors, extending Tang’s framework.
• Employs singular value decomposition on a small submatrix to approximate the dominant singular subspace, then inverts the singular values to estimate A⁻¹b.

Experimental results

Research questions

• RQ1 Can classical algorithms achieve sublinear time complexity for solving low-rank linear systems, matching the logarithmic runtime of the HHL quantum algorithm?
• RQ2 Under what sampling assumptions on A and b can one efficiently sample from A⁻¹b or estimate its entries without reading the full matrix?
• RQ3 Can the dequantization framework be extended beyond recommendation systems to general linear systems with low-rank structure?
• RQ4 How does the condition number κ and Frobenius norm ∥A∥F affect the query and time complexity of classical solvers for low-rank systems?
• RQ5 What is the trade-off between approximation error ϵ and computational cost in classical solvers under sampling access?

Key findings

• The proposed algorithms achieve O(poly(k, κ, ∥A∥F, 1/ϵ) polylog(m, n)) query and time complexity, enabling sublinear computation for low-rank linear systems.
• The sampling algorithm produces a distribution ϵ-close to DA⁻¹b in total variation distance with success probability 1−δ.
• The query algorithm estimates any entry (A⁻¹b)(i) with additive error ϵ and success probability 1−δ, using the same complexity bound.
• When A is positive semidefinite, the sampling assumption on b can be removed, allowing direct access to b via its entries.
• The algorithms are robust to approximation errors: by rescaling error parameters, the total error is bounded by ϵ, with explicit dependence on k, κ, ∥A∥F, and ∥b∥.
• The framework supports efficient estimation of quadratic forms like x†Mx using the same sampling primitives, extending its applicability to broader machine learning tasks.