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Sparsity-dependent Complexity Lower Bound of Quantum Linear System Solvers

Hitomi Mori, Yuta Kikuchi, Marcello Benedetti, Matthias Rosenkranz

TL;DR

This work advances the understanding of quantum linear system solver complexity by establishing a sparsity-dependent lower bound $Ω(κ sqrt{s})$ for QLS under sparse-access. It also provides a detailed reduction-based derivation of the standard bound $Ω(κ log(1/ε))$ via a reduction from the unbounded-error PARITY problem, clarifying historical uncertainties around the Harrow–Kothari result. The constructions relate the QLS problem to Boolean-query tasks, yielding κ ≈ Θ(n) and s ≈ Θ(m) in the reductions, and show that current techniques separate the ε, κ, and s dependencies. The paper highlights open questions toward a joint lower bound in all three parameters and outlines potential directions for achieving a comprehensive characterization of QLS complexity.

Abstract

Quantum linear system (QLS) solvers are a fundamental class of quantum algorithms used in many potential quantum computing applications, including machine learning and solving differential equations. The performance of quantum algorithms is often measured by their query complexity, which quantifies the number of oracle calls required to access the input. The main parameters determining the complexity of QLS solvers are the condition number $κ$ and sparsity $s$ of the linear system, and the target error $ε$. To date, the best known query-complexity lower bound is $Ω(κ\log(1/ε))$, which establishes the optimality of the most recent QLS solvers. The original proof of this lower bound is attributed to Harrow and Kothari, but their result is unpublished. Furthermore, when discussing a more general lower bound including the sparsity $s$ of the linear system, it has become folklore that it should read as $Ω( κ\sqrt{s}\log(1/ε))$. In this work, we establish the rigorous lower bound capturing the sparsity dependence of QLS. We prove the lower bound of $Ω(κ\sqrt{s})$ for any quantum algorithm that solves QLS with constant error. While the dependence on all parameters $κ,s,ε$ remains an open problem, our result provides a crucial stepping stone toward the complete characterization of QLS complexity.

Sparsity-dependent Complexity Lower Bound of Quantum Linear System Solvers

TL;DR

This work advances the understanding of quantum linear system solver complexity by establishing a sparsity-dependent lower bound for QLS under sparse-access. It also provides a detailed reduction-based derivation of the standard bound via a reduction from the unbounded-error PARITY problem, clarifying historical uncertainties around the Harrow–Kothari result. The constructions relate the QLS problem to Boolean-query tasks, yielding κ ≈ Θ(n) and s ≈ Θ(m) in the reductions, and show that current techniques separate the ε, κ, and s dependencies. The paper highlights open questions toward a joint lower bound in all three parameters and outlines potential directions for achieving a comprehensive characterization of QLS complexity.

Abstract

Quantum linear system (QLS) solvers are a fundamental class of quantum algorithms used in many potential quantum computing applications, including machine learning and solving differential equations. The performance of quantum algorithms is often measured by their query complexity, which quantifies the number of oracle calls required to access the input. The main parameters determining the complexity of QLS solvers are the condition number and sparsity of the linear system, and the target error . To date, the best known query-complexity lower bound is , which establishes the optimality of the most recent QLS solvers. The original proof of this lower bound is attributed to Harrow and Kothari, but their result is unpublished. Furthermore, when discussing a more general lower bound including the sparsity of the linear system, it has become folklore that it should read as . In this work, we establish the rigorous lower bound capturing the sparsity dependence of QLS. We prove the lower bound of for any quantum algorithm that solves QLS with constant error. While the dependence on all parameters remains an open problem, our result provides a crucial stepping stone toward the complete characterization of QLS complexity.
Paper Structure (7 sections, 6 theorems, 33 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower bound in condition number and target error
  4. Lower bound in condition number and sparsity
  5. Discussion
  6. Alternative proof of Lemma \ref{['lem:parity_or_unbounded']}
  7. Construction of the oracle

Key Result

Lemma 1

The bounded-error quantum query complexity $\mathsf{Q}$ of $\textsc{PARITY}\IfNoValueTF{n}{}{_{\,\mathclap{n}\,}}$ is $\mathsf{Q}( \textsc{PARITY}\IfNoValueTF{n}{}{_{\,\mathclap{n}\,}}\xspace) = \Theta(n)$.

Figures (1)

  • Figure 1: Quantum circuit for implementing $\mathcal{P}_B^{\rm val}$\ref{['eq:P_B^val']}. The registers with subscript "scr" serve as scratch registers that store intermediate results and are uncomputed at the end. See the main text for the definition of each box representing a quantum operation. In the circuit diagram, we omit the final write-out to $\ket{z}$ followed by uncomputation.

Theorems & Definitions (11)

  • Definition 1: Sparse-matrix oracles Berry_2015Berry_2012
  • Lemma 1: Tight bound of bounded-error $\textsc{PARITY}\IfNoValueTF{-NoValue-}{}{_{\,\mathclap{-NoValue-}\,}}$Farhi_1998
  • Lemma 2: Tight bound of unbounded-error $\textsc{PARITY}\IfNoValueTF{-NoValue-}{}{_{\,\mathclap{-NoValue-}\,}}$Farhi_1998
  • Lemma 3: Tight bound of bounded-error $\textsc{PARITY}\IfNoValueTF{-NoValue-}{}{_{\,\mathclap{-NoValue-}\,}}\xspace\,\circ\, \textsc{OR}\IfNoValueTF{-NoValue-}{}{_{-NoValue-}}\xspace$Farhi_1998Grover_1996reichardtReflectionsQuantumQuery2010
  • Lemma 4: Tight bound of unbounded-error $\textsc{PARITY}\IfNoValueTF{-NoValue-}{}{_{\,\mathclap{-NoValue-}\,}}\xspace\,\circ\, \textsc{OR}\IfNoValueTF{-NoValue-}{}{_{-NoValue-}}\xspace$
  • proof
  • Theorem 1: Lower bound of QLS with target error Costa_2023
  • proof
  • Theorem 2: Lower bound of QLS with sparsity
  • proof
  • ...and 1 more