Towards identifying possible fault-tolerant advantage of quantum linear system algorithms in terms of space, time and energy

TL;DR

This paper estimates the resource requirements for fault-tolerant quantum algorithms solving linear systems, identifying potential quantum advantages over classical methods at large problem sizes and highlighting the parameters influencing quantum advantage.

Contribution

It provides a detailed resource estimation for fault-tolerant quantum linear system algorithms, mapping quantum-classical boundaries and conditions for potential advantages.

Findings

Quantum advantage possible at problem sizes N ≈ 2^{33} to 2^{48}.

Requires around 10^5 physical qubits and 10^{12} to 10^{13} Joules.

Quantum advantage depends on parameters like condition number, sparsity, and precision.

Abstract

Quantum computing, a prominent non-Von Neumann paradigm beyond Moore's law, can offer superpolynomial speedups for certain problems. Yet its advantages in efficiency for tasks like machine learning remain under investigation, and quantum noise complicates resource estimations and classical comparisons. We provide a detailed estimation of space, time, and energy resources for fault-tolerant superconducting devices running the Harrow-Hassidim-Lloyd (HHL) algorithm, a quantum linear system solver relevant to linear algebra and machine learning. Excluding memory and data transfer, possible quantum advantages over the classical conjugate gradient method could emerge at $N \approx 2^{33} \sim 2^{48}$ or even lower, requiring ${O}(10^5)$ physical qubits, ${O}(10^{12}\sim10^{13})$ Joules, and ${O}(10^6)$ seconds under surface code fault-tolerance with three types of magic state distillation (15-1, 116-12, 225-1). Key parameters include condition number, sparsity, and precision $\kappa, s\approx{O}(10\sim100)$, $\epsilon\sim0.01$, and physical error $10^{-5}$. Our resource estimator adjusts $N, \kappa, s, \epsilon$, providing a map of quantum-classical boundaries and revealing where a practical quantum advantage may arise. Our work quantitatively determine how advanced a fault-tolerant quantum computer should be to achieve possible, significant benefits on problems related to real-world.