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

TL;DR

This work tackles identifying possible fault-tolerant quantum advantage for quantum linear system algorithms (QLSA) by performing a detailed, end-to-end resource estimation for HHL on superconducting hardware. It combines logical-level circuit analysis (HS1 bottleneck, Trotter and QFT error modeling) with a surface-code-based fault-tolerance resource estimator (distillation protocols, code distance) and an energy model rooted in cooling power to compare against classical CG. The authors derive explicit gate-count scales and demonstrate that a quantum advantage could emerge around matrix sizes of $N \approx 2^{33} \sim 2^{48}$, requiring ${O}(10^{5})$ physical qubits and significant energy, while noting the dependence on hardware assumptions and the omission of data-upload/download costs. The work highlights a practical pathway to quantify space-time-energy trade-offs for QLSA and outlines future directions for broader hardware contexts and more refined energy accounting.

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 $κ, s\approx{O}(10\sim100)$, $ε\sim0.01$, and physical error $10^{-5}$. Our resource estimator adjusts $N, κ, s, ε$, 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.

Figures (14)

• Figure 1: Comparison of Quantum Linear System Algorithm (QLSA) and Classical Conjugate Gradient Algorithm. In the plots, $N$ represents matrix size, $s$ is the sparsity, $\kappa$ is the condition number, $\epsilon$ is the precision tolerance in terms of vector-2 norm. (a) & (b): Heatmaps showing the ratio of classical runtime to quantum runtime and classical energy consumption to quantum energy consumption under different problem parameters. Blue regions indicate a possible quantum advantage. (c) & (d): Runtime and energy comparison between quantum and classical algorithms, assuming condition number and sparsity scale as $\log (N)$, and $\epsilon$ to be 0.01. (e): Runtime and qubit count scaling for the quantum algorithm over larger matrix sizes.
• Figure 2: Quantum circuit for one sparse Hamiltonian simulation.
• Figure 3: Quantum circuit for $e^{iFt}$.
• Figure 4: Controlled phase gate.
• Figure 5: Quantum circuit implementing the swap operator $\tilde{W}$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof