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Detailed Error Analysis of the HHL Algorithm

Xinbo Li Christopher Phillips

TL;DR

The paper provides a rigorous error analysis of the HHL quantum linear-system algorithm, focusing on how quantum phase estimation, eigenvalue inversion, and clock-register dynamics affect the final solution state. It derives explicit bounds showing the error between ideal and practical implementations scales as $O(\kappa/t_0)$ and demonstrates how post-selection on the flag register preserves these error orders under various conditioning scenarios. By analyzing the amplitude behavior $|\alpha_{k|j}|$ and proving Lipschitz continuity of the flag state $|h(\lambda)\rangle$, the work connects the Hamiltonian-simulation duration $t_0$ and clock-register size $T$ to the accuracy of the computed solution. The results correct and elaborate on prior supplementary material, providing practical guidance for choosing $t_0$ and $T$ to achieve targeted accuracy, and offering insights for developing improved HHL variants.

Abstract

We reiterate the contribution made by Harrow, Hassidim, and Llyod to the quantum matrix equation solver with the emphasis on the algorithm description and the error analysis derivation details. Moreover, the behavior of the amplitudes of the phase register on the completion of the Quantum Phase Estimation is studied. This study is beneficial for the comprehension of the choice of the phase register size and its interrelation with the Hamiltonian simulation duration in the algorithm setup phase.

Detailed Error Analysis of the HHL Algorithm

TL;DR

The paper provides a rigorous error analysis of the HHL quantum linear-system algorithm, focusing on how quantum phase estimation, eigenvalue inversion, and clock-register dynamics affect the final solution state. It derives explicit bounds showing the error between ideal and practical implementations scales as and demonstrates how post-selection on the flag register preserves these error orders under various conditioning scenarios. By analyzing the amplitude behavior and proving Lipschitz continuity of the flag state , the work connects the Hamiltonian-simulation duration and clock-register size to the accuracy of the computed solution. The results correct and elaborate on prior supplementary material, providing practical guidance for choosing and to achieve targeted accuracy, and offering insights for developing improved HHL variants.

Abstract

We reiterate the contribution made by Harrow, Hassidim, and Llyod to the quantum matrix equation solver with the emphasis on the algorithm description and the error analysis derivation details. Moreover, the behavior of the amplitudes of the phase register on the completion of the Quantum Phase Estimation is studied. This study is beneficial for the comprehension of the choice of the phase register size and its interrelation with the Hamiltonian simulation duration in the algorithm setup phase.
Paper Structure (16 sections, 3 theorems, 84 equations, 3 figures)

This paper contains 16 sections, 3 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algorithm Details
  3. The Behavior of the Amplitudes $\lvert\alpha_{k|j}\rvert$
  4. Error Analysis
  5. Proof of \ref{['eq: error bound, no post-selection']}: Error Bound in the Final State Without Post-Selection
  6. Proof of \ref{['eq: error bound, post selection to well and ill']}: Error Bound in the Solution State with Flag Register Post-Selected in $\{\ket{well}, \ket{ill}\}$
  7. Proof of \ref{['eq: error bound, post selection to well']}: Error Bound in the Solution State with Flag Register Post-Selected in $\{\ket{well}\}$
  8. Summary
  9. Conclusion
  10. The inner product between ideal and non-ideal final states
  11. The mapping $\lambda \to \ket{h(\lambda)}$ is $O(\kappa)$-Lipschitz.
  12. Model $\ket{h(\lambda)}$ as a Vector-Valued Function
  13. The Study for the Mapping $\lambda \to \ket{h(\lambda)}$
  14. Simplified expressions of $\alpha_{k|j}$ and $\lvert\alpha_{k|j}\rvert$
  15. The upper bound for $\lvert\alpha_{k|j}\rvert$ when $\lvert\delta\rvert > 2\pi$
  16. ...and 1 more sections

Key Result

Theorem 1

Figures (3)

  • Figure 1: The diagram illustration of the HHL circuit.
  • Figure 2: $\lvert\alpha\rvert$ versus $\delta$ for a) small eigenvalue $\frac{T-1}{T} \frac{1}{\kappa}$, b) moderate eigenvalue $\frac{T-1}{2T}$, and c) large eigenvalue $\frac{T-1}{T}$ with $t_0 = 2\pi T, T = \kappa + 1.$
  • Figure 3: Data similarly illustrated as in Figure \ref{['fig: 2. |alpha| versus delta for various eigenvalues, t0=2piT']} with $t_0 = \pi T, T = 2\kappa +1$.

Theorems & Definitions (3)

  • Theorem 1: The HHL Error Bound
  • Lemma 2: The continuity of the mapping $\lambda \to \ket{h(\lambda)}$
  • Lemma 3: Upper bound of $(\tilde{f}_k - f_j)^2 + (\tilde{g}_k-g_j)^2$