A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm
Junhyung Lyle Kim, Nai-Hui Chia, Anastasios Kyrillidis
TL;DR
This paper introduces a quantum algorithm for solving linear systems that leverages the proximal point algorithm to improve conditioning and control over runtime and accuracy, marking a novel iterative approach in quantum linear system solving.
Contribution
It presents the first iterative quantum algorithm for QLSP using PPA, enabling better conditioning and tunable parameters for runtime and error trade-offs.
Findings
Mitigates condition number dependence in quantum linear system algorithms.
Provides a tunable parameter for balancing runtime and approximation error.
First to incorporate proximal point algorithm principles into quantum linear system solving.
Abstract
Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but the advantage is bottlenecked by condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing \texttt{QLSP\_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size , the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that…
Peer Reviews
Decision·Submitted to ICLR 2025
This paper introduces a novel framework that enhances the dependence on the condition number $\kappa$ in quantum algorithms, while incorporating additional problem-dependent parameters. In the worst-case scenario, it achieves a significant constant-factor improvement over the existing state-of-the-art algorithm. The framework also includes a tunable parameter $\eta$, allowing users to adjust the balance between runtime and approximation error.
1. The meta algorithm itself is relatively simple, and the techniques used to analyze the problem is not extremely complicated. 2. The algorithm introduces additional problem-specific parameters, such as, $\Psi$ and $d\coloneqq ||x_0-x^*||$. Moreover, the paper did not provide a very thorough discussion on these parameters.
- This paper exploits the proximal point algorithm to reformulate the original linear system problem as an approximate optimization problem, where the new problem is parametrized by the "step size" $\eta$. This new parameter provides a continuous interpolation from the original problem (for $\eta = +\infty$) to a pre-conditioned problem (for small $\eta$). This is an interesting point of view and also the first proposal to combine the proximal point algorithm with a quantum linear system solver
- My main concern is that the proximal point algorithm proposed in this paper can have a **single** iteration. This feature makes this algorithm less useful in practice. The main difficulty is that the state preparation oracle (see Definition 2) is hard (or maybe impossible) to construct for subsequent steps. - While it is discussed that a multi-step PPA can be realized by implementing different powers of the modified matrix, it is not clear why this would lead to asymptotic speedup because inv
The proposed approach is easy to plug in any quantum linear systems solver. Improving the conditional number is one of the most important tasks in QLSA.
As demonstrated in Figure 1, kappa^hat is less than kappa/2. According to lemma 1, parameter eta would be less than kappa-2 (or simply kappa). According to the eta choice in theorem 3, if we simply use eta less than kappa, then we have epsilon_2 larger than d. Recall that epsilon_2 is the accuracy for x_{t+1}-x^* as in equation 14, which is the accuracy people care about in the classical setting. This means, if someone wants to eventually read out the quantum solution, ignoring the error from th
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