Iterative Partition Search Variational Quantum Algorithm for Solving Shortest Vector Problem

TL;DR

This work tackles solving the $SVP$ on near-term quantum devices by introducing Iterative Partition Search Algorithm (IPSA), which merges a $1$-tailed search strategy with a dynamic, stack-driven iterative process to ensure valid lattice-basis updates and to use smaller, adaptive search spaces. IPSA significantly outperforms the previous PSA and IQOAP approaches, achieving higher success rates and better solution quality while markedly reducing circuit depth and two-qubit gate counts on real superconducting hardware. Hardware experiments on the Baihua processor (via Quafu) with $n=4$ demonstrate a $≈14$-fold increase in success rate over PSA and an ≈$82.7 ext{%}$ depth reduction and ≈$91 ext{%}$ fewer CZ gates than IQOAP, with IPSA also delivering ≈$2.5 imes$ higher success rate than IQOAP. Numerical simulations corroborate hardware results and extend the analysis to larger or harder instances, confirming IPSA’s practical advantages and suggesting broader applicability to other lattice-based or post-quantum tasks.

Abstract

The Partition Search Algorithm (PSA) and Iterative Quantum Optimization with an Adaptive Problem (IQOAP) are leading variational quantum algorithms for solving Shortest Vector Problem (SVP). However, each has limitations that restrict its practical impact. IQOAP suffers from ineffective iterations that fail to update the lattice basis, whereas PSA's static partitioning leads to oversized search spaces. In this work, we propose the Iterative Partition Search Algorithm (IPSA), which systematically addresses these drawbacks by integrating a "1-tailed search spaces" with a dynamic, stack-managed iterative process. Specifically, the "1-tailed" strategy ensures that every successful execution yields an effective lattice basis update, thereby eliminating the ineffective iterations associated with IQOAP. Concurrently, the dynamic iterative process reduces the required qubit count, thereby avoiding the limitation of an oversized search space inherent to PSA. We validate IPSA on the Baihua superconducting quantum processor via the Quafu platform. Small-scale real hardware experiments demonstrate that, compared to PSA, IPSA achieves a 14-fold increase in success rate at a cost of less than double the total circuit depth. Conversely, compared to IQOAP, IPSA reduces the total circuit depth by 82.7% while achieving approximately 2.5 times its success rate. Furthermore, we also conduct numerical simulations whose results are in good agreement with the experimental findings and extend our analysis.

Figures (7)

• Figure 1: Decomposition of logical gates required by the QAOA ansatz. An $\text{R}_{\text{ZZ}}(\theta)$ rotation gate is decomposed into a sequence involving two CZ gates. A SWAP gate is decomposed into three CZ gates.
• Figure 2: Visualization of the compilation overhead for different ansatz structures. The logical circuit diagrams are shown alongside their corresponding physical circuits after transpilation for a linear connectivity topology. The all-to-all connectivity requirement of the QAOA ansatz, which is optimized by Qiskit's transpiler, still results in a deep physical circuit. The HEA, designed to match the linear topology, undergoes minimal change during transpilation.
• Figure 3: Comparative analysis of IPSA (blue), PSA (purple), and IQOAP (red) on the 50 SVP instances at $n=4$, executed on the Baihua superconducting quantum processor. The panels show (a) Success Rate (SR), (b) Approximation Ratio (AR), (c) Total Circuit Depth ($D_{\text{total}}$), and (d) Total CZ Count ($C_{\text{total}}$). For the data points, the error bars indicate the interquartile range (25th to 75th percentile), and the markers denote the median values.
• Figure 4: Comparative analysis of IPSA-HEA (blue) and IPSA-QAOA (orange) on the set of 600 SVP instances for dimensions $n \in \{4, 5, 6\}$. The four panels show (a) Success Rate (SR), (b) Approximation Ratio (AR), (c) Total Circuit Depth ($D_{\text{total}}$), and (d) Total CZ Count ($C_{\text{total}}$). The y-axis values for panels (c) and (d) are presented in units of $10^6$. In all plots, the shaded areas represent the interquartile range (25th to 75th percentiles), and the cross markers indicate the median values.
• Figure 5: The qubit connectivity topology of the Baihua superconducting quantum processor. Blue circles indicate available qubits, while light gray circles represent unavailable qubits. Blue arrows denote available connections for two-qubit gates.

Theorems & Definitions (1)

• Theorem 1