Iterative Partition Search Variational Quantum Algorithm for Solving Shortest Vector Problem

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.

💡 Research Summary

The paper introduces a new variational quantum algorithm, the Iterative Partition Search Algorithm (IPSA), designed to solve the Shortest Vector Problem (SVP) on noisy intermediate‑scale quantum (NISQ) devices. Existing approaches—Partition Search Algorithm (PSA) and Iterative Quantum Optimization with an Adaptive Problem (IQOAP)—each suffer from critical drawbacks. PSA uses a static partitioning of the coefficient space, which requires a large number of qubits and creates an oversized search space that often traps the optimizer in poor local minima. IQOAP reduces qubit requirements by limiting the coefficient range, but its iterative scheme can produce “ineffective iterations”: a shorter vector may be found but discarded because it does not satisfy the lattice‑basis update condition, wasting computational resources.

IPSA addresses both issues through two complementary innovations: (1) 1‑tailed search spaces and (2) a stack‑managed dynamic iterative process.
The 1‑tailed search space Y_i is defined as Y_i = { (y₁,…,y_{i‑1}, 1, 0,…,0)ᵀ ∈ ℤⁿ }. By fixing the i‑th coefficient to 1, the algorithm eliminates the need to encode that coefficient into qubits, thereby reducing the total qubit count. More importantly, this constraint guarantees that any vector v found within Y_i can replace the i‑th basis vector b_i while preserving the lattice structure—a property formally proved in Theorem 1. Consequently, every successful VQA run yields a vector that is automatically eligible for a basis update, eliminating the ineffective iterations that plague IQOAP.

The second innovation is a stack‑based control flow. Initially, the basis vectors are sorted by norm and the partitions Y_n … Y₁ are pushed onto an empty stack. The algorithm repeatedly pops the top partition, solves the SVP for that partition using a hardware‑efficient ansatz (HEA), and, if a shorter vector is obtained, replaces the corresponding basis vector. After a replacement, all partitions from the updated index up to the smallest index are re‑pushed onto the stack, ensuring that the algorithm immediately refines the lattice with the newly improved basis before moving on to larger partitions. This dynamic, depth‑first style reduces the number of costly large‑partition optimizations and leads to a markedly lower overall circuit depth compared with the fixed‑partition PSA.

From a circuit‑design perspective, the authors deliberately avoid the Quantum Approximate Optimization Algorithm (QAOA) because its all‑to‑all entangling gates (RZZ) would require many SWAP and CZ gates on the limited‑connectivity superconducting processor. Instead, they employ a Hardware‑Efficient Ansatz (HEA) whose entangling blocks are aligned with the native topology, dramatically cutting the transpilation overhead. Table I shows that a 6‑qubit HEA implementation needs only 7 logical layers (5 entangling gates) versus 12 layers (15 entangling gates) for QAOA; after transpilation the HEA depth is roughly one‑third of the optimized QAOA depth.

Experimental validation was performed on the Baihua superconducting quantum processor accessed via the Quafu cloud platform. The authors benchmarked IPSA against PSA and IQOAP on small‑scale SVP instances (effectively 4‑dimensional lattices encoded with 6 qubits). Results indicate that, relative to PSA, IPSA achieves a 14‑fold increase in success probability while incurring less than double the circuit depth. Compared with IQOAP, IPSA reduces the total circuit depth by 82.7 % and improves the success probability by approximately 2.5×. Numerical simulations under ideal, noiseless conditions reproduce these trends, confirming that the observed advantages stem from algorithmic design rather than hardware quirks.

The paper acknowledges several limitations. The experimental evaluation is confined to low‑dimensional lattices; scalability to higher dimensions (n ≥ 10) remains untested. The 1‑tailed restriction, while guaranteeing basis‑update feasibility, may exclude some optimal vectors in certain lattice structures, suggesting a possible trade‑off between search‑space size and completeness. Future work is proposed to explore multi‑tailed partitions (fixing multiple coefficients), adaptive selection of which coefficient to fix, integration with error‑mitigation techniques, and extensive benchmarking on larger lattices and alternative quantum hardware.

In summary, IPSA combines a qubit‑saving 1‑tailed partitioning with a stack‑driven dynamic iteration to overcome the principal shortcomings of PSA and IQOAP. The algorithm delivers substantially higher success rates with markedly reduced circuit depths, making it a promising candidate for practical SVP solving on near‑term quantum devices and for assessing the security of post‑quantum cryptographic schemes against realistic quantum attacks.