Efficient Lattice Hamiltonian Encoding for the Shortest Vector Problem

TL;DR

This work targets the shortest-vector problem on structured lattices by introducing a quantum encoding that leverages cyclic and nega-cyclic symmetries to compress the problem into reduced Hamiltonians. By expressing the SVP-length as $|\vec{b}|^2=\sum_{ij} G_{ij} n_i n_j$ and reformulating it as a Hamiltonian with integer spectra, the authors construct kernel-restricted encodings via a constraints matrix $A$, yielding $\hat{H}_r=\sum_{ij} F_{ij} \hat{N}_i \hat{N}_j$ with $F=A^{\dagger} G A$. This approach reduces qubit counts and circuit depth while maintaining or improving the likelihood of locating short vectors, as demonstrated by VQE benchmarks on 200 nega-cyclic lattices where reduced models often outperform full representations. The results suggest practical quantum-resource gains for SVP-oriented lattice cryptography and motivate hybrid quantum-classical strategies for scalable, symmetry-aware lattice reduction on near-term devices.

Abstract

The advent of quantum computing necessitates the transition of worldwide cryptosystems to post-quantum cryptography (PQC), which is founded upon the problem of finding short vectors in high-dimensional structured lattices. It is assumed that the structure of these lattices cannot be exploited by quantum or classical algorithms attempting to find short vectors. In this work, we focus on the structure of the lattices used in PQC protocols - nega-cyclic (and cyclic)lattices - and provide a quantum algorithmic framework that efficiently encodes the structured lattices into Hamiltonians by exploiting their underlying symmetries. The efficient encoding substantially reduces the dimension of the corresponding Hilbert space by limiting it to a relevant subspace where short vectors are likely to be found - leading to significant savings in quantum resources (e.g. qubit count and circuit depth) required to implement a quantum algorithm for finding short vectors. We analytically prove the efficient encoding procedure and benchmark the proposed framework using the variational quantum eigensolver, demonstrating improved results with reduced quantum resources.

Figures (4)

• Figure 1: Schematic solutions of Eq. \ref{['eq:kernelcyclic']} for the case of $N=3$ on the left and for $N=15$ on the right. On the left, a unit circle with 3 equally distributed phases across it is depicted. All three phases are marked with a circle, meaning there is only one solution. All three circles have the same color (red) meaning their corresponding integer coefficients must be equal. On the right, next to each phase there are two shapes - circles and triangles, meaning two sets of solutions are depicted. For each solution set separately, all integer coefficients corresponding to the phases marked with shapes with the same color must be equal.
• Figure 2: A schematic solution for Eq. \ref{['eq:kernel,nega-cyclic,q=0']} with $N=3$. Three phases are equally distributed in the bottom half of the unit circle. Next to each phase, a red circle is drawn, indicating that the integer coefficient associated with the phases must be equal in their absolute value. The yellow cross within the circle associated with the phase indexed $1$ indicates the negation of corresponding coefficient, i.e., $n_1 \rightarrow -n_1$, and the addition of $\pi$ radians to the corresponding phase. The flip of the phase, which is indicated by the dashed black line, results in the same distribution of phases across the entire unit circle as depicted in Fig. \ref{['fig:cyclic symmetries']} for the cyclic case.
• Figure 3: One layer of the variational quantum circuit for circuits with 6 qubits. The layer contains applications of the $RY$ and $RZ$ single qubit gates on all qubits, followed by circular ladder of entangling $CNOT$ gates.
• Figure 4: Histogram of the ratio between the energy values of the outputs of the VQE algorithms for the reduced problem Hamiltonians and the full problem Hamiltonians. Two hundred lattices are examined. Bars left to the dashed black line indicate lattices for which the VQE for the reduced problem Hamiltonian results in a lower excited-state than the VQE for the full problem Hamiltonian. This is the case for 130 out of the 200 lattices.

Theorems & Definitions (22)

• Theorem 1
• proof
• Lemma 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 2