Quantum Computing Inspired Approach for Self-Avoiding Walk (SAWs): 2D lattice and 3D lattice SAWs for single chain enumeration

TL;DR

This work addresses the computationally hard problem of exactly enumerating self-avoiding walks on 2D and 3D lattices by introducing a quantum amplitude estimation (QAE) framework. The method prepares uniform superpositions of possible walks, marks valid SAWs with a quantum oracle, and uses QAE with a $QFT^{\dagger}$-based diffusion step to count valid configurations, achieving a quadratic speedup over classical approaches. Empirically, 2D SAWs are enumerated up to $N=71$ in about $26.92$ minutes (versus ≈231 hours classically), and 3D SAWs up to $N=40$ in about $13.05$ minutes (versus ≈250 hours for classical up to $N=36$), with zero-noise extrapolation and hybrid simulation strategies mitigating hardware constraints. These results demonstrate the practical potential of quantum-inspired techniques for lattice-path enumeration and polymer-model problems, while outlining current limitations and avenues for scaling with future quantum hardware.

Abstract

We investigate the application of quantum computing algorithms to enhance the efficiency of enumerating self-avoiding walks (SAWs), utilizing quantum properties such as superposition and interference. A Quantum Amplitude Estimation (QAE)-based algorithm is developed to enumerate SAWs on both 2D and 3D lattices. In case of 2D square lattice, SAWs up to N=71 steps are successfully enumerated within 26.9 minutes - significantly improving upon the classical algorithm, which required approximately 231 hours(Jensen et al., 2012, J. Phys. A: Math. Theor. 45, 115202). The algorithm is further extended to 3D cubic lattices, where SAWs up to N=40 steps are enumerated in 13.06 minutes, compared to the classical result of N=36 in 250 hours (Schram et al., 2011, J. Stat. Mech. P06019). These results demonstrate a substantial reduction in computational time, highlighting the potential of quantum computing for combinatorial enumeration problems.

Figures (8)

• Figure 1: Quantum circuit for state preparation of $N=1$ step SAW, with 2 qubits over $4^1$ = 4 configurations. Red boxes $(H)$ represent Hadamard Gates for initializing superposition. Grey boxes represent the Z-basis for the measurement at the end of the computation.
• Figure 2: Quantum circuit for state preparation of $N=1$ step SAW, with 3 qubits over $6^1$ = 6 configurations. Red boxes $(H)$ represent Hadamard Gates for initializing superposition. Grey boxes represent the Z-basis for the measurement at the end of the computation.
• Figure 3: Quantum circuit for the fundamental function of $QFT^{\dagger}$ through Hadamard Gate $(H)$, Controlled Phase Shift Gate , and Swap Gate
• Figure 4: Schematic Flowchart for QAE Implementation by applying superposition state with Grover's Iteration encoded with $QFT^\dagger$ that is further applied in QAE for the enumeration of N-steps.
• Figure 5: Quantum Circuit demonstration for N=2 in 2D Lattice. Red boxes $(H)$ represent Hadamard Gates for initializing superposition. Blues $(+)$ indicates NOT $X$ gate for phase flip. Red $(+)$ symbols denote the Toffoli $(CCX)$ gate for controlled condition. The yellow line represents the Controlled-Z $(CZ)$ gate, which introduces a phase shift. Grey boxes represent the Z-basis for the measurement at the end of the computation.