[Paper Review] A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding

Mark Fingerhuth, Tomáš Babej+1 more|arXiv (Cornell University)|Oct 31, 2018

Quantum Computing Algorithms and Architecture24 references46 citations

TL;DR

This paper presents a variational quantum algorithm (QAOA) for lattice protein folding on universal gate-based quantum computers, introducing a one-hot encoded turn representation and hard/soft constraint separation via tailored mixer Hamiltonians. It analyzes planar and cubic lattice encodings and compares multiple mixer designs for improved ground-state sampling.

ABSTRACT

Gate-based universal quantum computers form a rapidly evolving field of quantum computing hardware technology. In previous work, we presented a quantum algorithm for lattice protein folding on a cubic lattice, tailored for quantum annealers. In this paper, we introduce a novel approach for solving the lattice protein folding problem on universal gate-based quantum computing architectures. Lattice protein models are coarse-grained representations of proteins that have been used extensively over the past thirty years to examine the principles of protein folding and design.These models can be used to explore a vast number of possible protein conformations and to infer structural properties of more complex atomistic protein structures. We formulate the problem as a quantum alternating operator ansatz, a member of the wider class of variational quantum/classical hybrid algorithms. To increase the probability of sampling the ground state, we propose splitting the optimization problem into hard and soft constraints. This enables us to use a previously under-utilised component of the variational algorithm to constrain the search to the subspace of solutions that satisfy the hard constraints.

Motivation & Objective

• Motivate and address the challenge of finding minimum-energy lattice protein folds on quantum devices.
• Introduce a one-hot encoded turn representation to enable flexible mixer design and constraint handling.
• Develop and compare multiple mixer Hamiltonians (XY and XZ variants) that enforce hard constraints and manage soft constraints.
• Apply variational quantum algorithms (QAOA) to lattice protein folding and assess feasibility on near-term hardware.
• Explore planar lattice implementations to align with current quantum hardware connectivity constraints.

Proposed method

• Encode lattice protein folding using a one-hot turn representation requiring O(N) qubits (6N-17 for cubic; planar reduces to 4N-10).
• Formulate the problem as a quantum alternating operator ansatz (QAOA) with cost Hamiltonian H_C and mixer Hamiltonian H_M.
• Split constraints into hard (feasibility such as one-hot per turn) and soft (energy-related) to guide the search within feasible subspace.
• Design two primary mixer families (XY and XZ) and variants (simple, short-range overlap, long-range overlap) to enforce hard constraints and reduce soft constraint burden.
• Define H_C as H_overlap + H_pair to penalize overlaps and capture pairwise amino-acid interactions (HP or MJ models).
• Adapt encodings for planar lattices to reflect near-term hardware connectivity, reducing per-turn qubits and preserving feasibility checks.

Experimental results

Research questions

• RQ1 Can a one-hot encoded turn representation support effective quantum optimization for lattice protein folding on gate-based quantum computers?
• RQ2 How do different mixer Hamiltonians (XY vs XZ; simple, short-range, long-range variants) impact sampling of ground states under QAOA for planar and cubic lattices?
• RQ3 Does separating hard from soft constraints in the mixer improve adherence to feasible subspaces and ground-state probabilities?
• RQ4 What are the resource requirements (qubits) for planar vs cubic lattice encodings in this framework?
• RQ5 How do overlap penalties (H_overlap) and interaction terms (H_pair) influence optimization performance across mixer designs?

Key findings

• Ground-state sampling probabilities vary across mixer definitions; pure X mixer shows median ~0.036 with max ~0.112, while some XY simple and XZ simple variants achieve higher maxima (~0.196–0.201) in p=1 planar experiments.
• In baseline setups, mixers preserving Hamming weight (XY, simple) did not outperform the traditional X mixer in all metrics, but certain configurations yielded higher maximum ground-state probabilities.
• Introducing hard constraint enforcement in mixers (e.g., XY_short, XZ_short) shifts overlap handling from the cost to the mixer, reducing the cost term complexity.
• Long-range overlap mixers (XY_long, XZ_long) integrate overlap handling into the mixer with distance-dependent factors to encourage transitions toward folded states.
• The study assesses planar lattices (4N-10 qubits) and cubic lattices (6N-17 qubits), discussing how encoding choices align with current NISQ hardware constraints.