[Paper Review] A quantum algorithm to count weighted ground states of classical spin Hamiltonians

Bhuvanesh Sundar, Roger Paredes+3 more|arXiv (Cornell University)|Aug 5, 2019

Quantum Computing Algorithms and Architecture28 references5 citations

TL;DR

This paper presents modified adiabatic quantum optimization (AQO) and quantum approximate optimization algorithm (QAOA) to count weighted ground states of classical spin Hamiltonians, including edge covers on graphs. By reweighting the final quantum state to importance-sample ground states, the method enables sub-quadratic quantum speedup over classical optimal Monte Carlo simulation (OMCS) when ground state weights are small, offering a path to quantum advantage on NISQ devices for combinatorial counting problems.

ABSTRACT

Ground state counting plays an important role in several applications in science and engineering, from estimating residual entropy in physical systems, to bounding engineering reliability and solving combinatorial counting problems. While quantum algorithms such as adiabatic quantum optimization (AQO) and quantum approximate optimization (QAOA) can minimize Hamiltonians, they are inadequate for counting ground states. We modify AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them. As a concrete example, we show how our method can be used to count the weighted fraction of edge covers on graphs, with user-specified confidence on the relative error of the weighted count, in the asymptotic limit of large graphs. We find the asymptotic computational time complexity of our algorithms, via analytical predictions for AQO and numerical calculations for QAOA, and compare with the classical optimal Monte Carlo algorithm (OMCS), as well as a modified Grover's algorithm. We show that for large problem instances with small weights on the ground states, AQO does not have a quantum speedup over OMCS for a fixed error and confidence, but QAOA has a sub-quadratic speedup on a broad class of numerically simulated problems. Our work is an important step in approaching general ground-state counting problems beyond those that can be solved with Grover's algorithm. It offers algorithms that can employ noisy intermediate-scale quantum devices for solving ground state counting problems on small instances, which can help in identifying more problem classes with quantum speedups.

Motivation & Objective

• To address the longstanding challenge of counting ground states of classical spin Hamiltonians, which is essential for applications in statistical physics, engineering reliability, and combinatorial optimization.
• To extend AQO and QAOA beyond ground state minimization to enable accurate estimation of weighted ground state counts with user-specified confidence in relative error.
• To demonstrate the feasibility of achieving quantum speedup in counting problems on near-term noisy intermediate-scale quantum (NISQ) devices.
• To compare the asymptotic time complexity of the proposed quantum algorithms with classical optimal Monte Carlo simulation (OMCS) and Grover-based methods.
• To identify conditions under which QAOA provides a sub-quadratic speedup over classical algorithms for counting problems with small total ground state weights.

Proposed method

• Modify AQO and QAOA by introducing a non-uniform initial state and a tailored Hamiltonian evolution that ensures the final quantum state amplitude is proportional to the weight of each classical ground state.
• Use a variational parameter search in QAOA to optimize angles αj and βj such that the final state amplitude correctly reflects the desired weight distribution over ground states.
• Implement a statistical estimation technique involving repeated runs of AQO or QAOA to estimate the total weight of ground states via sampling the final state.
• For edge cover counting, map the problem to a classical Ising-like Hamiltonian whose ground states correspond to valid edge covers, with weights assigned via a parameter q.
• Analyze the time complexity of AQO analytically and QAOA numerically, accounting for both the algorithmic evolution time and the number of required sampling iterations.
• Use a modified Grover oracle framework to enable amplitude estimation, ensuring the quantum state preparation can be used for efficient counting.

Experimental results

Research questions

• RQ1 Can AQO and QAOA be adapted to estimate the weighted count of ground states of arbitrary classical spin Hamiltonians, rather than just finding a single ground state?
• RQ2 Does the modified QAOA achieve a quantum speedup over classical optimal Monte Carlo simulation (OMCS) for counting problems with small total ground state weights?
• RQ3 What is the asymptotic time scaling of the modified AQO and QAOA algorithms, and how does it compare to classical and Grover-based methods?
• RQ4 Under what conditions does QAOA exhibit a sub-quadratic speedup in estimating the total weight of ground states?
• RQ5 Can the proposed algorithms be implemented on current NISQ devices, and what are the resource requirements for the variational parameter search?

Key findings

• AQO with a linear schedule does not provide a quantum speedup over classical OMCS for a fixed relative error and confidence, as its time scaling is slower than OMCS.
• QAOA achieves a sub-quadratic speedup over classical OMCS when the total weight on the ground states is small, scaling as T_QAOA ∼ 1/√P, where P is the total weight of ground states.
• The total computational time for QAOA is asymptotically less than that of OMCS, even when including the overhead from the variational parameter search and the count estimation step.
• Numerical simulations confirm that QAOA exhibits a quadratic speedup in T_QAOA for random graphs and various weighting parameters, supporting the expectation of sub-quadratic scaling for general graphs.
• The method enables importance-sampling of ground states with probabilities proportional to their assigned weights, making it suitable for constrained sampling and counting in engineering and physical systems.
• The approach is applicable to a broad class of #P-hard counting problems that cannot be solved efficiently by existing quantum counting algorithms, opening new avenues for quantum advantage.