Improved approximation algorithms for bounded-degree local Hamiltonians

TL;DR

This paper introduces shallow quantum circuits that enhance the approximation of ground state energies for bounded-degree local Hamiltonians, surpassing product state optimizations and extending to entangled states.

Contribution

It presents a novel family of shallow quantum circuits that improve energy approximations for local Hamiltonians beyond product state methods.

Findings

Energy improvement proportional to variance squared over number of qubits

Applicable to initial random product states, generalizing classical algorithms

Extends results to k-local Hamiltonians and entangled initial states

Abstract

We consider the task of approximating the ground state energy of two-local quantum Hamiltonians on bounded-degree graphs. Most existing algorithms optimize the energy over the set of product states. Here we describe a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an $n$-qubit product state $|v\rangle$ with mean energy $e_0=\langle v|H|v\rangle$ and variance $\mathrm{Var}=\langle v|(H-e_0)^2|v\rangle$, and outputs a state with an energy that is lower than $e_0$ by an amount proportional to $\mathrm{Var}^2/n$. In a typical case, we have $\mathrm{Var}=\Omega(n)$ and the energy improvement is proportional to the number of edges in the graph. When applied to an initial random product state, we recover and generalize the performance guarantees of known algorithms for bounded-occurrence classical constraint satisfaction problems. We extend our results to $k$-local Hamiltonians and entangled initial states.