Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations

TL;DR

This paper establishes concentration bounds for various quantum states, including those from shallow circuits and dense Hamiltonian evolutions, and demonstrates limitations of QAOA in solving dense spin models at super-constant levels.

Contribution

It introduces new concentration bounds for quantum states from shallow circuits, matrix product states, and dense Hamiltonian evolutions, and shows QAOA's limitations on dense instances at super-constant levels.

Findings

States are close to local operators, leading to concentration of measurement distributions.

QAOA has negligible success probability on dense spin models at super-constant levels.

Improves understanding of QAOA limitations beyond previous results.

Abstract

We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [DPMRF22]; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form $e^{\iota H^{(p)}} \cdots e^{\iota H^{(1)}} |\psi_0\rangle$ for any $n$-qubit product state $|\psi_0\rangle$, where each $H^{(i)}$ can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates. An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level $p = o(\log \log n)$, assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [BGMZ22].