On the gap of Hamiltonians for the adiabatic simulation of quantum circuits

TL;DR

This paper investigates the limitations of spectral gap amplification in adiabatic quantum circuit simulation, showing that under standard assumptions, the gap cannot be improved beyond certain bounds, which impacts the efficiency of such simulations.

Contribution

The paper establishes upper bounds on the spectral gap for adiabatic Hamiltonians simulating quantum circuits, highlighting fundamental limitations in gap amplification.

Findings

Spectral gap decreases at most polynomially with circuit size

Frustration-free Hamiltonians have even smaller upper bounds on the gap

Improving the spectral gap beyond established bounds is fundamentally challenging

Abstract

The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of order 1/L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upper bound is of order 1/L^2. Our proofs use recent results on adiabatic state transformations, spectral gap amplification, and the simulation of continuous-time quantum query algorithms. They also consider a reduction from the unstructured search problem, whose lower bound in the oracle cost translates into the upper bounds in the gaps. The impact of our results is that improving the gap beyond that of standard constructions (i.e., 1/L^2), if possible, is challenging.