Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits

TL;DR

This paper introduces a new quantum adiabatic theorem applicable to unbounded Hamiltonians, providing bounds on adiabatic timescales relevant for superconducting circuits, and demonstrates its implications for qubit leakage during quantum annealing.

Contribution

It presents a novel adiabatic theorem that avoids exponential dependence on the number of qubits and applies to unbounded Hamiltonians in superconducting circuits.

Findings

Bound on adiabatic timescale independent of Hilbert space cutoff

Leakage out of qubit subspace increases as tunneling barrier is raised

Explicit dependence of timescale on circuit parameters for flux qubits

Abstract

We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by unbounded Hamiltonians. Our bound is geared towards the qubit approximation of superconducting circuits, and presents a sufficient condition for remaining within the $2^n$-dimensional qubit subspace of a circuit model of $n$ qubits. The novelty of this adiabatic theorem is that unlike previous rigorous results, it does not contain $2^n$ as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit, and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a $2^n\times 2^n$ effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters.