[Paper Review] Quantum adiabatic theorem for unbounded Hamiltonians, with applications to superconducting circuits.

Evgeny Mozgunov, Daniel A. Lidar|arXiv (Cornell University)|Nov 16, 2020

Quantum Computing Algorithms and Architecture3 references4 citations

TL;DR

This paper presents a novel quantum adiabatic theorem that rigorously bounds the adiabatic timescale for unbounded Hamiltonians, particularly applicable to superconducting qubit circuits. Unlike prior results, it eliminates the $2^n$ factor in the timescale and provides a cutoff-independent expression, enabling accurate prediction of qubit subspace leakage during quantum annealing, especially as tunneling barriers increase.

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 tunneling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a $2^n imes 2^n$ effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters.

Motivation & Objective

• To develop a rigorous adiabatic theorem applicable to unbounded Hamiltonians in superconducting circuit models.
• To eliminate the $2^n$ factor in adiabatic timescale bounds, which previously limited practical applicability in multi-qubit systems.
• To provide a cutoff-independent expression for the adiabatic timescale, independent of the truncation of the infinite-dimensional Hilbert space.
• To analyze leakage out of the $2^n$-dimensional qubit subspace during quantum annealing in flux qubits.
• To derive an effective $2^n \times 2^n$ Hamiltonian that best approximates the true dynamics under slowly varying control parameters.

Proposed method

• Derives a new adiabatic condition based on the spectral properties of unbounded Hamiltonians, avoiding dependence on finite-dimensional truncation.
• Introduces a bound on the adiabatic timescale that scales inversely with the minimum energy gap and the rate of change of control parameters, without a $2^n$ factor.
• Applies the theorem to a superconducting flux qubit model, explicitly computing the dependence of the timescale on circuit parameters such as tunneling barrier height.
• Uses a projection formalism to derive an effective $2^n \times 2^n$ Hamiltonian that captures the dominant dynamics within the qubit subspace under slow parameter variation.
• Analyzes the role of the infinite-dimensional Hilbert space by showing that leakage out of the qubit subspace becomes inevitable as the tunneling barrier is increased.

Experimental results

Research questions

• RQ1 Can a rigorous adiabatic theorem be formulated for unbounded Hamiltonians in superconducting circuits without dependence on the Hilbert space cutoff?
• RQ2 Does the absence of a $2^n$ factor in the adiabatic timescale bound improve the practical feasibility of quantum annealing in multi-qubit systems?
• RQ3 What is the explicit dependence of the adiabatic timescale on circuit parameters such as the tunneling barrier in a flux qubit?
• RQ4 Is leakage out of the qubit subspace unavoidable during quantum annealing as the tunneling barrier is raised?
• RQ5 How can an effective $2^n \times 2^n$ Hamiltonian be systematically derived to approximate the true dynamics of a slowly varying superconducting circuit?

Key findings

• The proposed adiabatic theorem provides a timescale bound independent of the Hilbert space cutoff, removing the $2^n$ factor present in previous rigorous results.
• For a superconducting flux qubit, the adiabatic timescale depends explicitly on circuit parameters such as the tunneling barrier and level spacing, with a quantitative expression derived.
• Leakage out of the $2^n$-dimensional qubit subspace is inevitable as the tunneling barrier is raised toward the end of a quantum anneal, even under adiabatic evolution.
• The effective $2^n \times 2^n$ Hamiltonian derived provides a best-approximation to the true dynamics under slow parameter variation, enabling accurate modeling of qubit behavior.
• The analysis confirms that unbounded Hamiltonians in superconducting circuits require a refined adiabatic condition that accounts for infinite-dimensional effects, which standard bounds fail to capture.