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Quantum adiabatic theorem for unbounded Hamiltonians, with applications to superconducting circuits.

Evgeny Mozgunov, Daniel A. Lidar

Published Nov 16, 2020• 3 citations
OpenAlex

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 $2^n\times 2^n$ effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters.

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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 2n2^n2n-dimensional qubit subspace of a circuit model of nnn qubits. The novelty of this adiabatic theorem is that unlike previous rigorous results, it does not contain 2n2^n2n 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 2n×2n2^n\times 2^n2n×2n effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters.
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 originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the [Formula: see text]-dimensional qubit subspace of a circuit model of [Formula: see text] qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain [Formula: see text] 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 [Formula: see text] effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
We provide rigorous bounds for the error of the adiabatic approximation of quantum mechanics under four sources of experimental error: perturbations in the initial condition, systematic time-dependent perturbations in the Hamiltonian, coupling to low-energy quantum systems, and decoherent time-dependent perturbations in the Hamiltonian. For decoherent perturbations, we find both upper and lower bounds on the evolution time to guarantee that the adiabatic approximation performs within a prescribed tolerance. Our results include explicit definitions of constants, and we apply them to the spin-1/2 particle in a rotating magnetic field and to the superconducting flux qubit. We compare the theoretical bounds on the superconducting flux qubit to simulation results.
This thesis explores two mathematical aspects of adiabatic quantum computation. Adiabatic quantum computation depends on the adiabatic theorem of quantum mechanics, and (a) we provide a rigorous formulation of the adiabatic theorem with explicit definitions of constants, and (b) we bound error in the adiabatic approximation under conditions of noise and experimental error. We apply the new results to a standard example of violation of the adiabatic approximation, and to a superconducting flux qubit. Further, adiabatic quantum computation requires large ground-state energy gaps throughout a Hamiltonian evolution if it is to solve problems in polynomial time. We identify a class of random Hamiltonians with non-nearest-neighbor interactions and a ground-state energy gap of O(1/n)\mathcal{O}(1/\sqrt{n})O(1/n​), where nnn is the number of qubits. We also identify two classes of Hamiltonians with non-nearest-neighbor interactions whose ground state can be found in polynomial time with adiabatic quantum computing. We then use the Jordan-Wigner transformation to derive equivalent results for Hamiltonians defined using Pauli operators.
The adiabatic approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditions are simple yet general, providing a suitable instrument to investigate adiabaticity for arbitrary initial mixed states evolving under time local master equations. We first illustrate our results by showing that the adiabatic approximation for open systems is compatible with the description of quantum thermodynamics at thermal equilibrium, where irreversible entropy production is vanishing. We also apply our sufficient conditions as a tool in quantum control, evaluating the adiabatic behavior for the Hamiltonians of both the Deutsch algorithm and the Landau-Zener model under decoherence.
The adiabatic theorem and shortcuts to adiabaticity for time-dependent open quantum systems are explored in this paper. Starting from the definition of dynamical stable decoherence-free subspace, we show that, under a compact adiabatic condition, the quantum state remains in the time-dependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may not be adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar to that for closed quantum systems, except that the operators required to change slowly are the Lindblad operators. We also show that the adiabatic evolution of decoherence-free subspaces depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems be engineered according to the incoherent control protocol. In addition, shortcuts to adiabaticity for adiabatic decoherence-free subspaces are also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broadband squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.
In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.