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arXiv:1107.4034 · Journal of Physics A Mathematical and Theoretical

Relationship between minimum gap and success probability in adiabatic quantum computing

M Cullimore, M. J. Everitt, M A Ormerod, John H. Samson, R D Wilson, Alexandre M. Zagoskin

Published Nov 28, 2012• 7 citations
arXivHTMLPDF

Abstract

We explore the relationship between two figures of merit for an adiabatic quantum computation process: the success probability P and the minimum gap Δ min between the ground and first excited states, investigating to what extent the success probability for an ensemble of problem Hamiltonians can be fitted by a function of Δ min and the computation time T . We study a generic adiabatic algorithm and show that a rich structure exists in the distribution of P and Δ min . In the case of two qubits, P is to a good approximation a function of Δ min , of the stage in the evolution at which the minimum occurs and of T . This structure persists in examples of larger systems.

DOI
10.1088/1751-8113/45/50/505305

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We explore the relationship between two figures of merit for an adiabatic quantum computation process: the success probability P and the minimum gap Δ min between the ground and first excited states, investigating to what extent the success probability for an ensemble of problem Hamiltonians can be fitted by a function of Δ min and the computation time T . We study a generic adiabatic algorithm and show that a rich structure exists in the distribution of P and Δ min . In the case of two qubits, P is to a good approximation a function of Δ min , of the stage in the evolution at which the minimum occurs and of T . This structure persists in examples of larger systems.
Abstract Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.
Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyze a particular class of quantum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum-energy gap, which governs the time required for a successful evolution, is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.
In the standard approach to adiabatic quantum computing (AQC), quantum information stored on qubits are adia- batically evolved to find the lowest energy state of a problem Hamiltonian. Here we investigate a variation of AQC where spin ensembles are used in place of qubits. The use of ensembles duplicates the quantum information, and allows errors to be suppressed during the adiabatic evolution. We show that there are two distinct types of problem Hamiltonians under this mapping, where the first excited state is a single particle perturbation on the ground state (Type I); or a macroscopically distinct fully polarized state (Type II). For Type I problems, we find that the minimum gap for large ensembles is well predicted by mean-field theory and the AQC performance improves with ensemble size, realizing error-suppression. For Type II problems, the performance of the AQC is mixed, and the gap can increase or decrease with ensemble size depending on the problem instance. The incidences of the Type II problems are greatly suppressed with ensemble size, and consequently for randomly chosen problem instances the success probability of the scheme improves with the ensemble size. Our approach shows that it is possible to perform AQC without the necessity of controlling individual qubits, which allows for an alternative route towards implementing AQC.
Towards better understanding of how to design efficient adiabatic quantum algorithms, we study how the adiabatic gap depends on the spectra of the initial and final Hamiltonians in a natural family of test-bed examples. We show that perhaps counter-intuitively, changing the energy in the initial and final Hamiltonians of only highly excited states (we do this by assigning all eigenstates above a certain cutoff the same value), can turn the adiabatic algorithm from being successful to failing. Interestingly, our system exhibits a phase transition; when the cutoff in the spectrum becomes smaller than roughly n/2n/2n/2, nnn being the number of qubits, the behavior transitions from a successful adiabatic process to a failed one. To analyze this behavior, and provide an upper bound on both the minimal gap as well as the success of the adiabatic algorithm, we introduce the notion of "escape rate", which quantifies the rate by which the system escapes the initial ground state (a related notion was also used by Ilin and Lychkovskiy in arXiv:1805.04083). Our results indicate a phenomenon that is interesting on its own right: an adiabatic evolution may be robust to bounded-rank perturbations, even when the latter closes the gap or makes it exponentially small.
Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.
In the context of adiabatic quantum computation (AQC), it has been argued that first-order quantum phase transitions (QPTs) due to localization phenomena cause AQC to fail by exponentially decreasing the minimal spectral gap of the Hamiltonian along the annealing path as a function of the qubit number. The vanishing of the spectral gap is often linked to the localization of the ground state in a local minimum, requiring the system to tunnel into the global minimum at a later stage of the annealing. Recent methods have been proposed to avoid this phenomenon by carefully designing the involved Hamiltonians. However, it remains a challenge to formulate a comprehensive theory of the effect of the various parameters and the conditions under which QPTs make the AQC algorithm fail. Equipped with concepts from graph theory, in this work we link graph quantities associated with the Hamiltonians along the annealing path with the occurrence of QPTs. These links allow us to derive bounds on the location of the minimal spectral gap along the annealing path, augmenting the toolbox for the analysis of strategies to improve the runtime of AQC algorithms.