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arXiv:1906.02581

How the High-energy Part of the Spectrum Affects the Adiabatic Computation Gap

Yosi Atia, Dorit Aharonov

Published Jun 6, 2019• 0 citations
arXivHTMLPDF

Abstract

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 $n/2$, $n$ 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.

DOI
10.48550/arxiv.1906.02581

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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.
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.
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The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such.
We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real time axis, that some number of its time-derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is non-degenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time-derivative of the Hamiltonian, divided by the cube of the minimal gap.
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.
We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that, for problems that have an exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not suitable for adiabatic quantum computation.