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arXiv:2104.13884 · Europhysics Letters (EPL)

Universal witnesses of vanishing energy gap

Konrad Szymański, Karol Życzkowski

Published Nov 1, 2021• 3 citations
arXivHTMLPDF

Abstract

Abstract Energy gap, the difference between the energy of the ground-state of a given Hamiltonian and the energy of its first excited state, is a parameter of a critical importance in analysis of phase transitions and adiabatic quantum computation. We present a concrete technique to determine an upper bound for the energy gap of a Hamiltonian H 0 based on properties of the set of expectation values of H 0 and an additional auxiliary Hamiltonian V . This formalism can be applied to obtain an effective criterion of gaplessness, which we illustrate with a concrete example of the XY model —a physical system with vanishing energy gap.

DOI
10.1209/0295-5075/ac35f4

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Abstract Energy gap, the difference between the energy of the ground-state of a given Hamiltonian and the energy of its first excited state, is a parameter of a critical importance in analysis of phase transitions and adiabatic quantum computation. We present a concrete technique to determine an upper bound for the energy gap of a Hamiltonian H 0 based on properties of the set of expectation values of H 0 and an additional auxiliary Hamiltonian V . This formalism can be applied to obtain an effective criterion of gaplessness, which we illustrate with a concrete example of the XY model —a physical system with vanishing energy gap.
We show that the rate of closing of the energy gap between the ground state and the first excited state, as a function of system size, behaves in many qualitatively different ways at first-order quantum phase transitions of the infinite-range quantum XY model. Examples include polynomial, exponential and even factorially fast closing of the energy gap, all of which coexist along a single axis of the phase diagram representing the transverse field. This variety emerges depending on whether or not the transverse field assumes a rational number as well as on how the series of system size is chosen toward the thermodynamic limit. We conclude that there is no generically applicable rule to relate the rate of gap closing and the order of quantum phase transitions as is often implied in many studies, in particular in relation to the computational complexity of quantum annealing in its implementation as quantum adiabatic computation.
Abstract Estimating energy gaps, i.e. the energy difference between two different states, in quantum systems is crucial for understanding their properties. Conventionally, spectral gap estimation relies on independently computing the ground-state and first-excited-state energies and then taking their difference. This work introduces an alternative procedure for estimating spectral gaps on digital quantum devices using the adiabatic preparation technique to create a specific superposition state. The expectation values of observables measured on such a state exhibit time-dependent fluctuations which, through a fitting process, can be used to estimate the energy gap. We successfully test our method on the 1D and 2D Ising models, and H 2 and He 2 molecules, implementing relatively shallow circuits both on noiseless and noisy simulators. The robustness of the approach is corroborated by additional experiments on the real IonQ Aria device for the 1D Ising model up to 20 qubits, demonstrating the applicability of the proposed method for currently available digital quantum devices and paving the way for more complex energy gap calculation requiring deeper circuits in the fault-tolerant era to come.
Estimating energy gaps, i.e. the energy difference between two different states, in quantum systems is crucial for understanding their properties. Conventionally, spectral gap estimation relies on independently computing the ground-state and first-excited-state energies and then taking their difference. This work introduces an alternative procedure for estimating spectral gaps on digital quantum devices using the Adiabatic Preparation technique to create a specific superposition state. The expectation values of observables measured on such a state exhibit time-dependent fluctuations which, through a fitting process, can be used to estimate the energy gap. We successfully test our method on the 1D and 2D Ising models, and H2 and He2 molecules, implementing relatively shallow circuits both on noiseless and noisy simulators. The robustness of the approach is corroborated by additional experiments on the real IonQ Aria device for the 1D Ising model up to 20 qubits, demonstrating the applicability of the proposed method for currently available digital quantum devices and paving the way for more complex energy gap calculation requiring deeper circuits in the fault-tolerant era to come.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians HI{H}_{\mathrm{I}}HI​ and HF{H}_{\mathrm{F}}HF​. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of HF{H}_{\mathrm{F}}HF​ from the ground state of HI{H}_{\mathrm{I}}HI​. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
We show for the one-dimensional quantum XYXYXY model with s=1/2s=1/2s=1/2 that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum XYXYXY model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum XYXYXY model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.