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Open Systems & Information Dynamics

On the Speed Limit for Imaginary-Time Schrödinger Equation with Application to Quantum Searches

Jie Sun, Songfeng Lu

Published Jun 1, 2021• 0 citations
Publisher

Abstract

Recently, Okuyama and Ohzek [1] derived a speed limit for the imaginary-time Schrödinger equation, which is inspired by the prior work of Kieu, who had shown a new class of time–energy uncertainty relations suitable for actually evaluating the speed limit of quantum dynamics. In this paper, we apply the result of Okuyama and Ohzek to obtain a generalized speed limit for Grover’s search in imaginary-time quantum annealing. An estimate of the lower bound on the computational time is shown, from which the role of the coefficient function corresponding to the final Hamiltonian played in the quantum dynamics for the problem is sticking out. However, when trying to apply the speed limit to the analogue of Grover’s problem, we find that not only the coefficient of the target Hamiltonian is related to the time complexity of the algorithm, but also the coefficient of the initial Hamiltonian is crucial for determining the time complexity. This is new and generalizes one of the results in our previous work.

DOI
10.1142/s1230161221500074

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Recently, Okuyama and Ohzek [1] derived a speed limit for the imaginary-time Schrödinger equation, which is inspired by the prior work of Kieu, who had shown a new class of time–energy uncertainty relations suitable for actually evaluating the speed limit of quantum dynamics. In this paper, we apply the result of Okuyama and Ohzek to obtain a generalized speed limit for Grover’s search in imaginary-time quantum annealing. An estimate of the lower bound on the computational time is shown, from which the role of the coefficient function corresponding to the final Hamiltonian played in the quantum dynamics for the problem is sticking out. However, when trying to apply the speed limit to the analogue of Grover’s problem, we find that not only the coefficient of the target Hamiltonian is related to the time complexity of the algorithm, but also the coefficient of the initial Hamiltonian is crucial for determining the time complexity. This is new and generalizes one of the results in our previous work.
Recently, Kieu proposed a new class of time-energy uncertainty relations for time-dependent Hamiltonians, which is not only formal but also useful for actually evaluating the speed limit of quantum dynamics. Inspired by this work, Okuyama and Ohzeki obtained a similar speed limit for the imaginary-time Schrödinger equation. In this paper, we refine the latter one to make it be further like that of Kieu formally. As in the work of Kieu, only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave like functions, which would demand a full solution for a time-dependent system, are required for our optimized speed limit. It turns out to be more helpful for estimating the speed limit of an actual quantum annealing driven by the imaginary-time Schrödinger equation. For certain case, the refined speed limit given here becomes the only useful tool to do this estimation, because the one given by Okuyama and Ohzeki cannot do the same job.
We present the analog analogue of Grover’s problem as an example of the time-independent Hamiltonian for applying the speed limit of the imaginary-time Schrödinger equation derived by Okuyama and Ohzeki and the new class of energy-time uncertainty relation proposed by Kieu. It is found that the computational time of the imaginary-time quantum annealing of this Grover search can be exponentially small, while the counterpart of the quantum evolution driven by the real-time Schrödinger equation could only provide square root speedup, compared with classic search. The present results are consistent with the cases of the time-dependent quantum evolution of the natural Grover problem in previous works. We once again emphasize that the logarithm and square root algorithmic performances are generic in imaginary-time quantum annealing and quantum evolution driven by real-time Schrödinger equation, respectively. Also, we provide evidences to search deep reasons why the imaginary-time quantum annealing can lead to exponential speedup and the real-time quantum annealing can make square root speedup.
The quantum speed limit (QSL), or the energy-time uncertainty relation, gives a fundamental speed limit for quantum dynamics. Recently, Kieu [arXiv:1702.00603] derived a new class of QSL which is not only formal but also suitable for actually evaluating the speed limit. Inspired by his work, we obtain a similar speed limit for the imaginary-time Schrödinger equation. Using this new bound, we show that the optimal computational time of the Grover problem in imaginary-time quantum annealing is bounded from below by log⁡N\log NlogN, which is consistent with a result of previous study.
Recently, Kieu derived a new class of time–energy uncertainty relations, which is not only formal but also suitable for evaluating the speed limit of quantum dynamics. However, when this time–energy uncertainty relation was applied to study the quantum adiabatic Grover’s search, some approximation had been done for reaching the right conclusion. That is, the final state being orthogonal to the initial state in this problem was assumed which may cause confusion, while the truth is that the overlap between these two states is indeed very small when the problem size is very big but not equal to zero exactly. In this paper, we modify the time–energy uncertainty relation of Kieu applied to two quantum evolutions of Grover’s problem, and find that the resulting computational complexities match well with the previous works on this topic but without the need of making the assumption like that in the work of Kieu.
We derive a Geometric quantum speed limit (QSL) for imaginary-time evolution, where the dynamics is governed by a non-unitary Schrödinger equation. By introducing a cost function based on the angular distance between the normalized evolving state and the initial state, we obtain a lower bound on the evolution time expressed as the ratio between this angle and the time-averaged energy dispersion. Our bound is analytical, general, and applicable to arbitrary time-independent Hamiltonians. We analytically evaluate this bound for two physically motivated cases. First, we apply it to a two-level system and derive an expression for the minimal time. Second, we analyze the imaginary-time version of Grover search problem and rigorously reproduce the well-known logarithmic scaling T=O(log⁡N)T=\mathcal{ O}(\log N)T=O(logN) within our QSL framework.
We confirmed the annealing time of Grover's search which is required to obtain desired success probability for quantum annealing by the imaginary-time and the real-time Schrödinger equation with two kinds of schedulings; one linearly decreases the quantum fluctuation and the other tunes the evolution rate of the Hamiltonian based on the adiabatic condition. With linear scheduling, the required annealing time for quantum annealing by the imaginary-time Schrödinger equation is of order log⁡N\log NlogN, which is very different from O(N)O(N)O(N) required for the quantum annealing by the real-time Schrödinger equation. With the scheduling based on the adiabatic condition, the required annealing time is of order N\sqrt{N}N​, which is identical to the annealing time for quantum annealing by the real-time Schrödinger equation. Although the scheduling based on the adiabatic condition is optimal for the quantum annealing by the real-time Schrödinger equation, it is inefficient for the quantum annealing by the imaginary-time Schrödinger equation. This result implies that the optimal scheduling for the quantum annealing by the imaginary-time and the real-time Schrödinger equation is very different, and the efficient scheduling considered with the quantum Monte Carlo methods, which is based on imaginary-time Schrödinger equation, is not necessarily effective to improve the performance of quantum annealing by the real-time Schrödinger equation. We discuss the efficient scheduling for quantum annealing by the imaginary-time Schrödinger equation with respect to the exponential decay of excited states.