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arXiv:1702.00603 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

A class of time-energy uncertainty relations for time-dependent Hamiltonians

Tien D. Kieu

Published Jun 1, 2019• 12 citations
arXivHTMLPDF

Abstract

A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave funct- ions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.

DOI
10.1098/rspa.2019.0148

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A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave funct- ions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.
A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and the energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a He+{\mathrm{He}}^{+}He+ ion moving in a uniform magnetic field loses its electron.
Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.
We derive exact relations and general inequalities that extend the usual time-energy uncertainty relations from the domain of unitary Hamiltonian dynamics to that of dissipative dynamics as described by a broad class of linear and nonlinear evolution equations for the density operator. For non-dissipative dynamics, by using the Schroedinger inequality instead of the Heisenberg-Robertson inequality, we obtain a general exact time-energy uncertainty relation which is sharper than the usual Mandelstam-Tamm-Messiah relation τFΔH≥ℏ/2τ_FΔ_H\ge \hbar/2τF​ΔH​≥ℏ/2. For simultaneous unitary/dissipative dynamics, the usual time-energy uncertainty relation is replaced by a less restrictive relation that depends on the characteristic time of dissipation, τττ, and the uncertainty associated with the generalized nonequilibrium Massieu-function operator which defines the structure of the dissipative part of the assumed class of evolution equations. Within the steepest-entropy-ascent dissipative quantum dynamics of an isolated system introduced earlier by this author, we obtain the interesting time-energy and time-entropy uncertainty relation (2τFΔH/ℏ)2+(τFΔS/kBτ)2≥1(2τ_FΔ_H/ \hbar)^2+ (τ_FΔ_S/k_Bτ)^2 \ge 1(2τF​ΔH​/ℏ)2+(τF​ΔS​/kB​τ)2≥1. We illustrate this result and various other inequalities by means of numerical simulations.
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 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.
Hamiltonian simulations are key subroutines in adiabatic quantum computation and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. Previous studies have shown that the low-energy assumption can reduce the resource requirements of standard time-independent Hamiltonian simulation algorithms. However, whether such advantages extend to time-dependent Hamiltonian simulation remains open. In this paper, we consider the adiabatic regime where the relevant low-energy subspace is spanned by a fixed number of low-energy eigenstates and separated from the rest of the spectrum by a gap. We show that, for simulating spin Hamiltonians by product formulas, the explicit system size dependence in the leading commutator-scaling term can be replaced by a low-energy scale up to logarithmic factors. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further conduct numerical experiments on adiabatic state preparation of an illustrative example system to support our theoretical findings. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.