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arXiv:1912.11979 · Physical Review Research

Performance evaluation of adiabatic quantum computation via quantum speed limits and possible applications to many-body systems

Keisuke Suzuki, Kazutaka Takahashi

Published Jul 15, 2020• 46 citations
arXivHTMLPDF

Abstract

The quantum speed limit specifies a universal bound of the fidelity between the initial state and the timeevolved state.We apply this method to find a bound of the fidelity between the adiabatic state and the timeevolved state.The bound is characterized by the counterdiabatic Hamiltonian and can be used to evaluate the worst case performance of the adiabatic quantum computation.The result is improved by imposing additional conditions and we examine several models to find a tight bound.We also derive a different type of quantum speed limit that is meaningful even when we take the thermodynamic limit.By using solvable spin models, we study how the performance and the bound are affected by phase transitions.

DOI
10.1103/physrevresearch.2.032016

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The quantum speed limit specifies a universal bound of the fidelity between the initial state and the timeevolved state.We apply this method to find a bound of the fidelity between the adiabatic state and the timeevolved state.The bound is characterized by the counterdiabatic Hamiltonian and can be used to evaluate the worst case performance of the adiabatic quantum computation.The result is improved by imposing additional conditions and we examine several models to find a tight bound.We also derive a different type of quantum speed limit that is meaningful even when we take the thermodynamic limit.By using solvable spin models, we study how the performance and the bound are affected by phase transitions.
Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. Most known quantum speed limits, including the celebrated Mandelstam-Tamm and Margolus-Levitin ones, are general bounds applicable to arbitrary initial states. However, when applied to mixed states of many-body systems, they, as a rule, dramatically overestimate the speed of quantum evolution and fail to provide meaningful bounds in the thermodynamic limit. Here we derive a quantum speed limit for a closed system initially prepared in a thermal state and evolving under a time-dependent Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. In a broad class of many-body setups it proves to be drastically stronger than general quantum speed limits.
We provide two inequalities for estimating adiabatic fidelity in terms of two other more handily calculated quantities, i.e., generalized orthogonality catastrophe and quantum speed limit. As a result of considering a two-dimensional subspace spanned by the initial ground state and its orthogonal complement, our method leads to stronger bounds on adiabatic fidelity than those previously obtained. One of the two inequalities is nearly sharp when the system size is large, as illustrated using a driven Rice-Mele model, which represents a broad class of quantum many-body systems whose overlap of different instantaneous ground states exhibits orthogonality catastrophe.
Establishing quantitative adiabaticity criteria at finite temperature remains substantially less developed than in the pure-state setting, even though realistic quantum systems are never at absolute zero. Here, by combining a mixed-state quantum speed limit with mixed-state fidelity susceptibility within the Liouville-space formulation of quantum mechanics, we derive rigorous bounds on the Hilbert-Schmidt fidelity between mixed states. Focusing on protocols that drive an initial Gibbs state toward a quasi-Gibbs target, these bounds yield an explicit threshold driving rate for the onset of nonadiabaticity. For a broad class of local Hamiltonians in gapped phases, we show that, in the thermodynamic limit, the threshold driving rate factorizes into a system-size contribution that recovers the zero-temperature scaling and a universal temperature-dependent factor. The latter is exponentially close to unity at low temperature, whereas at high temperature it is linear in temperature. We verify the predicted scaling in several spin-1/2 chains by obtaining closed-form expressions for the threshold driving rate. Our results provide a practical and largely model-independent criterion for finite-temperature adiabaticity in driven many-body systems under closed-system unitary evolution.
We discuss bounds for nonadiabatic transitions from the viewpoints of the adiabatic perturbation theory and the quantum speed limit. We show that the amount of nonadiabatic transitions from the nthn\mathrm{th}nth level to the mthm\mathrm{th}mth level is bounded by a function of the quantum geometric tensor for the mthm\mathrm{th}mth level. We analyze this bound from the viewpoint of the adiabatic perturbation theory. In addition, this bound and the viewpoint of the quantum speed limit suggest a nontrivial relationship between the dynamical transformation and the adiabatic transformation. We also derive a universal bound for any nonadiabatic transition. This bound is written in terms of the counterdiabatic Hamiltonian.
Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control.While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive.Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones.We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable.This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics.Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new trade-off relation between time and the quantum phase difference.Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound.We also find that, beyond expectation values of macroscopic observables, the speed of macroscopic quantum coherence can be bounded from above by our general approach.The newly obtained bounds are verified in transport phenomena in particle systems and nonequilibrium dynamics in many-body spin systems.We also demonstrate that our strategy can be applied for finding new speed limits for macroscopic transitions in stochastic systems, including quantum ones, where the bounds are expressed by the entropy production rate.Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.
The speed limit provides an upper bound for the dynamical evolution time of a quantum system. Here, we introduce the notion of quantum acceleration limit for unitary time evolution of quantum systems under time-dependent Hamiltonian. We prove that the quantum acceleration is upper bounded by the fluctuation in the derivative of the Hamiltonian. This leads to a universal quantum acceleration limit (QAL) which answers the question: What is the minimum time required for a quantum system to be accelerated from arbitrary initial state to final state? We illustrate the quantum acceleration limit for a two-level quantum system and show that the bound is indeed tight. This notion can have important applications in adiabatic quantum computing, quantum control and quantum thermodynamics.