Discretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution

Motivated by recent progress of quantum technologies making small-scale programmable quantum computing possible, here we study a discretized quantum adiabatic process for a one-dimensional free-fermion system described by a variational wave function, i.e., a parametrized quantum circuit. The wave function is composed of M layers of two elementary sets of time-evolution operators, each set being decomposed into commutable local operators acting on neighboring sites. The evolution time of each time-evolution operator is treated as a variational parameter so as to minimize the expectation value of the energy. We show that the exact ground state is reached by applying the layers of time-evolution operators as many as a quarter of the system size, implying that at least in this case, the state is exactly prepared in a quantum circuit with linear depth. This is the minimum number MB of layers set by the limit of speed, i.e., the Lieb-Robinson bound, for propagating quantum entanglement via the local time-evolution operators. Indeed, we show the mutual information of the variational wave function that reveals a causality-cone-like structure in the propagation of quantum entanglement. Quantities such as the energy E and the entanglement entropy S of the optimized variational wave function with the number M of layers less than MB are independent of the system size L but fall into some universal functions of M, indicating that the entanglement generated in this variational ansatz with a finite M is bounded, as in the case of the matrix product states with a finite bond dimension. Furthermore, in this case, we find that these two quantities behave asymptotically as E/L−ɛ∞∼M−2 (ɛ∞: the exact ground-state energy per site in the thermodynamic limit) and S≈13lnM. The development of the entanglement in the variational ansatz through the discretized quantum adiabatic process is further manifested in the progressive propagation of single-particle orbitals in the variational wave function. We also find that the optimized variational parameters converge systematically to a smooth function of the discretized time, which provides the optimum scheduling function in the quantum adiabatic process, with the effective total evolution time of the variational ansatz to the exact ground state being proportional to the system size L. This is a drastic improvement compared to the evolution time proportional to L2 for the continuous-time quantum adiabatic process with a linear scheduling and is attributed to diabaticity of the discretized quantum adiabatic process represented in the variational ansatz. Finally, we investigate the imaginary-time evolution counterpart of this variational wave function, where the causality relation is absent due to the nonunitarity of the imaginary-time evolution operators, and thus the norm of the wave function is no longer conserved. We find that the convergence to the exact ground state is exponentially fast, despite that the system is at the critical point, suggesting that implementation of the nonunitary imaginary-time evolution in a quantum circuit is highly promising to further shallow the circuit depth, provided that the local nonunitary operators are represented with a reasonable amount of unitary operators.15 MoreReceived 27 August 2020Revised 4 December 2020Accepted 7 December 2020DOI:https://doi.org/10.1103/PhysRevResearch.3.013004Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAdiabatic quantum optimizationQuantum algorithmsQuantum computationQuantum gatesQuantum InformationCondensed Matter, Materials & Applied Physics