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arXiv:2008.07168 · Physical Review Research

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

Tomonori Shirakawa, Kazuhiro Seki, Seiji Yunoki

Published Jan 4, 2021• 16 citations
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Abstract

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 M B 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 M B 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 -ε ∞ ∼ M -2 (ε ∞ : the exact ground-state energy per site in the thermodynamic limit) and S ≈ 1 3 ln M. 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 L 2 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.

DOI
10.1103/physrevresearch.3.013004

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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 M B 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 M B 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 -ε ∞ ∼ M -2 (ε ∞ : the exact ground-state energy per site in the thermodynamic limit) and S ≈ 1 3 ln M. 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 L 2 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.
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.
We introduce a quantum-circuit algorithm for performing imaginary-time evolution on infinite one-dimensional lattice systems. The method uses a parameterized quantum circuit to represent a uniform matrix product state ansatz. We derive the ITE algorithm using the time-dependent variational principle and employ the quantum Lanczos algorithm to improve the ground-state energy estimate. As a benchmark, we simulate the transverse-field Ising model using both classical simulators and IBM Quantum devices. Our analysis includes a statistical study of the distributions of the cost function and energy density obtained from quantum measurements, illustrating the effects of finite-sampling noise on convergence.
Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the locality of the operators comes at the expense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic degrees of freedom, one must satisfy a set of constraints. In this work, we introduce quantum circuits that exactly satisfy these stringent constraints. We demonstrate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advantageous to simulate the time evolution operator of fermionic systems in d>1 dimensions. We also discuss how these families of circuits can be used as variational quantum states, focusing on two approaches: a first one based on general constant-fermion-number gates, and a second one based on the Hamiltonian variational ansatz where the eigenstates are represented by parametrized time-evolution operators. We apply our methods to the problem of finding the ground state and time-evolved states of the ttt-VVV model.
Implementing time evolution operators on quantum circuits is important for quantum simulation. However, the standard way, Trotterization, requires a huge numbers of gates to achieve desirable accuracy. Here, we propose a local variational quantum compilation (LVQC) algorithm, which allows to accurately and efficiently compile a time evolution operators on a large-scale quantum system by the optimization with smaller-size quantum systems. LVQC utilizes a subsystem cost function, which approximates the fidelity of the whole circuit, defined for each subsystem as large as approximate causal cones brought by the Lieb-Robinson (LR) bound. We rigorously derive its scaling property with respect to the subsystem size, and show that the optimization conducted on the subsystem size leads to the compilation of whole-system time evolution operators. As a result, LVQC runs with limited-size quantum computers or classical simulators that can handle such smaller quantum systems. For instance, finite-ranged and short-ranged interacting LLL-size systems can be compiled with O(L0)O(L^0)O(L0)- or O(log⁡L)O(\log L)O(logL)-size quantum systems depending on observables of interest. Furthermore, since this formalism relies only on the LR bound, it can efficiently construct time evolution operators of various systems in generic dimension involving finite-, short-, and long-ranged interactions. We also numerically demonstrate the LVQC algorithm for one-dimensional systems. Employing classical simulation by time-evolving block decimation, we succeed in compressing the depth of a time evolution operators up to 404040 qubits by the compilation for 202020 qubits. LVQC not only provides classical protocols for designing large-scale quantum circuits, but also will shed light on applications of intermediate-scale quantum devices in implementing algorithms in larger-scale quantum devices.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm.