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Predicting ground state, excited states and long-time evolution of many-body systems from short-time evolution on a quantum computer

Edgar Andres Ruiz Guzman, Denis Lacroix A

Published Apr 16, 2021• 2 citations
OpenAlex

Abstract

The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. With the constraint of minimizing the quantum resources needed on near-term quantum computers, we show that the evaluation of the generating function can be made with only one extra ancillary qubit. Due to the limited stability of actual quantum devices, the function $F(t)$ can a priori be estimated on a restricted time interval limiting automatically the orders of the moments that could be computed. Despite of this current limitation and even with a small number of moments, a post-processing on classical computers can be used to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the $t$-expansion approach that is based on imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models. Possible extensions using symmetry breaking and restoration as well as a multi-reference framework are discussed.

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The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the $t$-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.

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The generating function of a Hamiltonian HHH is defined as F(t)=⟨e−itH⟩F(t)=\langle e^{-itH}\rangleF(t)=⟨e−itH⟩, where ttt is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian ⟨HK⟩\langle H^{K}\rangle⟨HK⟩ at various orders KKK. With the constraint of minimizing the quantum resources needed on near-term quantum computers, we show that the evaluation of the generating function can be made with only one extra ancillary qubit. Due to the limited stability of actual quantum devices, the function F(t)F(t)F(t) can a priori be estimated on a restricted time interval limiting automatically the orders of the moments that could be computed. Despite of this current limitation and even with a small number of moments, a post-processing on classical computers can be used to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the ttt-expansion approach that is based on imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models. Possible extensions using symmetry breaking and restoration as well as a multi-reference framework are discussed.
The generating function of a Hamiltonian HHH is defined as F(t)=⟨e−itH⟩F(t)=\langle e^{-itH}\rangleF(t)=⟨e−itH⟩, where ttt is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian ⟨HK⟩\langle H^{K}\rangle⟨HK⟩ at various orders KKK. The real and imaginary parts of F(t)F(t)F(t) can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time ttt. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the ttt-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.
We propose a quantum-classical hybrid algorithm of the power method, here dubbed as the quantum power method, to evaluate Ĥn |ψ with quantum computers, where n is a non-negative integer, Ĥ is a time-independent Hamiltonian of interest, and |ψ is a quantum state.We show that the number of gates required for approximating Ĥn scales linearly in the power and the number of qubits, making it a promising application for near-term quantum computers.Using numerical simulation, we show that the power method can control systematic errors in approximating the Hamiltonian power Ĥn for n as large as 100.As an application, we combine our method with a multireference Krylov-subspace-diagonalization scheme to show how one can improve the estimation of ground-state energies and the ground-state fidelities found using a variational-quantum-eigensolver scheme.Finally, we outline other applications of the quantum power method, including several moment-based methods.We numerically demonstrate the connectedmoment expansion for the imaginary-time evolution and compare the results with the multireference Krylov-subspace diagonalization.
Abstract Most quantum algorithms designed to generate or probe properties of the ground state of a quantum many-body system require as input an initial state with a large overlap with the desired ground state. One approach for preparing such a ground state is Imaginary Time Evolution (ITE). Recent work by [Motta, M. , Sun, C. , Tan, A.T.K. et al (2020)] introduced an algorithm—which we will refer to as Quantum Imaginary Time Evolution (QITE)—that shows how ITE can be approximated by a sequence of unitary operators, making QITE potentially implementable on early fault-tolerant quantum computers. In this work, we provide a heuristic study of the capabilities of the QITE algorithm in approximating the ITE of lattice and molecular electronic structure Hamiltonians. We numerically study the performance of the QITE algorithm when provided with a good classical initial state for a large class of systems, some of which are of interest to industrial applications, and check if QITE is able to qualitatively replicate the ITE behavior and improve over a classical mean-field solution. The systems we consider in this work range from one- and two-dimensional lattice systems of various lattice geometries displaying short- and long-range interactions, to active spaces of molecular electronic structure Hamiltonians. In addition to the comparison of QITE and ITE, we explicitly show how imaginary time evolved fermionic Gaussian states can serve as initial states which can be efficiently computed on classical computers and efficiently implemented on quantum computers for generic spin Hamiltonians in arbitrary lattice geometries and dimensions, which can be of independent interest.
Abstract Imaginary-time evolution plays an important role in algorithms for computing ground-state and thermal equilibrium properties of quantum systems, but can be challenging to simulate on classical computers. Many quantum algorithms for imaginary-time evolution have resource requirements that are prohibitive for current quantum devices and face performance issues due to noise. Here, we propose a new algorithm for computing imaginary-time evolved expectation values on quantum computers, inspired by probabilistic error cancellation, an error-mitigation technique. Our algorithm works by decomposing a Trotterization of imaginary-time evolution into a probabilistic linear combination of operations, each of which is then implemented on a quantum computer. The measurement data is then classically post-processed to obtain the expectation value of the imaginary-time evolved state. Our algorithm requires no ancillary qubits and can be made noise-resilient without additional error-mitigation. It is well-suited for estimating thermal expectation values by making use of the notion of a thermal pure quantum state. We demonstrate our algorithm by performing numerical simulations of thermal pure quantum state preparation for the 1D Heisenberg Hamiltonian on 8 qubits, and by using an IBM quantum computer to estimate the energy of the same Hamiltonian on 2 qubits. We observe promising results compared to the exact values, illustrating the potential of our algorithm for probing the physics of quantum many-body systems on current hardware.
This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.
We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional postprocessing. The final goal is to develop configuration interaction techniques based on many-body trial states preoptimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and the quantum Krylov approaches for the postprocessing. The latter method combined with the quantum variation after projection leads to very fast convergence toward the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing Hamiltonian.