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Europhysics Letters (EPL)

Iteration-free digital quantum simulation of imaginary-time evolution based on the approximate unitary expansion

Jingwei Wen, Chao Zheng, Zhiguo Huang, Ling Qian

Published Mar 1, 2023• 3 citations
Publisher

Abstract

Abstract Imaginary-time evolution plays an important role in many areas of quantum physics and has been widely applied to the ground-state determination of various Hamiltonian in the quantum computation field. In this work, we propose an iteration-free quantum algorithm in a full gate-based frame using the approximate unitary expansion to simulate the imaginary-time evolution operator, avoiding the resource overhead caused by repeated measurement for state reconstruction or complex pre-calculations in the classical computers. We detail the algorithm and analyze the complexity and related characteristics including a lower bound for ancillary qubits at a given success probability. Then an application demonstration of the algorithm in quantum chemistry with hydrogen molecule under noiseless and noisy conditions is offered. In addition, we present another imaginary-time evolution simulation method based on similar construction schemes. Our algorithms can serve as the alternative proposals for the imaginary-time evolution realization in the future fault-tolerant quantum computers.

DOI
10.1209/0295-5075/acc052

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Abstract Imaginary-time evolution plays an important role in many areas of quantum physics and has been widely applied to the ground-state determination of various Hamiltonian in the quantum computation field. In this work, we propose an iteration-free quantum algorithm in a full gate-based frame using the approximate unitary expansion to simulate the imaginary-time evolution operator, avoiding the resource overhead caused by repeated measurement for state reconstruction or complex pre-calculations in the classical computers. We detail the algorithm and analyze the complexity and related characteristics including a lower bound for ancillary qubits at a given success probability. Then an application demonstration of the algorithm in quantum chemistry with hydrogen molecule under noiseless and noisy conditions is offered. In addition, we present another imaginary-time evolution simulation method based on similar construction schemes. Our algorithms can serve as the alternative proposals for the imaginary-time evolution realization in the future fault-tolerant quantum computers.
Imaginary-time evolution is a powerful tool for obtaining the ground state of a quantum system, but the complexity of classical algorithms designed for simulating imaginary-time evolution will increase significantly as the size of the quantum system becomes larger. Here, a probabilistic quantum algorithm based on Taylor expansion for implementing imaginary-time evolution is introduced. For Hamiltonians composed of Pauli product terms, the quantum circuit requires only a single ancillary qubit and is exclusively constructed using elementary single-qubit and two-qubit gates. Furthermore, similar principles are used to extend the algorithm to the case where the Hamiltonian takes a more general form. The algorithm only requires negligible precomputed numerical calculations, without the need for complex classical pre-mathematical calculations or optimization loops. We demonstrate the algorithm by solving the ground state energy of hydrogen molecules and Heisenberg Hamiltonians. Moreover, we conducted experiments on real quantum computers through the quantum cloud platform to find the ground state energy of Heisenberg Hamiltonians. Our work extends the methods for realizing imaginary-time evolution on quantum computers, and our algorithm exhibits potential for implementation on near-term quantum devices, particularly when the Hamiltonian consists of Pauli product terms.
Abstract Imaginary-time evolution is fundamental for analyzing quantum many-body systems, with applications spanning quantum chemistry, condensed matter physics, and quantum field theory, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size scaling, existing methods rely on heuristic techniques with measurement precision or success probability that deteriorates as evolution time increases. We present a quantum algorithm that prepares normalized imaginary-time evolved states using an adaptive normalization factor to maintain a stable success probability over long imaginary-time intervals. Our algorithm approximates the target state with error polynomially small in the inverse imaginary time using a polynomial number of elementary quantum gates and a single ancilla qubit, with success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial query complexity in the system size. To our knowledge, this is the first quantum algorithm for imaginary-time evolution with provably polynomial resource scaling in evolution time. Numerical experiments validate our theoretical analysis for evolution time up to 50, demonstrating the algorithm’s effectiveness for long-time evolution. Building on this technique, we further develop imaginary-time-evolution-based algorithms for ground-state-related problems and for simulating open quantum systems. These algorithms can reduce circuit depth in certain regimes compared with existing methods, at the expense of higher total query complexity, advancing the practical feasibility of quantum simulation on early fault-tolerant devices.
Imaginary-time evolution is a powerful tool in the study of quantum physics. However, existing classical algorithms for simulating imaginary-time evolution suffer from high computational complexity as the dimension of the quantum system increases. In this study we propose a quantum algorithm for implementing imaginary-time evolution using nonunitary quantum circuits with one ancillary qubit. The success probability of our algorithm is a polynomial function of the output error and can be enhanced by reorganizing the terms of the Hamiltonian. To illustrate the practicality of our algorithm on current quantum devices, we conduct a demonstration on superconducting and trapped-ion quantum processors to calculate the ground-state energy and determine the most stable molecular structure of H2{\mathrm{H}}_{2}H2​. Additionally, we validate the feasibility of our algorithm by numerically simulating the ground-state energies of LiH molecules and the quantum Ising chain. In contrast to existing algorithms, our method provides a systematic approach to construct the required nonunitary circuits using universal quantum gates, making it suitable for experimental implementation. Our algorithm opens up possibilities for exploring other physical phenomena such as finite-temperature properties and non-Hermitian systems.
Imaginary time evolution is a powerful tool applied in quantum physics, while existing classical algorithms for simulating imaginary time evolution suffer high computational complexity as the quantum systems become larger and more complex. In this work, we propose a probabilistic algorithm for implementing imaginary time evolution based on non-unitary quantum circuit. We demonstrate the feasibility of this method by solving the ground state energy of several quantum many-body systems, including H2, LiH molecules and the quantum Ising chain. Moreover, we perform experiments on superconducting and trapped ion cloud platforms respectively to find the ground state energy of H2 and its most stable molecular structure. We also analyze the successful probability of the algorithm, which is a polynomial of the output error and introduce an approach to increase the success probability by rearranging the terms of Hamiltonian.
There is increasing interest in quantum algorithms (QAs) that are based on the imaginary time evolution (ITE), a successful classical numerical approach to obtain ground states. However, most of the proposals so far require heavy postprocessing computational steps on a classical computer, such as solving linear equations. Here we provide an alternative approach to implement ITE. A key feature in our approach is the use of an orthogonal basis set: the propagated state is efficiently expressed in terms of orthogonal basis states at every step of the evolution. We argue that the number of basis states needed at those steps to achieve an accurate solution can be kept on the order of nnn, the number of qubits, by controlling the precision (number of significant digits) and the imaginary time increment. The number of quantum gates per imaginary time step is estimated to be polynomial in nnn. Additionally, while in many QAs the locality of the Hamiltonian is a key assumption, in our algorithm this restriction is not required. This characteristic of our algorithm renders it useful for studying highly nonlocal systems, such as the occupation-representation nuclear shell model. We illustrate our algorithm through numerical implementation on an IBM quantum simulator.
Evolution in imaginary time is a prominent technique for finding the ground state of quantum many-body systems, and the heart of a number of numerical methods that have been used with great success in quantum chemistry, condensed matter, and nuclear physics. We propose an algorithm to implement imaginary time propagation on a quantum computer. Our algorithm is devised in the context of an efficient encoding into an optimized gate, drawing on the underlying characteristics of the quantum device of a unitary operation in an extended Hilbert space. However, we prove that for simple problems it can also be successfully applied to standard digital quantum machines. This work paves the way for porting quantum many-body methods based on imaginary-time propagation to near-term quantum devices, enabling the future quantum simulation of the ground states of a broad class of microscopic systems.