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arXiv:2407.14163 · Physical Review Research

Subspace-based local compilation of variational quantum circuits for large-scale quantum many-body simulation

Shota Kanasugi, Yuichiro Hidaka, Yuya O. Nakagawa, Shoichiro Tsutsui, Norifumi Matsumoto, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato

Published May 21, 2025• 4 citations
arXivHTMLPDF

Abstract

Simulation of quantum many-body systems is one of the most promising applications of quantum computers. It is crucial to efficiently implement the time-evolution operator as a quantum circuit to execute such simulations on near-term quantum computing devices with limited computational resources. However, standard approaches such as Trotterization sometimes require a deep quantum circuit, which is hard to implement on near-term quantum computers. Here, we propose a hybrid quantum-classical algorithm, called local subspace variational quantum compilation (LSVQC), for compiling the time-evolution operator of quantum many-body systems. The LSVQC performs a variational optimization to reproduce the action of the target time-evolution operator within a physically reasonable subspace. The optimization is performed for small local subsystems based on the Lieb-Robinson bound, which allows us to execute the cost function evaluation using small-scale quantum devices and/or classical computers. We demonstrate the validity of the LSVQC algorithm through numerical simulations of a simple spin-lattice model and an effective model of a parent compound of cuprate superconductors, Sr 2 CuO 3 , constructed by the downfolding method. It is shown that the LSVQC achieves a 95% reduction of the circuit depth for simulating quantum many-body dynamics compared to the Trotterization at best while maintaining the same computational accuracy. We also demonstrate that the restriction to a subspace leads to a substantial reduction of required resources and improved accuracy compared to the case of considering the entire Hilbert space. Furthermore, we estimate the gate count needed to execute the quantum simulations using the LSVQC on near-term quantum computing architectures in the noisy intermediate-scale or early fault-tolerant quantum computing era. Our estimation suggests that the acceptable physical gate error rate for the LSVQC can be about one order of magnitude larger than that for the Trotterization.

DOI
10.1103/kb94-tf7t

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Simulation of quantum many-body systems is one of the most promising applications of quantum computers. It is crucial to efficiently implement the time-evolution operator as a quantum circuit to execute such simulations on near-term quantum computing devices with limited computational resources. However, standard approaches such as Trotterization sometimes require a deep quantum circuit, which is hard to implement on near-term quantum computers. Here, we propose a hybrid quantum-classical algorithm, called local subspace variational quantum compilation (LSVQC), for compiling the time-evolution operator of quantum many-body systems. The LSVQC performs a variational optimization to reproduce the action of the target time-evolution operator within a physically reasonable subspace. The optimization is performed for small local subsystems based on the Lieb-Robinson bound, which allows us to execute the cost function evaluation using small-scale quantum devices and/or classical computers. We demonstrate the validity of the LSVQC algorithm through numerical simulations of a simple spin-lattice model and an effective model of a parent compound of cuprate superconductors, Sr 2 CuO 3 , constructed by the downfolding method. It is shown that the LSVQC achieves a 95% reduction of the circuit depth for simulating quantum many-body dynamics compared to the Trotterization at best while maintaining the same computational accuracy. We also demonstrate that the restriction to a subspace leads to a substantial reduction of required resources and improved accuracy compared to the case of considering the entire Hilbert space. Furthermore, we estimate the gate count needed to execute the quantum simulations using the LSVQC on near-term quantum computing architectures in the noisy intermediate-scale or early fault-tolerant quantum computing era. Our estimation suggests that the acceptable physical gate error rate for the LSVQC can be about one order of magnitude larger than that for the Trotterization.
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.
Computation of the Green's function is crucial to study the properties of quantum many-body systems such as strongly correlated systems. Although the high-precision calculation of the Green's function is a notoriously challenging task on classical computers, the development of quantum computers may enable us to compute the Green's function with high accuracy even for classically-intractable large-scale systems. Here, we propose an efficient method to compute the real-time Green's function based on the local variational quantum compilation (LVQC) algorithm, which simulates the time evolution of a large-scale quantum system using a low-depth quantum circuit constructed through optimization on a smaller-size subsystem. Our method requires shallow quantum circuits to calculate the Green's function and can be utilized on both near-term noisy intermediate-scale and long-term fault-tolerant quantum computers depending on the computational resources we have. We perform a numerical simulation of the Green's function for the one- and two-dimensional Fermi-Hubbard model up to $4\ifmmode\times\else\texttimes\fi{}4$ sites lattice (32 qubits) and demonstrate the validity of our protocol compared to a standard method based on the Trotter decomposition. We finally present a detailed estimation of the gate count for the large-scale Fermi-Hubbard model, which also illustrates the advantage of our method over the Trotter decomposition.
Quantum dynamics compilation is an important task for improving quantum simulation efficiency: It aims to synthesize multiqubit target dynamics into a circuit consisting of as few elementary gates as possible. Compared to deterministic methods such as Trotterization, variational quantum compilation (VQC) methods employ variational optimization to reduce gate costs while maintaining high accuracy. In this work, we make a paradigm shift in VQC’s framework design. First, we use a cost function from recently established out-of-distribution generalization results in quantum machine learning: By learning the action of a given many-body dynamics on a small dataset of product states, we can obtain a unitary circuit that generalizes to highly entangled states such as the Haar random states. Moreover, a key observation is that the low complexity in the data states allows us to use tensor-network methods to compress such time-evolved product states by exploiting their low entanglement features. Utilizing warm start methods, our approach exceeds state-of-the-art compilation results in both system size and accuracy in one dimension. For the first time, we extend VQC to systems to two-dimensional systems, demonstrating a significant resource advantage over standard Trotterization methods. The flexibility of our framework highlights the method's promise for advancing quantum simulation tasks on near-term quantum processors.
Implementing time-evolution operators in shallow quantum circuits is important for quantum simulations. The standard method of Trotterization requires a large number of gates to achieve practical accuracy. Variational Quantum Simulation (VQS) is an algorithm that calculates the time evolution of a quantum state and can be executed with shallower circuits than Trotterization. However, the operator obtained by VQS evolves only a fixed initial state and is not the time evolution operator itself. In this paper, we propose Variational Quantum Operator Simulation (VQOS), a method to realize time evolution operators in shallow quantum circuits. This method is based on the variational principle for operators and does not require the implementation of the desired Trotter decomposition of the time evolution operator. We performed numerical simulations of the VQOS algorithm and successfully implemented the time evolution operator for closed systems in a quantum circuit that is up to 5 times shallower than the Trotterization. By providing a more practical way to implement time evolution operators, VQOS increases the applicability of near-term quantum computers.
The time evolution of quantum many-body systems is one of the most promising applications for near-term quantum computers. However, the utility of current quantum devices is strongly hampered by the proliferation of hardware errors. The minimization of the circuit depth for a given quantum algorithm is therefore highly desirable, since shallow circuits generally are less vulnerable to decoherence. Recently, it was shown that variational circuits are a promising approach to outperform current state-of-the-art methods such as Trotter decomposition, although the optimal choice of parameters is a computationally demanding task. In this work, we demonstrate a simplification of the variational optimization of circuits implementing the time evolution operator of local Hamiltonians by directly encoding constraints of the physical system under consideration. We study the expressibility of such constrained variational circuits for different models and constraints. Our results show that the encoding of constraints allows a reduction of optimization cost by more than one order of magnitude and scalability to arbitrary large system sizes, without loosing accuracy in most systems. Furthermore, we discuss the exceptions in locally constrained systems and provide an explanation by means of an restricted lightcone width after incorporating the constraints into the circuits.
Simulating the time evolution of quantum systems is a classically challenging and often intractable task due to the exponential scaling of its constituent operators. This scaling makes it a promising application for quantum computation, especially for correlated systems. Constructing an approximate time evolution operator can be done using the Trotter-Suzuki decomposition with an error that scales at most polynomially in the target simulation time and can be decreased at the cost of longer circuit depth. Variational approaches to classically optimize this approximation have shown significant improvement in accuracy for large quantum systems in both gate count and accuracy by several orders of magnitude. However, it is unclear if a similar improvement can be obtained for strongly correlated systems. We extend this variational approach to two example systems with strong correlations: a one-dimensional Fermi-Hubbard chain and an extended Fermi-Hubbard chain. The encoding between qubits and fermionic modes in these systems is generally nonlocal and so we propose an encoding scheme for these Hamiltonians with minimal overall operator distance between nearest neighbor connected qubits. We demonstrate that this variational method grants a similarly strong improvement in both gate count and accuracy and examine the performance of these optimized gate sequences using realistic noise models to estimate their performance on noisy quantum hardware. Finally, we discuss an approach to combine the optimized nonlocal gates of small chains to larger systems without the need for additional computational resources.