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arXiv:2311.06347 · Physical review. B./Physical review. B

Optimal compression of constrained quantum time evolution

Maurits S. J. Tepaske, David J. Luitz, Dominik Hahn

Published May 15, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.205134

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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.
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.
The simulation of time evolution of large quantum systems is a classically challenging and often intractable task, making it a promising application for quantum computation. A Trotter-Suzuki approximation yields an implementation thereof, where a certain desired accuracy can be achieved by raising the gate count adequately. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of large quantum systems. Our strategy improves on the Trotter-Suzuki ansatz in accuracy and gate count by several orders of magnitude. In a NISQ-friendly setting, we can either significantly reduce the approximation error while using the same amount of gates as the Trotter-Suzuki ansatz or achieve a comparable accuracy with significantly fewer gates. We find, that our strategy outperforms the Trotter-Suzuki benchmark in translation invariant systems as well as for open boundaries.
Many optimally scaling quantum simulation algorithms employ controlled time evolution of the Hamiltonian, which is typically the major bottleneck for their efficient implementation. This work establishes a compression protocol for encoding the controlled time evolution operator of translationally invariant, local Hamiltonians into a quantum circuit. It achieves a near-optimal in time ttt scaling for circuit depth O(t polylog(tN/ε))\mathcal{O}(t \text{ polylog}(t N/ε))O(t polylog(tN/ε)), while reducing the control overhead from a multiplicative to an additive factor. We report that this compression protocol enables the implementation of Iterative Quantum Phase Estimation with as few as 414 CNOT gates for a frustrated quantum spin system on a 6$\times$6 triangular lattice and delivers ground state energy errors below 1% (with ±\pm± 1.5% variation, calculated with a hardware noise aware pipeline) on a 4$\times$4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.
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.
The advent of near-term digital quantum computers could offer us an exciting opportunity to investigate quantum many-body phenomena beyond that of classical computing. To make the best use of the hardware available, it is paramount that we have methods that accurately simulate Hamiltonian dynamics for limited circuit depths. In this paper, we propose a method to classically optimize unitary brickwall circuits to approximate quantum time evolution operators. Our method is scalable in system size through the use of tensor networks. We demonstrate that, for various three-body Hamiltonians, our approach produces quantum circuits that can outperform trotterization in both their accuracy and the quantum circuit depth needed to implement the dynamics, with the exact details being dependent on the Hamiltonian. We also explain how to choose an optimal time step that minimizes the combined errors of the quantum device and the brickwall circuit approximation. Published by the American Physical Society 2024