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arXiv:1910.02719 · Physical Review Applied

Resource Estimation for Quantum Variational Simulations of the Hubbard Model

Zhenyu Cai

Published Jul 21, 2020• 77 citations
arXivHTMLPDF

Abstract

As the advances in quantum hardware bring us into the noisy intermediate-scale quantum (NISQ) era, one possible task we can perform without quantum error correction using NISQ machines is the variational quantum eigensolver (VQE) due to its shallow depth. A specific problem that we can tackle is the strongly interacting Fermi-Hubbard model, which is classically intractable and has practical implications in areas like superconductivity. In this paper, we outline the details about the gate sequence, the measurement scheme, and the relevant error-mitigation techniques for the implementation of the Hubbard VQE on a NISQ platform. We perform resource estimation for both silicon spin qubits and superconducting qubits for a 50-qubit simulation, which cannot be solved exactly via classical means, and find similar results. The number of two-qubit gates required is on the order of 20 000. Hence, to suppress the mean circuit-error count to a level such that we can obtain meaningful results with the aid of error mitigation, we need to achieve a two-qubit gate error rate of approximately ${10}^{\ensuremath{-}4}$. When searching for the ground state, we need a few days for one gradient-descent iteration, which is impractical. This can be reduced to around 10 min if we distribute our task among hundreds of quantum-processing units. Hence, implementing a 50-qubit Hubbard model VQE on a NISQ machine can be on the brink of being feasible in near term, but further optimization of our simulation scheme, improvements in the gate fidelity, improvements in the optimization scheme and advances in the error-mitigation techniques are needed to overcome the remaining obstacles. The scalability of the hardware platform is also essential to overcome the runtime issue via parallelization, which can be done on one single silicon multicore processor or across multiple superconducting processors.

DOI
10.1103/physrevapplied.14.014059

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Abstract Quantum computers promise to solve electronic structure, lattice dynamics, and strongly correlated problems beyond classical reach. However, near-term noisy intermediate scale quantum (NISQ) processors require systematic benchmarking to establish where variational algorithms with error mitigation can achieve chemical accuracy and cor rectly predict physical phenomena. Here we present 28 quantum material simula tions implemented with Qiskit, spanning molecular hydrogen (H 2 ) through the iron molybdenum cofactor (FeMoco), one-dimensional Hubbard chains, Kitaev honeycomb spin liquids, topological insulators (SSH model), BCS superconductors, and disordered Anderson insulators. Using the variational quantum eigensolver (VQE) with UCCSD ansatz, we achieve chemical accuracy (±1.6 mHa) for H 2 across the full dissociation curve and recover exact ground-state energies for H4 within 9 mHa after zero-noise extrapolation (ZNE) — reducing raw VQE errors from 152 mHa to 9 mHa. We accu rately identify the transverse-field Ising quantum critical point (h c = J) with 4 qubits, resolve Majorana Dirac dispersions in the Kitaev spin liquid, and reproduce the Mott transition in the Fermi-Hubbard model at U c = 6t. Resource estimation for FeMoco reveals O(N 2.5 ) scaling, projecting chemical accuracy with ∼200 logical qubits. Across all benchmarks, we quantify the accuracy-circuit depth tradeoff, demonstrating that depth-6 VQE ans¨atze achieve chemical accuracy for small molecules while depth-10 is required for correlated lattices. These results establish a comprehensive benchmark suite for evaluating quantum hardware, error mitigation protocols, and variational al gorithms for materials science on NISQ processors.
Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a nonzero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently large, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.
Variational quantum algorithms have promising applications in noisy intermediate-scale quantum (NISQ) devices. These algorithms rely on a classical optimization outer loop that minimizes a parameterized quantum circuit function. The optimization in variational quantum eigensolver (VQE) is NP-hard, meaning that finding the optimal solution is infeasible in the worst-case scenario. One way to address this challenge is through parallel optimization of parameters using multiple-parameterized quantum circuits. However, this approach is unsuitable for cloud-based quantum processing unit utilization due to the increased number of quantum circuit executions. Although NISQ devices have limitations in terms of gate depth, their size has been growing in recent years. Therefore, implementing multiple-parameterized quantum circuits in NISQ devices can suppress the increase in the number of executions. In this study, we propose a parallel-VQE, which leverages the parallel execution of parameterized quantum circuits to perform parallel parameter optimization in VQE, achieving convergence to solutions closer to the ground state. We validate the effectiveness of parallel-VQE in solving the random weighted max-cut problem using numerical simulations and a real quantum device. We present the results of running up to six circuits in parallel (120 qubits) and demonstrate the advantages of using multiple units to improve computational accuracy. This study provides a potential method for solving eigenvalue problems and combinatorial optimization problems for future quantum devices.
Abstract We present a method to improve the convergence of variational algorithms based on hidden inverses (HIs) to mitigate coherent errors. In the context of error mitigation, this means replacing the hardware implementation of certain Hermitian gates with their inverses. Doing so results in noise cancellation and a more resilient quantum circuit. This approach improves performance in a variety of two-qubit error models where the noise operator also inverts with the gate inversion. We apply the mitigation scheme on superconducting quantum processors running the variational quantum eigensolver (VQE) algorithm to find the H 2 ground-state energy. When implemented on superconducting hardware we find that the mitigation scheme effectively reduces the energy fluctuations in the parameter learning path in VQE, reducing the number of iterations for a converged value. We also provide a detailed numerical simulation of VQE performance under different noise models and explore how HIs and randomized compiling affect the underlying loss landscape of the learning problem. These simulations help explain our experimental hardware outcomes, helping to connect lower-level gate performance to application-specific behavior in contrast to metrics like fidelity which often do not provide an intuitive insight into observed high level performance.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.