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arXiv:2003.12490 · Physical Review A

Entanglement production and convergence properties of the variational quantum eigensolver

Andreas J. C. Woitzik, Panagiotis Kl. Barkoutsos, Filip Wudarski, Andreas Buchleitner, Ivano Tavernelli

Published Oct 9, 2020• 20 citations
arXivHTMLPDF

Abstract

We perform a systematic investigation of variational forms (wave-function Ans\"atze), to determine the ground-state energies and properties of two-dimensional model fermionic systems on triangular lattices (with and without periodic boundary conditions), using the variational quantum eigensolver (VQE) algorithm. In particular, we focus on the nature of the entangler blocks which provide the most efficient convergence to the system ground state inasmuch as they use the minimal number of gate operations, which is key for the implementation of this algorithm in noisy intermediate-scale quantum computers. Using the concurrence measure, the amount of entanglement of the register qubits is monitored during the entire optimization process, illuminating its role in determining the efficiency of the convergence. Finally, we investigate the scaling of the VQE circuit depth as a function of the desired energy accuracy. We show that the number of gates required to reach a solution within an error $\ensuremath{\varepsilon}$ follows the Solovay-Kitaev scaling, $O[{log}_{10}^{c}(1/\ensuremath{\varepsilon})]$, with an exponent $c=1.31\ifmmode\pm\else\textpm\fi{}0.13$.

DOI
10.1103/physreva.102.042402

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Variational quantum eigensolvers (VQE) are one of the possible tasks that can be performed in noisy intermediate-scale quantum (NISQ) computers. While one of the limitations of NISQ platforms is their restricted number of available qubits, the theory of compressable matchgate circuits can be used to circumvent this limitation for certain types of Hamiltonians. We show how VQE algorithms can be used to find the ground state of quadratic fermionic Hamiltonians, providing an expressible ansatz in a logarithmic number of qubits. In particular, for systems of nnn orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with O(log⁡n)O\left(\log n\right)O(logn) sets of measurements. This result is invariant of the dimensions in which the nnn sites are arranged.
This work studies the variational quantum eigensolver (VQE) algorithm, which is designed to determine the ground state of a quantum mechanical system by combining classical and quantum hardware. Two methods of reducing the number of required qubit manipulations, prone to induce errors, for the variational quantum eigensolver are studied. First, we formally justify the multiple ℤ2 symmetry qubit reduction scheme first sketched by Bravyi et al. [arXiv:1701.08213 (2017)]. Second, we show that even in small, but non-trivial systems such as H2, LiH, and H2O, the choice of entangling methods (gate based or native) gives rise to varying rates of convergence to the ground state of the system. Through both the removal of qubits and the choice of entangler, the demands on the quantum hardware can be reduced. We find that in general, analyzing the VQE problem is complex, where the number of qubits, the method of entangling, and the depth of the search space all interact. In specific cases however, concrete results can be shown, and an entangling method can be recommended over others as it outperforms in terms of difference from the ground state energy.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
Abstract Variational quantum algorithms aim at harnessing the power of noisy intermediate-scale quantum (NISQ) computers, by using a classical optimizer to train a parameterized quantum circuit to solve tractable quantum problems. The variational quantum eigensolver (VQE) is one of the aforementioned algorithms designed to determine the ground-state of many-body Hamiltonians. Here, we apply the VQE to study the ground-state properties of N -component fermions. With such knowledge, we study the persistent current of interacting SU( N ) fermions, which is employed to reliably map out the different quantum phases of the system. Our approach lays out the basis for a current-based quantum simulator of many-body systems that can be implemented on NISQ computers.
In resource-efficient near-term quantum computers, development on the variational quantum eigensolver (VQE) algorithm that improves the accuracy of electronic structure calculations while maintaining low-requirement quantum gate counts is indispensable. We integrate Slater-type orbitals within the VQE framework, where the ground-state energy minimization with respect to the Slater exponents is shown as a second-order dependence on these parameters, enhancing accuracy compared with the standard methods while maintaining the same number of qubits. Using the full configuration interaction energy as the cost function, optimized parameters for first- and second-row elements are reported and related to the shielding constant, indicating a need to revise the previously established Slater rules. The active space is improved by substantially contracting the orbital exponents of extended basis sets relative to minimal ones, hence capturing additional electron-correlation effects. By considering the geometric structure of the parameter space, quantum Slater-exponent optimization (Q-SEO) demonstrates accelerated convergence to the target ground state, characterized by distinct parameter trajectories. The integration of a decision tree with Q-SEO effectively mitigates error in noisy environments.
The variational quantum eigensolver (VQE), which has attracted attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called Ans\"atze. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the Ans\"atze, it has been difficult to systematically analyze the performance of optimizers for the VQE. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achievable with a given Ansatz by using the idea of a parent Hamiltonian of a low-depth parametrized quantum circuit. We compare the convergence of several optimizers by varying the distance of the initial parameters from the solution and find that the converged energies showed a thresholdlike behavior depending on the distance. This work provides a systematic way to analyze optimizers for the VQE and contribute to the design of the Ansatz and its initial parameters.