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arXiv:2007.14338 · PRX Quantum

Quantifying the Efficiency of State Preparation via Quantum Variational Eigensolvers

Gabriel Matos, Sonika Johri, Zlatko Papić

Published Jan 15, 2021• 24 citations
arXivHTMLPDF

Abstract

Recently, there has been much interest in the efficient preparation of complex quantum states using low-depth quantum circuits, such as the quantum approximate optimization algorithm (QAOA).While it has been numerically shown that such algorithms prepare certain correlated states of quantum spins with surprising accuracy, a systematic way of quantifying the efficiency of the QAOA in general classes of models has been lacking.Here, we propose that the success of the QAOA in preparing ordered states is related to the interaction distance of the target state, which measures how close that state is to the manifold of all Gaussian states in an arbitrary basis of single-particle modes.We numerically verify this for several examples of nonintegrable quantum models, including Ising models with two-and three-spin interactions and the cluster model in an external field.Our results suggest that the structure of the entanglement spectrum, as witnessed by the interaction distance, correlates with the success of QAOA state preparation, and that this correlation also contains information about different phases present in the model.We conclude that the QAOA typically finds a solution that perturbs around the closest free-fermion state.

DOI
10.1103/prxquantum.2.010309

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Recently, there has been much interest in the efficient preparation of complex quantum states using low-depth quantum circuits, such as the quantum approximate optimization algorithm (QAOA).While it has been numerically shown that such algorithms prepare certain correlated states of quantum spins with surprising accuracy, a systematic way of quantifying the efficiency of the QAOA in general classes of models has been lacking.Here, we propose that the success of the QAOA in preparing ordered states is related to the interaction distance of the target state, which measures how close that state is to the manifold of all Gaussian states in an arbitrary basis of single-particle modes.We numerically verify this for several examples of nonintegrable quantum models, including Ising models with two-and three-spin interactions and the cluster model in an external field.Our results suggest that the structure of the entanglement spectrum, as witnessed by the interaction distance, correlates with the success of QAOA state preparation, and that this correlation also contains information about different phases present in the model.We conclude that the QAOA typically finds a solution that perturbs around the closest free-fermion state.
We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras, we show that the Quantum Approximate Optimization Algorithm (QAOA) on a one-dimensional lattice – with and without decoupled angles – is able to prepare all fermionic Gaussian states respecting the symmetries of the circuit. Leveraging these results, we numerically study the interplay between these symmetries and the locality of the target state, and find that an absence of symmetries makes nonlocal states easier to prepare. An efficient classical simulation of Gaussian states, with system sizes up to 80 and deep circuits, is employed to study the behavior of the circuit when it is overparameterized. In this regime of optimization, we find that the number of iterations to converge to the solution scales linearly with system size. Moreover, we observe that the number of iterations to converge to the solution decreases exponentially with the depth of the circuit, until it saturates at a depth which is quadratic in system size. Finally, we conclude that the improvement in the optimization can be explained in terms of better local linear approximations provided by the gradients.
Abstract The variational preparation of complex quantum states using the quantum approximate optimization algorithm (QAOA) is of fundamental interest, and becomes a promising application of quantum computers. Here, we systematically study the performance of QAOA for preparing ground states of target Hamiltonians near the critical points of their quantum phase transitions, and generating Greenberger–Horne–Zeilinger (GHZ) states. We reveal that the performance of QAOA is related to the translational invariance of the target Hamiltonian: without the translational symmetry, for instance due to the open boundary condition (OBC) or randomness in the system, the QAOA becomes less efficient. We then propose a generalized QAOA assisted by the parameterized resource Hamiltonian (PRH-QAOA), to achieve a better performance. In addition, based on the PRH-QAOA, we design a low-depth quantum circuit beyond one-dimensional geometry, to generate GHZ states with perfect fidelity. The experimental realization of the proposed scheme for generating GHZ states on Rydberg-dressed atoms is discussed. Our work paves the way for performing QAOA on programmable quantum processors without translational symmetry, especially for recently developed two-dimensional quantum processors with OBC.
The quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm intending to find the ground state of a target Hamiltonian. Theoretically, QAOA can obtain the approximate solution if the quantum circuit is deep enough. Actually, the performance of QAOA decreases practically if the quantum circuit is deep since near-term devices are not noise free, and the errors caused by noise accumulate as the quantum circuit increases. In order to reduce the depth of quantum circuits, we propose an ansatz dubbed as ``Shortcuts to QAOA'' (S-QAOA), S-QAOA provides shortcuts to the ground state of the target Hamiltonian by including more two-body interactions and releasing the parameter freedoms. To be specific, besides the existing ZZZZZZ interaction in the QAOA ansatz, other two-body interactions are introduced in the S-QAOA ansatz such that the approximate solutions could be obtained with smaller circuit depth. Considering the Max-Cut problem and the Sherrington-Kirkpatrick model, numerically computation shows the YYYYYY interaction has the best performance. The reason for this might arise from the counterdiabatic effect generated by YYYYYY interaction. On top of this, we release the freedom of parameters of two-body interactions, which a priori do not necessarily have to be fully identical, and numerical results show that it is worth paying the extra cost of having more parameter freedom since one has a greater improvement on success rate.
Assemblies of strongly interacting fermions, whether in a condensed-matter or a quantum chemistry context, range amongst the most promising candidate systems for which quantum computing platforms could provide an advantage. Near-term quantum state preparation is typically realized by means of the variational quantum eigensolver (VQE) algorithm. One of the main challenges to a successful implementation of VQE lies in the sensitivity to noise exhibited by deep variational circuits. On the other hand, sufficient depth must be allowed to be able to reach a good approximation to the target state. In this work, we present a refined VQE scheme that consists in topping VQE with state-informed updates of the elementary fermionic modes (spin-orbitals). These updates consist in moving to the natural-orbital basis of the current, converged variational state, a basis we argue eases the task of state preparation. We test the method on the Hubbard model in the presence of experimentally relevant noise levels. For a fixed circuit structure, the method is shown to enhance the capabilities of the circuit to reach a state close to the target state without incurring too much overhead from shot noise. Moreover, coupled with an adaptive VQE scheme that constructs the circuit on the fly, we evidence reduced requirements on the depth of the circuit as the orbitals get updated.
Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).