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arXiv:1908.09533

Short-depth trial-wavefunctions for the variational quantum eigensolver based on the problem Hamiltonian

Gian Salis, Nikolaj Moll

Published Aug 26, 2019• 2 citations
arXivHTMLPDF

Abstract

For the variational quantum eigensolver we propose to generate trial wavefunctions from a small amount of selected Pauli terms of the problem Hamiltonian. Two different approaches, one inspired by the quantum approximate optimization algorithm and the other by imaginary-time evolution, are proposed and studied in detail. Using numerical calculations, we study the efficiency of these trial wavefunctions for finding the ground-state energy of three molecules: H2, LiH and H2O. We find that only a small number of Pauli terms are needed to reach chemical accuracy, leading to short-depth quantum circuits with a small number of variational parameters. For the LiH molecule, the quantum circuit consists of 36 two-qubit gates, 45 one-qubit gates, and four variational parameters, with a favorable scaling for larger molecules.

DOI
10.48550/arxiv.1908.09533

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For the variational quantum eigensolver we propose to generate trial wavefunctions from a small amount of selected Pauli terms of the problem Hamiltonian. Two different approaches, one inspired by the quantum approximate optimization algorithm and the other by imaginary-time evolution, are proposed and studied in detail. Using numerical calculations, we study the efficiency of these trial wavefunctions for finding the ground-state energy of three molecules: H2, LiH and H2O. We find that only a small number of Pauli terms are needed to reach chemical accuracy, leading to short-depth quantum circuits with a small number of variational parameters. For the LiH molecule, the quantum circuit consists of 36 two-qubit gates, 45 one-qubit gates, and four variational parameters, with a favorable scaling for larger molecules.
In the noisy near-term quantum computing era variational quantum algorithms are promising to explore the boundaries of quantum advantage. One prevalent instance is the variational quantum eigensolver (VQE) employed in quantum chemistry to approximate ground-state energies of molecules. The performance of VQE depends on the design of the parametrized quantum circuit, as well as the optimization algorithm used to optimize those parameters. Circuit optimization methods have been developed to generate quantum circuits to reach accurate approximations to the ground-states. In this work, we conduct an empirical comparison between two prominent adaptive quantum circuit optimization algorithms, namely qubit-ADAPT-VQE and a reinforcement learning (RL) based method, on the problem of finding the ground state energy of Lithium Hydride (LiH). In our experiment, we focus on the circuit depth generated by each circuit optimization method and the number of two-qubit quantum gates in the best-found circuit.
Abstract We present a comprehensive quantum chemical study of nine molecules H₂, HeH⁺, LiH, BeH₂, H₂O, NH₃, CH₄, CO₂, and N₂ using the Variational Quantum Eigensolver (VQE) algorithm implemented in Qiskit 2.4.1. Molecular Hamiltonians are constructed in the STO-3G minimal basis set, transformed to qubit operators via the Jordan-Wigner mapping, and optimized using the n-local hardware-efficient ansatz with the SLSQP optimizer. Ground state energies are benchmarked against Full Configuration Interaction (FCI)/STO-3G reference values from authoritative literature sources. VQE achieves accuracies from 96.46% (HeH⁺) to 99.99% (H₂), average 98.68%. Potential energy surfaces are generated for all nine molecules and compared with high level ab initio literature PES references including the Kolos-Wolniewicz H₂ curve, the Polyansky water surface, and the Bytautas-Ruedenberg N₂ FCI surface. Equilibrium bond lengths agree with experiment to within 0.000–0.006 Å. Quantum ansatz circuits are analyzed in detail for all molecules. These results confirm the viability of near-term VQE algorithms for molecular quantum chemistry across diverse bond types and geometries.
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
Abstract The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the H2{H}_{2}H2​ H 2 and LiHLiHLiH L i H molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.
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.
Quantum computing hardware is now surpassing the 50-qubit level, which cannot be simulated with a classical computer. In practical use, generating the ground states of slightly different Hamiltonians (e.g.e.g.e.g. induced by the different atomic coordinates in a molecule) is often required, but here the traditional form of the variational quantum eigensolver is inefficient. Thus the authors discuss ``training'' a quantum circuit with a small number of Hamiltonians and then generalizing the output by interpolation, and also propose a purpose-built quantum circuit, to greatly reduce the time needed to find ground states with a near-term quantum computer.