Quantum Orbital Minimization Method for Excited States Calculation on a Quantum Computer

Abstract

We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parametrized ansatz circuits representing eigenstates, qOMM implements quantum circuits to represent the objective function in the orbital minimization method and adopts a classical optimizer to minimize the objective function with respect to the parameters in ansatz circuits. The objective function has an orthogonality constraint implicitly embedded, which allows qOMM to apply a different ansatz circuit to each input reference state. We carry out numerical simulations that seek to find excited states of H2, LiH, and a toy model consisting of four hydrogen atoms arranged in a square lattice in the STO-3G basis with UCCSD ansatz circuits. Comparing the numerical results with existing excited states methods, qOMM is less prone to getting stuck in local minima and can achieve convergence with more shallow ansatz circuits.

Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization

“…OptOrbVQE could easily be generalized to "Op-tOrbMC-VQE" or "OptOrbSSVQE" by modifying eq 9 to be a weighted sum of the transformed Hamiltonian with respect to mutually orthogonal parametrized states in the same manner as these methods. Orbital optimization could also be applied to the quantum Orbital Minimization Method (qOMM) 34 by modifying eq 9 in an analogous way. These methods all find low-lying excited states simultaneously through the minimization of a single objective function.…”

Qubit Count Reduction by Orthogonally Constrained Orbital Optimization for Variational Quantum Excited-State Solvers

“…For example, in a previous work by the authors, it was demonstrated that the convergence quality of state-averaged eigensolvers such as SSVQE in a fixed basis is highly sensitive to the ansatz expressiveness and choice of circuit initialization (compared to the ground-state VQE problem and the overlap-based qOMM excited-state solver). 32 In this work, we investigate the extent to which analogous observations hold true for the orbital-optimized case as well.…”

“…SSVQE is tested with CIS and CISD as well as an "excited Hartree−Fock" initialization used in a previous study by the authors. 32 This initialization applies single-particle Fermionic excitations to the Hartree−Fock state and chooses the resulting Slater determinants with the lowest energy to initialize the circuit. Such states are orthogonal and can thus be used with both MCVQE and SSVQE.…”

“…The optimization with respect to θ is handled via one of several known quantum excited-state solvers such as SSVQE, 35 MCVQE, 36 VQD, 37 or qOMM. 32 The optimization with respect to V ̂(keeping θ fixed) is carried out using an orthogonally constrained optimization procedure. In this work, we use an orthogonally constrained projected gradient method, 31 which has a parameter update step defined as…”

Improved Iterative Quantum Algorithm for Ground‐State Preparation

“…[8] Many works have focused on the ground state preparation of a Hamiltonian. [9][10][11][12][13] For instance, the earlier work using quantum phase estimation prepares the ground state by projecting a guess state onto the ground state. [14] However, the long coherence time and high-fidelity gates render it impractical for noisy intermediate-scale quantum (NISQ) devices.…”