Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer

Abstract

Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projectors to irreducible eigen-subspaces of available physical symmetries. Feasibility of symmetry projectors in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce higher numbers of terms for VQE measurement compare to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.

Digital quantum simulation of strong correlation effects with iterative quantum phase estimation over the variational quantum eigensolver algorithm: $\mathrm{H_4}$ on a circle as a case study

“…We also checked the error from the choice of mapping (Jordan-Wigner, Parity, and Bravyi-Kitaev schemes), and found that too to be well within 0.1 mH. This is not too surprising, as the transformed qubit Hamiltonians from each of these mappings are expected to be isospectral to the electronic Hamiltonian [34].…”

A systematic variational approach to band theory in a quantum computer

“…For all these reasons, VQE tends to be impractical for perfectly robust ansatz, and much of the literature focuses on methods for constructing effective ansatz accounting for system symmetries and hardware limitations. 15,[22][23][24][25] Because circuit depth and gate count are kept low, VQE is well-suited to NISQ devices.…”

Spin-projection for quantum computation: A low-depth approach to strong correlation

“…. ., which results in an exponential increase in the number of Pauli operators [19], hindering its practical usage.…”

“…Thus such spin-adapted (SA) methods [16,17] still incur Trotterization for large t ab i j . Some of the previous studies on this matter have suggested the use of a constrained approach [14,[18][19][20], wherein the Hamiltonian is augmented with a penalty term λ(Ŝ 2…”