Optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver*

Efficient measurement of the expectation value of the molecular electronic Hamiltonian is crucial for the variational quantum eigensolver to be practical. A widely used strategy is to partition the Hamiltonian into linear combinations of operators that can be measured simultaneously (fragments). The total number of measurements required to obtain the Hamiltonian expectation value is then proportional to the sum of fragment variances. Here, we introduce two new methods for lowering the fragments’ variances by exploiting the flexibility in the fragments’ form. One approach is based on adding Pauli products (ghosts) that are compatible with members of multiple fragments such that all ghosts sum to zero [1], while the other approach uses the idempotency of the fermion occupation number operators to turn some parts of two-electron fragments into one-electron fragments, which are then partially collected into a purely one-electron fragment [2]. In both approaches, the modifications do not affect the total Hamiltonian expectation value but have non-vanishing contributions to the variance of each fragment. The proposed algorithms minimize individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on several molecules show that the algorithms can lower the number of measurements by more than an order of magnitude.

[1] SC, TCY, AFI, arXiv:2208.06563 (2022).

[2] SC, IL, AFI, arXiv:2208.14490 (2022).