Fidelity overhead for non-local measurements in variational quantum algorithms

Abstract

Measuring quantum observables by grouping terms that can be rotated to sums of only products of Pauli ẑ operators (Ising form) is proven to be efficient in near term quantum computing algorithms. This approach requires extra unitary transformations to rotate the state of interest so that the measurement of a fragment's Ising form would be equivalent to measurement of the fragment for the unrotated state. These extra rotations allow one to perform a fewer number of measurements by grouping more terms into the measurable fragments with a lower overall estimator variance. However, previous estimations of the number of measurements did not take into account non-unit fidelity of quantum gates implementing the additional transformations. Through a circuit fidelity reduction, additional transformations introduce extra uncertainty and increase the needed number of measurements. Here we consider a simple model for errors introduced by additional gates needed in schemes involving grouping of commuting Pauli products. For a set of molecular electronic Hamiltonians, we confirm that the numbers of measurements in schemes using non-local qubit rotations are still lower than those in their local qubit rotation counterparts, even after accounting for uncertainties introduced by additional gates.

Optimizing the number of measurements for vibrational structure on quantum computers: coordinates and measurement schemes

“…The QWC scheme thus has a prima facie advantage in terms of circuit depth. For electronic Hamiltonians it has however been shown that this advantage is outweighed by other factors making the FC scheme preferable 18,19 in that case.…”

“…However, it has been shown empirically that the estimation of the electronic ground state of small molecules to chemical accuracy requires infeasible runtimes due to sampling overhead, which is called the measurement problem. 14,15 Different measurement schemes have been employed to reduce the sampling overhead, including the grouping of Pauli products into mutually commuting sets, [16][17][18][19][20][21][22][23][24][25] classical shadow tomography, [26][27][28][29] low-rank factorizations of the electronic Hamiltonian 30 and algebraic approaches. 31 Current studies have however only investigated electronic Hamiltonians.…”

Improving quantum measurements by introducing "ghost" Pauli products

“…It was shown 18 that M (ǫ) in the QWC techniques are typically higher than M (ǫ) in the FC techniques. Moreover, a recent study 22 compared M (ǫ) in FC and QWC techniques while accounting for the non-unit fidelities of the quantum gates implementing Ûα . This study demonstrated that even after considering fidelities of one-and two-qubit gates, the FC fragmentation methods usually have lower M (ǫ).…”