Measurement reduction in variational quantum algorithms

Abstract

Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10-30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual-basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.

Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

“…This is to be expected, as the commutativity structure is defined by the physical electronic Hamiltonian; there is no obvious reason to expect that the mapping technique used would significantly affect this. Our results here are in agreement with those in Reference [ 56 ], where the same scaling was observed for the dual problem (i.e., coloring the compatibility graph of the Hamiltonian).…”

“…The commutativity structure—represented by the incompatibility graph—can be efficiently found on a classical computer. Greedy coloring heuristics can be used to partition this graph into commuting or anticommuting sets, a technique which has been used recently in the context of term reduction in variational quantum algorithms [ 53 , 54 , 55 , 56 ]. It remains an open question as to how these term reduction techniques can be related to Trotter ordering strategies when implementing variational quantum algorithms.…”

“…There is no immediately obvious reason to suspect that coloring the Hamiltonian using a heuristic is problematic for the purposes of our analysis. Indeed, these heuristics have seen frequent recent use for partitioning electronic structure Hamiltonians, for the purpose of measurement reduction in variational quantum algorithms [ 53 , 54 , 55 , 56 ]. Nonetheless, it does suggest that caution should be used when considering the generalizability of the results presented.…”

“…For example, it is not immediately obvious whether it would be preferable to split the graph into few large independent sets, or many smaller ones. As mentioned above, various schemes of Hamiltonian term partitioning have been developed to reduce the cost of variational quantum algorithms, by combining sets of commuting or anticommuting terms [ 53 , 54 , 55 , 56 ]. It remains an open question as to whether Trotter ordering schemes based on graph coloring heuristics can be applied to Hamiltonians with terms combined in such a fashion.…”

Orbital optimized unitary coupled cluster theory for quantum computer

“…The costs and errors owing to the measurements can be reduced using simultaneous measurements and/or other techniques. Such techniques have been actively developed [65][66][67][68][69][70][71][72][73][74][75], and they can be used to alleviate the problems caused by measuring 1 and 2RDMs. In practice, OO-VQE repeatedly performs the VQE and the orbital optimization until convergence.…”