Unitary Partitioning Approach to the Measurement Problem in the Variational Quantum Eigensolver Method

Abstract

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as N 4 , where N is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anti-commuting terms of the Hamiltonian in a single series of single-qubit measurements. Compared to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity, the unitary partitioning provides at least an N -fold reduction in the number of measurable groups.

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

“…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.…”

Challenges and Opportunities of Near-Term Quantum Computing Systems

“…While the UCC approach offers a structured ansatz that maintains physical symmetries, the hardware-efficient circuits employ interactions that are native to the quantum hardware. Other important considerations for the practical implementation of VQE for quantum chemistry are qubit-efficient fermionic mapping schemes [126], robustness of classical optimizers to hardware noise, and most importantly, the effect of decoherence [32], [33], [37] and the measurement cost for molecular Hamiltonians [130], [131].…”