Abstract
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule. This increases the computational cost and can increase uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: (1) grouping terms into parts whose eigenstates have a single-qubit product structure, and (2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in severalfold reduction of separately measured terms are illustrated with regard to the H2 and LiH problems.
Highlights
The last step uses a hybrid quantum-classical technique where a classical computer suggests a trial unitary transformation U, and its quantum counterpart provides an energy expectation value of EU 1⁄4 hJ0|U†HqU|J0i, here |J0i is an initial qubit wavefunction
The variational quantum eigensolver (VQE) was successfully implemented on several quantum computers and used for few small molecules up to BeH2.13
In the conventional VQE scheme the Hq is separated into sums of qubit-wise commuting (QWC) terms, X
Summary
One of the most practical schemes for solving the electronic structure problem of current and near-future universal quantum computers is the variational quantum eigensolver (VQE) method.[1,2,3,4,5] This approach involves the following steps: (1) reformulating the electronic Hamiltonian (He) in the second quantized form, (2) transforming He to the qubit form (Hq) by applying iso-spectral fermion-spin transformations such as Jordan–Wigner (JW)[6,7] or more resource-efficient Bravyi–Kitaev (BK),[8,9,10,11,12] (3) solving the eigenvalue problem for Hq by variational optimization of unitary transformations for a qubit wavefunction. Due to the linear scaling of the variance sum with n and the inverse quadratic scaling of variances of individual terms with n, the overall scaling of the variance is inversely proportional to n and can be made arbitrarily small by choosing large enough n This follows from a wrong assumption that parts (Hq/n) are independent and covariances between them are zero. The number of non-commuting terms in Hq grows with the size of the original molecular problem, and the total uncertainty from the measurement of individual terms will increase This increase raises the standard deviation of the total measurement process and leads to a large number of measurements to reach convergence in the energy expectation value. Measurement of newly introduced terms requires the scheme appearing in the cluster-state quantum computing,[15,16] it is qubit-wise measurement with use of previous measurement results to de ne what single-qubit operators to measure
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