Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver

Abstract

Measuring the expectation value of the molecular electronic Hamiltonian is one of the challenging parts of the variational quantum eigensolver. A widely used strategy is to express the Hamiltonian as a sum of measurable fragments using fermionic operator algebra. Such fragments have an advantage of conserving molecular symmetries that can be used for error mitigation. The number of measurements required to obtain the Hamiltonian expectation value is proportional to a sum of fragment variances. Here, we introduce a new method for lowering the fragments' variances by exploiting flexibility in the fragments' form. Due to idempotency of the occupation number operators, some parts of two-electron fragments can be turned into one-electron fragments, which then can be partially collected in a purely one-electron fragment. This repartitioning does not affect the expectation value of the Hamiltonian but has non-vanishing contributions to the variance of each fragment. The proposed method finds the optimal repartitioning by employing variances estimated using a classically efficient proxy for the quantum wavefunction. Numerical tests on several molecules show that repartitioning of one-electron terms lowers the number of measurements by more than an order of magnitude.

Quantum Simulations of Fermionic Hamiltonians with Efficient Encoding and Ansatz Schemes

“…In particular, we could go beyond the minimum models for the NV – and VV 0 and tackle complex defects such as V Si – for the first time. Work is in progress to improve the efficiency of the measurements of ⟨ H ⟩ on quantum architectures, for example, by adopting advanced measurement techniques with different term groupings, , fragmentation procedures, , and classical shadows, , and to extend the applicability of our protocol to larger active spaces appropriate, e.g., to investigate adsorbates on surfaces or ions and nanostructures in solution. We finally note that establishing which algorithms are better suited to achieve quantum advantage in electronic structure calculations remains an open area of research.…”

Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms

“…For a summary of common quantum measurement techniques, with a focus on those employed in this work, we refer the reader to Appendix A. In this study, we consider the following popular measurement schemes: fully commuting (FC), qubit-wise commuting (QWC), and Majorana classical shadow (CS), , the derandomized extension of QWC classical shadow (Derand), FC and QWC sorted insertion (SI) algorithm, FC and QWC iterative measurement allocation (IMA), FC iterative coefficient splitting (ICS), and fluid Fermionic fragments (F 3 ) . Specifically, the “Full” version of F 3 based on the low-rank (LR) ,, decomposition was chosen due to its efficient combination of low classical computational cost with reduced quantum measurements.…”

“…The M (ϵ) metric varies significantly depending on the quantum measurement scheme employed to obtain Ĥ α and Û α . Many methods obtain Ĥ α and the corresponding Û α according to an optimized deterministic scheme. − ,− Alternatively, one can first sample Û α probabilistically, − then compose Ĥ α by taking a linear combination of every operator that is rotated into the measurable Ising form by Û α . The latter probabilistic approaches are commonly known as classical shadow-based techniques.…”

“…The latter probabilistic approaches are commonly known as classical shadow-based techniques. Typically, if one is interested in measuring the expectation value of a single operator (e.g., Ĥ e ), the deterministic methods have lower M (ϵ)’s. ,, On the other hand, if one wishes to measure the expectation values of many observables simultaneously [e.g., all k -body reduced density matrices ( k -RDM)], the classical shadow-based schemes are superior. , …”

“…Several studies exist that compare the performance of measurement techniques in ground state VQE (e.g., see refs , , , and ). Following our expectations, these studies suggest that deterministic measurement schemes that utilize the Hamiltonian ( Ĥ e ) and quantum state (|ψ⟩) information to minimize M (ϵ) outperform classical shadow techniques.…”