Hardware-Friendly Estimation of LCU-Based Expectation Values via Quantum-to-Classical Workload Shifting

Abstract

Estimating the expectation value of an observable $O$ for a quantum state evolved under a general, possibly non-unitary, operator $A$ is a fundamental task in quantum simulation, chemistry, and variational algorithms. Existing linear-combination-of-unitaries (LCU) approaches implement $A$ on hardware, requiring ancilla qubits, controlled unitaries and, in randomized variants, fresh circuit executions for every Monte Carlo sample. These costs are prohibitive on NISQ and early fault-tolerant devices. We propose HERMES (Hardware-Efficient Reusable Measurements for Estimating LCU-based expectation valueS), which moves the processing of $A$ entirely to the classical domain. Expanding $A$ and $O$ in the Pauli basis and sampling pairs of Pauli terms classically, HERMES rewrites $μ= \mathrm{Tr}(OAρA^{\dagger})/\mathrm{Tr}(AρA^{\dagger})$ as a ratio of weighted sums of Pauli expectation values on the unmodified state $ρ$, with no ancillas, controlled unitaries, or quantum implementation of $A$. A two-level measurement reduction scheme (Pauli deduplication and commuting-group partitioning) bounds the number of distinct measurement settings independently of the sample count, unlike randomized LCU methods whose circuit executions grow linearly with it. On ground-state energy estimation of $\mathrm{H}_2$ and $\mathrm{LiH}$ via imaginary-time evolution, HERMES matches or beats Pauli single-ancilla LCU baseline using shallow circuits with no two-qubit gates beyond state preparation under qubit-wise commuting grouping. It is also more robust to noise and completes a noisy $\mathrm{LiH}$ simulation infeasible for the baseline, at the cost of classical overhead growing with the number of Pauli terms in $A$.