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 $A$.

The paper studies estimation of $\mu=\mathrm{Tr}(OA\rho A^\dagger)/\mathrm{Tr}(A\rho A^\dagger)$ without implementing the possibly non-unitary operator $A$ on quantum hardware. HERMES expands $A$ and $O$ in the Pauli basis, samples pairs of terms from $A$ with probabilities proportional to coefficient magnitudes, and classically reduces their contributions to weighted Pauli expectations on the unchanged state $\rho$. Duplicate Pauli strings are removed and the remainder partitioned into commuting measurement groups. With qubit-wise commuting grouping, circuits require only state preparation, single-qubit basis rotations, and measurement, without ancillas or controlled operations. The number of distinct settings is bounded independently of the classical Monte Carlo sample count; this is not a fixed-precision shot-complexity guarantee. Numerical benchmarks compare against Pauli single-ancilla LCU for four-qubit $\mathrm{H}_2$ and twelve-qubit LiH at five bond lengths, using 10,000 sampled pairs and 10,000 shots per circuit. Evolution is approximated by $A=I-\tau H+\tau^2H^2/2$ with $\tau=100$, containing 24 and 25,542 Pauli terms, respectively. Against this approximation, noiseless mean absolute energy errors are 0.00111 versus 0.00109 Hartree for $\mathrm{H}_2$, and 0.00756 versus 0.03510 for LiH. Under a simulated IBM noise model, $\mathrm{H}_2$ errors are 0.124 versus 0.984 Hartree; noisy LiH baseline simulations could not be completed within available resources. Applicability requires manageable Pauli expansions and incurs classical overhead. The paper notes potentially difficult normalization when the denominator is small, shared evolution-truncation bias, and the absence of rigorous convergence and sample-complexity bounds.