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

Hari Hara Suthan Chittoor, Thirumalai Manimaran

arXiv:2610.32841v1quant-phcs.ET

TL;DR

LCU-based estimation of expectation values for general, possibly non-unitary operators requires quantum circuits with ancillas and controlled operations, and randomized variants execute fresh circuits per sample. HERMES instead processes the operator classically through Pauli expansions and reusable measurements; on H2 and LiH energy estimation it matches or improves the PSA-LCU baseline, with classical overhead that grows with the number of Pauli terms in A.

  • Problem

    LCU-based estimation implements A with ancillas and controlled gates, while randomized variants require fresh circuit executions for each Monte Carlo sample.

    Source: §I · p. 1, D. Single-Ancilla LCU · p. 4

  • Method

    HERMES expands A and O in the Pauli basis and classically samples A-term pairs to estimate weighted Pauli expectations on the unmodified input state.

    Source: §I · p. 2

  • Results

    Across H2 and LiH, HERMES matches or improves PSA-LCU; for ideal LiH, MAE falls by factors of 4.6 (Eaprx) and 2.3 (EF CI), and only HERMES yields noisy LiH results.

    Source: D. Discussion · p. 10

  • Takeaways & Limitations

    HERMES circuits have no two-qubit gates, and its measurement settings are bounded independently of Monte Carlo sample count; noisy LiH remained tractable when PSA-LCU could not complete.

    Source: D. Discussion · pp. 10–11, §V · p. 11

    HERMES incurs classical overhead that grows with the number of Pauli terms in A.

    Source: D. Discussion · p. 11

Abstract

from arXiv · show

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 $\mu = \mathrm{Tr}(OA\rho A^{\dagger})/\mathrm{Tr}(A\rho A^{\dagger})$ as a ratio of weighted sums of Pauli expectation values on the unmodified state $\rho$, 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 $A$.

I. INTRODUCTION

Estimating a scalar expectation value after applying a general, possibly non-unitary operator is a central task, but conventional LCU methods impose substantial hardware costs. HERMES shifts processing of the operator to classical Pauli-term sampling and reuses measurements, while retaining quantum hardware for expectation values on the input state.

I. INTRODUCTION

  • The target is the scalar μ for a state ρ, Hermitian observable O, and general operator A, rather than the full evolved state.The stated setting includes unitary Hamiltonian simulation, imaginary-time evolution, and quantum linear systems.Source: §I · p. 1
  • Standard LCU requires ancillas and multi-qubit controlled gates, making it difficult to use on NISQ and early fault-tolerant hardware.The standard method also uses amplitude amplification, which multiplies effective circuit depth.Source: §I · p. 1
  • Randomized LCU still requires coherent implementation of some unitaries, and each Monte Carlo sample requires a fresh circuit execution.For fixed shots per sample, circuit executions grow linearly with sample count.Source: §I · pp. 1–2
  • The Pauli measurement literature groups commuting terms, but had not integrated that measurement strategy with LCU estimation of μ, whose structure depends on both A and O.The introduction identifies this as a gap alongside the hardware demands of existing LCU approaches.Source: §I · p. 2
  • HERMES expands A and O in the Pauli basis and samples pairs of A terms classically, avoiding quantum implementation of A.It estimates the desired scalar from weighted sums of Pauli expectation values on the unmodified state.Source: §I · p. 2

1. Pauli Operators and Pauli Strings

Pauli strings provide an operator basis, and their commutation relations determine which expectation values can be measured together. Grouping commuting strings can reduce the number of measurement settings needed.

1. Pauli Operators and Pauli Strings

  • An n-qubit Pauli string is a tensor product of single-qubit Pauli operators, and Pauli strings form a basis for operators on the n-qubit space.Multiplying strings produces another Pauli string up to an overall phase.Source: §1 · p. 3

2. Commuting Pauli Operators

  • Two Pauli strings commute when the number of positions with anticommuting non-identity operators is even; otherwise they anticommute.The passages also give the equivalent commutation criterion in the displayed equations.Source: §2 · p. 3
  • Mutually commuting Pauli strings share a common eigenbasis and can be measured together after a basis transformation, reducing measurement settings.The passage connects this property to measurement optimization.Source: §2 · p. 3

B. Linear Combination of Unitaries

LCU represents an operator as a weighted sum of unitaries and implements that combination using ancilla-controlled operations. Postselection on the ancilla’s zero state yields the transformed state up to normalization, motivating lower-overhead alternatives.

B. Linear Combination of Unitaries

  • An operator A is expressed as a linear combination of M unitaries, with normalized coefficient magnitudes forming a probability distribution.The decomposition exists for operators of physical interest and can be computed with classical algorithms.Source: B. Linear Combination of Unitaries · p. 3
  • Standard LCU uses PREP and SELECT oracles with an ancilla register of ⌈log2 M⌉ qubits to implement the unitary combination.The ancilla encodes the coefficient distribution and controls application of the selected unitary.Source: B. Linear Combination of Unitaries · p. 3, Figure 1 · p. 4
  • Postselecting the zero-ancilla outcome prepares A|ψ⟩ up to normalization, while amplitude amplification boosts the probability of that outcome.The standard procedure also requires multi-qubit controlled operations and amplitude-amplification rounds that increase effective circuit depth.Source: B. Linear Combination of Unitaries · p. 3

C. Expectation Value Estimation under Operator Evolution

The target is the normalized expectation value after applying a possibly non-unitary operator, but estimating it requires only its numerator and denominator, not preparation of the evolved state. Single-ancilla LCU estimates those quantities with fewer ancillas than standard LCU, while retaining per-sample controlled-unitary circuits.

C. Expectation Value Estimation under Operator Evolution

  • The target µ is the expectation of O on the normalized state evolved under A, with µN and µD serving as its numerator and denominator.When A is unitary, µD = 1.Source: C. Expectation Value Estimation under… · p. 4
  • For non-unitary A, the normalization factor µD can be exponentially small in system size, making ratio estimation statistically challenging.The desired output is the scalar µ, so coherently preparing the evolved state is unnecessary.Source: C. Expectation Value Estimation under… · p. 4

D. Single-Ancilla LCU

  • Single-ancilla LCU estimates µN and µD without preparing the evolved state, using one ancilla instead of the multi-qubit register required by standard LCU.It estimates µD analogously with O = I and obtains µ by dividing the two estimates.Source: D. Single-Ancilla LCU · p. 4
  • Each single-ancilla LCU sample uses two controlled unitaries and requires at least GO circuit executions to measure O's commuting groups.A fresh circuit set is needed for every sampled pair, so total quantum execution cost grows as O(R · GO).Source: D. Single-Ancilla LCU · p. 4

III. PROPOSED ALGORITHM

HERMES estimates the expectation value through three stages: classical preprocessing, quantum measurement, and classical post-processing. Its preprocessing recasts the problem into Pauli-observable expectations, using Monte Carlo sampling because direct deterministic evaluation involves many Pauli combinations.

III. PROPOSED ALGORITHM

  • HERMES divides estimation into classical preprocessing, quantum processing, and classical post-processing.Preprocessing transforms the non-unitary expectation problem into a structured collection of Pauli observables without quantum hardware execution.Source: §III · p. 5, A. Classical Preprocessing · p. 5

1. Pauli Decomposition of operator A and observable O

  • Expanding A and Hermitian O in the Pauli basis makes the target numerator and denominator sums of Pauli combinations.The Pauli basis permits exact operator expansions; the coefficients of A may be complex, while those of O are real.Source: §1 · p. 5
  • The numerator contains O(M^2L) Pauli combinations and the denominator O(M^2), making direct deterministic evaluation computationally expensive.The resulting estimation problem reduces to finding Pauli-observable expectations on ρ, motivating Monte Carlo sampling.Source: §1 · p. 5

2. Construction of probability distribution and Monte Carlo sampling

HERMES samples pairs of Pauli terms from a distribution formed from A's coefficients, then builds numerator and denominator Pauli observables from those pairs. The resulting unbiased estimators require hardware only to measure the Pauli observables, not to implement A.

2. Construction of probability distribution and Monte Carlo sampling

  • Each Monte Carlo pair independently samples two indices from the probability distribution formed using A's coefficients.The sum of A's coefficient magnitudes is its l1 norm.Source: §2 · p. 5

3. Generation of numerator and denominator Pauli observables

  • For each unique sampled pair, HERMES collects Pauli strings and phase coefficients for the numerator and denominator observables.Numerator products combine Pk, Pl, and Pj; denominator products combine Pk and Pj.Source: §3 · pp. 5–6
  • The numerator and denominator estimators are unbiased, and quantum hardware measures their Pauli observables without implementing A or its Pauli strings.The estimation task therefore reduces to expectation values for observables in PN ∪ PD.Source: §3 · p. 6

4. Two level measurement reduction

HERMES reduces measurement overhead first by deduplicating Pauli strings and then by grouping commuting strings. This limits growth in measurement settings as sampling increases, unlike single-ancilla LCU, whose circuit executions continue to grow with sample count.

4. Two level measurement reduction

  • Deduplication ensures each distinct Pauli observable is measured only once, regardless of how many samples reference it.The preprocessing algorithm forms Punique from the distinct strings in PN ∪ PD.Source: §4 · p. 6
  • Commuting Pauli strings can be measured simultaneously using one basis-rotation circuit and computational-basis measurements.The algorithm partitions Punique into groups of commuting Paulis, with one measurement setting per group.Source: §4 · p. 6
  • Beyond a characteristic sample count R*, the number of groups G saturates, while single-ancilla LCU circuit executions keep growing with Monte Carlo samples.G is bounded by the Pauli support of A and O and may be further reduced by decomposition sparsity.Source: §4 · pp. 6–7

B. Quantum Processing

HERMES estimates the required Pauli expectations using grouped measurements of the unmodified state, then reconstructs the non-unitary expectation value classically. Reusing repeated sample contributions reduces classical reconstruction work.

B. Quantum Processing

  • For each commuting group, one basis rotation and computational-basis measurement estimate multiple Pauli expectations simultaneously; QWC grouping uses only single-qubit Clifford rotations.With QWC grouping, no two-qubit entangling gates are needed beyond state preparation.Source: B. Quantum Processing · p. 7

C. Classical Post-processing

  • Caching repeated Monte Carlo index-pair contributions avoids recomputation, while deduplication and commuting-group partitioning reduce distinct measurement settings and classical overhead.HERMES handles the structure associated with A classically and measures Pauli observables on the unmodified state.Source: C. Classical Post-processing · pp. 7–8
  • The post-processing stage combines the sampled contributions into numerator and denominator estimates, then returns their ratio as the estimate of the desired expectation value.Only real components contribute because physical expectation values are real-valued.Source: C. Classical Post-processing · p. 7

A. Benchmark and simulation setup

The evaluation compares HERMES with PSA-LCU on imaginary-time ground-state energy estimation for H2 and LiH. It uses a second-order Taylor approximation to the propagator and reports estimator error separately from deviation from the FCI reference.

A. Benchmark and simulation setup

  • The simulations cover five bond lengths for each molecule in STO-3G, with Hamiltonians of 15 Pauli terms on 4 qubits for H2 and 631 terms on 12 qubits for LiH.Both methods use the same Pauli expansion of A; PSA-LCU adds an ancilla and controlled Pauli gates, while HERMES uses state preparation and single-qubit basis rotations.Source: A. Benchmark and simulation setup · p. 8
  • Ground-state estimation uses imaginary-time evolution, for which the normalized evolved state approaches the ground state as τ increases when the initial state has ground-state overlap.The Hartree–Fock state is used as the initial state for both molecules.Source: A. Benchmark and simulation setup · p. 8
  • The propagator is approximated by a second-order Taylor truncation; its shared energy-reference bias is at most ≈0.03 Hartree for H2 and 0.02 Hartree for LiH.The reference energy is computed classically, but direct evaluation becomes intractable as system size grows.Source: A. Benchmark and simulation setup · p. 8
  • Both methods use 10,000 Monte Carlo samples and 10,000 shots per circuit; noisy simulations use the FakeSherbrooke noise model after optimization-level-3 transpilation.The two reported errors distinguish deviation from the approximate reference from end-to-end deviation from the FCI ground state.Source: A. Benchmark and simulation setup · p. 8

B. H2 ground-state energy

Across H2 and LiH, HERMES achieves comparable or improved estimation accuracy with simpler circuits and greater robustness under noise than PSA-LCU. Its measurement workload shifts repeated quantum executions into classical processing, which grows with the number of Pauli terms in A.

B. H2 ground-state energy

  • For H2, both methods reproduce Eaprx closely without noise; under noise HERMES retains the energy-profile shape while PSA-LCU fails across the scan.HERMES’s noisy error is concentrated at short bond lengths, reaching ≈0.45 Hartree at 0.50 Å; its minimum shifts from 0.74 to 0.90 Å.Source: B. H2 ground-state energy · p. 9

C. LiH ground-state energy

  • For LiH, ideal HERMES tracks Eaprx and has lower error than PSA-LCU at four of five bond lengths; under noise it preserves the profile except for increased error at 0.99 Å.The noisy LiH PSA-LCU simulation stopped after approximately 570 circuit executions; noiseless HERMES deviations from FCI are of the order of the truncation bias.Source: C. LiH ground-state energy · p. 10

D. Discussion

  • Against PSA-LCU, HERMES’s H2 ideal MAEs agree within 2%, noisy MAEs fall by factors 7.9 (εaprx) and 7.6 (εFCI), and only HERMES yields noisy LiH results.The MAEs cover five bond lengths per molecule; the noisy LiH PSA-LCU simulation did not complete with available resources.Source: D. Discussion · pp. 10–11, Table I · p. 10
  • After transpilation, HERMES circuits have no two-qubit gates and depth at most 3, while PSA-LCU reaches 14 two-qubit gates and depth 66 for H2, and 96 gates and depth 279 for LiH.Controlled-Pauli translation and routing ancilla interactions under limited connectivity contribute to PSA-LCU’s larger overhead and noise sensitivity.Source: D. Discussion · p. 10, Table II · p. 10
  • HERMES avoids quantum measurement executions that grow with Monte Carlo sample count, trading them for classical overhead that grows with the number of Pauli terms in A.The authors report this trade-off as favorable when quantum execution is the dominant computational bottleneck.Source: D. Discussion · p. 11

V. CONCLUSIONS

HERMES shifts processing of A to the classical domain and estimates expectation values using Pauli measurements on the unmodified input state. For H2 and LiH ground-state estimation, it achieves comparable or improved accuracy to PSA-LCU with shallow circuits and greater noise robustness, while retaining classical overhead that grows with A's Pauli-term count.

V. CONCLUSIONS

  • For H2 and LiH ground-state estimation, HERMES achieves comparable or improved accuracy to PSA-LCU, greater noise robustness, and feasible noisy LiH simulation.With QWC grouping, its circuits have no two-qubit gates beyond state preparation and much smaller depth; classical overhead grows with the number of Pauli terms in A.Source: §V · p. 11
  • HERMES estimates the target as a ratio of weighted sums of Pauli expectation values on the unmodified input state, with no quantum implementation of A.Its two-level measurement reduction bounds distinct measurement settings independently of Monte Carlo sample count, unlike randomized LCU methods whose circuit executions grow linearly with sample count.Source: §V · p. 11
  • The same pipeline applies to other LCU-based expectation-value problems when O and A, or a polynomial approximation of A, have manageable Pauli expansions.The authors identify rigorous convergence guarantees and explicit sample-complexity bounds as an open research direction; general commuting groups may reduce measurement settings but require additional two-qubit gates.Source: §V · p. 11

COMPETING INTERESTS

  • HHSC is an inventor on a patent application related to the methods, filed by QpiAI India Private Limited.Source: COMPETING INTERESTS · p. 11

AI DISCLOSURE

  • The authors report using GPT-6 and Claude Sonnet 5.5 for proofreading and minor editing suggestions.Source: AI DISCLOSURE · p. 11
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