Quantum Computed Moments (QCM)

• Quantum Computed Moments (QCM) is a hybrid quantum–classical method that estimates ground-state energies by combining low-order Hamiltonian moments with a Lanczos-based analytical reconstruction.
• It reduces ansatz complexity by using shallow trial state circuits while shifting computational effort to precise moment measurements and classical post‐processing.
• Demonstrations in quantum magnetism and ab initio chemistry highlight QCM’s noise robustness and its capacity to capture correlation effects beyond traditional variational approaches.

Quantum Computed Moments (QCM) is a hybrid quantum–classical, moment-based quantum/subspace-expansion method in which Hamiltonian moments with respect to a trial state, $\langle H^p\rangle$, are measured on a quantum computer and then combined through a cumulant or Lanczos cluster expansion to estimate ground-state energies and, in later extensions, other ground-state observables (Vallury et al., 2020, Jones et al., 2021). In its characteristic use on near-term hardware, QCM keeps the trial-state circuit shallow, shifts much of the complexity into the measurement of low-order moments and classical post-processing, and produces a non-variational correction to the direct expectation value $\langle H\rangle$ that can incorporate correlation beyond the prepared ansatz (Vallury et al., 2022).

1. Definition, scope, and historical placement

QCM emerged from the broader family of moment-based and connected-moment methods, including the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) functional, but it is distinguished by the use of quantum-computed Hamiltonian moments together with a Lanczos-based analytic reconstruction of the ground-state energy (Kowalski et al., 2020, Claudino et al., 2021). In the formulation that became standard in later work, a quantum processor prepares a trial state $|\phi\rangle$, measures $\langle H^p\rangle$ for a small number of orders $p$, and a classical routine transforms those moments into a corrected estimate of the ground-state energy (Vallury et al., 2020).

The central design objective is to reduce reliance on deep, highly expressive ansätze. Standard variational methods require the prepared state itself to approach the true ground state, whereas QCM uses the spectral information contained in the sequence of moments of a fixed or weakly parameterized trial state. In this sense, QCM trades ansatz complexity for measurement and post-processing complexity. The trial state must still have nonzero overlap with the target ground state, but the formalism is specifically intended to extract more information than is available from the first moment alone (Vallury et al., 2023, Karim et al., 10 Mar 2026).

The method has been developed and tested in several distinct settings. Initial demonstrations focused on quantum magnetism and lattices up to $5\times 5$ (25 qubits) on IBM Quantum hardware (Vallury et al., 2020). It was then adapted to ab initio chemistry, first with Hartree–Fock reference determinants for hydrogen chains up to H$_6$ (Jones et al., 2021), then with more general trial states and extensive reduced-density-matrix reconstructions for the water molecule (Jones et al., 2023, Jones et al., 13 Sep 2025). A parallel line of work extended QCM from energies to arbitrary ground-state observables through Hellmann–Feynman constructions (Vallury et al., 2023).

2. Moment formalism and Lanczos reconstruction

For a normalized trial state $|v_1\rangle$, QCM begins from the raw Hamiltonian moments

$$\mu_p \equiv \langle H^p\rangle = \langle v_1|H^p|v_1\rangle.$$

From these, one defines connected moments or cumulants recursively as

$$c_p = \langle H^p\rangle -\sum_{j=0}^{p-2} \binom{p-1}{j}\, c_{j+1}\,\langle H^{p-1-j}\rangle.$$

In particular, $\langle H\rangle$0, so the ordinary variational energy is the first term in the QCM construction (Jones et al., 2021, Vallury et al., 2023).

The reconstruction step uses Lanczos expansion theory. Rather than explicitly generating Krylov vectors on hardware, QCM exploits the fact that the sequence $\langle H\rangle$1 is encoded implicitly in the moments. The ground-state energy is expressed through an infimum formula involving analytic expansions of effective Lanczos coefficients $\langle H\rangle$2 and $\langle H\rangle$3, and truncation at fourth order produces the widely used closed-form estimator

$\langle H\rangle$4

Different papers denote this quantity as $\langle H\rangle$5, $\langle H\rangle$6, or $\langle H\rangle$7, but the underlying expression is the same fourth-order Lanczos-cluster approximation (Vallury et al., 2020, Jones et al., 2021, Jones et al., 2023).

This estimator is non-variational. It is not the expectation value of $\langle H\rangle$8 in any prepared state, and it is not constrained to lie above the exact ground-state energy. That non-variational character is essential to the method’s purpose: it allows the estimate to pass below the Hartree–Fock or ansatz energy and to capture correlation encoded in higher moments. It also creates a corresponding caveat: inaccurate or noisy moments can drive the estimate to unphysical values or below the exact energy (Jones et al., 2021, Karim et al., 10 Mar 2026).

The underlying interpretation is Krylov-theoretic. QCM is related to Lanczos, quantum subspace expansion, and other Krylov methods, but it does not require explicit preparation of $\langle H\rangle$9 states or the measurement of Hamiltonian and overlap matrices for a generated basis. The Krylov subspace is represented implicitly by the measured moments, and the diagonalization step is replaced by an analytic moment functional (Vallury et al., 2020, Vallury et al., 2023).

3. Measurement strategies and classical post-processing

The measurement layer depends strongly on the structure of the trial state and the Hamiltonian representation. In the generic spin-model formulation, one writes

$|\phi\rangle$0

with Pauli strings $|\phi\rangle$1, expands $|\phi\rangle$2, and groups the resulting Pauli strings into tensor product bases (TPBs) or qubit-wise commuting sets so that many expectation values can be extracted from one measurement basis (Vallury et al., 2020, Karim et al., 10 Mar 2026). This is the route used in the original lattice-model implementations, where moments up to fourth order were obtained directly from Pauli-string measurements on superconducting devices (Vallury et al., 2020).

In chemistry, more specialized reductions are possible. For hydrogen chains with a single Slater determinant as trial state, the expectation value of any excitation operator can be written as a determinant of submatrices of the one-body reduced density matrix (1-RDM). In that implementation, the authors expanded $|\phi\rangle$3 classically, measured only 1-RDM elements $|\phi\rangle$4 on hardware, reconstructed higher-body expectations through determinant identities, and then formed $|\phi\rangle$5 up to $|\phi\rangle$6 entirely in classical post-processing (Jones et al., 2021). Because the reference was a singlet Slater determinant, spin-degeneracy reduction and symmetry constraints further reduced qubit and measurement counts.

The water-molecule implementation used a different strategy because the trial state was a selected UCCD ansatz rather than a single determinant. There, the full 4-body reduced density matrix was reconstructed from 200 different Pauli bases with 25,000 shots each, readout error mitigation and symmetry verification were applied, and Wick’s theorem was used classically to evaluate $|\phi\rangle$7 and related mixed operators (Jones et al., 13 Sep 2025). The same infrastructure supported both ground-state energy estimation and the later extension to dipole moments.

Across these variants, the classical pipeline is consistent: measure a sufficient set of low-order correlators on hardware, assemble $|\phi\rangle$8, compute the cumulants $|\phi\rangle$9, and evaluate the fourth-order Lanczos formula. The method is therefore “classical-heavy” in two distinct senses: it requires symbolic or numerical preprocessing of moment expressions, and it pushes the energy reconstruction itself into post-processing rather than state preparation (Jones et al., 2023, Aulicino et al., 2021).

4. Noise robustness, purification, and mitigation layers

A defining claim of the QCM literature is that the fourth-order Lanczos estimator is unusually robust to noise on near-term hardware. In the hydrogen-chain study, raw QCM energies showed a much smaller upward shift than raw Hartree–Fock energies, and for H$\langle H^p\rangle$0 at $\langle H^p\rangle$1 Å the method recovered 97.1% of the molecular energy without error mitigation, compared with 78% for direct measurement of $\langle H^p\rangle$2 (Jones et al., 2021). In the deep-circuit magnetism study, QCM maintained reasonable energy estimates for instances up to 20 qubits and trial-state circuits of up to $\langle H^p\rangle$3 CNOTs, whereas direct VQE energies drifted toward their high-temperature limits; the authors argued that matching these results by VQE would require hardware improvement by about two orders of magnitude in error rates (Vallury et al., 2022).

An analytic account of this robustness was given in the study of deep noisy circuits. Under a global white-noise model, the fourth-order Lanczos estimator $\langle H^p\rangle$4 cancels the leading noise contribution that appears in $\langle H^p\rangle$5, so the first noise-sensitive term enters only at higher order in a small spectral-gap parameter. That analysis was presented as an explicit filtering of incoherent noise by the structure of the moment functional (Vallury et al., 2022).

Several implementations added post-processing mitigation. For single-determinant chemistry, the measured 1-RDM should be idempotent, $\langle H^p\rangle$6, and noise breaks this property. The hydrogen-chain work applied McWeeny purification,

$\langle H^p\rangle$7

to push the measured 1-RDM back toward the manifold of Slater-determinant density matrices before reconstructing all moments. The purified QCM estimate crossed the Hartree–Fock variational limit and reached within 99.9% of the exact electronic ground-state energy for H$\langle H^p\rangle$8, with dissociation-curve errors of order 10 mH for H$\langle H^p\rangle$9 and as low as 0.1 mH for H$p$0 in STO-3G (Jones et al., 2021).

Later work identified specific “pathological” noise regimes. Because the fourth-order formula contains both a square root,

$p$1

and a denominator,

$p$2

noise can make the square-root argument negative or the denominator singular. In fermionic problems, noisy moments can also shift the effective spectral weight into the wrong charge sector, so QCM reconstructs the ground state of the noisy effective matrix rather than the intended constrained problem (Karim et al., 10 Mar 2026). To address this, the FCQEM method applies a purely classical post-processing map to measured probability distributions, $p$3, before moment reconstruction. In the reported HeH$p$4, TFIM, and H$p$5O studies, FCQEM+QCM improved upon either method individually and, in the H$p$6O depolarization study, improved the correction by two orders of magnitude relative to QCM alone (Karim et al., 10 Mar 2026).

5. Applications in chemistry, magnetism, and arbitrary observables

QCM was first demonstrated for two-dimensional quantum magnetism models on lattices up to $p$7 (25 qubits), where the infimum estimate consistently outperformed the benchmark variational calculation for the same shallow trial state (Vallury et al., 2020). It was then adapted to ab initio chemistry, beginning with linear hydrogen chains H$p$8, H$p$9, and H$5\times 5$0, where moments with respect to the Hartree–Fock determinant were sufficient to recover dynamical correlation beyond the Hartree–Fock limit on superconducting hardware (Jones et al., 2021).

The most precise chemistry energy demonstration reported for a real molecule was the 8-qubit water calculation on IBM hardware. Using a 4-excitation UCCD-inspired trial circuit of depth 25 with 22 CNOTs, QCM combined with readout mitigation, symmetry verification, reduced-density-matrix rescaling, and reference-state calibration produced a ground-state energy within $5\times 5$1 mHa of exact diagonalisation in the 14 spin-orbital basis (Jones et al., 2023). The paper characterized this as chemically relevant accuracy for a non-trivial molecular system on noisy superconducting hardware.

A major extension was the generalization from energies to arbitrary ground-state observables. The method introduces a perturbed Hamiltonian

$5\times 5$2

uses QCM to estimate the ground-state energy of $5\times 5$3, and then applies the Hellmann–Feynman theorem. In the finite-difference form used in practice,

$5\times 5$4

Because the $5\times 5$5 dependence is introduced in classical post-processing, the same quantum data can be reused for $5\times 5$6 and $5\times 5$7 (Vallury et al., 2023).

This framework was first demonstrated for magnetization and spin–spin correlations in Heisenberg models, where QCM tracked exact ground-state observables across parameter regions in which direct trial-state expectations deviated strongly (Vallury et al., 2023). It was then applied experimentally to the electric dipole moment of the water molecule. In that study, the QCM-based Hellmann–Feynman estimate agreed with full configuration interaction within $5\times 5$8 debye $5\times 5$9, whereas direct expectation-value determination had errors on the order of 0.07 debye $_6$0, even in noiseless statevector calculations (Jones et al., 13 Sep 2025). This established that the moment-based improvement is not restricted to the energy itself.

6. Relation to other methods, limitations, and outlook

QCM is closely related to VQE, quantum subspace expansion, quantum Lanczos, CMX, and quantum phase estimation, but it occupies a distinct regime. Relative to VQE, it keeps the ansatz simple and fixed or weakly optimized, then uses higher moments to generate a non-variational correction rather than seeking all accuracy through state preparation (Vallury et al., 2020, Jones et al., 2021). Relative to explicit Krylov or subspace methods, it does not prepare $_6$1 states or measure Hamiltonian and overlap matrices in a generated basis; the Krylov information is encoded implicitly in the moment sequence (Vallury et al., 2023). Relative to QPE, it is designed for shallow circuits and NISQ devices rather than deep coherent evolution and fault-tolerant operation (Jones et al., 2021, Vallury et al., 2022).

The main limitations recur across the literature. The method requires nonzero ground-state overlap of the trial state, and finite-order truncation means that accuracy still depends on the quality of that reference (Vallury et al., 2023, Claudino et al., 2021). Higher-order moments are more sensitive to shot noise and device noise, so most practical implementations stop at fourth order (Jones et al., 2021, Jones et al., 2023). In chemistry, the symbolic or tensorial preprocessing of $_6$2 can become the dominant bottleneck, particularly for conventional molecular-orbital Hamiltonians with $_6$3 terms (Jones et al., 2021, Jones et al., 13 Sep 2025). Some implementations also rely on special structure, such as a single Slater determinant or singlet symmetry, to reduce the measurement problem (Jones et al., 2021).

The non-variational character of QCM is both its strength and its principal caution. It enables energies below Hartree–Fock and beyond the direct ansatz limit, but it also means that noisy or mispurified moments can produce ill-conditioned expressions, complex branches, or energies that correspond to the wrong physical sector (Karim et al., 10 Mar 2026). The later literature therefore emphasizes physically constrained purification, better Hamiltonian representations, improved error mitigation, and integration with more flexible trial states as the central directions for further development (Jones et al., 13 Sep 2025, Karim et al., 10 Mar 2026).

Taken together, the published work presents QCM as a family of Lanczos-cluster, moment-based reconstruction methods that are specifically adapted to shallow-circuit quantum hardware. Its established domain is low-order, noise-robust recovery of ground-state energies and observables from imperfect trial states; its open problem is how far that strategy can be scaled, both computationally and chemically, as Hamiltonian representations, measurement reductions, and mitigation methods improve.