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Say NO to Optimization: A Non-Orthogonal Quantum Eigensolver

Unpil Baek, Diptarka Hait, James Shee, Oskar Leimkuhler, William J. Huggins, Torin F. Stetina, Martin Head-Gordon, K. Birgitta Whaley

arXiv:2205.09039v1quant-phcond-mat.str-elphysics.chem-ph

TL;DR

Strongly correlated molecular systems require balanced descriptions of static and dynamic correlation, which classical multiconfigurational methods find challenging. NOQE correlates non-orthogonal reference states and evaluates the resulting energies at polynomial quantum cost. The method captures a significant portion of the exact wavefunction compactly and can compute selected low-lying states together.

  • Problem

    Classical multiconfigurational methods face a challenge in balancing static and dynamic correlation for systems with nearly degenerate low-lying states.

    Source: §I · p. 1

  • Method

    NOQE applies classically determined UCC-like operators to non-orthogonal reference states and diagonalizes the correlated subspace without variational optimization.

    Source: §I · p. 2, §II · p. 4, §VI · p. 21

  • Results

    NOQE captures a significant portion of the exact wavefunction compactly, with resulting energy evaluation possible at polynomial cost on a quantum computer.

    Source: §VI · p. 21

  • Takeaways & Limitations

    For H4, NOUCC(2) computes relative ordering and energy gaps for selected low-lying states in one calculation without additional measurement overhead.

    Source: §VI · p. 22

Abstract

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A balanced description of both static and dynamic correlations in electronic systems with nearly degenerate low-lying states presents a challenge for multi-configurational methods on classical computers. We present here a quantum algorithm utilizing the action of correlating cluster operators to provide high-quality wavefunction ans\"atze employing a non-orthogonal multireference basis that captures a significant portion of the exact wavefunction in a highly compact manner, and that allows computation of the resulting energies and wavefunctions at polynomial cost with a quantum computer. This enables a significant improvement over the corresponding classical non-orthogonal solver, which incurs an exponential cost when evaluating off-diagonal matrix elements between the ansatz states, and is therefore intractable. We implement the non-orthogonal quantum eigensolver (NOQE) here with an efficient ansatz parameterization inspired by classical quantum chemistry methods that succeed in capturing significant amounts of electronic correlation accurately. By taking advantage of classical methods for chemistry, NOQE provides a flexible, compact, and rigorous description of both static and dynamic electronic correlation, making it an attractive method for the calculation of electronic states of a wide range of molecular systems.

I. INTRODUCTION

Strong correlation makes accurate descriptions of low-lying electronic states difficult for classical multiconfigurational methods. NOQE uses classically informed, non-orthogonal correlated references to address this challenge without variational optimization.

I. INTRODUCTION

  • Strongly correlated systems can have closely spaced spin states requiring multiconfigurational wavefunctions, while chemically accurate predictions remain challenging for classical methods.These systems are important in contexts including bond-breaking reactions and magnetically coupled transition-metal catalysts.Source: §I · p. 1
  • VQE avoids the large circuit depth required by QPE, but UCC circuit depth, qubit coherence, measurement burden, and high-dimensional optimization remain constraints.NOVQE increases wavefunction expressivity without increasing circuit depth, but requires more measurements.Source: §I · p. 2
  • NOQE applies perturbative UCC-like operators to unrestricted determinants, then diagonalizes their non-orthogonal subspace to capture static and dynamic correlation without variational optimization.Subspace diagonalization can provide ground and low-lying excited states with approximately correct spin quantum numbers.Source: §I · p. 2

II. BACKGROUND

  • NOCI can approximate spin purification, but evaluating off-diagonal elements between correlated non-orthogonal coupled-cluster states has exponential classical cost.The non-orthogonal setting prevents Slater–Condon simplifications from eliminating many terms.Source: §II · pp. 3–4

III. THEORY

  • The theory section introduces preliminary analysis of NOUCC(2) and NOUCJ reference-state ansätze before presenting the NOQE protocol.NOUCC(2) uses MP2 amplitudes with a UCC ansatz; NOUCJ adds a Cluster Jastrow correlator.Source: §III · p. 4, §II · p. 4

Notation

  • The notation defines N as the number of spin-orbitals and qubits, η as the electron count, r as the number of radical sites, and I and J as reference-state indices.Lowercase indices distinguish occupied, virtual, and general spin-orbitals.Source: Notation · p. 4

A. Preliminaries

The preliminaries motivate MP2 amplitudes as a first-order approximation to UCCD amplitudes and describe a low-rank circuit construction. The construction permits rank truncation and avoids Jordan–Wigner strings through basis rotations.

1. MP2 amplitudes as first-order approximation to UCCD amplitudes

  • MP2 amplitudes are a first-order Newton–Raphson approximation to UCCD amplitudes, and the resulting wavefunction energy is correct through third order in MP perturbation theory.The protocol also predicts zero single-excitation amplitudes from unrestricted Hartree–Fock references, consistent with Brillouin’s theorem.Source: §1 · p. 5
  • The authors explore spin-component-scaled, scaled-opposite-spin, and uniform rescalings of MP2 amplitudes as possible improvements to NOUCC(2).The passage motivates testing these scalings because standard MP2 may underestimate opposite-spin correlation.Source: §1 · p. 5

2. Single-reference state generation: Quantum circuit ansatz based on the low-rank decomposition

  • Low-rank decomposition permits truncating small singular values to reduce circuit blocks and depth, while number-operator bases remove Jordan–Wigner strings that add CNOT layers.The decomposition’s approximation comes from Trotter error, which can be reduced by increasing the expansion order.Source: §2 · pp. 5–6

3. Single-reference state generation: adding Cluster-Jastrow correlators

The section introduces a cluster-Jastrow variant and a non-orthogonal quantum eigensolver (NOQE) that combines correlated UHF reference states without variationally optimizing the NOQE coefficients. It describes how the states and matrix elements are constructed and how the approach can be extended or made more compact.

3. Single-reference state generation: adding Cluster-Jastrow correlators

  • Adding Jastrow correlations to MP2-approximated UCCD yields NOUCJ, a more expressive but equally compact ansatz that may allow lower tensor truncation and shorter circuits.The UCJ form relates to the double-SVD decomposition; its K and J matrix elements are classically optimized, and fewer terms have been reported to recover high accuracy comparable to exact results.Source: §3 · p. 6
  • For STO-3G H2, NOUCJ(L = 1) achieves accuracy comparable to untruncated (L = N^2) NOUCC(2), without further Trotterization.The authors attribute this result to variational optimization of the Jastrow parameters compensating for the small number of imposed terms.Source: §3 · p. 6

B. Non-Orthogonal Quantum Eigensolver, NOQE

  • NOQE uses multiple unrestricted Hartree-Fock references, each paired with its basis-specific doubles MP2 tensor, unlike prior NOVQE work using a single restricted reference.The differing reference bases require tracking orbital-basis transformations; the authors state that UHF references provide greater flexibility for strongly correlated systems.Source: B. Non-Orthogonal Quantum Eigensolver,… · p. 7
  • Classical diagonalization of the Hamiltonian in the non-orthogonal reference subspace determines the NOQE coefficients, provided the overlap matrix is numerically stable.The paper says thresholding overlap-matrix singular values can also handle near-singular cases, including relatively large perturbations to the matrices.Source: B. Non-Orthogonal Quantum Eigensolver,… · p. 7
  • A modified Hadamard-test circuit evaluates off-diagonal Hamiltonian and overlap elements between ansatz states, using 2N + 1 qubits in the general circuit.A variant reduces the requirement to N + 1 qubits but approximately doubles circuit depth; measurements on hardware would require repeated circuit runs.Source: B. Non-Orthogonal Quantum Eigensolver,… · pp. 7–9
  • The circuit rotates states prepared in different UHF orbital bases into a common basis so their matrix elements can be evaluated consistently.The basis rotation can be implemented with Givens rotations and single-qubit gates, including on linear-connectivity devices.Source: B. Non-Orthogonal Quantum Eigensolver,… · p. 8
  • The circuit can use a UHF reference for an excited configuration to target higher-lying electronic states beyond the ground and low-lying states considered here.The paper gives an excited determinant formed by moving an electron from an occupied orbital to an unoccupied orbital as an example.Source: B. Non-Orthogonal Quantum Eigensolver,… · p. 8

C. Technical Details

Reference determinants are generated by optimizing, localizing, and permuting radical-site spins, then optimizing each configuration. The calculations use an ms = 0 subspace and ideal, noiseless matrix elements.

C. Technical Details

  • Reference states are generated by optimizing a high-spin restricted open-shell Hartree-Fock solution, localizing orbitals on radical sites, permuting spins, and optimizing each configuration.The initial state is triplet H2 or quintet H4; the subsequent permutations produce two H2 determinants or six H4 determinants at ms = 0.Source: C. Technical Details · p. 9
  • The reference-generation procedure resembles classical spin-flip NOCI, but allows full orbital relaxation because no orbitals must be frozen.Orbital localization is used to avoid spurious Hartree-Fock solutions with delocalized spins, especially for H4.Source: C. Technical Details · p. 9
  • The calculations use d radical sites and d electrons in the ms = 0 subspace, with d/2 up-spin and d/2 down-spin electrons.The paper states that this subspace can describe states with different ⟨S^2⟩ values and gives an advantageous cancellation of energy-gap errors.Source: C. Technical Details · p. 9
  • Benchmark matrix elements are evaluated from simulated ansatz-state vectors by direct inner products, without modeling quantum measurement or circuit noise.The authors frame this choice as establishing ideal values possible under NOQE.Source: C. Technical Details · p. 9

IV. RESULTS

For H2, NOQE with NOUCC(2) improves on classical NOCI across internuclear distances and qualitatively agrees with FCI. Scaling MP2 amplitudes further reduces energy errors, while the reference-state choice limits T1 calculations to distances beyond the CF point.

A. Hydrogen molecule

  • The chosen references yield T1 only beyond the CF point, where two references are available; below it, only the RHF reference is used.The authors consider this restriction not particularly limiting because T1 is very high in energy relative to S0 in that regime.Source: A. Hydrogen molecule · p. 10
  • Across H2 internuclear distances, NOQE with NOUCC(2) improves on classical NOCI for both S0 and T1 and qualitatively agrees with FCI.The NOQE S0 result also removes the spurious second local minimum just beyond the CF point that appears in classical NOCI.Source: Figure 4 · p. 11, A. Hydrogen molecule · p. 11
  • The S0 absolute-energy error exceeds chemical accuracy at most distances and reaches about 8 mHa near the CF point.Errors fall below chemical accuracy only at larger separations, where H2 increasingly resembles two independent atoms.Source: A. Hydrogen molecule · p. 11, Figure 5 · p. 12
  • Near the CF point, the singlet-triplet gap error is less than 4 mHa, and its maximum decreases as the basis grows from STO-3G to 6-31G to 6-311G.For 6-311G, the gap error is lower than the absolute-energy errors at every internuclear distance.Source: A. Hydrogen molecule · pp. 11–12, Figure 5 · p. 12
  • With MP2 amplitudes scaled by s = 1.2 or 1.3, singlet-triplet gaps are within chemical accuracy of FCI at all internuclear distances.For s = 1.3, the maximum S0 absolute-energy error is nearly halved.Source: Figure 6 · p. 12, A. Hydrogen molecule · p. 12
  • NOQE reproduces the S0 and T1 states with at least about 99% fidelity, with maximum infidelity near 1% at the CF point before amplitude scaling.The maximum error in ⟨S2⟩ relative to FCI is 3×10^-7 across the same distance range.Source: A. Hydrogen molecule · pp. 12–13

B. Square H4 system

Square H4 tests NOQE on a strongly correlated system with competing low-lying spin states. NOQE reproduces the S0 and T1 energies closely, while accuracy for the higher-energy Q1 state is limited near the Coulson-Fischer analogue; a separate H2 test compares NOUCJ and NOUCC(2).

B. Square H4 system

  • Square H4 switches from a triplet ground state at short side lengths to a singlet at 0.82 Å, with six UHF references spanning distinct spin arrangements.At 1.105 Å, four references undergo a collapse toward closed-shell states; this is a close, but not exact, analogue of H2's Coulson-Fischer point.Source: B. Square H4 system · pp. 13–14
  • T1 stays within chemical accuracy at every H4 side length, while S0 has a maximum error of about 7 mHa near the Coulson-Fischer analogue.NOQE agrees closely with FCI for both states; classical NOCI is inadequate for S0 at shorter side lengths.Source: Figure 9 · p. 14, B. Square H4 system · pp. 14–15
  • The NOUCC(2) subspace poorly captures Q1 below 1.105 Å, where the highest-energy state can have ⟨S^2⟩ around 4–5 instead of 6.Above 1.25 Å, the highest-energy state reaches ⟨S^2⟩ ≥ 5.95 and can reasonably be labeled Q1.Source: B. Square H4 system · p. 15
  • Uniformly scaling MP2 amplitudes by s = 1.3 brings the H4 S0 energy error below chemical accuracy at every side length.Scaled NOQE states also show greater overlap with FCI than unscaled states; the unscaled ansatz already exceeds 99% fidelity across side lengths.Source: B. Square H4 system · pp. 15–16
  • The H4 S0 state has ⟨S^2⟩ below 0.01 even with unscaled amplitudes, and the T1 state's maximum deviation is 5 × 10^-6.The higher-energy Q1 state is less spin-pure, especially at shorter side lengths; SCS- and SOS-MP2 scaling can worsen its spin contamination.Source: B. Square H4 system · pp. 15–16

C. NOUCJ for H2/STO-3G

  • For H2/STO-3G, NOUCJ(L = 1) has lower maximum S0 error than unscaled NOUCC(2), but remains above chemical accuracy until about 2 Å.Scaling NOUCC(2) amplitudes by 1.3 performs better; increasing NOUCJ's L offers a route to systematic improvement, with lower resource cost motivating further study.Source: C. NOUCJ for H2/STO-3G · p. 16, Figure 11 · p. 16

D. Low-rank truncation for H2 and H4

The section examines rank and norm-based truncation of NOQE cluster tensors to reduce circuit costs while retaining energy accuracy. The results show that truncation effectiveness depends on system, geometry, and the distribution of singular values.

D. Low-rank truncation for H2 and H4

  • For H2, truncating the cluster tensor to L = 11 rather than its full rank of 16 keeps the equilibrium energy within chemical accuracy of FCI.At the Coulson-Fischer point, truncated energies converge more quickly to the full-rank result, whose energy lies slightly outside chemical accuracy; stretched-bond orbital localization lowers the effective tensor rank.Source: D. Low-rank truncation for H2 and H4 · p. 16, Figure 12 · p. 17
  • Norm-based truncation selects per-reference ranks so the multi-reference energy remains within 1.6 mHa of the full-rank result.The procedure sweeps an ℓ2-norm threshold and omits circuit blocks associated with discarded singular values; it is used when multi-reference cluster tensors have differing full ranks.Source: D. Low-rank truncation for H2 and H4 · p. 17
  • The reported cost savings depend on the truncation criterion; changing the ℓp norm changes the estimated retained ranks and reduction percentages.Total cost reduction estimates track gate count, whereas single-reference circuit-depth reductions more directly relate to required coherence times.Source: D. Low-rank truncation for H2 and H4 · pp. 17–18
  • Truncation is more effective for the larger H2 basis than for H4, and circuit reductions tend to be greater at larger bond distances.The authors expect larger savings when many singular values are small, as may occur for well-separated, nearly orthogonal orbitals in long saturated hydrocarbons.Source: D. Low-rank truncation for H2 and H4 · p. 18
  • Total circuit cost falls by 18.8% at H2 equilibrium and 15.6% at its Coulson-Fischer point under the selected ℓ2-norm thresholds.For H4, the reference states are grouped by edge versus diagonal spin configurations when estimating circuit-depth reductions.Source: D. Low-rank truncation for H2 and H4 · p. 17, Table IV · p. 19

V. CIRCUIT RESOURCE ESTIMATION

The resource analysis estimates NOUCC(2) gate counts as system size and radical-site count vary, under stated circuit assumptions. It compares low-rank UCC with the lower-cost UCJ option and reports distinct asymptotic scaling for the two ansätze.

V. CIRCUIT RESOURCE ESTIMATION

  • The total classical diagonalization cost grows exponentially with the number of radical sites because the number of NO reference states grows binomially.The resource analysis counts M diagonal and M(M − 1)/2 off-diagonal matrix elements for M reference states.Source: §V · p. 19
  • The gate-count estimates assume full device connectivity; a linear architecture would require additional SWAP gates for non-neighboring number-operator rotations.The analysis also counts CNOTs and single-qubit rotations for the circuit's basis changes and controlled swaps.Source: §V · p. 19
  • Adding one radical site increases the estimated two-qubit gate count by approximately one order of magnitude, while eigenvalue truncation produces sizable reductions.The plotted estimates cover systems with 2, 4, and 6 radical sites.Source: Figure 14 · p. 20, §V · p. 20
  • For two radical sites and 100 spin orbitals, estimated CNOT counts are 6.3 × 10^7 for NOUCC(2) and 1.3 × 10^5 for NOUCJ(L = 1).The NOUCC(2) estimate assumes L = 0.04N^2; the authors describe the UCJ resource advantage as promising but say it remains to be verified for general molecular systems.Source: §V · p. 20
  • At fixed reference-state count, NOQE's CNOT and Rz counts scale as O(N^2L); UCJ(L = 1) is quadratic in spin orbitals, versus quartic for Takagi/SVD-decomposed UCC amplitudes.The stated scaling assumes first-order Trotter-Suzuki decomposition.Source: §V · p. 21

VI. DISCUSSION AND OUTLOOK

NOQE combines non-orthogonal multireference states with cluster-operator correlation to describe strongly correlated molecular systems. The discussion highlights its quantum-computational advantages, comparisons with classical methods, and scaling boundaries.

VI. DISCUSSION AND OUTLOOK

  • NOQE correlates non-orthogonal reference states with cluster operators, compactly capturing strong and dynamic correlation while enabling polynomial-cost quantum energy evaluation.The corresponding classical evaluation of off-diagonal matrix elements can scale exponentially.Source: §VI · p. 21
  • Classical coupled-cluster methods can require O(N^(d+4)) cost for arbitrary strong correlation, limiting their practical use largely to d ≤ 4.The discussion contrasts this with the potential for quantum NOQE implementations on strongly correlated dimetallic species.Source: §VI · p. 21
  • NOQE and CASPT2/NEVPT2 both use exponentially growing diagonalization spaces, but NOQE's non-orthogonal references give a smaller subspace and avoid iterative CASSCF optimization.NOQE's ground-state energy is bounded from below by the FCI ground-state energy, and its diagonal Hamiltonian elements are correct to third order in perturbation theory.Source: §VI · pp. 21–22
  • A single NOQE calculation can determine relative ordering and gaps for selected low-lying H4 states without additional measurement overhead.The authors also describe NOQE amplitudes as compact guesses for FCI eigenstates in QPE, while noting that the overall state-preparation cost benefit needs further investigation.Source: §VI · p. 22
  • NOQE's one-shot, chemistry-specific multireference ansatz avoids iterative quantum-device optimization, reducing gates and measurements while forfeiting VQE's intrinsic error mitigation.The authors point to randomized compiling as an error-mitigation option for NOQE on NISQ devices.Source: §VI · p. 22
  • The demonstrated UCCD-like ansatz uses second-order perturbation amplitudes, with scaling procedures needed to reach chemical accuracy for H2 and H4.The authors also report that a preliminary Jastrow-correlated NOUCJ ansatz appears to reduce circuit cost at comparable accuracy for H2.Source: §VI · p. 22
  • The number of NO reference states still grows exponentially with radical-site count, although the authors identify systems with d < 8 as suitable targets given appropriate hardware.Examples include metalloenzyme subunits, polyacenes, metal complexes, and iron-sulfur clusters.Source: §VI · p. 23
  • The authors conclude that NOQE can compute energy gaps between low-lying states across internuclear distances, supporting potential-energy surfaces for ground and excited states.This conclusion is framed for strongly correlated molecular systems.Source: §VI · p. 23

Appendix A: Orbitals in quantum chemistry

This appendix introduces quantum-chemistry orbital and basis-set terminology, then explains how orbital choices and Hartree-Fock symmetry breaking relate to molecular descriptions.

Appendix A: Orbitals in quantum chemistry

  • The appendix introduces terminology needed to follow the paper's quantum-chemistry discussion.It focuses on orbitals, basis sets, molecular orbitals, and the Coulson-Fischer point.Source: Appendix A · p. 28

1. Restricted, Unrestricted and General Many-electron Wavefunctions

  • Spin-orbitals combine spatial position and spin projection; RHF uses a more constrained orbital form common in closed-shell ground-state calculations, whereas UHF uses a factorizable form better suited to unpaired-electron systems or excited states.Source: §1 · p. 28

2. Basis Functions and Atomic Orbitals

  • Atom-centered orbitals use spatial basis functions, commonly Gaussian combinations, and a basis set specifies those functions for each atomic shell.The paper uses 6-31G and 6-311G basis sets as double- and triple-zeta examples.Source: §2 · pp. 28–29

3. Molecular Orbitals and Coefficient Matrices

  • Molecular orbitals are linear combinations of atom-centered orbitals, with Hartree-Fock optimizing coefficients to minimize energy through the occupied orbitals.The coefficient matrix maps atom-centered orbitals to molecular orbitals.Source: §3 · pp. 29–30

4. Coulson-Fischer Point

  • The Coulson-Fischer point marks where a bond-stretched Hartree-Fock solution can lower its energy by breaking spin symmetry, a model artifact rather than a true-eigenstate feature.At this point, the restricted solution ceases to be an energy minimum against spin-symmetry-breaking perturbations.Source: §4 · p. 30

5. Summary of terms

The terms distinguish orbital types, spatial basis functions, and the mathematical objects used to construct molecular orbitals.

5. Summary of terms

  • A spin-orbital accounts for both position and spin, while a spatial orbital accounts only for position.Spin-general orbitals can mix spin-up and spin-down components; spin-unrestricted orbitals factor into spatial and spin components; restricted orbitals constrain paired spin orbitals to share a spatial function.Source: §5 · p. 31
  • An atom-centered orbital is centered at a nucleus and decays away from it; its spatial part is a basis function.The basis function is predefined and fixed in standard quantum-chemistry calculations.Source: §5 · p. 31
  • A basis set collects the basis functions assigned to the shells of the atoms in a system.These predefined functions provide a spatial discretization centered on the atoms.Source: §5 · p. 31
  • A molecular orbital is generally a linear combination of atom-centered orbitals, and a coefficient matrix maps between those representations.The glossary notes that N linearly independent atom-centered orbitals can yield N orthonormal molecular orbitals.Source: §5 · p. 31
  • The Coulson-Fischer point is where breaking spin symmetry becomes energetically favorable in mean-field models such as Hartree-Fock or DFT.Source: §5 · p. 31

Appendix B: Details of the low-rank factorization

This appendix outlines a low-rank factorization route for the UCCD ansatz, including symmetrization, matrix decompositions, operator rewriting, and circuit-depth reduction.

Appendix B: Details of the low-rank factorization

  • The factorization procedure symmetrizes the coefficient supermatrix before applying a singular-value decomposition and defining operators from its singular values.The symmetrization is needed for Takagi factorization and may increase the number of nonzero singular values by up to a factor of two.Source: Appendix B · p. 32
  • Takagi factorization uses normal operators and reduces the number of operators per singular value from four to two.The matrix of coefficients is then diagonalized, with a unitary matrix and eigenvalues defining a rotated basis.Source: Appendix B · pp. 32–33
  • The operators are expressed using fermionic number operators after rotating the qubit register, and neighboring basis rotations are grouped to reduce accumulated circuit depth by roughly half.Source: Appendix B · p. 33

Appendix C: Details of the NOQE circuit

The NOQE circuit measures non-orthogonal overlap and Hamiltonian matrix elements through ancilla-conditioned measurements. The error analysis identifies the first factorization as the source of Trotter and truncation errors.

Appendix C: Details of the NOQE circuit

  • The circuit prepares UHF ansatz and vacuum registers, applies controlled swaps and cluster operators, and rotates both ansatz states into a common basis.A second controlled swap leaves register 2 in the vacuum state, which can then be ignored.Source: Appendix C · p. 33
  • Ancilla measurements yield the real parts of overlap and Hamiltonian matrix elements, while preparing the ancilla in the +Y eigenstate gives their imaginary parts.The real-part procedure combines the ancilla outcome shotwise with the conditioned Hamiltonian measurement.Source: Appendix C · p. 34

Appendix D: Trotter error bounds

  • Trotter and low-rank truncation errors arise at the first factorization step; the later number-operator terms commute exactly.The authors describe their error-bound analysis as cursory and expect the derived bound to overestimate eigenvalue error.Source: Appendix D · p. 34
  • The norm analysis considers one-body-squared operators and uses singular-vector sparsity; in the Takagi factorization, the paired vectors are identical.The index k combines the l and μ indices.Source: Appendix D · p. 34

1. Trotter error

The appendix bounds errors from Trotterization and low-rank truncation, then propagates matrix-element errors through the generalized eigenproblem. The bounds have stated approximations and may overestimate actual eigenvalue error.

1. Trotter error

  • The Trotter operator-difference bound retains leading non-cancelling terms and uses a commutator expansion when σmax/p ≪ 1.The estimate neglects Taylor-series factorial factors and possible commutator cancellations, so the authors note it may overestimate the error.Source: §1 · pp. 34–35

2. Low-rank truncation error

  • Low-rank truncation discards singular values beyond rank R using a fixed ℓp-norm threshold, with the bound retaining O(σtol) terms when σtol ≪ 1.The analysis neglects terms of O(σtol^2).Source: §2 · p. 35

3. Multi-reference error propagation

  • Matrix-element error bounds for H and S are propagated through the generalized eigenproblem using the Mathias-Li eigenangle bound to set a required Trotter order.The bound is a worst-case estimate and is likely loose: the operator-difference and matrix-element bounds may overstate actual error, particularly when UHF references have low overlap with relevant singular vectors.Source: §3 · pp. 35–36

Appendix E: H2 results with STO-3G and 6-31G basis sets

The appendix plots NOQE energy errors and infidelities relative to FCI for H2 using STO-3G and 6-31G. The supplied captions identify the plotted comparisons but do not report curve values, except that the STO-3G T1 state is predicted exactly.

1. STO-3G

  • For H2 with STO-3G, Figure 15 plots singlet and triplet energy errors relative to FCI against the scaling s of MP2 amplitudes, marking the CF point.The caption does not state the plotted error values or which state performs better.Source: Figure 15 · p. 37
  • For H2 with STO-3G, Figure 16 plots FCI-state infidelity against scaled MP2 amplitudes and notes that the T1 state is predicted exactly by NOQE.The caption attributes the extremely low T1 infidelity to numerical noise.Source: Figure 16 · p. 37

2. 6-31G

  • For H2 with 6-31G, Figures 17 and 18 plot singlet and triplet energy errors and FCI-state infidelities, respectively, against scaled MP2 amplitudes and mark the CF point.The captions do not report plotted values or identify a better-performing state.Source: Figure 17 · p. 38, Figure 18 · p. 38
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