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Three independent checks find that the condition number of linearized coupled-cluster matrices grows only polylogarithmically with system size, supporting exponential quantum advantage over classical solvers for the LCC problem.

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2026-07-10 11:06 UTC pith:DM5FY33A

load-bearing objection Solid multi-reference QLS-LCC formulation and consistent small-system κ diagnostics; the exponential-advantage claim still rests on unproven asymptotic extrapolation from n_v ≤ 40. the 2 major comments →

arxiv 2607.08220 v1 pith:DM5FY33A submitted 2026-07-09 quant-ph physics.atom-phphysics.chem-ph

Quantum linear solvers for quantum chemistry: prospects of exponential quantum advantage

Peniel Bertrand Tsemo , Kenji Sugisaki , Ishita Bhattacharjee , V. S. Prasannaa This is my paper
classification quant-ph physics.atom-phphysics.chem-ph
keywords quantum linear solverslinearized coupled clustercondition number scalingquantum advantagemulti-reference methodsHHL algorithmsparsityquantum chemistry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends quantum linear solvers from single-reference linearized coupled cluster equations to an internally contracted multi-reference form that can treat strong correlation. It then shows, with three complementary diagnostics ranging from direct eigenvalue calculations to a diagonal-ratio proxy and an adapted structural conjecture, that the condition number of the governing matrix grows only polylogarithmically with system size. Combined with the sub-linear growth of sparsity, this scaling implies an exponential runtime advantage for quantum linear solvers relative to classical conjugate gradient. Numerical HHL simulations on small multi-reference molecules recover ground-state energies to within 0.009 percent of classical benchmarks. The result matters because it identifies a concrete quantum-chemistry problem class where the notoriously hard condition-number barrier appears to be favourable.

Core claim

For both single-reference and internally contracted multi-reference linearized coupled cluster equations, the condition number κ of the coefficient matrix A scales polylogarithmically with system size. When this is joined to sparsity that grows only as a sub-linear power of N, quantum linear solvers (HHL for excitation ranks above doubles, CKS already at doubles) deliver an exponential separation in runtime from classical conjugate gradient for the LCC problem.

What carries the argument

Three complementary diagnostics for κ: (i) explicit diagonalization of A, (ii) the ratio of largest to smallest diagonal entries as a cheap proxy justified by Gershgorin and Weyl bounds, and (iii) adaptation of the edge-spawning conjecture that reads diffuse nonzero patterns in A as predictors of slow κ growth.

Load-bearing premise

The polylogarithmic κ trend seen on four small molecules with at most about forty virtual orbitals, and the diffuse heat-map patterns that support it, continues to hold for chemically realistic system sizes and higher excitation ranks.

What would settle it

Compute the condition number of A for a systematic series of larger molecules or longer atomic chains (for example hydrogen chains with dozens of atoms or a molecule with several hundred virtual orbitals) and test whether κ remains polylogarithmic in N or eventually turns polynomial.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Quantum linear solvers become usable for both dynamical and static correlation once the multi-reference formulation is in place.
  • Near-optimal solvers such as CKS already yield exponential advantage at the LCCSD level, lowering the excitation-rank barrier.
  • A natural embedding of QLS-LCC outside a small active space with QPE-CASCI inside it can treat larger orbital spaces while retaining high accuracy where it is needed most.
  • Classical pre-processing that builds A and b remains the dominant remaining bottleneck that must be quantized for an end-to-end advantage.
  • Higher excitation ranks strengthen the exponential separation for HHL because sparsity stagnates while system size continues to grow.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the diffuse matrix pattern persists for molecules with hundreds of orbitals, practical quantum advantage for ground-state energies could appear earlier for LCC than for full configuration interaction or pure phase estimation.
  • The same structural diagnostics may apply to other internally contracted multi-reference methods whose working equations are linear systems.
  • Classical preconditioners that exploit the observed near-diagonal dominance or diffuse structure could further reduce κ for both classical and quantum solvers.
  • Fault-tolerant resource estimates for HHL-LCC should focus on T-count reductions inside Hamiltonian simulation, because the condition-number contribution is already favourable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript extends quantum linear solvers (QLSs) from single-reference linearized coupled cluster (SRLCC) to an internally contracted multi-reference form (QLS-icMRLCC), so that the same linear-system framework can address strong correlation. It then assesses whether the condition number κ of the LCC matrix A scales favourably enough for exponential advantage over classical conjugate gradient. Three diagnostics—direct diagonalization of A, the ratio d_max/d_min of diagonal entries (supported by two theorems), and an adaptation of the edge-spawning conjecture to heat-map patterns of A—are applied to small molecules (LiH, H4, BeH2+, BeH2) and short atomic chains with n_v ≤ 40. All three indicate polylogarithmic κ growth. Combined with the claim that sparsity s scales as N^{2/E}, the authors obtain runtime ratios (Eqs. 37–40) that exhibit exponential separation for HHL when E > 2 and for CKS already at E = 2. Proof-of-concept HHL-icMRLCCSD simulations on PECs recover energies to within 0.009 % of classical benchmarks.

Significance. If the polylogarithmic κ scaling persists at chemically relevant sizes, the work would establish a concrete, chemistry-native setting in which QLSs can deliver exponential runtime advantage over CG for the LCC problem, including multi-reference regimes. The derivation of the icMRLCC linear system (Sec. II.C, Eqs. 20–25), the two theorems that relate κ to d_max/d_min (Sec. III.B.2), the explicit sparsity argument (Sec. III.C), and the numerical energy benchmarks (Sec. IV, Table A2) are genuine contributions that go beyond the earlier SRLCC-only study. The paper also correctly flags classical pre-processing as an end-to-end bottleneck, which is useful for framing future work.

major comments (2)
  1. The central claim of exponential advantage (Abstract; Sec. III.D; Eqs. 37–40) rests on κ remaining polylogarithmic in N (or n_v) for both SRLCC and icMRLCC. All three diagnostics are demonstrated only for n_v ≤ 40 on four small molecules and short chains (Sec. III.B.1–3, Figs. 2–5, A1–A4, A8–A16). Direct κ shows staircase plateaus whose height still grows slowly; Theorem 2 requires ||V|| ≪ d_min, which is never quantified outside this window; and the adapted edge-spawning conjecture assumes that a diffuse pattern observed at small sizes remains diffuse at all larger sizes. None of these steps rules out a later crossover to poly(N) growth once many irreps and higher excitation ranks are populated. The manuscript should either (i) supply additional evidence (larger n_v, higher E, or a rigorous bound) that makes the asymptotic extrapolation more secure, or (ii) restate the claim as a finite
  2. Theorem 2 (Sec. III.B.2) is used to justify d_max/d_min as a proxy for κ when diagonal dominance fails. The proof is correct under the hypothesis ||V|| ≪ d_min, yet the paper never reports ||V|| or any related norm for the matrices studied (Figs. 2–5, A5–A7). Without that check, the proxy remains an empirical observation rather than a controlled approximation. A short numerical verification of ||V||/d_min versus n_v for at least one molecule would make the second diagnostic load-bearing rather than circumstantial.
minor comments (5)
  1. The edge-spawning conjecture is adapted from unweighted graph Laplacians (Ref. [26], overlapping authors) to weighted LCC matrices by setting all nonzero entries to 1. A brief remark on why the unweighted pattern is still expected to control κ for the weighted case would strengthen Sec. III.B.3.
  2. Figs. 7–8 report two-qubit gate counts for the isometry and full HHL unitary, but the asymptotic T-count estimate in Sec. III.E is only schematic (~2^{n_r} N_s^4 log(1/ζ)). Clarifying that these are order-of-magnitude illustrations rather than full resource estimates would avoid over-interpretation.
  3. Notation for the number of virtuals switches between n_v and n_vir without a single definition; a short glossary entry or consistent choice would help.
  4. The redundancy threshold ε = 10^{-6} (Sec. III.F) is stated without a sensitivity check; a one-sentence remark on how κ or energies change under modest variations of ε would be useful.
  5. Table A2 and Fig. 10 report energy differences in mHa; converting the largest HHL-icMRLCCSD deviations into the 0.009 % figure quoted in the abstract would make the accuracy claim easier to verify.

Circularity Check

1 steps flagged

No circular derivation of the polylog-κ claim; self-citation of the edge-spawning conjecture supplies only a qualitative third diagnostic whose patterns are independently observed on the constructed A matrices.

specific steps
  1. self citation load bearing [Sec. III.B.3 (Invoking the edge spawning conjecture)]
    "Finally, we employ an indirect method that is computationally much simpler, to predict κ scaling with system size. In particular, we invoke the edge spawning conjecture that was proposed in Ref. [26] for unweighted graph adjacency matrices. In particular, in our adaptation of the conjecture, we map matrix elements of the A matrix to vertices of a graph… Sub-figures (c) and (d) of Figs. 2, 3, 4, and 5 display a diffuse pattern, which according to our extended edge spawning conjecture, indicates a slow κ growth…"

    The inference from observed diffuse heat-map patterns to polylogarithmic κ rests on a conjecture whose authors overlap with the present paper. While the patterns themselves are new data and the conjecture is only one of three consistent diagnostics (not the sole support), the qualitative step is not externally independent; if the conjecture is false the pattern-to-κ link collapses. This is a minor, non-load-bearing self-citation rather than a definitional loop.

full rationale

The load-bearing numerical evidence for polylogarithmic κ is obtained by explicit construction of the SRLCC and icMRLCC A matrices (via HF/CASSCF integrals + Slater-Condon or redundancy-removal steps) followed by direct diagonalization at successive n_v (or n_A) values up to ~40; the resulting staircase data are then fitted (Figs. 2–5, A1–A4). The d_max/d_min proxy is derived from two self-contained theorems (Thm. 1 under diagonal dominance, Thm. 2 via Weyl) whose assumptions are checked against the same matrices, not fitted to force the target scaling. Sparsity scaling follows from the two-body nature of the Hamiltonian and is independent of the κ diagnostics. The only self-citation that enters the κ argument is the adaptation of the edge-spawning conjecture of Ref. [26] (overlapping authors); it is used solely as a qualitative pattern check on heat-maps and is not required for the direct or proxy results. Prior HHL-SRLCC papers [11,20] supply the single-reference baseline that is re-derived and extended, not a uniqueness theorem that forces the multi-reference conclusions. Consequently the central claim does not reduce by construction to its inputs; the remaining uncertainty is ordinary extrapolation risk outside the computed window, not circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 1 invented entities

The advantage claim rests on standard QLS runtime formulas, empirical κ trends on small systems, two theorems under structural assumptions on A, an adapted unproven graph conjecture, fixed-occupation scaling N~n_v^E, and classical chemistry constructions of A and b. Free parameters are numerical thresholds and HHL ancilla choices, not fitted to force polylog κ. No new physical entities are postulated.

free parameters (3)
  • redundancy eigenvalue threshold ε = 10^{-6}
    ε=10^{-6} used to slice the overlap matrix S when removing linearly dependent excitations in icMRLCC (Sec. III.F); changes n' and thus A.
  • HHL clock qubits n_r and rotation constant C = n_r=6–9; C=λ_min
    n_r chosen per molecule (6–9) until energies look good; C set to λ_min (Appendix A). These control numerical accuracy of proof-of-concept energies, not the κ scaling claim.
  • virtual-orbital increment and irrep ordering
    System size grown by adding one virtual at a time ordered by irrep; plateau structure of κ depends on this growth protocol (Sec. III.B.1).
axioms (6)
  • standard math Standard HHL/CKS and CG asymptotic runtimes (Eqs. 1–3) with efficient loading of A and preparation of |b|.
    Sec. II.A; used throughout runtime-ratio analysis.
  • domain assumption LCC A matrices are real symmetric (Hermitian) with positive diagonal entries observed empirically.
    Remark 1 after Theorem 1; required for κ=λ_max/λ_min and Gershgorin/Weyl arguments.
  • ad hoc to paper Edge-spawning conjecture: diffuse nonzero patterns imply polylog κ growth and remain diffuse at all sizes (adapted from unweighted graph Laplacians to LCC A).
    Sec. III.B.3; third diagnostic; conjecture not proven for weighted chemistry matrices.
  • domain assumption With n_occ (and n_a) fixed, N ~ n_v^E and s ~ n_v^2 ~ N^{2/E} from Slater–Condon two-body structure.
    Sec. III.A, III.C; enters runtime ratios Eqs. 37–40.
  • domain assumption Neglect of the fourth term in the icMRLCC projected equations (MRCEPA(0)-like approximation) still yields a useful linear system for strong correlation.
    Sec. II.C.d, Eq. (19)→(20); defines the multi-reference A they analyze.
  • ad hoc to paper ∥V∥ ≪ d_min so that κ(A) ≈ d_max/d_min when diagonal dominance fails (Theorem 2).
    Sec. III.B.2; not checked by computing ∥V∥; justified only by observed agreement of κ and d_max/d_min.
invented entities (1)
  • QLS-icMRLCC framework (HHL/CKS applied to internally contracted multi-reference LCC linear systems) independent evidence
    purpose: Extend quantum linear solvers from weakly correlated SRLCC to strongly correlated regimes while keeping a linear A x = b form.
    Methodological packaging of existing icMRLCC/MRCEPA(0) equations for QLS; not a new physical object. Independent evidence is the numerical energy curves vs FCI/icMRCI.

pith-pipeline@v1.1.0-grok45 · 37194 in / 4088 out tokens · 41624 ms · 2026-07-10T11:06:38.018728+00:00 · methodology

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read the original abstract

Quantum linear solvers (QLSs) can offer the potential for exponential quantum advantage in solving quantum chemical problems, but its assessment hinges on determining the condition number ($\kappa$) scaling, which itself is computationally challenging. While a recent work applied the Harrow-Hassidim-Lloyd (HHL) algorithm to single-reference linearized coupled cluster equations (SRLCC), the validity of the HHL-SRLCC framework is restricted to weakly correlated regimes. A general treatment requires a formulation that can access strongly correlated regions. We thus begin by extending the QLS-SRLCC framework to its multi-reference form, which is based on the internally contracted multi-reference LCC method (QLS-icMRLCC). We then analyze $\kappa$ scaling using three complementary diagnostics that range from explicit computations to use of indirect structural indicators: (i) direct calculations of $\kappa$, (ii) scaling of the ratio of maximum to minimum diagonal entries of an A matrix, and (iii) structural analyses of the A matrices based on a recently proposed conjecture, which we adapt to the QLS-LCC problem. The three approaches yield consistent predictions, indicating a polylogarithmic $\kappa$ scaling in system size. This finding, when combined with our arguments on sub-linear scaling of sparsity, supports the prospects of exponential advantage using QLSs for the LCC problem. Finally, numerical calculations on potential energy curves of model systems containing up to four atoms recover the ground state energies with errors relative to benchmark classical methods not exceeding 0.009$\%$.

Figures

Figures reproduced from arXiv: 2607.08220 by Ishita Bhattacharjee, Kenji Sugisaki, Peniel Bertrand Tsemo, V. S. Prasannaa.

Figure 1
Figure 1. Figure 1: FIG. 1. Overview of the present work and organization (Section II onwards). Section II derives the working equations of single [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Sub-figure (a) shows the scaling of condition number, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sub-figure (a) illustrates the scaling of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a): Scaling of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Diagram of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Figure showing the growth of the number of two-qubit gates incurred in the isometry unitary for the case of (a) LiH, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Figure showing the growth of the number of two-qubit gates incurred in the HHL-LCCSD unitary for the case of (a) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. This figure summarizes the procedure used to construct the [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Figure showing the energy difference between the total energy and the FCI energy of LiH, H [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗

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    Finally, we measure the HHL ancilla in the computational basis and post-select the state outcome|1⟩. The obtained normalized solution appears on the state register as follows: |˜x⟩= 1r P i biC ˜λi 2 X i biC ˜λi |ui⟩.(A8) FIG. A1. The figure depicts the following: (a)κversusn v, (b)d max dmin versus number of atoms,n A, and (c) theAmatrix heatmap atn A = 6...

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