Captured weight and boundary leakage bound the error of sample-based spectral functions

AI summary

Overview

Research area: Quantum computing applied to quantum many-body physics — specifically sample-based quantum diagonalization (SQD) and the reconstruction of frequency-resolved spectral functions (Green's functions, dynamical structure factors) from measured bitstrings, plus the theory of how wrong such reconstructions are.

Technical level: Advanced. The paper assumes familiarity with Rayleigh–Ritz diagonalization, Lehmann spectral representations, Krylov subspaces, Hubbard models and determinant (configuration-interaction) bases. The prose is unusually direct, but the content is research-grade.

Scope (one sentence): The paper proves a two-sided, a posteriori error bound for spectral functions built on bitstring-sampled subspaces — indexed on the missed Born weight and the Hamiltonian coupling across the subspace boundary — and then measures, across Hubbard rings at L = 4 to 14 and nineteen molecular active spaces, where that bound is actually informative.

What This Paper Is About

A sample-based quantum calculation returns a list of bitstrings; diagonalizing the Hamiltonian in the span of the configurations those bitstrings name gives energies, and — as this paper and others show — a frequency-resolved response as well. What it does not give is any statement of how wrong that response is. The paper asks the question a user faces after the shots are spent: given the subspace those shots populated, how wrong is the spectral function built on it? Its central claim is that the obvious diagnostic — the fraction of the probe's squared norm captured in the subspace — cannot answer this, and that the answer requires a second quantity: how strongly the Hamiltonian couples the retained subspace to what it left out.

Key Contributions

1. A proof that captured Born weight alone cannot bound the error. An analytic counterexample forces any bound indexed only on the captured weight w to its trivial value. The paper backs this with a concrete demonstration: on Hubbard rings at broadening η = 0.18 t, a subspace holding the probe's entire Born support (w = 1) still yields a relative L₁ error of 0.43 at L = 6 and 0.35 at L = 8 (and 0.28 on the open chain of Sec. IV, a different lattice). The mechanism is that Rayleigh–Ritz does not delete the poles a subspace misses — it displaces the poles it keeps.

2. Theorem 1: a two-sided bound on the relative L₁ error, for any orthogonal projector. Given the exact ground state, the error lies between 1 − w and min{1 + w, (1 + √w)(√(1 − w) + Λ̂(η)/η)}, where Λ̂(η) is the Hamiltonian coupling across the subspace boundary at resolution η. This leakage term is a posteriori: it has a closed form in the Ritz pairs the reconstruction already produces and in the retained probe, requiring one matrix–vector product per Ritz vector and no knowledge of the exact spectral function.

3. An exact statement of what subspace structure does guarantee, plus a shot count. Theorem 2 shows a subspace containing the order-K Krylov space reproduces spectral moments through order 2K + 1 and no further, and this reach is measured on coordinate subspaces. Proposition 1 gives a non-circular count of the shots needed to capture every high-probability configuration.

4. An empirical map of where the certificate is informative, and evidence that no a priori predictor can replace it. The bound is tested across five sector sizes up to 10,306,296 determinants, sixteen (L, η) cells at L = 6–12, sixteen momentum channels, and nineteen molecular active spaces. Separately, Theorem 3 and Proposition 2 show that on fibers of a pairing model every orbital-rotation invariant — including the efficiently measurable fermionic non-Gaussianity measure ℱ₁ = 4 tr[γ(1 − γ)] — is held fixed while the boundary leakage is not, so no such invariant can substitute for the certificate.

Main Findings

• The operating regime is not certified. At every operating fraction of the resource scan — the acceptance fractions at which the infinite-shot ranking meets the paper's 5% accuracy criterion (relative L₁ < 0.05 at η = 0.15 t) — the leakage branch of the bound is worse than the trivial bound 1 + w, at all five sizes up to a sector of 10,306,296 determinants, by factors of 1.66 to 8.61. At L = 12 that verdict is converged in Krylov depth.

• Certifying the criterion is expensive. To certify the 5% criterion at η = 0.18 t, the bound needs at least 0.98 of the sector at L = 8. The bound becomes informative only well above the operating fractions.

• Where informative, it is calibrated. Over sixteen (L, η) cells at L = 6–12, the bound first crosses below the trivial bound at a true relative error between 1.2 × 10⁻³ and 7.3 × 10⁻³ — a spread of 5.9, over true errors spanning five decades. An unproven two-constant fit in the missed weight, fitted on some sizes and tested on others, localizes this crossing more tightly on eleven of fourteen splits; the paper explicitly labels that fit a regularity of these systems with no guarantee, in contrast to the proven inequality.

• It is non-vacuous but loose on the Born-ranked A(k, ω). At 85% of its sector, the bound is non-vacuous in all sixteen momentum channels, with relative bound 0.77 to 1.75 against the trivial 2 — but that is 350 to 900 times the true error.

• The reconstruction itself can be very accurate. Reconstructed from 5.2 × 10⁴ simulated shots at L = 6 (open chain, U/t = 8, η = 0.15 t), the addition branch matches exact diagonalization to relative L₁ of 0.017. The error falls by a factor of 32 across thirteen Krylov orders, from 0.59 at K = 0 to 0.0185 at K = 12, while the mean subspace grows from 75 to 242 determinants of the 300-dimensional sector, with one non-monotone step at K = 3. The Born-ranked A(k, ω) at L = 8 reaches a mean error of 2.36 × 10⁻³ (maximum 3.27 × 10⁻³) over all eight momenta.

• Finite-shot cost is measured, not extrapolated. At L = 8, η = 0.15 t, 2.60 × 10⁶ shots over 19 snapshots cover 0.835 ± 0.0025 of the sector at relative L₁ = (4.81 ± 0.27) × 10⁻³. For the A(k, ω) configuration at η = 0.18 t, 5.2 × 10⁶ shots per channel reach (2.16 ± 0.23) × 10⁻³ — but on a larger subspace, 0.882 of the addition sector against the 0.850 of the infinite-shot panel. Half that budget yields (6.16 ± 0.42) × 10⁻³, a factor of 2.61 worse.

• Leakage is basis-dependent. In the determinant basis, zero leakage at full weight requires every symmetry-allowed determinant: the probe's measured Krylov support at L = 6 to 14 lies on every determinant that reflection about the seed site permits. Sampling in plane-wave orbitals instead confines a momentum probe's support to one momentum block, 1/L of an exponentially growing sector — but the rotation costs about L(L − 1) further two-qubit gates, whose effect on the sampled distribution is unmeasured.

• Orbital invariants track the determinant count but cannot bound the leakage. Within each of the six (L, N) strata of the Hubbard interaction sweep, the rank correlation of ℱ₁ with the determinant support is exactly −1, with exact permutation p = 3.35 × 10⁻¹³ — a null the paper notes is not satisfied, since six strata share one mechanism. The sign reverses between families, as it must when an orbital invariant is compared with a basis-dependent count.

• Device runs are execution records only. On ibm_fez (L = 6, U/t = 4, η = 0.15 t, 3.5 × 10⁵ shots over seven circuits), the retained subspace was the entire 300-determinant sector, so agreement with exact diagonalization is exact by coverage rather than a test of the method. An independent implementation recovers the scan's acceptance fractions to under 2% and confirms the 5% criterion on the published subspaces: the reconstructions are reproduced, and what they lack is a certificate.

• No quantum advantage is claimed. Every system studied is classically tractable: Hubbard rings at L ≤ 14 and nineteen active spaces with a largest minimal bond dimension of 42.

Methodology in Plain English

The pipeline has one quantum step and a purely classical rest. A circuit prepares a "probe" state — the ground state acted on by the operator whose response you want (add an electron, remove one, or apply a density or spin operator) — and then time-evolves it a few steps, repeating this for K + 1 snapshots and reading out computational-basis bitstrings each time. The distinct bitstrings collected define a subspace. The Hamiltonian is then projected onto that subspace, diagonalized classically, and a continued-fraction resolvent turns the resulting Ritz pairs into a broadened spectral curve. Which operator was used as the seed determines which observable comes out: A(ω) and A(k, ω) for charged seeds, S(q, ω) and S^zz(q, ω) for neutral ones. Crucially, the projection is a Rayleigh–Ritz projection, not a truncation — retained spectral weight is moved in frequency rather than kept in place, which is the source of the whole problem.

To test the bound, the authors replace hardware sampling with exact classical Born draws from the exactly time-evolved probe, and separately with the infinite-shot limit (ranking configurations by accumulated Born weight). They then check the bound's two ingredients — the missed weight and the leaked coupling — against the true error computed by exact diagonalization. The systems are half-filled one-dimensional Hubbard models at U/t = 8 on rings and open chains from L = 4 to 14, plus nineteen molecular active spaces, all small enough for exact references. Two device runs are reported but flagged as execution records, not tests. To probe whether any cheaper predictor could substitute, they construct families of states on a pairing model's fibers where all orbital-rotation invariants are frozen but the leakage is not, and they correlate one such invariant, the antiflatness ℱ₁, against determinant counts.

Why This Matters

For research, the paper draws a sharp line between two things routinely conflated in the sample-based quantum diagonalization literature: a reconstruction that is accurate and a reconstruction that is certified. It shows the field's natural figure of merit — captured Born weight — is structurally incapable of certifying anything, and supplies a proven substitute that costs one matrix–vector product per Ritz vector. It also places an honest cost on certification: at the operating points that meet the field's own accuracy targets, the certificate is vacuous, and making it informative requires upward of 0.98 of a symmetry sector. The negative results on a priori predictors (orbital invariants, compactly measurable non-Gaussianity monotones) close off an attractive shortcut.

Real-world applications implied by the channels the paper constructs:

• Photoemission and ARPES interpretation: A(k, ω) for correlated electron systems, where the paper computes a Hubbard-ring reference showing dispersing removal and addition bands and a Mott gap.
• Neutron scattering and resonant inelastic X-ray scattering: the S(q, ω) and S^zz(q, ω) channels, built from number-conserving density and spin seeds with no extra circuit machinery.
• Dynamical mean-field theory: Green's functions for impurity problems, the target of the concurrent sampled-Krylov construction the paper cites.
• Molecular spectroscopy: the nineteen active spaces at dissociation, including the N₂ CAS(6,6) dissociation curve, where frequency-resolved response is the observable of interest.

Industry relevance is indirect but real: any near-term quantum workflow whose output is a spectrum rather than an energy needs a stopping rule for how many shots and how large a subspace to buy. This paper shows the current stopping rule is not a certificate, and quantifies what one would cost.

Future Directions

• A certificate below full weight is open. A second-order certificate follows at full captured weight, but whether one exists below it — where every operating fraction of the resource scan lies — is stated as open. This is the paper's central unresolved question.
• Closing the gap between proof and practice. The proven bound is loose (350 to 900 times the true error on the Born-ranked A(k, ω)) while an unproven two-constant fit in the missed weight is tighter on eleven of fourteen splits. Turning that empirical regularity into a theorem, or explaining why it holds in these systems, is a natural next step.
• The basis question is measured only halfway. Rotating to plane-wave orbitals shrinks a momentum probe's support from an exponentially growing sector to one block of size 1/L, at a cost of about L(L − 1) two-qubit gates. The effect of that rotation on the sampled distribution is explicitly unmeasured, and the block, like the sector, still grows exponentially.
• Whether a priori predictors can ever work. The paper rules out orbital-rotation invariants on pairing-model fibers and leaves open whether some other efficiently measurable quantity, plus a hardness argument the paper does not establish, could serve as a pre-circuit predictor. Related: the basis-dependent bond dimension tracks the determinant support, which — wherever that bond dimension is small — argues for a tensor network rather than a sampler.

Target Audience

Researchers in quantum simulation and quantum information working on sample-based and subspace diagonalization methods, quantum Krylov algorithms and near-term spectral-function algorithms; condensed-matter and materials theorists interested in what a quantum device can and cannot certify about a Green's function or structure factor; and method developers in quantum chemistry who need to decide how large a determinant subspace and shot budget to buy. It is also valuable reading for anyone benchmarking hybrid quantum–classical pipelines, because its central lesson — that reproducing a reference spectrum and certifying one are different achievements — applies well beyond the Hubbard models studied here.

Authors’ abstract

Rayleigh-Ritz on a subspace from sample-based quantum diagonalization yields the spectral function of a probe state; we ask what bounds its error. The captured Born weight w does not do so alone: an analytic counterexample forces any bound indexed on it to its trivial value. On Hubbard rings at broadening $η=0.18t$, the probe's whole support (w=1) still has relative error 0.35--0.43: Rayleigh-Ritz displaces the poles a subspace keeps. The error is bounded instead by the missed weight and the Hamiltonian coupling across the subspace boundary. Given the exact ground state, we prove for any orthogonal projector that the relative $L_1$ error lies between 1-w and $\min\{1+w,(1+\sqrt{w})(\sqrt{1-w}+Λ(η)/η)\}$, with $Λ$, that coupling at resolution $η$, computed from the Ritz pairs and the retained probe. We test it on L-site rings with classical Born draws of an exactly time-evolved probe, or their infinite-shot ranking; our device runs are execution records only. On the ranked subspaces the bound is vacuous at every operating fraction of the resource scan, at all five sizes up to a 10306296-determinant sector: its leakage branch exceeds the trivial bound 1+w by factors 1.66--8.61. It is informative only well above those fractions, where it is calibrated: at L=6--12 it first beats the trivial bound at a true error of (1.2--7.3)$\times10^{-3}$, though an unproven fit in 1-w is tighter on 11 of 14 splits. For the Born-ranked $A(k,ω)$ at 85% of its sector, the bound is 350--900 times the true error. In the determinant basis, zero leakage at full weight needs every symmetry-allowed determinant: the probe's measured Krylov support at L=6--14. Plane-wave orbitals reduce a momentum probe's support to one block, 1/L of an exponentially growing sector. At full weight a second-order certificate follows; below it, where every operating fraction lies, one is open.