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Repairability of Inexact Solvers in Recursive State Estimation with Machine Learning

Yanjun Ji, Dennis Willsch, Orkun Şensebat, Priyanka Arkalgud Ganeshamurthy, Zhi Pei, M. Sahnawaz Alam, Ivelina Stoyanova, Frank K. Wilhelm, Bo Zhao, Chao Wang, Kristel Michielsen

arXiv:2609.28425v1quant-phcs.LG

TL;DR

Approximate gain errors in recursive estimation affect later covariance and gain computations, so local accuracy alone does not characterize their finite-horizon effect. The paper develops repairability, certification, and response analyses for a fixed linear Kalman model; its power-grid study reports reduced certified solver depth with learned correction and selected zero-fallback settings with complete residual acceptance.

  • Problem

    Approximate gain errors alter posterior covariance and subsequent gain systems, making their recursive finite-horizon effects a central question alongside local solve accuracy.

    Source: §I · p. 1

  • Method

    The paper separates executed defects from inherited gain drift and combines finite-horizon response analysis with local repairability and learner-independent certification.

    Source: §I · p. 3, A. Covariance mismatch and… · p. 6

  • Results

    The power-grid study reports reduced certified solver depth with learned correction, and residual acceptance reaches 100% at selected zero-fallback iterations.

    Source: §I · p. 3, B. Recursive execution and… · p. 13

  • Takeaways & Limitations

    The analysis evaluates correction and execution through independent certification and quantifies the finite-horizon consequences of the gain defect actually executed.

    Source: §VI · pp. 19–20

    The theory assumes a fixed linear-Gaussian model and matched initialization; changes in model, topology, measurement configuration, or covariance specification are outside its scope.

    Source: §VI · p. 19

Abstract

from arXiv · show

Recursive state estimation often executes approximate numerical solutions inside a feedback loop, where highly accurate local steps do not guarantee better overall results. For a fixed linear Kalman model, we characterize when a correction within a prescribed subspace and norm budget can meet a local admissibility tolerance, and how the defects actually executed affect the finite-horizon covariance response. Centering each defect on the exact gain for the implemented covariance separates current solve error from inherited gain drift. Expanding the exact residual-drift identity reveals opposing quartic contributions beyond the quadratic response: innovation-covariance inflation enters positively, while local-gain reoptimization enters subtractively. Under matched initialization, an absolute sixth-order remainder bound, uniform over bounded defect sequences at fixed horizon, gives sufficient conditions for quadratic under- or overprediction. Machine learning proposes bounded corrections, while a learner-independent residual certificate and verified fallback govern execution of classical and quantum candidates without changing the reference estimator. In a power-grid tolerance study, learned correction lowers the minimum conjugate-gradient iteration count for deployment without fallback relative to uncorrected solves under the same residual certificate. Gains reconstructed from a variational quantum linear solver and from an annealing-based binary encoding, with small-scale terminal measurements on superconducting hardware and sampling on a quantum annealer, are executed through the same interface. By linking local repairability to nonlinear error propagation, the framework evaluates approximate solvers and learned corrections through independent certification and finite-horizon response, providing a practical basis for studying hybrid quantum--classical computation.

I. INTRODUCTION

Approximate gain computation is a repair-and-execution problem because executed defects alter covariance and subsequent gains. The paper separates current solve error from inherited gain drift, characterizes local repairability, and analyzes finite-horizon response under a fixed linear model.

I. INTRODUCTION

  • The analysis isolates computational approximation under a fixed linear model; state-dependent updates, topology changes, and data-dependent covariance adaptation require relaxed assumptions.The exact reference gains can be computed offline, so the construction does not establish an online computational requirement for the reference estimator.Source: §I · p. 2
  • The finite-horizon identity separates current solve error from inherited gain drift; covariance feedback adds opposing quartic terms beyond the quadratic response.Innovation-covariance inflation contributes positively and local-gain reoptimization subtractively; under matched initialization, the absolute sixth-order remainder is uniform over bounded defect sequences at fixed horizon.Source: §I · pp. 2–3
  • A necessary-and-sufficient boundary determines whether a correction in the prescribed subspace and norm budget can meet the local admissibility tolerance.It distinguishes unavailable correction directions from an insufficient correction budget, independently of how the correction is constructed.Source: §I · p. 3
  • A learner proposes bounded corrections, while an independent residual certificate and verified fallback determine which gain is executed.Classical and reconstructed quantum candidates use the same interface; only gains satisfying the execution criterion enter the recursion.Source: §I · p. 3
  • Quantum candidates enter only after their solutions are reconstructed or decoded into explicit gains whose represented residuals can be recomputed.The paper demonstrates this interface with a variational quantum linear solver and QUBO-encoded candidates, without claiming quantum advantage.Source: §I · p. 3

A. Reference and implemented recursions

  • The reference is a correctly specified conditional-mean Kalman estimator with a predeclared covariance schedule and matched initialization for the analyzed horizon.Its gain sequence is measurement independent in this setting; nonmatched initialization adds an initial-response contribution outside the defect-only result.Source: A. Reference and implemented recursions · p. 4

B. Gain hierarchy and computational defects

The framework distinguishes the current computational defect from gain drift inherited through covariance mismatch, then uses residuals to assess candidates for local admissibility.

B. Gain hierarchy and computational defects

  • The executed gain differs from the reference gain by the current executed defect plus inherited gain drift; a correction acts on the current defect.The local gain is the exact solution of the current implemented gain system with the implemented covariance fixed, whereas the reference gain corresponds to the reference covariance.Source: B. Gain hierarchy and computational… · p. 5

C. Solver residual interface and local admissibility

  • The residual certificate is sufficient, but not necessary, for local admissibility of the executed gain.Satisfying the criterion guarantees the executed defect is within tolerance; failure of the criterion does not establish inadmissibility.Source: C. Solver residual interface and local… · p. 5
  • Approximate solutions of transposed gain systems are assembled into candidate gains, with zero-start conjugate gradient providing the primary classical implementation.Any candidate-generation method can use the interface if it yields an explicit represented gain whose residual can be evaluated.Source: C. Solver residual interface and local… · p. 5

III. FINITE-HORIZON RESPONSE TO INEXACT GAIN COMPUTATION

The section gives an exact finite-horizon accounting identity that separates current gain-solve error from inherited gain drift. It then shows how covariance feedback modifies the leading quadratic response through competing quartic terms, with a uniform sixth-order remainder under fixed-horizon bounded-defect conditions.

A. Covariance mismatch and residual-drift identity

  • The local-centered covariance identity makes positive semidefiniteness explicit, while the reference-centered form isolates the subtractive gain-drift term used in finite-horizon accounting.The section states that the excess estimation-error risk is nonnegative under the stated result.Source: A. Covariance mismatch and… · p. 6
  • The exact recentered identity makes the residual contribution an upper bound on finite-horizon response, with recursively induced gain drift providing the subtractive slack.Centering each defect on the exact gain for the current implemented covariance separates current computational error from inherited drift.Source: A. Covariance mismatch and… · p. 6

B. Signed-quartic finite-horizon response

  • Under matched initialization and fixed-horizon bounded defects, the expansion has a uniform absolute sixth-order remainder, and the signed quartic term can certify quadratic under- or overprediction when it exceeds that bound.If the net quartic correction exceeds twice the remainder bound, including it is guaranteed to reduce absolute prediction error.Source: B. Signed-quartic finite-horizon… · pp. 6–7
  • With matched initialization, covariance mismatch and induced gain drift are second order in defect size, so recursive feedback first contributes at quartic order.The fixed-reference response is quadratic in executed defects.Source: B. Signed-quartic finite-horizon… · pp. 6–7
  • The quartic correction combines nonnegative innovation-covariance inflation and local-gain reoptimization terms with opposing signs, so its net sign is not fixed.Inflation increases residual-side exposure, while reoptimization removes response as the exact local gain adapts to displaced covariance.Source: B. Signed-quartic finite-horizon… · pp. 6–7
  • At δexec = 1/8, quadratic and signed-quartic relative prediction errors are 11.803% and 3.144%, respectively; at δexec = 1/16, they are 3.221% and 0.229%.Both errors are normalized by |RM_T| along one prescribed defect-direction sequence; retaining the signed-quartic term improves prediction of the exact recursive response in this instance.Source: Figure 2 · p. 7, B. Signed-quartic finite-horizon… · p. 8

C. Residual-response bounds and finite-horizon control

This section connects local residual checks to sufficient finite-horizon response control and gives task-aware and radial ways to allocate that control. Its statistical interpretation relies on measurement-independent execution and a correctly specified conditional-mean reference estimator.

C. Residual-response bounds and finite-horizon control

  • The statistical identification result assumes execution is independent of the physical initial state and complete disturbance trajectory, with the reference recursion correctly specified as a conditional-mean estimator.Under these conditions, the executed gain sequence is determined by the independent execution environment.Source: C. Residual-response bounds and… · p. 8
  • Equal Frobenius residual norms can produce different finite-horizon contributions because the task-aware exposure retains directional state-side weighting.The radial upper bound instead depends on the residual through its Frobenius norm.Source: C. Residual-response bounds and… · p. 8
  • Under the exact covariance model, applying the local residual criterion to each executed gain, including any fallback, connects per-step acceptance to a sufficient finite-horizon response bound.The connection requires the exact-accounting assumptions, including matched initialization; the represented-system implication additionally requires separate control of matrix-formation and covariance-propagation errors.Source: C. Residual-response bounds and… · p. 8

IV. LOCAL REPAIRABILITY AND MACHINE-LEARNING CORRECTION

The section separates whether a bounded correction exists from whether a learner finds one and whether its candidate passes an independent residual check. It characterizes exact one-step feasibility for a correction subspace and budget, then uses a bounded learned map to propose corrections within that class.

IV. LOCAL REPAIRABILITY AND MACHINE-LEARNING CORRECTION

  • The section frames three distinct preexecution questions: whether the correction class contains an admissible action, whether the learned map attains one, and whether the candidate passes independent residual checking.A candidate can lie outside the admissible ball even when an admissible correction exists.Source: §IV · pp. 8–9, A. Correction class and local… · p. 9

A. Correction class and local repairability

  • Theorem 2 gives an exact feasibility test: an admissible correction exists within the declared class if and only if the minimum attainable post-correction defect εk is at most δadm.The minimization is a projection problem over a norm-bounded correction subspace; feasibility, learner attainment, and residual acceptance are separate questions.Source: A. Correction class and local… · p. 9
  • In Figure 3, the harmless fraction is independent of correction-subspace fraction, and subspace obstruction vanishes at d = nm, leaving only budget obstruction.The plot classifies 4096 held-out instances after three zero-start, unpreconditioned CG iterations, with a fixed independently calibrated correction budget.Source: Figure 3 · p. 9
  • Subspace obstruction cannot be removed by increasing the correction budget at fixed subspace, whereas budget obstruction can be removed by sufficient budget.For a raw-inadmissible defect, the orthogonal component determines whether the subspace can reach the admissible ball.Source: A. Correction class and local… · p. 9

B. Bounded machine-learning correction

  • The learned map proposes a bounded correction within the declared class, while the admissibility criterion and execution decision remain independent of the learner.A checkpoint-specific basis, ridge map, and correction radius are frozen before deployment; if no directions are retained, no learned correction is applied.Source: B. Bounded machine-learning correction · p. 10

C. Residual certification and verified fallback

Residual certification and verified fallback connect local gain admissibility to recursive response, while experiments compare learned-corrected and monitored policies across tolerances and benchmark settings. On IEEE 14, learned correction reaches zero-fallback frontiers at shallower CG iterations, though shallower frontiers do not guarantee smaller state mismatch across networks.

C. Residual certification and verified fallback

  • The residual certificate accepts a candidate only when its verified residual upper bound meets the verified admissibility bound; otherwise, a Cholesky-based fallback is checked and may be executed.The represented-system guarantee additionally requires a lower bound on the current matrix’s smallest singular value, and transferring it to the exact-model defect requires matrix-formation and covariance-propagation controls.Source: C. Residual certification and verified… · p. 11

V. RESULTS

  • The execution framework evaluates learned corrections and reconstructed quantum-solver gains alongside certified solver effort, runtime, and recursive response.The reported quantum components include terminal VQLS measurements on superconducting hardware and QUBO sampling on a quantum annealer.Source: §V · p. 11

A. Experimental design and evaluation protocol

  • The benchmark uses synthetic linear-Gaussian power-network systems, and learned corrections are frozen, bounded proposals evaluated under a shared residual certificate and fallback rule.Policies share a commissioned reference state estimate and covariance; the tolerance scale is fixed within each configuration across η, CG iteration, and policy.Source: A. Experimental design and evaluation… · pp. 11–12

B. Recursive execution and finite-horizon consequence

  • Across all nine admissibility parameters, LC-CG reaches zero deployment fallback two to four CG iterations earlier than M-CG, with smaller normalized mismatch at each policy-specific frontier.The frontier measures certified solver depth, while mismatch measures the state-space consequence of the executed policy.Source: Figure 4 · p. 12, B. Recursive execution and… · pp. 12–13
  • At η = 0.001, 0.01, and 0.1, LC-CG has smaller deployment-averaged normalized RMS mismatch and total finite-horizon response than M-CG at each policy’s zero-fallback frontier.The corresponding M-CG and LC-CG zero-fallback iterations are (10, 7), (7, 4), and (3, 1).Source: Figure 5 · p. 13, B. Recursive execution and… · p. 13
  • Every retained learned candidate in the repairable subset attains local admissibility, but pre-frontier residual acceptance ranges from 81.2% to 98.0% and reaches 100% at selected zero-fallback iterations.The observed gap is between learned attainment and residual acceptance; the certificate is sufficient but not necessary for local admissibility.Source: B. Recursive execution and… · p. 13, Table II · p. 14

C. Measurement-dimension, network, and horizon dependence

  • Across measurement dimensions on IEEE 14, LC-CG reaches zero fallback one to three iterations earlier with smaller frontier mismatch, but at small measurement dimensions its shallower frontier has larger mismatch in several larger networks.The horizon study reports a substantially shallower LC-CG frontier without a monotonic benefit from increasing the commissioning window; mismatch dependence on commissioning protocol varies by network.Source: C. Measurement-dimension, network, and… · pp. 13–14

D. Computational cost and tolerance dependence

Learned correction lowers measured online runtime relative to selected M-CG in the IEEE 14 tolerance study, while quantum-generated gains are evaluated through the same correction, certification, fallback, and recursive-response framework. The discussion emphasizes that the theory concerns fixed-model computational defects and that candidate generation remains distinct from certified execution.

D. Computational cost and tolerance dependence

  • At the smallest measurement dimensions, LC-CG’s shallower zero-fallback frontier has larger state mismatch than M-CG in several larger IEEE systems.The commissioning protocol is network dependent, with especially visible realization-to-realization variation for IEEE 57.Source: Figure 7 · p. 15
  • Selected LC-CG reduced measured online runtime versus selected M-CG by 14.3%, 17.2%, 14.8%, and 24.6% at η = 10−3, 10−2, 10−1, and 0.2, respectively.The reduction is attributed to lower candidate-generation cost exceeding the added learner, correction, and residual-recomputation costs.Source: D. Computational cost and tolerance… · p. 15
  • Warm-start CG and PCG are faster at the first three tolerances, while both direct-solve baselines are faster at all four.LC-CG has lower measured online runtime at η = 0.2; its one-time setup cost is reported separately from online cost.Source: D. Computational cost and tolerance… · pp. 15–16, Table III · p. 17

E. Quantum solver candidates

  • Quantum candidates enter as reconstructed complete gains, then bounded correction, residual checking, and verified fallback determine the gain executed and evaluated recursively.Candidate generation may involve incomplete optimization, finite sampling, hardware imperfections, or encoding resolution.Source: E. Quantum solver candidates · pp. 16–17, §2 · p. 19

1. VQLS execution across simulated and hardware backends

  • Across Statevector, finite-shot Sampler, and Origin VQLS realizations, LC-VQLS reduced response and state mismatch; it executed without fallback in the latter two.The Sampler and Origin corrected policies certified all 60 deployment candidates, despite raw candidates requiring correction or fallback.Source: §1 · p. 17, Figure 9 · p. 18

2. QUBO execution on quantum annealing hardware

  • RI_60 fell from 2.99 × 10−4 for M-QUBO to 1.04 × 10−5 for LC-QUBO, a factor of 28.8; mismatch fell from 8.53 × 10−2 to 9.52 × 10−3, a factor of 9.0.Both policies executed without fallback at this operating point.Source: Figure 10 · p. 19, §2 · p. 19

VI. DISCUSSION AND CONCLUSIONS

  • The framework links local repairability and independent execution certification to finite-horizon response, which depends on the defect of the gain actually executed.The response expansion identifies opposing quartic contributions from innovation-covariance inflation and local-gain reoptimization.Source: §VI · pp. 19–20
  • The theory isolates computational approximation under a fixed linear-Gaussian model and matched initialization; topology, measurement, and covariance changes remain outside its scope.The m = 2 one-qubit quantum circuit also does not probe multiqubit connectivity or entangling-gate costs; broader model and precision extensions are future work.Source: §VI · pp. 19–20

Appendix A: Notation and Covariance Identities

The appendix establishes covariance identities used to express recursive estimation error and response. It develops the residual–drift decomposition and the uniform fixed-horizon remainder underlying the signed-quartic response result.

1. Notation table

  • The notation appendix collects symbols used for recursive estimation, constrained correction, residual certification, and finite-horizon response.The Joseph identities begin by defining the innovation covariance and exact Kalman gain for a prediction covariance.Source: §1 · p. 21, §2 · p. 21, Table IV · p. 22

2. Joseph identities

  • The Joseph covariance at an arbitrary gain differs from the exact-gain update by the positive-semidefinite injection ESk(P)ET when the gain is Kk(P) + E.At the exact Kalman gain, the Joseph update reduces to the standard Kalman covariance update.Source: §2 · p. 21

3. Covariance mismatch and gain drift

  • The covariance-mismatch derivation inserts the exact local gain to separate the executed-gain injection from the remaining covariance and gain-drift terms.The appendix states that the local-centered identity has positive-semidefinite terms under its stated covariance condition.Source: §3 · pp. 21–22

1. Exact residual-response identity

  • Unrolling the reference-centered recursion yields an exact finite-horizon response identity with nonnegative gain-drift contributions.The proof establishes nonnegativity from positive-semidefinite task weights and positive-definite innovation covariances.Source: §1 · p. 23

2. Proof of the signed-quartic response

  • The signed-quartic proof bounds the response expansion’s unresolved contribution by an absolute sixth-order remainder, uniformly over bounded defect sequences at fixed horizon.The proof obtains fixed-horizon bounds by deterministic recursive estimates under matched initialization.Source: §2 · p. 23

3. Consequences of the signed-quartic response

The section describes consequences and qualifications of the signed-quartic response expansion. It distinguishes exact task-nullness, limits on quadratic approximation, and structural properties of the response in defect coordinates.

3. Consequences of the signed-quartic response

  • Exact global task-nullness concerns the full prescribed defect sequence and does not rule out higher-order interactions between a locally task-null defect and other non-null defects.The section’s corollary states the global nullity condition, while the accompanying qualification limits its interpretation.Source: §3 · pp. 24–25
  • The quadratic normal form does not ensure uniform relative accuracy when its positive-semidefinite quadratic term can become arbitrarily small or vanish.A restricted class with a lower bound on that term yields a uniform O(δ^2) relative error; exact global task-nullness makes the term undefined.Source: §3 · p. 25
  • The response is blockwise even in intrinsic defect coordinates, so any nonzero Taylor monomial has even degree in each defect block and cross-time monomials have degree at least four.These properties need not hold for solver iterations, stopping tolerances, correction parameters, or fallback decisions.Source: §3 · p. 25

4. Statistical identification under measurement-independent execution

This section defines separate information structures for physical measurements and numerical execution. Under measurement-independent execution, conditioning on execution information leaves the reference estimator’s conditional moments unchanged and supports identification of covariance response with physical error moments.

4. Statistical identification under measurement-independent execution

  • The measurement sigma-field records physical information available through observations, while the execution sigma-field contains auxiliary information and randomness governing numerical computation.Conditioning on execution information holds numerical execution fixed while leaving physical states, disturbances, and measurements random.Source: §4 · p. 25
  • Under the stated independence hypotheses, adjoining execution information does not alter conditional moments of the correctly specified reference estimator.The execution environment is jointly independent of the physical initial state and disturbance trajectory.Source: §4 · p. 26
  • With matched initialization and the stated hypotheses, the implemented covariance is a second moment conditional on execution information, and the task-response identities follow by summing the conditional error-moment relations.The appendix distinguishes this conditional second moment from a posterior covariance conditioned on the measurement history.Source: §4 · p. 26

5. Residual magnitude and task dependence

The section relates task-weighted exposure to both defect magnitude and direction, and connects finite-horizon response bounds to local admissibility. A zero fixed-reference quadratic exposure for the prescribed defect sequence also implies zero exact finite-horizon response.

5. Residual magnitude and task dependence

  • Frobenius magnitude alone determines fixed-reference quadratic exposure on U_k exactly when the compressed task operator is a scalar multiple of the identity.The extrema are attained at each nonzero Frobenius radius.Source: §5 · p. 27
  • If the full prescribed intrinsic defect sequence has zero fixed-reference quadratic exposure, its exact finite-horizon response also vanishes.Source: §5 · p. 27

6. Finite-horizon response control

  • The finite-horizon bound links local admissibility to a radial allocation of executed residual tolerances when the stated maximum eigenvalue is positive.Equation (10) is equivalent to that local allocation under this condition.Source: §6 · p. 28

7. Uniform-in-time bounds under strict Euclidean contraction

Uniform-in-time bounds require additional all-time assumptions, including strict Euclidean contraction, and remain conservative sufficient bounds. The section also states a local criterion for whether a correction can bring a defect into the admissible region.

7. Uniform-in-time bounds under strict Euclidean contraction

  • The all-time bounds require a bounded reference innovation covariance and strict Euclidean contraction, which is stronger than Schur stability.The executed path must also satisfy an all-time assumption; successful verification at one step does not guarantee a candidate or fallback later.Source: §7 · p. 28
  • These bounds do not control infinite-horizon accumulated risk and exclude some Schur-stable systems with substantial nonnormal transient amplification.They are conservative sufficient extensions of the finite-horizon theory, not runtime certificates.Source: §7 · p. 29

1. Repairability boundary and minimum correction budget

  • A correction within the declared subspace and budget reaches local admissibility if and only if ε_k ≤ δ_adm.When the raw defect norm exceeds δ_adm, no correction in that subspace can make the gain locally admissible.Source: §1 · p. 29

2. Correction basis and learned model

The learned correction uses a frozen, checkpoint-specific basis and ridge map, with an offline target that cancels the supported defect component subject to a radius constraint. The benchmark uses synthetic covariance and the classical solver recomputes the represented gain residual independently.

2. Correction basis and learned model

  • The correction basis, feature scales, correction radius, and ridge map are frozen before deployment, and the offline target cancels the supported defect component subject to the radius constraint.The target is not selected to minimize represented residual norm or finite-horizon task response.Source: §2 · p. 30

1. Benchmark construction

  • The benchmark measurement covariance is fixed over the recursive chronology and is synthetic rather than field-calibrated.Its voltage and branch-current nominal noise scales are 2.5 × 10^-3 and 6.0 × 10^-3, respectively.Source: §1 · p. 31

2. Classical solver implementations

  • The execution interface recomputes the residual from the assembled represented gain rather than relying only on the recurrence-updated CG residual.The stated CG worst-case energy-norm bound does not imply monotonic decrease of the represented matrix residual at every iteration.Source: §2 · pp. 31–32

3. Runtime comparison protocol

The runtime protocol applies a common represented-residual certificate and fallback while allowing each policy to evolve using its executed gains. VQLS and QUBO candidates are reconstructed or decoded into complete gains before residual checking and recursive execution.

3. Runtime comparison protocol

  • All runtime policies start from the same commissioning state and covariance, process the same observations, and use the represented-residual certificate with verified fallback upon rejection.They need not have identical gain systems or equal achieved accuracy.Source: §3 · p. 32

a. VQLS reconstruction

  • VQLS solutions are reconstructed for each right-hand side, assembled into a complete gain, and then independently residual-checked before correction, certification, fallback, or execution.For the Origin hardware realization, terminal X- and Z-measurements use 64 shots per setting, with no hardware variational optimization.Source: a. VQLS reconstruction · pp. 32–33

b. Binary encoding of the QUBO formulation

  • The QUBO encoding increases representation resolution with bit depth but also increases the number of binary variables; higher resolution does not guarantee a better finite-time optimizer solution.The D-Wave procedure decodes all right-hand-side solutions, assembles the gain, and recomputes its represented residual.Source: b. Binary encoding of the QUBO… · pp. 33–34
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