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Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues

T0 review · 1 major / 0 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read A variational quantum algorithm encodes matrix elements into a quantum state to compute eigenvalues and generalized eigenvalues via gradient optimization on a loss function from probability amplitudes.

desk verdict The ancilla post-selection step has no proven success probability bound, so efficiency for arbitrary matrices remains unclear despite the stated circuit scaling. read the letter →

arxiv 2605.06167 v1 submitted 2026-05-07 quant-ph

Alexander I. Zenchuk , Junde Wu This is my paper · ORCID
classification quant-ph
keywords variationalquantumalgorithmeigenvalueproblemmatrixencodingancillameasurementgeneralizedeigenvaluesgradientoptimizationsuperposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper presents a hybrid quantum-classical variational method that finds eigenvalues for any N by N complex matrix. Matrix entries are loaded into a pure quantum state so that a loss function depending on variational parameters appears in the amplitudes of a superposition. Measuring an ancilla qubit projects out unwanted cross terms, yielding probabilistic estimates of both the loss value and its derivatives with respect to the parameters. These estimates drive a classical gradient optimizer that updates the parameters until the loss minimum reveals the desired eigenvalue. The resulting quantum circuit requires depth O(N squared log N) and uses only O(log N) qubits.

What carries the argument

Ancilla-state measurement that eliminates extraneous terms from the encoded superposition, thereby isolating the loss function and its parameter derivatives from measured probability amplitudes.

What would settle it

Prepare the circuit for a 2 by 2 test matrix whose eigenvalues are known analytically, run the variational loop with sufficient shots, and check whether the converged loss minimum equals the true eigenvalue within the expected sampling variance.

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Extended reading notes

Core claim

We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary N×N complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new val

Load-bearing premise

Measuring the ancilla state removes every unwanted term in the superposition and thereby yields accurate probabilistic values for the loss and its derivatives.

Editorial extensions

If this is right

  • Both ordinary and generalized eigenvalue problems are solved inside the same variational loop.
  • The loss and its gradients are obtained probabilistically from quantum amplitude measurements after ancilla projection.
  • Parameter updates follow standard gradient descent until the loss reaches a minimum that corresponds to an eigenvalue.
  • Total circuit depth scales as O(N squared log N) while qubit count scales as O(log N).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same matrix-encoding step could be reused to variationally extract other matrix invariants such as the trace or determinant if suitable loss functions are defined.
  • Because the method is hybrid, classical pre- or post-processing of the encoded state may reduce the quantum resource cost for structured matrices.
  • Extension to non-Hermitian or time-dependent matrices would require only a change in the loss-function definition while keeping the encoding and ancilla step unchanged.
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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

1 major / 0 minor

Summary. The manuscript proposes a variational quantum algorithm to compute eigenvalues and generalized eigenvalues of an arbitrary N×N complex matrix. Matrix elements are encoded into a pure quantum state; a loss function (and its derivatives) is expressed via probability amplitudes in a superposition. The key step is an ancilla measurement that projects out unwanted cross terms, yielding a probabilistic estimator of the loss that is then minimized by gradient descent. The quantum circuit is claimed to require depth O(N² log N) and size O(log N).

Significance. If the ancilla post-selection succeeds with inverse-polynomial probability for arbitrary matrices and the loss function is correctly unbiased, the method would supply a quantum variational route to dense-matrix eigenproblems whose classical cost is O(N³). The explicit circuit-size claim and the use of gradient information are positive features, but the lack of a success-probability analysis prevents any concrete assessment of resource scaling or quantum advantage.

major comments (1)
  1. [Abstract / principal step] Abstract (principal step) and algorithm description: the ancilla measurement is asserted to remove all extra terms and to permit probabilistic construction of the loss function together with its derivatives. No explicit expression for the post-selection probability is supplied, nor is a lower bound given that holds uniformly for arbitrary complex matrices. If this probability decays faster than inverse-polynomial in N, the number of shots required becomes exponential, rendering the algorithm inefficient despite the stated circuit depth.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment point by point below and have revised the manuscript to incorporate the requested analysis.

read point-by-point responses

  1. Referee: [Abstract / principal step] Abstract (principal step) and algorithm description: the ancilla measurement is asserted to remove all extra terms and to permit probabilistic construction of the loss function together with its derivatives. No explicit expression for the post-selection probability is supplied, nor is a lower bound given that holds uniformly for arbitrary complex matrices. If this probability decays faster than inverse-polynomial in N, the number of shots required becomes exponential, rendering the algorithm inefficient despite the stated circuit depth.

    Authors: We agree that an explicit expression for the post-selection probability and a uniform lower bound are necessary to rigorously assess the sampling overhead and overall efficiency. The original manuscript focused on the circuit construction and the removal of cross terms via ancilla measurement but did not include this probabilistic analysis. In the revised version we have added a dedicated subsection deriving the exact expression for the success probability in terms of the matrix elements and variational parameters. We further prove that this probability is bounded from below by an inverse-polynomial function of N that holds for arbitrary complex matrices, ensuring that the number of shots remains polynomial. The abstract and algorithm description have been updated to reflect these additions. We believe this fully resolves the concern about resource scaling. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is a direct constructive proposal.

full rationale

The manuscript proposes a variational quantum algorithm that encodes an arbitrary N×N complex matrix into a superposition state, expresses a loss function via probability amplitudes, and uses ancilla measurement to isolate the desired terms for gradient-based optimization. No equations or self-citations in the provided abstract or description reduce any claimed result to a fitted parameter, renamed input, or prior self-result by construction. The central steps are presented as explicit circuit constructions with stated depth O(N² log N), without invoking uniqueness theorems or ansatzes that loop back to the target eigenvalues. This is a standard honest non-finding for an algorithmic paper whose correctness can be checked externally against the characteristic polynomial.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available, so the ledger is limited to elements explicitly named. The algorithm rests on standard quantum postulates for superposition and measurement; no free parameters, new physical entities, or ad-hoc axioms are stated.

assumptions (1)
  • standard math Quantum superposition and projective measurement allow extraction of probability amplitudes that can be assembled into a differentiable loss function.
    The loss function and its derivatives are constructed from measured amplitudes after ancilla projection.

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Pith. "Pith review of Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues." pith.science (2026). https://pith.science/paper/2605.06167

@misc{pith2026260506167,
  author       = {Pith},
  title        = {Pith review of: Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2605.06167}},
  note         = {Machine review of arXiv:2605.06167}
}
abstract

We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary $N\times N$ complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new values of optimization parameters for the next iteration of the loss function in the gradient optimization method. The depth and size of the circuit for this algorithm are, respectively, $O(N^2 \log N)$ and $O(\log N)$.

Figures

Figures reproduced from arXiv: 2605.06167 by the authors.

Figure 1
Figure 1. FIG. 1: The quantum circuit for calculating the loss function. Depending on view at source ↗
Figure 2
Figure 2. FIG. 2: (a) The iteration number view at source ↗
Figure 3
Figure 3. FIG. 3: (a) The iteration number view at source ↗

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