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arxiv: quant-ph/9511026 · v1 · submitted 1995-11-20 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Quantum measurements and the Abelian Stabilizer Problem

A.Yu.Kitaev (L.D.Landau Institute for Theoretical Physics, Moscow)

Pith reviewed 2026-05-13 05:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum algorithmAbelian stabilizer problemfactoringdiscrete logarithmquantum Fourier transformeigenvalue measurementunitary operatorphase estimation
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0 comments X

The pith

A quantum algorithm solves the Abelian stabilizer problem in polynomial time, covering factoring and discrete logarithm.

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

The paper introduces a quantum procedure that finds the stabilizer of an Abelian group action by measuring eigenvalues of the corresponding unitary operator. This yields an efficient algorithm for any problem reducible to the Abelian stabilizer task, including integer factoring and discrete logarithm computation. The same eigenvalue measurement also produces a quantum Fourier transform over an arbitrary finite Abelian group. A sympathetic reader would care because the result shows quantum computers can handle a natural class of algebraic problems beyond what Shor's earlier algorithms covered. The work includes a self-contained introduction to the basics of quantum computation.

Core claim

There exists a polynomial-time quantum algorithm for the Abelian stabilizer problem. The algorithm works by repeatedly measuring the eigenvalues of a unitary operator that encodes the group action; the measured phases reveal the stabilizer subgroup. The same eigenvalue measurement technique immediately supplies a quantum Fourier transform for any finite Abelian group.

What carries the argument

A procedure that measures an eigenvalue of a unitary operator by using controlled applications of the operator and an ancillary register to extract phase information.

If this is right

  • Factoring and discrete logarithm are solvable in polynomial time on a quantum computer.
  • A quantum Fourier transform can be performed over any finite Abelian group in polynomial time.
  • Any algebraic problem that reduces to finding the stabilizer of an efficiently implementable Abelian action inherits a polynomial quantum algorithm.
  • Quantum phase estimation becomes a reusable primitive for designing new algorithms.

Where Pith is reading between the lines

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

  • The eigenvalue measurement technique may generalize to non-Abelian groups if suitable unitary representations can be constructed efficiently.
  • Problems whose solutions are hidden in the eigenspectrum of implementable unitaries become candidates for similar quantum speedups.
  • The method separates the algebraic structure of the problem from the details of the quantum circuit, suggesting a modular approach to algorithm design.

Load-bearing premise

The unitary operator corresponding to the group action or function can be realized by an efficient quantum circuit.

What would settle it

An explicit superpolynomial lower bound on the quantum circuit complexity of either integer factoring or the Abelian stabilizer problem would show the algorithm cannot exist.

read the original abstract

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a polynomial-time quantum algorithm for the Abelian stabilizer problem (encompassing factoring and discrete logarithm) via eigenvalue measurement of the unitary operator realizing the group action. It also derives a quantum Fourier transform over arbitrary finite Abelian groups and includes a detailed introductory overview of quantum computation.

Significance. If the central claims hold, the work provides a unifying framework that generalizes Shor's algorithms and introduces the eigenvalue-measurement primitive as a reusable tool for quantum algorithms on group problems. The derivation follows directly from standard quantum postulates and circuit constructions under the efficient-oracle assumption, which is the conventional model for these problems.

minor comments (3)
  1. [Abstract] The abstract states the algorithm is 'polynomial' but does not explicitly note the dependence on the group order or the precision parameter; a single clarifying sentence would improve readability.
  2. [Section on eigenvalue measurement] In the description of the eigenvalue measurement procedure, the analysis of the number of repetitions needed to achieve sufficient precision for stabilizer extraction is sketched but could be expanded with an explicit bound on the failure probability.
  3. [Introduction and preliminaries] Notation for the group action unitary and its eigenvectors is introduced without a consolidated table of symbols; adding one would aid readers new to the stabilizer formulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, accurate summary of the manuscript, and recommendation to accept. The significance assessment aligns with our intent to provide a unifying framework via eigenvalue measurement that generalizes Shor's algorithms.

Circularity Check

0 steps flagged

No significant circularity; algorithm derived from standard quantum postulates and efficient unitary oracle

full rationale

The paper's central construction is the eigenvalue measurement procedure for a unitary operator U (via controlled powers and inverse QFT), applied to the group-action unitary to extract stabilizer characters. This follows directly from the quantum circuit model and the assumption that the group action is realized by an efficient unitary (the standard oracle model). No equations reduce to fitted parameters, no self-definitional loops, and no load-bearing self-citations; the derivation is self-contained against the stated assumptions and does not rename or smuggle prior results by the same authors. The extension of Shor's algorithm is presented as a generalization, not a circular reuse.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies only on standard quantum postulates and the computational assumption of an efficient group-action oracle; no free parameters, new physical entities, or ad-hoc axioms are introduced.

axioms (2)
  • standard math Standard postulates of quantum mechanics (unitary evolution and projective measurement)
    Invoked in the derivation of the eigenvalue measurement procedure in the main algorithm section.
  • domain assumption The group action admits an efficient quantum circuit implementation of the corresponding unitary
    Required for the polynomial-time claim; stated as the standard oracle model for the stabilizer problem.

pith-pipeline@v0.9.0 · 5355 in / 1303 out tokens · 81443 ms · 2026-05-13T05:03:44.059564+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 26 Pith papers

  1. [1]

    Quantum mechanical Hamiltonian models of Tu ring machines

    P. Benioff, “Quantum mechanical Hamiltonian models of Tu ring machines”, J. Stat. Phys. 29, 515 (1982)

    work page 1982

  2. [2]

    Reversible logic and quantum computers

    A. Peres, “Reversible logic and quantum computers”, Phys. Rev. A 32 , 3266 (1985)

    work page 1985

  3. [3]

    Quantum mechanical computers

    R. P. Feynman, “Quantum mechanical computers”, Optics N ews, February 1985, 11, p. 11

    work page 1985

  4. [4]

    Quantum theory, the Church-Turing princip le and the universal quantum computer

    D. Deutsch, “Quantum theory, the Church-Turing princip le and the universal quantum computer”, Proc. R. Soc. Lond. A 400 , 97 (1985)

    work page 1985

  5. [5]

    Quantum computational networks

    D. Deutsch, “Quantum computational networks”, Proc. Roy. Soc. Lond. A 425, 73 (1989)

    work page 1989

  6. [6]

    Quantum Circuit Complexity

    A. C.-C. Yao, “Quantum Circuit Complexity”, Proceedings of the 34th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1993), p. 352

    work page 1993

  7. [7]

    Algorithms for quantum computation: discre te log and factoring

    P. W. Shor, “Algorithms for quantum computation: discre te log and factoring”, Pro- ceedings of the 35th Annual Symposium on the Foundations of C omputer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994), p. 124

    work page 1994

  8. [8]

    Quantum complexity theor y

    E. Bernstein and U. Vazirani, “Quantum complexity theor y”, Proceedings of the 25th Annual ACM Symposium on Theory of Computing , (ACM Press, New York, 1993), pp. 11 – 20

    work page 1993

  9. [9]

    Testing shift equivalence of polynomial s using quantum machines

    D. Grigoriev, “Testing shift equivalence of polynomial s using quantum machines” (to ap- pear)

    work page

  10. [10]

    On the power of quantum computation

    D. Simon, “On the power of quantum computation”, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994), p. 116

    work page 1994

  11. [11]

    Rapid solution of problems by quantum computation

    D. Deutsch, and R. Jozsa, “Rapid solution of problems by quantum computation”, Pro- ceedings of the Royal Society , London, A439, 1992, 553–558

    work page 1992

  12. [12]

    An approximate Fourier transform use ful in quantum factoring

    D. Coppersmith, “An approximate Fourier transform use ful in quantum factoring”, IBM Research Report RC19642 (1994)

    work page 1994

  13. [13]

    Riemann’s hypothesis and tests for prima rity

    G. L. Miller, “Riemann’s hypothesis and tests for prima rity”, J. Comp. Sys. Sci , 13, 300–317 (1976)

    work page 1976

  14. [14]

    Machines de Turing reversibles. Recursi ve insolubilite en nǫN de l’equation u =θ n ou θ est un “isomorphism de codes

    Yves Lecerf, “Machines de Turing reversibles. Recursi ve insolubilite en nǫN de l’equation u =θ n ou θ est un “isomorphism de codes”. Comptes Rendus 257, 2597–2600 (1963)

    work page 1963

  15. [15]

    Logical reversibility of computation

    C. H. Bennett, “Logical reversibility of computation” , IBM Journal of Research and De- velopment 17, 525 (1973)

    work page 1973

  16. [16]

    Elementary gates for quantum computation

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation”, quant-ph/9503016 22

    work page arXiv