Pith. sign in

REVIEW 2 major objections 4 minor 39 references

Quantum Algorithms for Matrix Operations Based on Unitary Transformations and Ancillary State Measurements

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper constructs quantum circuits that add rows, swap rows, and compute traces of amplitude-encoded matrices in $O(\log N)$ gates, and transpose in $O(m)$ gates, using Toffoli gates, single-qubit operations, and ancilla measurements.

desk verdict The row-operation circuits are correct and worth knowing, but the trace algorithm only yields |tr|^2 with exponential sampling overhead, so the headline O(log N) trace claim does not hold end-to-end. read the letter →

arxiv 2501.15137 v2 pith:F6BR5TFO submitted 2025-01-25 quant-ph

Yu-Hang Liu , Yuan-Hong Tao , jing-Run Lan , Shao-Ming Fei This is my paper · ORCID
classification quant-ph
keywords quantumalgorithmmatrixoperationrowadditionswappingtracetransposeToffoligateancillameasurement
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 tries to establish that basic matrix operations—adding one row to another, swapping two rows, computing the trace, and transposing—can be performed directly on a matrix encoded in the amplitudes of a quantum pure state, using only multi-qubit Toffoli gates, single-qubit gates, and measurements of ancilla qubits. If correct, the gate counts for the row operations and the trace grow as $O(\log N)$ as the number of rows doubles, and transposition costs $O(m)$ swap gates with unit success probability, making these building blocks inexpensive inside larger quantum algorithms. The algorithms work by labeling the useful part of the encoded state with ancillas, swapping row labels where needed, superposing terms with Hadamard gates, and projecting out the unwanted terms. The complexity counts start from an already-prepared normalized input state, so the logarithmic scaling is a claim about the transformation gates themselves, not about state preparation or final readout.

What carries the argument

The carrying object is the pure-state encoding $|\Psi_1\rangle=\sum_{ij}a_{ij}|i\rangle|j\rangle$, in which matrix entries are probability amplitudes indexed by row and column registers. The mechanism is a four-part recipe: ancilla qubits with controlled operations label the amplitudes that should survive; controlled-SWAP gates move labels or rows into the right places; Hadamard gates superpose the labels; and a projective measurement on the ancillas discards redundant components. The complexity bound rests on the multi-qubit Toffoli gate, a controlled-NOT with many control qubits, and on the fact that such a gate can be decomposed into $O(n)$ ordinary Toffoli gates when there are $n$ control qubits, which turns the needed $n$-qubit controls into $O(n)$-gate operations.

What would settle it

For a known $2\times 2$ matrix, prepare the input state $|\Psi_1\rangle$, run the row-addition circuit, and compare the measured ancilla success probability and the amplitudes of the post-selected state with the formulas $G^2/8$ and $(a_{lj}+a_{kj})/G$; a reproducible mismatch would show the algorithm does not realize the claimed transformation.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that unitary control plus ancilla measurement is enough to realize four matrix operations on an amplitude-encoded state. For row addition, an auxiliary register $|\Psi_2\rangle=(|k\rangle+|l\rangle)/\sqrt{2}$ labels the source row; controlled operators mark the term where the row register is $|k\rangle$, a controlled-SWAP exchanges the row register with the source label, a Hadamard superposes the labels, and projecting the ancillas onto $|000\rangle$ yields an output state whose amplitudes are $(a_{lj}+a_{kj})/G$ on the target row, normalized by $G$. Row swapping uses the same construction with three labeled terms and has success probability $1/24$. Trace is obtained by marking the diagonal terms where the row and column registers agree, applying a Hadamard transform to all registers, and reading the diagonal sum from the amplitude of the projected state. Transpose is a single SWAP between the column register and an ancilla register, so it succeeds with probability 1. The central complexity claim is that the multi-qubit controls involved decompose into $O(n)=O(\log N)$ Toffoli gates, making the row operations and trace logarithmic in the row count and transpose linear in $m$.

Load-bearing premise

The algorithms assume the matrix is already available as a normalized pure state with its entries as amplitudes, and the logarithmic gate counts include only the operations applied after that state exists; if preparing or reading out that state costs more than logarithmically in the matrix size, the end-to-end cost is not logarithmic.

Editorial extensions

If this is right

  • For an $N\times M$ matrix with $N=2^n$, the row-addition and row-swapping circuits use $O(\log N)$ gates after the input state exists, so their gate counts grow logarithmically as the number of rows doubles.
  • The transpose circuit uses $O(m)$ SWAP gates and succeeds with probability 1, so transposition is deterministic once the amplitude-encoded input state is available.
  • The row-addition success probability $G^2/8$ and the row-swap success probability $1/24$ do not depend on matrix size, while the trace success probability $|\sum_i a_{ii}|^2/2^{3n}$ decays with dimension; the paper notes that amplitude amplification can be used to boost these probabilities.
  • The outputs remain pure states with the desired transformed matrix held in their amplitudes, so they can feed directly into later quantum computations without a classical readout of every entry.

Reading between the lines

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

  • Because the paper counts only gates applied after the amplitude-encoded input state exists, an end-to-end logarithmic claim would also require state preparation and readout costs that grow at most polylogarithmically; the paper explicitly leaves those costs out of scope.
  • The same label, swap, superpose, and project recipe is likely to generalize to other linear operations on amplitude-encoded data, such as linear combinations of rows with coefficients or column versions of the same elementary transformations.
  • The trace algorithm's success probability $|\sum_i a_{ii}|^2/2^{3n}$ implies that, without amplitude amplification, estimating the trace to fixed precision would require many repetitions, so the practical cost includes a dimension-dependent factor despite the logarithmic gate count.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper presents quantum circuits for four matrix operations on amplitude-encoded matrices: adding one row to another (Section 2.A), swapping two rows (Section 2.B), computing the trace (Section 3.A), and transposing (Section 3.B). The circuits use multi-qubit Toffoli-type controlled operations, C-SWAP gates, Hadamard transforms, and postselection on ancilla measurements. The authors report gate complexities of O(log N) for the row operations and trace, O(m) for transpose, with success probabilities G^2/8, 1/24, |sum_i a_ii|^2/2^{3n}, and 1, respectively. The row-operation and transpose derivations are explicit and internally consistent. The trace algorithm, however, does not deliver the trace as claimed.

Significance. The row-addition and row-swapping circuits are concrete, fully specified constructions that correctly implement the intended transformations, with success probabilities independent of matrix dimension; this is a useful incremental contribution in the line of reference [20]. The transpose circuit is correct and deterministic. A notable strength is that the circuits are given explicitly and the amplitude calculations can be checked step by step; there are no fitted parameters. The trace algorithm, however, only places the global phase of sum_i a_ii on a fixed basis state, and the only measurable quantity is the exponentially small probability |sum_i a_ii|^2/2^{3n}; it therefore does not compute the trace, and the claimed O(log N) cost is not the end-to-end cost. This substantially reduces the scope of the paper's claims.

major comments (2)
  1. [3.A] Section 3.A, Eqs. (20)-(22) and the Conclusion: the trace algorithm does not actually compute the trace. After Step 6 the postselected state is |Phi5> = (sum_i a_ii / |sum_i a_ii|) |0>_R|0>_C|0>_A|1>_B1, in which the trace appears only as an unobservable global phase. The paper states that the trace is obtained from the measurement probability |sum_i a_ii|^2 / 2^{3n}, which yields only |tr S|^2 scaled by 2^{3n} and loses the phase of tr S. Since |sum_i a_ii|^2 <= N by Cauchy-Schwarz, this probability is at most 1/N^2, so estimating it to constant relative error requires Omega(N^2) repetitions classically or Omega(N) with amplitude amplification; both are exponential in n = log N. Therefore the advertised O(log N) complexity is not the end-to-end cost of obtaining the trace, contrary to the central claim.
  2. [3.A / Conclusion] The Conclusion claims that in all algorithms the final result is stored in the probability amplitude of certain quantum states. For the trace algorithm this is not the case: the result is encoded as a global phase in Eq. (22), and the only accessible quantity is the exponentially small postselection probability. The authors should either modify the algorithm to recover the phase of the trace (for example by using an interferometric reference) or explicitly restate the contribution as an algorithm for estimating |tr S|^2 with exponential sample complexity; as written, the trace algorithm is not a trace-calculation algorithm.
minor comments (4)
  1. [2.A, Step 1] The phrase 'the elements of thek + 1and the l + 1rows' is garbled; it should read 'the elements of the (k+1)-th and (l+1)-th rows of Z'.
  2. [2.B, Step 5] The statement that W^(4)_{B1B2B3} has 'one control qubit' is inaccurate: the projections P^(i) act on the three qubits B1 and B2, so the control register has three qubits; the complexity O(1) is unaffected.
  3. [Throughout] There are numerous typographical issues: 'SW AP' and 'C-SW AP' have misplaced spaces, 'swaping' and 'swaped' should be 'swapping' and 'swapped', and '2th' in Appendix 1 should be '2nd'.
  4. [References] Several references have formatting errors, e.g., [1] '563591' and [23] 'V .47'; please correct these.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the circuits are derived explicitly from stated gates, ancillas, and measurements, with no fitted parameters or predictions that reduce to inputs.

full rationale

The paper is a constructive circuit-synthesis result. Starting from the explicitly stated premise that the matrix has already been encoded in a normalized pure state, each algorithm applies explicit multi-qubit Toffoli-type control operations, C-SWAP gates, Hadamard gates, and ancilla postselection, and the output states are derived algebraically (e.g., Eqs. (1)-(8) for row addition, Eqs. (9)-(16) for row swapping, Eqs. (17)-(22) for trace, and Eqs. (23)-(25) for transpose). No parameter is fitted to data, no empirical prediction is made, and no load-bearing conclusion is imported from a self-citation: the cited prior work [20] supplies the general idea of unitary transformations plus ancilla measurements, but the specific circuits and their stated complexities are worked out in the text. The skeptical concern about Section 3.A is a correctness and end-to-end complexity issue, not a circularity: Eq. (22) leaves the trace only in an unobservable global phase factor, and the measurable branch probability gives |tr(S)|^2/2^{3n} rather than tr(S), so the claimed O(log N) trace computation is not fully supported. That gap is a mismatch between the algorithm and its advertised output, not a reduction of a prediction to its own input. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims are built on standard quantum-computing assumptions: amplitude encoding, powers-of-two dimensions, Kitaev's Toffoli decomposition, and projective postselection. There are no fitted constants or invented entities; the only inputs are the matrix entries and the chosen row indices.

assumptions (5)
  • domain assumption The matrix to be processed is already encoded in a normalized pure quantum state with amplitudes equal to the matrix entries.
    Stated in Section 1 as the premise; used to build every circuit. Violating normalization changes the physical state.
  • domain assumption Matrix dimensions are powers of two, N=2^n and M=2^m, so row and column indices fit in n and m qubits.
    Used in Sections 2 and 3 whenever registers R and C are introduced.
  • standard math A multi-controlled single-qubit operation with n control qubits can be implemented with O(n) Toffoli gates (Kitaev's construction).
    Invoked in Sections 2 and 3 for complexity claims, citing reference [23].
  • standard math Projective measurement on the ancilla qubits and postselection on the marked outcome is an allowed operation.
    Step 7 of every algorithm relies on measurement to discard the junk component.
  • domain assumption The desired rows k and l are known classically and can be used to construct the projections P_R2, P_R1B1, and similar operators.
    The circuits are parameterized by k and l; without knowing them, the controlled projections cannot be built.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Algorithms for Matrix Operations Based on Unitary Transformations and Ancillary State Measurements." pith.science (2026). https://pith.science/paper/F6BR5TFO

@misc{pith2026250115137,
  author       = {Pith},
  title        = {Pith review of: Quantum Algorithms for Matrix Operations Based on Unitary Transformations and Ancillary State Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6BR5TFO}},
  note         = {Machine review of arXiv:2501.15137}
}
read the original abstract

MQuantum algorithms of matrix operations are of great significance in many fields in science and technology. In this paper, by leveraging multi-qubit Toffoli gates and basic single-qubit operations, the quantum algorithms of matrix operations of row addition, row swapping, trace calculation and transpose are obtained. In particular, the complexities of these quantum algorithms are presented, too.

Figures

Figures reproduced from arXiv: 2501.15137 by the authors.

Figure 1
Figure 1. FIG.1. This algorithm requires two pure states and four [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The quantum circuit diagram for swapping two [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Quantum circuit for calculating matrix trace. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Quantum Circuit for Matrix Transpose. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 36 canonical work pages

  1. [20]

    Algorithms for quantum computation: discrete logarithms and factoring

    Shor P W. Algorithms for quantum computation: discrete logarithms and factoring. Proceedings 35th Annual Sym- posium on Foundations of Computer Science. IEEE, 1994: 124134

    work page 1994

  2. [1]

    In 1980 Benioff and Manin

    INTRODUCTION Quantum computations offer exponential speedups over their classical counterparts by leveraging the prin- ciples of quantum mechanics. In 1980 Benioff and Manin

    work page 1980

  3. [2]

    sender-receiver

    proposed the concept of quantum computing, and Feynman [2] introduced the theoretical framework of a quantum computer in 1982. In 1985 Deutsch [3] pro- posed the quantum parallel algorithm, showcasing the sig- nificant computational advantages of quantum computing over classical methods. In 1994 Shor [4, 5] presented a quantum algorithm for prime factoriz...

    arXiv 1982

  4. [3]

    Construction of the initial state: matrix elements are first encoded into the probability amplitudes of quantum pure states by using tensor product operations

    work page

  5. [4]

    Information separation and labeling: combining multi-qubit control operators with ancillary states, where the ancillary states are used to accurately label the useful and redundant parts of the information, achieving effec- tive separation of the data

    work page

  6. [5]

    State swaping operations: depending on the specific algorithms, C-SW AP operations are flexibly employed to precisely exchange the states of two quantum bits

    work page

  7. [6]

    State superposition processing: Hadamard operators are applied to induce superposition states among quantum bits

    work page

  8. [7]

    This paper focuses on the core algorithm layer of quan- tum computers and aims to design a set of quantum cir- cuit schemes for implementing basic matrix operations

    Measurement on the ancillary states: auxiliary mea- surement techniques are used to eliminate redundant in- formation, ensuring that the desired results are accurately reflected in the probability amplitudes of the final state. This paper focuses on the core algorithm layer of quan- tum computers and aims to design a set of quantum cir- cuit schemes for i...

    work page

Show all 39 references
  1. [8]

    TWO TYPES OF ELEMENTARY ROW TRANSFORMA TIONS This section presents quantum algorithms for two types of elementary row transformations of an N × M matrix A = {aij} (N = 2 n, M = 2 m), including adding one row to another and swapping two rows. A. Adding one row of a matrix to an...

  2. [9]

    In the first sum, extract all elements except for those in the (k + 1)th and (l + 1)th rows

  3. [10]

    In the second sum, extract all elements in the(l+1)th row

  4. [11]

    In the third sum, extract all elements in the(k + 1)th row

  5. [12]

    For the terms in |Φ1⟩ labeled by |1⟩B1, extract all terms except those where R1 is in the states |k⟩R1 and |l⟩R1

  6. [13]

    For the two sums in|Φ1⟩ labeled by |0⟩B1, extract the terms where R1 is in the state |l⟩R1 and the terms where R1 is in the state |k⟩R1. We introduce an auxiliary qubit B2 in state |0⟩B2, and use the following control operators, W (1) = PR1R2 ⊗ σ(x) B2 1 + (IR1R2 − PR1R2 ) ⊗ I...

  7. [14]

    Lecture Notes on Quantum Computing

    Kockum A F , Soro A, Garc´ıa- ´Alvarez L, et al. Lecture Notes on Quantum Computing. arXiv: 2311.08445 [quant-ph]

    arXiv

  8. [15]

    MA TRIX TRACE CALCULA TION AND TRANSPOSE In this section we present quantum algorithms for trace calculation of an N × N matrix and transposition of an N × M matrix. A. Matrix Trace Calculation The quantum circuit for calculating the trace of the ma- trix S is shown in FIG.3. ...

  9. [16]

    The operational mechanism of all these algorithms is as follows: First, construct the ini- tial state of the entire system

    CONCLUSION We have developed quantum algorithms for two types of elementary row transformations of matrices, the trace and transpose of matrices. The operational mechanism of all these algorithms is as follows: First, construct the ini- tial state of the entire system. Second,...

  10. [17]

    The computer as a physical system: A micro- scopic quantum mechanical Hamiltonian model of comput- ers as represented by Turing machines.J

    Benioff P. The computer as a physical system: A micro- scopic quantum mechanical Hamiltonian model of comput- ers as represented by Turing machines.J. Stat. Phys., 1980, 22(5): 563591

    1980

  11. [18]

    Computable and noncomputable (in Russian)

    Manin Y I. Computable and noncomputable (in Russian) . Sov. Radio, 1980, 13–15

    1980

  12. [19]

    Quantum theory, the Church–Turing principle and the universal quantum computer

    Deutsch D. Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A, 1985, 400(1818): 97117

    1985

  13. [21]

    Polynomialtime algorithms for prime factor- ization and discrete logarithms on a quantum computer

    Shor P W. Polynomialtime algorithms for prime factor- ization and discrete logarithms on a quantum computer. SIAM Rev., 1999, 41(2): 303332

    1999

  14. [22]

    A fast quantum mechanical algorithm for database search

    Grover L K. A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing. 1996: 212219

    1996

  15. [23]

    Quantum algorithm for linear systems of equations

    Harrow A W, Hassidim A, Lloyd S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 2009, 103(15): 150502

    2009

  16. [24]

    The overall computational complexity of the algo- rithm is O(n), mainly determined by W (1) R2C2B1 and W (2) R1R2C2B2

    This probability is inde- pendent of the matrix dimension. The overall computational complexity of the algo- rithm is O(n), mainly determined by W (1) R2C2B1 and W (2) R1R2C2B2. For specific examples of this algorithm, please refer to Appendix 1

  17. [25]

    Preconditioned quantum linear system algorithm

    Clader B D, Jacobs B C, Sprouse C R. Preconditioned quantum linear system algorithm. Phys. Rev. Lett., 2013, 110(25): 250504

    2013

  18. [26]

    Matrix multiplication via arithmetic progressions

    Coppersmith D, Winograd S. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 1990, 9(3): 251–280

    1990

  19. [27]

    Imple- mentation of the quantum Fourier transform

    Weinstein Y S, Pravia M A, Fortunato E M, et al. Imple- mentation of the quantum Fourier transform. Phys. Rev. Lett., 2001, 86(9): 1889

    2001

  20. [28]

    Quantum Computation and Quan- tum Information

    Nielsen M A, Chuang I. Quantum Computation and Quan- tum Information. Cambridge: Cambridge University Press, 2000

    2000

  21. [29]

    Measurementbased quantum phase estimation algorithm for finding eigenvalues of non- unitary matrices

    Wang H, Wu L, Liu Y , et al. Measurementbased quantum phase estimation algorithm for finding eigenvalues of non- unitary matrices. Phys. Rev. A, 2010, 82(6):062303

    2010

  22. [30]

    Low rank represen- tations for quantum simulation of electronic structure

    Motta M, Ye E, McClean J R, et al. Low rank represen- tations for quantum simulation of electronic structure. Npj Quantum Inf., 2021, 7: 83

    2021

  23. [31]

    Quantum algorithms for linear algebra and machine learning

    Prakash A. Quantum algorithms for linear algebra and machine learning. Ph.D. Thesis, UC Berkeley, 2014

    2014

  24. [32]

    On the product of semi-groups of operators.P

    Trotter H F. On the product of semi-groups of operators.P. Am. Math. Soc., 1959, 10(4): 545-551

    1959

  25. [33]

    Quantumassisted Gaussian process regression

    Zhao Z, Fitzsimons J K, Fitzsimons J F. Quantumassisted Gaussian process regression. Phys. Rev. A, 2019, 99(5): 052331

    2019

  26. [34]

    Computing scalar products via a twoterminal quantum transmission line

    Stolze J, Zenchuk A I. Computing scalar products via a twoterminal quantum transmission line. Phys. Lett. A, 2019, 383(34): 125978

    2019

  27. [35]

    Wentao Q, Alexander I Z, Asutosh K, Junde W, Quantum algorithms for matrix operations and linear systems of equations. Commun. Theor. Phys., 2024, 76: 035103

    2024

  28. [36]

    Quantum Inf

    Alexander I Z, Wentao Q, Asutosh K, Junde W,Matrix ma- nipulations via unitary transformations and ancilla state measurements. Quantum Inf. Comput., 2024, 24(13) : 1099-1109

    2024

  29. [37]

    arXiv: 2401.16619 [quant-ph]

    Alexander I Z, Wentao Q, Asutosh K, Junde W, Polynomi- aldepth quantum algorithm for computing matrix determi- nant. arXiv: 2401.16619 [quant-ph]. 11

    arXiv

  30. [38]

    arXiv:2504.00015 [quant-ph]

    Alexander I Z, Wentao Q, Junde W, Quantum Her- mitian conjugate and encoding unnormalized matrices. arXiv:2504.00015 [quant-ph]

    arXiv

  31. [39]

    APPENDIX 1

    Kitaev A Y , Shen A H, Vyalyi M N, Classical and Quan- tum Computation, Graduate Studies in Mathematics, V .47, American Mathematical Society, Providence, Rhode Island (2002). APPENDIX 1. SW APPING TWO ROWS OF A MA TRIX For the quantum algorithm that swaps two rows of a matrix...

    2002

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.