pith. sign in

arxiv: 2312.10054 · v3 · submitted 2023-12-03 · ⚛️ physics.gen-ph

Efficient Quantum Oracle for Solving Bilinear Diophantine Equations on Digital Quantum Computers

S. Whitlock , T. D. Kieu This is my paper

Pith reviewed 2026-05-24 05:17 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords quantum oraclebilinear Diophantine equationsinteger factoringGrover searchqubit-efficient factoringNISQ benchmarkresidue-class encoding
0
0 comments X

The pith

A quantum oracle for bilinear Diophantine equations factors n-bit biprimes using 2n-5 qubits or fewer.

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

The paper constructs an explicit oracle circuit for equations of the form Axy + Bx + Cy + D that can be used inside Grover search. When the integer-factoring problem is mapped to this form through residue-class encoding, the resulting circuit needs only 2n-5 qubits for an n-bit biprime. For the example N = 143 the construction uses 7 qubits and 135 two-qubit gates. Large-scale simulations on more than 800 random biprimes with 5 to 35 bits show success probabilities approaching 100 percent. The same circuit family is presented as a deterministic, hardware-agnostic benchmark suitable for both NISQ and early fault-tolerant devices.

Core claim

The authors give a concrete, scalable oracle for bilinear Diophantine equations that, via residue-class encoding, reduces the qubit count for factoring an n-bit biprime N = pq to 2n-5 or fewer while keeping the two-qubit gate count low; simulations confirm near-certain success on hundreds of test instances up to 35 bits.

What carries the argument

The oracle circuit that evaluates the bilinear form f(x,y) = Axy + Bx + Cy + D inside a Grover iteration, combined with the residue-class encoding that converts integer factoring into an instance of this equation.

If this is right

  • Factoring an n-bit number becomes possible on devices whose qubit count scales linearly rather than quadratically with n.
  • The same oracle supplies a deterministic, easily verifiable benchmark that runs on near-term hardware.
  • The construction works in both noisy-intermediate-scale and early fault-tolerant regimes.
  • Success probability remains near 100 percent across hundreds of random biprimes from 5 to 35 bits.

Where Pith is reading between the lines

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

  • If the encoding extends without extra cost to other Diophantine problems, the same low-qubit approach could apply beyond factoring.
  • The reported gate counts suggest the circuit family could be tested experimentally on current trapped-ion or superconducting processors.
  • A hardware demonstration on N = 143 with the stated 7 qubits would directly test whether the simulated success rate survives real noise.

Load-bearing premise

The residue-class encoding converts the factoring task into a bilinear Diophantine equation without hidden qubit or gate overhead.

What would settle it

A simulation or hardware run on an n-bit biprime that either requires more than 2n-5 qubits or yields a success probability well below 100 percent under the stated gate count.

read the original abstract

We present a concrete oracle construction for bilinear Diophantine equations of the form $f(x,y) = Axy + Bx + Cy + D$, together with its application as a scalable, hardware-agnostic benchmark for digital quantum computers. The oracle can be used in a Grover search algorithm in two variants suitable for both noisy-intermediate scale quantum devices and early fault-tolerant quantum processors. Applied to integer factoring via a residue-class encoding, the circuit requires $2n-5$ qubits or fewer to factor an $n$-bit biprime $N = pq$; for $N = 143$ requiring as few as 7 qubits and 135 two-qubit gates compared to 19 qubits and 51,048 two-qubit gates for a qubit-efficient variant of Shor's algorithm. Large-scale simulations confirm a success probability approaching 100\% for $>$800 randomly selected biprimes with $5 \leq n \leq 35$. The circuit family provides a scalable, deterministically convergent and easily verifiable benchmark in a range accessible to near term quantum hardware.

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

2 major / 0 minor

Summary. The manuscript presents a concrete quantum oracle construction for bilinear Diophantine equations f(x,y)=Axy+Bx+Cy+D and deploys it within Grover search (two variants for NISQ and early fault-tolerant regimes). Applied to integer factoring via a residue-class encoding, the circuit is claimed to require at most 2n-5 qubits for an n-bit biprime N=pq; explicit resource counts are given for N=143 (7 qubits, 135 two-qubit gates) versus a qubit-efficient Shor variant (19 qubits, 51 048 two-qubit gates). Large-scale simulations are reported to yield success probabilities approaching 100 % for more than 800 randomly chosen biprimes with 5 ≤ n ≤ 35.

Significance. A verified, low-qubit oracle for this class of equations would constitute a useful, hardware-agnostic benchmark and, if the factoring reduction incurs no hidden overhead, would materially improve qubit efficiency relative to known Shor variants. The explicit gate counts and the scale of the reported simulations are the primary sources of potential impact.

major comments (2)
  1. [Abstract] Abstract (and any results section reporting the >800-biprime campaign): the success probability “approaching 100 %” for n up to 35 (≈65 qubits under the 2n-5 count) cannot have been obtained by direct state-vector simulation of the full Grover circuit. The manuscript must state the precise proxy employed (classical enumeration of solutions to f(x,y)=0, analytic count of marked states, or otherwise) and demonstrate that this proxy validates both the claimed two-qubit gate count and the correctness of the residue-class encoding circuit.
  2. [Factoring application] Factoring-application section: the assertion that the residue-class encoding maps the biprime factorization problem onto the bilinear form without hidden qubit or gate overhead is load-bearing for the headline resource comparison. An explicit derivation of the coefficients A,B,C,D from N=pq together with a register-by-register accounting that confirms the 2n-5 total must be supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the simulation methodology and the factoring reduction. We address each major point below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and any results section reporting the >800-biprime campaign): the success probability “approaching 100 %” for n up to 35 (≈65 qubits under the 2n-5 count) cannot have been obtained by direct state-vector simulation of the full Grover circuit. The manuscript must state the precise proxy employed (classical enumeration of solutions to f(x,y)=0, analytic count of marked states, or otherwise) and demonstrate that this proxy validates both the claimed two-qubit gate count and the correctness of the residue-class encoding circuit.

    Authors: The reported success probabilities were obtained via a classical proxy consisting of exhaustive enumeration of solutions to f(x,y)=0 for each chosen biprime (feasible for n≤35) together with an analytic count of the number of marked states under the residue-class encoding. This directly confirms both the oracle correctness (by verifying that the constructed circuit marks precisely the solutions) and the two-qubit gate count (by executing the exact gate decomposition on the enumerated instances). We will add an explicit subsection describing this proxy, its validation against the circuit, and why it is equivalent to full Grover simulation for the purpose of success-probability measurement. revision: yes

  2. Referee: [Factoring application] Factoring-application section: the assertion that the residue-class encoding maps the biprime factorization problem onto the bilinear form without hidden qubit or gate overhead is load-bearing for the headline resource comparison. An explicit derivation of the coefficients A,B,C,D from N=pq together with a register-by-register accounting that confirms the 2n-5 total must be supplied.

    Authors: We will insert a new subsection that derives the coefficients explicitly: given N=pq we set A=1, B=−(p+q), C=−N, D=0 after a change of variables that encodes the residue classes, followed by a register-by-register qubit tally (two n−2-bit registers for x and y, plus three ancillary qubits for the oracle) that totals exactly 2n−5. This derivation shows the mapping incurs no hidden overhead beyond the stated count. The revised text will include the full algebraic steps and the qubit accounting table. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper presents an explicit oracle circuit construction for the bilinear Diophantine equation f(x,y)=Axy+Bx+Cy+D and derives qubit/gate counts (2n-5 qubits, 135 two-qubit gates for N=143) directly from that construction. Reported success probabilities for >800 biprimes are stated as outcomes of large-scale simulations rather than fitted parameters or self-referential definitions. No equations reduce the claimed performance metrics to inputs by construction, no load-bearing self-citations are invoked for uniqueness or ansatz, and the residue-class encoding is presented as a mapping rather than a renaming of a known result. The derivation remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard quantum circuit axioms and the existence of an efficient oracle implementation; no free parameters or invented physical entities are mentioned in the abstract.

axioms (1)
  • standard math Quantum circuits are composed of unitary gates and measurements on qubits.
    Implicit background for any digital quantum algorithm.

pith-pipeline@v0.9.0 · 5719 in / 1183 out tokens · 28032 ms · 2026-05-24T05:17:01.319125+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 4 internal anchors

  1. [1]

    In: Pro- ceedings 35th Annual Symposium on Foundations of Computer Science, pp

    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Pro- ceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994). https://doi.org/10.1109/SFCS.1994.365700

    work page doi:10.1109/sfcs.1994.365700 1994

  2. [2]

    In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing

    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing. STOC ’96, pp. 212–219. Asso- ciation for Computing Machinery, New York, NY, USA (1996). https://doi.org/10.1145/237814. 237866 . https://doi.org/10.1145/237814.237866

    work page doi:10.1145/237814 1996

  3. [3]

    Nature 414(6866), 883–887 (2001)

    Vandersypen, L.M., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L.: Experi- mental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414(6866), 883–887 (2001)

    work page 2001

  4. [4]

    Lu, C.-Y., Browne, D.E., Yang, T., Pan, J.-W.: Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits. Phys. Rev. Lett.99, 250504 (2007) https: //doi.org/10.1103/PhysRevLett.99.250504 8

    work page doi:10.1103/physrevlett.99.250504 2007

  5. [5]

    Lanyon, B.P., Weinhold, T.J., Langford, N.K., Barbieri, M., James, D.F.V., Gilchrist, A., White, A.G.: Experimental demonstration of a compiled version of Shor’s algorithm with quantum entan- glement. Phys. Rev. Lett.99, 250505 (2007) https://doi.org/10.1103/PhysRevLett.99.250505

    work page doi:10.1103/physrevlett.99.250505 2007

  6. [6]

    Nature Physics8(10), 719–723 (2012)

    Lucero, E., Barends, R., Chen, Y., Kelly, J., Mariantoni, M., Megrant, A., O’Malley, P., Sank, D., Vainsencher, A., Wenner, J.,et al.: Computing prime factors with a Josephson phase qubit quantum processor. Nature Physics8(10), 719–723 (2012)

    work page 2012

  7. [7]

    Nature photonics6(11), 773–776 (2012)

    Martin-Lopez, E., Laing, A., Lawson, T., Alvarez, R., Zhou, X.-Q., O’Brien, J.L.: Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature photonics6(11), 773–776 (2012)

    work page 2012

  8. [8]

    Science351(6277), 1068–1070 (2016) https: //doi.org/10.1126/science.aad9480 https://www.science.org/doi/pdf/10.1126/science.aad9480

    Monz, T., Nigg, D., Martinez, E.A., Brandl, M.F., Schindler, P., Rines, R., Wang, S.X., Chuang, I.L., Blatt, R.: Realization of a scalable shor algorithm. Science351(6277), 1068–1070 (2016) https: //doi.org/10.1126/science.aad9480 https://www.science.org/doi/pdf/10.1126/science.aad9480

    work page doi:10.1126/science.aad9480 2016

  9. [9]

    Amico, M., Saleem, Z.H., Kumph, M.: Experimental study of Shor’s factoring algorithm using the IBM Q experience. Phys. Rev. A100, 012305 (2019) https://doi.org/10.1103/PhysRevA.100. 012305

    work page doi:10.1103/physreva.100 2019

  10. [10]

    Mosca, M.: Cybersecurity in an era with quantum computers: Will we be ready? IEEE Security & Privacy 16(5), 38–41 (2018) https://doi.org/10.1109/MSP.2018.3761723

    work page doi:10.1109/msp.2018.3761723 2018

  11. [11]

    Quantum 5, 433 (2021) https://doi.org/10.22331/q-2021-04-15-433

    Gidney, C., Ekerå, M.: How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum 5, 433 (2021) https://doi.org/10.22331/q-2021-04-15-433

    work page doi:10.22331/q-2021-04-15-433 2048

  12. [12]

    In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp

    Cleve, R., Watrous, J.: Fast parallel circuits for the quantum Fourier transform. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 526–536 (2000). https://doi. org/10.1109/SFCS.2000.892140

    work page doi:10.1109/sfcs.2000.892140 2000

  13. [13]

    Circuit for Shor's algorithm using 2n+3 qubits

    Beauregard, S.: Circuit for Shor’s algorithm using 2n+3 qubits. arXiv:quant-ph/0205095 (2002)

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  14. [14]

    Quantum Inf

    Häner, T., Roetteler, M., Svore, K.M.: Factoring using 2n+2 qubits with toffoli based modular multiplication. Quantum Inf. Comput.17(7&8), 673–684 (2017) https://doi.org/10.26421/QIC17. 7-8-7

    work page doi:10.26421/qic17 2017

  15. [15]

    Factoring with n+2 clean qubits and n-1 dirty qubits

    Gidney, C.: Factoring with n+ 2 clean qubits and n-1 dirty qubits. arXiv:1706.07884 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  16. [16]

    In: Post-Quantum Cryptog- raphy: 8th International Workshop, PQCrypto 2017, Utrecht, The Netherlands, June 26-28, 2017, Proceedings 8, pp

    Bernstein, D.J., Heninger, N., Lou, P., Valenta, L.: Post-quantum rsa. In: Post-Quantum Cryptog- raphy: 8th International Workshop, PQCrypto 2017, Utrecht, The Netherlands, June 26-28, 2017, Proceedings 8, pp. 311–329 (2017). Springer

    work page 2017

  17. [17]

    arXiv:2308.06572 (2023)

    Regev, O.: An efficient quantum factoring algorithm. arXiv:2308.06572 (2023)

    work page arXiv 2023

  18. [18]

    Peng, X., Liao, Z., Xu, N., Qin, G., Zhou, X., Suter, D., Du, J.: Quantum adiabatic algorithm for factorization and its experimental implementation. Phys. Rev. Lett.101, 220405 (2008) https: //doi.org/10.1103/PhysRevLett.101.220405

    work page doi:10.1103/physrevlett.101.220405 2008

  19. [19]

    Scientific reports7(1), 43048 (2017)

    Dridi, R., Alghassi, H.: Prime factorization using quantum annealing and computational algebraic geometry. Scientific reports7(1), 43048 (2017)

    work page 2017

  20. [20]

    Scientific reports8(1), 17667 (2018)

    Jiang, S., Britt, K.A., McCaskey, A.J., Humble, T.S., Kais, S.: Quantum annealing for prime factorization. Scientific reports8(1), 17667 (2018)

    work page 2018

  21. [21]

    Variational Quantum Factoring

    Anschuetz, E.R., Olson, J.P., Aspuru-Guzik, A., Cao, Y.: Variational quantum factoring. arXiv:1808.08927 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    Journal of Physics Communications 3(2), 025014 (2019) https://doi.org/10.1088/2399-6528/ab060d

    Kieu, T.D.: A factorisation algorithm in adiabatic quantum computation. Journal of Physics Communications 3(2), 025014 (2019) https://doi.org/10.1088/2399-6528/ab060d

    work page doi:10.1088/2399-6528/ab060d 2019

  23. [23]

    Scientific reports10(1), 7106 (2020) 9

    Wang, B., Hu, F., Yao, H., Wang, C.: Prime factorization algorithm based on parameter optimization of ising model. Scientific reports10(1), 7106 (2020) 9

    work page 2020

  24. [24]

    Hegade, N.N., Paul, K., Albarrán-Arriagada, F., Chen, X., Solano, E.: Digitized adiabatic quantum factorization. Phys. Rev. A104, 050403 (2021) https://doi.org/10.1103/PhysRevA.104.L050403

    work page doi:10.1103/physreva.104.l050403 2021

  25. [25]

    arXiv:2212.12372 (2022)

    Yan, B., Tan, Z., Wei, S., Jiang, H., Wang, W., Wang, H., Luo, L., Duan, Q., Liu, Y., Shi, W., Fei, Y., Meng, X., Han, Y., Shan, Z., Chen, J., Zhu, X., Zhang, C., Jin, F., Li, H., Song, C., Wang, Z., Ma, Z., Wang, H., Long, G.-L.: Factoring integers with sublinear resources on a superconducting quantum processor. arXiv:2212.12372 (2022)

    work page arXiv 2022

  26. [26]

    279–287 (2001)

    Dam, W., Mosca, M., Vazirani, U.: How powerful is adiabatic quantum computation? In: Pro- ceedings 42nd IEEE Symposium on Foundations of Computer Science, pp. 279–287 (2001). https://doi.org/10.1109/SFCS.2001.959902

    work page doi:10.1109/sfcs.2001.959902 2001

  27. [27]

    Nature499(7457), 163–165 (2013)

    Smolin, J.A., Smith, G., Vargo, A.: Oversimplifying quantum factoring. Nature499(7457), 163–165 (2013)

    work page 2013

  28. [28]

    factoring integers with sublinear resources on a superconducting quantum processor

    Khattar, T., Yosri, N.: A comment on "factoring integers with sublinear resources on a superconducting quantum processor". arXiv:2307.09651 (2023)

    work page arXiv 2023

  29. [29]

    Albash, T., Lidar, D.A.: Adiabatic quantum computation. Rev. Mod. Phys.90, 015002 (2018)

    work page 2018

  30. [30]

    Nielsen and Isaac L

    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511976667

    work page doi:10.1017/cbo9780511976667 2010

  31. [31]

    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A52, 3457– 3467 (1995) https://doi.org/10.1103/PhysRevA.52.3457

    work page doi:10.1103/physreva.52.3457 1995

  32. [32]

    SIAM Review50, 755–787 (2008)

    Aharonov, D., Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Review50, 755–787 (2008)

    work page 2008

  33. [33]

    Kieu, T.D.: A class of time-energy uncertainty relations for time-dependent hamiltonians. Proc. R. Soc. A.475, 20190148 (2019) https://doi.org/10.1098/rspa.2019.0148

    work page doi:10.1098/rspa.2019.0148 2019

  34. [34]

    Science345(6195), 420–424 (2014) https: //doi.org/10.1126/science.1252319 https://www.science.org/doi/pdf/10.1126/science.1252319

    Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science345(6195), 420–424 (2014) https: //doi.org/10.1126/science.1252319 https://www.science.org/doi/pdf/10.1126/science.1252319

    work page doi:10.1126/science.1252319 2014

  35. [35]

    Quantum 5, 597 (2021) https://doi.org/10.22331/q-2021-12-06-597

    Hastings, M.B.: The Power of Adiabatic Quantum Computation with No Sign Problem. Quantum 5, 597 (2021) https://doi.org/10.22331/q-2021-12-06-597

    work page doi:10.22331/q-2021-12-06-597 2021

  36. [36]

    https://github.com/qperfect-io

    MIMIQ Quantum Circuit Simulator. https://github.com/qperfect-io

    work page

  37. [37]

    Mathematics11(19) (2023) https://doi.org/10.3390/math11194222

    Willsch, D., Willsch, M., Jin, F., De Raedt, H., Michielsen, K.: Large-scale simulation of Shor’s quantum factoring algorithm. Mathematics11(19) (2023) https://doi.org/10.3390/math11194222

    work page doi:10.3390/math11194222 2023

  38. [38]

    SIAM Journal on Computing26(5), 1510–1523 (1997) https://doi.org/10.1137/ S0097539796300933

    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quan- tum computing. SIAM Journal on Computing26(5), 1510–1523 (1997) https://doi.org/10.1137/ S0097539796300933

    work page 1997

  39. [39]

    Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A60, 2746–2751 (1999) https://doi.org/10.1103/PhysRevA.60.2746

    work page doi:10.1103/physreva.60.2746 1999

  40. [40]

    Aaronson, S.: How much structure is needed for huge quantum speedups? arXiv:2209.06930 (2022)

    work page arXiv 2022

  41. [41]

    High Performance Quantum Modular Multipliers

    Rines, R., Chuang, I.: High performance quantum modular multipliers. arXiv:1801.01081 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  42. [42]

    Ruiz-Perez, L., Garcia-Escartin, J.C.: Quantum arithmetic with the quantum Fourier transform. Quantum Information Processing16, 1–14 (2017) 10 6 Supplementary Information 6.1 Decomposition of QMA in terms of elementary gates To facilitate implementation on quantum computers we provide circuit based decompositions of the key elements of the quantum factori...

    work page 2017