These collected entries accompany all four mathematical treatments, in their original order and numbering.
Entries and citation numbers follow the retained manuscript. Citation keys keep links stable. These links do not establish what a cited source proves.
[1] q:Ramanujan1918
S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Transactions of the Cambridge Philosophical Society 22, no. 13 (1918), 259–276. Original-paper scan index: https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper21/page1.htm.
[2] q:Vaidyanathan2014FIR
P. P. Vaidyanathan, Ramanujan-sum expansions for finite duration (FIR) sequences, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4933–4937. https://doi.org/10.1109/ICASSP.2014.6854540. Author manuscript: https://systems.caltech.edu/dsp/PPVSquare.pdf.
[3] q:Vaidyanathan2014I
P. P. Vaidyanathan, Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals, IEEE Transactions on Signal Processing 62(16) (2014), 4145–4157. https://doi.org/10.1109/TSP.2014.2331617.
[4] q:Vaidyanathan2014II
P. P. Vaidyanathan, Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications, IEEE Transactions on Signal Processing 62(16) (2014), 4158–4172. https://doi.org/10.1109/TSP.2014.2331624.
[5] q:KomorebiStratified16
Komorebi P., Stratified Universality for Resolved Representations, v1.16.0, Block 16, internal manuscript, 2026. Proposition 21.11 and Boundary 21.12, printed pp. 189–190. Preserved source: Komorebi_Stratified_Universality_Foundations_v1.16.0_Block16.pdf.
[6] s:AaronsonGottesman2004
S. Aaronson and D. Gottesman, Improved Simulation of Stabilizer Circuits, Physical Review A 70 (2004), 052328.
[7] s:AaronsonAmbainis2018
S. Aaronson and A. Ambainis, Forrelation: A Problem that Optimally Separates Quantum from Classical Computing, SIAM Journal on Computing 47 (2018), 982–1038.
[8] s:BallSimoens2026
S. Ball and R. Simoens, Thirty-six quantum officers are entangled, arXiv:2603.02334, 2026.
[9] s:BernsteinVazirani1997
E. Bernstein and U. Vazirani, Quantum Complexity Theory, SIAM Journal on Computing 26 (1997), 1411–1473.
[10] s:JozsaMiyake2008
R. Jozsa and A. Miyake, Matchgates and Classical Simulation of Quantum Circuits, Proceedings of the Royal Society A 464 (2008), 3089–3106.
[11] s:MustoVicary2016
B. Musto and J. Vicary, Quantum Latin Squares and Unitary Error Bases, Quantum Information and Computation 16 (2016), 1318–1332.
[12] s:RatherEtAl2022
S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski, Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem, Physical Review Letters 128 (2022), 080507.
[13] s:SchurWeyl
Standard Schur–Weyl duality and representation-theoretic decomposition results are used in their finite-dimensional form.
[14] s:Shor1997
P. W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing 26 (1997), 1484–1509.
[15] s:Simon1997
D. Simon, On the Power of Quantum Computation, SIAM Journal on Computing 26 (1997), 1474–1483.
[16] s:Valiant2002
L. G. Valiant, Quantum Circuits that Can Be Simulated Classically in Polynomial Time, SIAM Journal on Computing 31 (2002), 1229–1254.
[17] s:TransferRecords
K. P., Quantum Transfer Blocks and Quantum Polyphase Transfer Calculus, project research records, 2026.
[18] g:AcuavivaEtAl2026
A. Acuaviva, A. Acuaviva, and P. Acuaviva, CNOT-Distance is NP-complete under all-to-all connectivity, arXiv:2608.03825, 2026.
[19] g:AmyMosca2019
M. Amy and M. Mosca, \(T\)-count optimization and Reed–Muller codes, IEEE Trans. Inform. Theory 65 (2019), 4771–4784.
[20] g:AmyEtAl2013
M. Amy, D. Maslov, M. Mosca, and M. Roetteler, A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32 (2013), 818–830.
[21] g:Bataille2022
M. Bataille, Quantum circuits of CNOT gates: optimization and entanglement, Quantum Inf. Process. 21 (2022), Article 269.
[22] g:BravyiKitaev2005
S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71 (2005), 022316.
[23] g:BuFanJoo2025
A. Bu, E. Fan, and R. Joo, Minimum synthesis cost of CNOT circuits, Quantum Inf. Process. 24 (2025), Article 208.
[24] g:Carter1972
R. W. Carter, Simple Groups of Lie Type, Wiley, 1972.
[25] g:ChristensenEtAl2025
J. E. Christensen, S. F. Jørgensen, A. Pavlogiannis, and J. van de Pol, On exact sizes of minimal CNOT circuits, in Reversible Computation 2025, Lecture Notes in Comput. Sci. 15716, Springer, 2025, pp. 71–88.
[26] g:DehaeneDeMoor2003
J. Dehaene and B. De Moor, Clifford group, stabilizer states, and linear and quadratic operations over \(\mathrm{GF}(2)\), Phys. Rev. A 68 (2003), 042318.
[27] g:GongYu2026
S. Gong and A. Yu, Explicit matrices over \(\mathbb Z_2\) with CNOT and row complexity \(4n-o(n)\) and local logic gates, arXiv:2607.28598, 2026.
[28] g:Gottesman1997
D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology, 1997.
[29] g:GoubaultEtAl2021
T. Goubault de Brugière, M. Baboulin, B. Valiron, S. Martiel, and C. Allouche, Gaussian elimination versus greedy methods for the synthesis of linear reversible circuits, ACM Trans. Quantum Comput. 2 (2021), Article 11.
[30] g:Hartshorne1977
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, 1977.
[31] g:Humphreys1975
J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Springer, 1975.
[32] g:JiangEtAl2020
J. Jiang, X. Sun, S.-H. Teng, B. Wu, K. Wu, and J. Zhang, Optimal space-depth trade-off of CNOT circuits in quantum logic synthesis, in Proceedings of SODA 2020, SIAM, 2020, pp. 213–229.
[33] g:Jorgensen2026
S. F. Jørgensen, Lower bounds for the CNOT-complexity of linear reversible operators, arXiv:2607.22248, 2026.
[34] g:KliuchnikovEtAl2013
V. Kliuchnikov, D. Maslov, and M. Mosca, Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and \(T\) gates, Quantum Inf. Comput. 13 (2013), 607–630.
[35] g:Landsberg2012
J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, 2012.
[36] g:NielsenChuang2010
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
[37] g:Oxley2011
J. Oxley, Matroid Theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, 2011.
[38] g:PatelMarkovHayes2008
K. N. Patel, I. L. Markov, and J. P. Hayes, Optimal synthesis of linear reversible circuits, Quantum Inf. Comput. 8 (2008), 282–294.
[39] g:RatherEtAl2023
S. A. Rather, N. Ramadas, V. Kodiyalam, and A. Lakshminarayan, Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler, Phys. Rev. A 108 (2023), 032412.
[40] g:SaeediMarkov2013
M. Saeedi and I. L. Markov, Synthesis and optimization of reversible circuits—a survey, ACM Comput. Surv. 45 (2013), Article 21.
[41] g:Selinger2013
P. Selinger, Quantum circuits of \(T\)-depth one, Phys. Rev. A 87 (2013), 042302.
[42] g:Serre1977
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, vol. 42, Springer, 1977.
[43] g:ShaikVanDePol2025
I. Shaik and J. van de Pol, CNOT-optimal Clifford synthesis as SAT, in 28th International Conference on Theory and Applications of Satisfiability Testing, LIPIcs 341 (2025), Article 28.
[44] g:WebsterEtAl2025
M. Webster, S. Koutsioumpas, and D. E. Browne, Heuristic and optimal synthesis of CNOT and Clifford circuits, arXiv:2503.14660, 2025.
[45] g:WeedbrookEtAl2012
C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84 (2012), 621–669.
[46] c:foundations
Komorebi P. Foundations: restored computation and complete discrepancies. docs/FOUNDATIONS.md, source release v1.2.0, September 21, 2026. Supplied repository source.
[47] c:theory
Komorebi P. Complete discrepancy covers, grade extraction, and exact gluing. docs/THEORY.md, source release v1.2.0, September 21, 2026. Supplied repository manuscript.
[48] c:claims
Komorebi P. Claim register. docs/CLAIMS.md and docs/claims.json, source release v1.2.0. Authoritative active groups C01–C08.
[49] c:boundary
Komorebi P. Proof and computation boundary. PROOF_BOUNDARY.md, source release v1.2.0. Controlling model and evidence qualifications.
[50] c:audit
Komorebi P. Release audit: 1.2.0. evidence/lean-block02/RELEASE_AUDIT.md, September 21, 2026. Scoped archived formal verification. This edition also supplies fresh computational and document validation records.
[51] c:reproduction
Komorebi P. Reproduction contract. docs/REPRODUCIBILITY.md, source release v1.2.0. Frozen finite results: evidence/baseline/completeness.json, projection.json, and fingerprint.json in the same baseline directory.
[52] c:soeken
Mathias Soeken. Determining the Multiplicative Complexity of Boolean Functions using SAT. arXiv:2005.01778v1, 2020. Background XOR–AND model and degree-bound context. https://arxiv.org/abs/2005.01778v1
[53] c:findboyar
Joan Boyar and Magnus Gausdal Find. Multiplicative Complexity of Vector Valued Boolean Functions. arXiv:1407.6169v3, 2018 revision of the 2014 preprint. Background affine-restriction and quadratic-complexity context. https://arxiv.org/abs/1407.6169v3
[54] c:beckli
Chris Beck and Yuan Li. Represent MOD function by low degree polynomial with unbounded one-sided error. arXiv:1304.0713v1, 2013. Background one-sided polynomial-immunity terminology. https://arxiv.org/abs/1304.0713v1