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Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

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Pith's one-line read Four restricted Clifford+T gate sets are exactly characterized by four rings of matrix entries: $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$.

desk verdict Four new exact-synthesis characterizations for restricted Clifford+T gate sets, with a solid proof structure; the terse column-induction step holds up under scrutiny. read the letter →

arxiv 1908.06076 v3 pith:SJ2LTTWS submitted 2019-08-16 quant-ph

Matthew Amy , Andrew N. Glaudell , Neil J. Ross This is my paper · ORCID
classification quant-ph
keywords exactsynthesisClifford+Tcircuitsrestrictedgatesetsnumber-theoreticcharacterizationunitarymatricesoverringsToffoliHadamardancilla-free
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

This paper proves exact-synthesis theorems for four families of quantum circuits that lie between classical reversible logic and the full Clifford+T gate set. For each family, a $2^n\times 2^n$ unitary can be built exactly, with a single ancilla, from the stated gates if and only if its entries lie in a specified subring of $\mathbb{Z}[1/\sqrt{2},i]$: $\mathbb{Z}[1/2]$ for $\{X,CX,CCX,H\otimes H\}$, $\mathbb{Z}[1/\sqrt{2}]$ for $\{X,CX,CCX,H,CH\}$, $\mathbb{Z}[1/i\sqrt{2}]$ for $\{X,CX,CCX,F\}$, and $\mathbb{Z}[1/2,i]$ for $\{X,CX,CCX,\omega H,S\}$. Here $\omega=e^{i\pi/4}$ and $F$ is a square root of $iH$. The ring condition is a direct check on matrix entries, so the results turn exact compilation of these universal gate sets into a number-theoretic membership test, and the proof supplies the circuit. Two corollaries extend the characterization to the gate sets $\{X,CX,CCX,H\}$ and $\{X,CX,CCX,H,S\}$, where the allowed matrices are $W/\sqrt{2}^{\,q}$ with $W$ over $\mathbb{Z}$ or $\mathbb{Z}[i]$.

What carries the argument

The load-bearing mechanism is a column lemma for each ring. Writing a vector as $u/p^q$ with $u$ over the relevant integer ring and $q$ the $p$-denominator exponent, the lemma uses residue arithmetic modulo $2$, $2i\sqrt{2}$, or $1+i$ to group entries into pairs or quadruples that are congruent to $1$ modulo $2$, then applies $H$, $H\otimes H$, $F$, or $\omega H$ to make the new entries divisible by $p$, lowering $q$. Iterating reduces any unit vector to a standard basis vector, and applying the reduction column by column expresses the full unitary as a product of realizable multi-level matrices. The distinct move in the dyadic case is the four-level $(H\otimes H)$ gate, which substitutes for the two-level Hadamard moves used elsewhere.

What would settle it

Run the paper's column-reduction algorithm on a small unitary with entries in $\mathbb{Z}[1/2]$, say a $4\times4$ or $8\times8$ example, and inspect each step: if any denominator-lowering move for a later column acts on a row fixed by an earlier column, that matrix is a counterexample to the constructive direction.

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Extended reading notes

Core claim

The central claim is that the obvious necessary condition is also sufficient: for each of the four gate sets, having all entries in the ring forces the unitary to be exactly representable. The proof is constructive and proceeds by column reduction: every unit vector over the ring is reduced to a standard basis vector by one-, two-, and four-level operations that the gate set can realize, and repeating this column by column expresses the whole unitary as a product of realizable operations. In the imaginary and Gaussian cases, the ancilla-free version is settled for $n\ge4$: a matrix in $U_{2^n}(\mathbb{Z}[1/i\sqrt{2}])$ or $U_{2^n}(\mathbb{Z}[1/2,i])$ has an ancilla-free circuit exactly when its determinant is $1$. The same machinery yields the two corollaries for gates with a single-qubit Hadamard instead of the two-qubit or scaled variants.

Load-bearing premise

The proof assumes that when later columns are reduced, the generators chosen act only on rows that have not yet been fixed, so earlier columns are not disturbed; this step is asserted rather than demonstrated, and the constructive direction of all four characterizations rests on it.

Editorial extensions

If this is right

  • Every unitary in $U_{2^n}(\mathbb{Z}[1/2])$ compiles exactly over $\{X,CX,CCX,H\otimes H\}$ with one ancilla, and similarly for the other three rings and gate sets.
  • For the $F$ and $\omega H$ gate sets on $n\ge4$ qubits, ancilla-free synthesis is equivalent to determinant $1$; for $n<4$ the determinant condition can be dropped.
  • Replacing $H\otimes H$ by $H$ widens the integral characterization to matrices $W/\sqrt{2}^{q}$ with $W$ an integer matrix, and replacing $\omega H$ by $H$ widens the Gaussian characterization to $W/\sqrt{2}^{q}$ with $W$ over $\mathbb{Z}[i]$.
  • Each characterization gives an exact synthesis algorithm, so membership in these ring groups is decidable and the compiled circuits use at most one ancilla.

Reading between the lines

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

  • Beyond the paper: the same column-reduction template should characterize other subrings in the lattice of subrings of $\mathbb{Z}[1/\sqrt{2},i]$, since each ring needs only a residue lemma matching a gate to the relevant modulus.
  • Beyond the paper: the determinant-1 obstruction for ancilla-free imaginary and Gaussian circuits suggests that the open real and integral ancilla-free cases will also be controlled by a phase invariant, and small-dimension searches could test the paper's conjecture of a strict subgroup.
  • Beyond the paper: because the proofs are constructive, they make these restricted but universal gate sets usable as compilation targets for subroutines that must avoid $T$ gates, with the number ring serving as a quick pre-check on exact representability.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. This paper proves number-theoretic characterizations for four restricted but universal Clifford+T gate sets. The main theorem states that an n-qubit unitary V can be exactly represented over {X,CX,CCX,H⊗H}, {X,CX,CCX,H,CH}, {X,CX,CCX,F}, and {X,CX,CCX,ωH,S} if and only if V lies respectively in U_{2^n}(Z[1/2]), U_{2^n}(Z[1/√2]), U_{2^n}(Z[1/(i√2)]), and U_{2^n}(Z[1/2,i]). The proof adapts the Giles–Selinger column-reduction framework to these subrings, using one-, two-, and four-level generators and explicit circuit identities for the required multi-level operators. The paper also derives corollaries for {X,CX,CCX,H} and {X,CX,CCX,H,S}, and determinant-one ancilla-free characterizations for n≥4 in the real-imaginary and Gaussian cases. The 'only if' directions are immediate from the entries of the generators; the substantive work is the constructive 'if' direction, which is carried out by reducing unit vectors to standard basis vectors and then iterating over columns.

Significance. If the main theorem holds, it gives clean algebraic classifications of several natural universal gate sets, directly extending the Kliuchnikov–Maslov–Mosca and Giles–Selinger characterizations. The result places these gate sets in a lattice of subgroups of U_{2^n}(Z[1/√2,i]) and contributes to the program of classifying universal extensions of classical reversible gates. The paper is constructive: it provides explicit circuits for the multi-level generators, proves the key denominator-reduction lemmas in detail for the D and D[i] cases, and carefully states the ancilla overhead. The ancilla-free corollaries and the super-integral/super-Gaussian variants are useful refinements. The overall strategy is convincing, and the few compressed or omitted arguments appear to be routine analogues rather than substantive gaps.

minor comments (6)
  1. [Section 1 and throughout] The main theorem and the corollaries state '2n×2n unitary matrix' for an n-qubit circuit, but an n-qubit unitary is 2^n×2^n. The same mismatch appears in Corollaries 5.6, 5.11, 5.16, 5.21, 5.27, and 5.31, where Section 5 uses 'n-dimensional' for the matrix dimension while the corollaries speak of n-qubit circuits. Please adopt a consistent notation (e.g., N = 2^n for matrix dimension and n for qubit count) throughout the statements.
  2. [Section 5, Theorem 5.5] The proof of Theorem 5.5 is the single sentence 'By iteratively applying Lemma 5.4 to the columns of V.' This is sound but too terse; please add a sentence explaining the induction: after the first k columns have been reduced to e_1,...,e_k, unitarity forces the remaining lower-right block to be unitary, and the generators used for the next column act only on the remaining rows and can be embedded as identity on the already fixed rows.
  3. [Sections 5.2 and 5.3, Lemmas 5.14–5.16, 5.19, and 5.25] Several lemmas, including the column-reduction lemma and the main factorization theorem for the D[√2] and D[i√2] cases, are stated without proof, with the text saying they are 'established like the corresponding ones in the previous section.' The analogy is plausible, but for self-containedness please provide a proof sketch or appendix with the exact base case (n < 4) and the parity/denominator-exponent reduction for these cases.
  4. [Lemma 5.18] The equation '(-2)^q = Σ u_j†u_j' is incorrect: the unitarity condition gives 2^q = Σ u_j†u_j, since |i√2|^2 = 2. The sign error is harmless for the parity argument that follows, but it should be corrected.
  5. [Lemma 5.8] In the proof of Lemma 5.8, the base case uses X[0,j] and X[1,j'], but indices are elsewhere in [n] = {1,...,n}; this appears to be an indexing typo. The target vector in the lemma statement is also typeset with three explicit entries; it should be displayed as an n-vector.
  6. [Corollaries 5.16 and 5.21] There is a stray brace in 'U_{2n}(D[√2]{' and 'U_{2n}(D[i√2]{'; these should read U_{2^n}(D[√2]) and U_{2^n}(D[i√2]).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the characterizations are proved constructively from ring arithmetic and standard circuit identities; self-citations are not load-bearing.

full rationale

The derivation is self-contained with respect to the claimed characterizations. For each of the four rings the 'if' direction is proved by first showing that every unit vector over the ring can be reduced to a standard basis vector using the abstract generators (Lemmas 5.4, 5.14, 5.19, 5.25), with the residue arguments carried out directly in the relevant quotient rings (Propositions 3.4-3.6, Lemmas 5.1-5.3, 5.12-5.13, 5.17-5.18, 5.23-5.24), and then iterating over columns in the standard Giles-Selinger manner (Theorems 5.5, 5.15, 5.20, 5.26). The Giles-Selinger column-reduction framework is external prior work and is re-derived here in adapted form rather than assumed. The abstract generators are realized by explicit circuit identities over the target gate sets (Propositions 4.4-4.9), which rely on standard Barenco et al. constructions for multiply-controlled gates. The super-integral and super-Gaussian extensions reduce to the base cases by multiplying by H or omega (Theorems 5.10 and 5.30). No parameter is fitted, no target theorem is used as an induction hypothesis, and no load-bearing step rests on a self-citation: self-references [3-6] appear as background literature. The terse statement 'By iteratively applying Lemma 5.4 to the columns of V' in Theorem 5.5 is an under-specified but sound induction, since previously fixed columns are orthonormal to the remaining submatrix and the generators can be chosen on the unfixed rows. Some omitted proofs of analogous lemmas and a likely determinant-sign issue in Corollaries 5.22/5.28 are correctness or exposition concerns, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard ring-theoretic facts and standard quantum circuit constructions; no free parameters or invented entities are introduced.

assumptions (3)
  • domain assumption The group U_{2^n}(D[ω]) contains all exact Clifford+T unitaries, i.e., matrices with entries in Z[1/√2,i].
    Background result from Kliuchnikov-Maslov-Mosca and Giles-Selinger, used to motivate the restricted rings and gate sets.
  • domain assumption Multi-controlled gates can be implemented with at most one dirty ancilla using only {X,CX,CCX}.
    Barenco et al. construction invoked in Propositions 4.1-4.9 to control ancilla overhead.
  • standard math Residue properties of the quotient rings Z[√2]/(2), Z[i√2]/(2), Z[i√2]/(2i√2), and Z[i]/(2) as stated in Propositions 3.5-3.6.
    Elementary ring theory used in the column-reduction lemmas; these are proved or verified by residue tables in the paper.

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Pith. "Pith review of Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits." pith.science (2026). https://pith.science/paper/SJ2LTTWS

@misc{pith2026190806076,
  author       = {Pith},
  title        = {Pith review of: Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJ2LTTWS}},
  note         = {Machine review of arXiv:1908.06076}
}
abstract

Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+$T$ circuits by considering unitary matrices over subrings of $\mathbb{Z}[1/\sqrt{2},i]$. We focus on the subrings $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$, and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates $\{X, CX, CCX\}$ with an analogue of the Hadamard gate and an optional phase gate.

Figures

Figures reproduced from arXiv: 1908.06076 by the authors.

Figure 1
Figure 1. Some subgroups of Un(D [ω]). To the left of the cube, in yellow, the symmetric group Sn corresponds to circuits over the gate set {X, CX, CCX}. On the bottom face of the cube, in blue, are generalized symmetric groups, and on the top face of the cube, in red, are universal subgroups of Un(D [ω]). The edges of the lattice denote inclusion. The gates labeling the edges are sufficient to extend the expressive power of … view at source ↗

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Works this paper leans on

43 extracted references · 23 canonical work pages

  1. [1]

    Aaronson, D

    S. Aaronson, D. Grier, and L. Schaeffer. The classification of reversible bit operations. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference , volume 67 of LIPIcs, pages 23:1–23:34, 2017. DOI: 10.4230/LIPIcs.ITCS.2017.23. Also available from arXiv:1504.05155

    arXiv 2017

  2. [2]

    Aharonov

    D. Aharonov. A simple proof that Toffoli and Hadamard are quantum universal. Preprint available from arXiv:quant-ph/0301040, Jan. 2003

    arXiv 2003

  3. [3]

    Amy and M

    M. Amy and M. Mosca. T-count optimization and Reed-Muller codes. IEEE Transactions on Information Theory, 65(8):4771–4784, 2019. DOI: 10.1109/TIT.2019.2906374. Also available from arXiv:1601.07363

    arXiv 2019

  4. [4]

    M. Amy, D. Maslov, M. Mosca, and M. Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 32(6):818–830, 2013. DOI: 10.1109/TCAD.2013.2244643. Also available from arXiv:1206.0758

    arXiv 2013

  5. [5]

    M. Amy, D. Maslov, and M. Mosca. Polynomial-time T-depth optimization of Clifford+T circuits via matroid partitioning. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems , 33(10):1476–1489, 2014. DOI: 10.1109/TCAD.2014.2341953. Also available from arXiv:1303.2042

    arXiv 2014

  6. [6]

    M. Amy, J. Chen, and N. J. Ross. A finite presentation of CNOT-dihedral operators. In Proceedings of the 14th International Conference on Quantum Physics and Logic , QPL ’17, pages 84–97, 2017. DOI: 10.4204/EPTCS.266.5

    doi:10.4204/eptcs.266.5 2017

  7. [7]

    M. Artin. Algebra. Prentice Hall, 1991

    work page 1991

  8. [8]

    Backens and A

    M. Backens and A. Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. In Proceedings of the 15th International Conference on Quantum Physics and Logic, QPL ’18, pages 23–42, 2018. DOI: 10.4204/EPTCS.287.2

    doi:10.4204/eptcs.287.2 2018

Show all 43 references
  1. [9]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Physical Review A , 52: 3457–3467, 1995. DOI: 10.1103/PhysRevA.52.3457. Also available from arXiv:quant-ph/9503016

    1995 arXiv

  2. [10]

    X. Bian. Private communication, July 2019

    2019

  3. [11]

    Bian and P

    X. Bian and P. Selinger. Relations for the group of 2-qubit Clifford+T operators. Talk given at the Quantum Programming and Circuits Workshop. Slides available from https://www.mathstat. dal.ca/˜xbian/talks/slide_cliffordt2.pdf, June 2015

    2015

  4. [12]

    Bocharov, Y

    A. Bocharov, Y. Gurevich, and K. M. Svore. Efficient decomposition of single-qubit gates into V basis circuits. Physical Review A , 88:012313, 2013. DOI: 10.1103/PhysRevA.88.012313. Also available from arXiv:1303.1411. 17

    2013 arXiv

  5. [13]

    Bocharov, M

    A. Bocharov, M. Roetteler, and K. M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Physical Review A, 91:052317, 2015. DOI: 10.1103/PhysRevA.91.052317. Also avail- able from arXiv:1409.3552

    2015 arXiv

  6. [14]

    Bouland and S

    A. Bouland and S. Aaronson. Generation of universal linear optics by any beam splitter. Physical Review A, 89:062316, 2014. DOI: 10.1103/PhysRevA.89.062316. Also available from arXiv:1310. 6718

    2014 doi

  7. [15]

    De Vos, R

    A. De Vos, R. Van Laer, and S. Vandenbrande. The group of dyadic unitary matrices. Open Systems & Information Dynamics , 19(01):1250003, 2012. DOI: 10.1142/S1230161212500035

    2012 doi

  8. [16]

    Forest, D

    S. Forest, D. Gosset, V. Kliuchnikov, and D. McKinnon. Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets. Journal of Mathematical Physics , 56(8):082201, 2015. DOI: 10.1063/1.4927100. Also available from arXiv:1501.04944

    2015 arXiv

  9. [17]

    Giles and P

    B. Giles and P. Selinger. Exact synthesis of multiqubit Clifford+T circuits. Physical Review A, 87: 032332, 2013. DOI: 10.1103/PhysRevA.87.032332. Also available from arXiv:1212.0506

    2013 arXiv

  10. [18]

    Giles and P

    B. Giles and P. Selinger. Remarks on Matsumoto and Amano’s normal form for single-qubit Clif- ford+T operators. Preprint available from arXiv:1312.6584, Dec. 2013

    2013 arXiv

  11. [19]

    Gosset, V

    D. Gosset, V. Kliuchnikov, M. Mosca, and V. Russo. An algorithm for the T-count. Quantum In- formation & Computation , 14(15-16):1261–1276, 2014. DOI: 10.26421/QIC14.15-16. Also available from arXiv:1308.4134

    2014 arXiv

  12. [20]

    S. Greylyn. Generators and relations for the group U4(Z[1/ √ 2,i ]). Master’s thesis. Available from arXiv:1408.6204, 2014

    2014 arXiv

  13. [21]

    Grier and L

    D. Grier and L. Schaeffer. The classification of stabilizer operations over qubits. Preprint available from arXiv:1603.03999, 2016

    2016 arXiv

  14. [22]

    A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman. Real randomized benchmarking. Quantum, 2:85, 2018. DOI: 10.22331/q-2018-08-22-85. Also available from arXiv:1801.06121

    2018 arXiv

  15. [23]

    L. E. Heyfron and E. T. Campbell. An efficient quantum compiler that reduces T count. Quantum Science and Technology, 4(1):015004, 2018. DOI: 10.1088/2058-9565/aad604. Also available from arXiv:1712.01557

    2018 arXiv

  16. [24]

    Jeandel, S

    E. Jeandel, S. Perdrix, and R. Vilmart. Y-calculus: A language for real matrices derived from the ZX-calculus. In Proceedings of the 14th International Conference on Quantum Physics and Logic , QPL ’17, pages 23–57, 2017. DOI: 10.4204/EPTCS.266.2

    2017 doi

  17. [25]

    Kliuchnikov and J

    V. Kliuchnikov and J. Yard. A framework for exact synthesis. Preprint available from arXiv: 1504.04350, April 2015

    2015 arXiv

  18. [26]

    Kliuchnikov, D

    V. Kliuchnikov, D. Maslov, and M. Mosca. Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and T gates. Quantum Information & Computation , 13(7-8):607–630, 2013. DOI: 10.26421/QIC13.7-8. Also available from arXiv:1206.5236

    2013 arXiv

  19. [27]

    Kliuchnikov, A

    V. Kliuchnikov, A. Bocharov, and K. M. Svore. Asymptotically optimal topological quantum com- piling. Physical Review Letters , 112:140504, 2014. DOI: 10.1103/PhysRevLett.112.140504. Also available from arXiv:1310.4150

    2014 arXiv

  20. [28]

    Kliuchnikov, D

    V. Kliuchnikov, D. Maslov, and M. Mosca. Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits. IEEE Transactions on Computers , 65(1):161–172,

  21. [29]

    Matsumoto and K

    K. Matsumoto and K. Amano. Representation of quantum circuits with Clifford and π/8 gates. Preprint available from arXiv:0806.3834, June 2008

    2008 arXiv

  22. [30]

    Meuli, M

    G. Meuli, M. Soeken, and G. D. Micheli. SAT-based {CNOT, T } quantum circuit synthesis. In Proceedings of the 10th International Conference on Reversible Computation, RC ’17, pages 175–188,

  23. [31]

    M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information . Cam- bridge Series on Information and the Natural Sciences. Cambridge University Press, 2000. ISBN 9780521635035. DOI: 10.1017/CBO9780511976667

    doi

  24. [32]

    Parzanchevski and P

    O. Parzanchevski and P. Sarnak. Super-Golden-Gates for PU(2). Advances in Mathematics , 327: 869 – 901, 2018. DOI: https://doi.org/10.1016/j.aim.2017.06.022. Special volume honoring David Kazhdan. Also available from arXiv:1704.02106

    2018 arXiv

  25. [33]

    N. J. Ross. Optimal ancilla-free Clifford+V approximation of z-rotations. Quantum Information & Computation , 15(1112):932950, 2015. DOI: 10.26421/QIC15.11-12. Also available from arXiv: 1409.4355. 18

    2015 arXiv

  26. [34]

    N. J. Ross and P. Selinger. Optimal ancilla-free Clifford+T approximation of z-rotations. Quantum Information & Computation , 16(11-12):901–953, 2016. DOI: 10.26421/QIC16.11-12. Also available from arXiv:1403.2975

    2016 arXiv

  27. [35]

    Rudolph and L

    T. Rudolph and L. Grover. A 2 rebit gate universal for quantum computing. Preprint available from arXiv:quant-ph/0210187, Nov. 2002

    2002 arXiv

  28. [36]

    Selinger

    P. Selinger. Generators and relations for n-qubit Clifford operators. Logical Methods in Computer Science, 11(10):1–17, 2015. DOI: 10.2168/LMCS-11(2:10)2015. Also available from arXiv:1310. 6813

    2015 doi

  29. [37]

    Y. Shi. Both Toffoli and controlled-NOT need little help to do universal quantum computing. Quantum Information & Computation , 3(1):84–92, 2003. DOI: 10.26421/QIC3.1. Also available from arXiv:quant-ph/0205115

    2003 arXiv

  30. [38]

    R. Vilmart. A ZX-calculus with triangles for Toffoli-Hadamard, Clifford+T, and beyond. In Proceed- ings of the 15th International Conference on Quantum Physics and Logic , QPL ’18, pages 313–344,

  31. [39]

    Welch, A

    J. Welch, A. Bocharov, and K. M. Svore. Efficient approximation of diagonal unitaries over the Clif- ford+T basis. Quantum Information & Computation, 16(1-2):87–104, 2016. DOI: 10.26421/QIC16.1-

    2016 doi

  32. [41]

    DOI: 10.4204/EPTCS.287.18

    doi

  33. [43]

    Also available from arXiv:1412.5608. A Ancilla-Free Circuit Constructions A.1 The D [ i √ 2 ] case In this appendix, we give ancilla-free constructions of the two-level operators XZ , ZX = (XZ )†, FZ , and ZF over{X,CX,CCX,F }. We progressively build up to the necessary operat...

    arXiv

  34. [2016]

    Also available from arXiv:1212.6964

    DOI: 10.1109/TC.2015.2409842. Also available from arXiv:1212.6964

    2015

  35. [2018]

    DOI: 10.1007/978-3-319-99498-7˙12

    doi

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