Lower bounds for the CNOT-complexity of linear reversible operators

S{\o}ren Fuglede J{\o}rgensen

REVIEW 2 major objections 5 minor 1 cited by

The paper proves that an explicit family of invertible matrices over F2, built from parity-check matrices of BCH codes, requires at least 4N − o(N) CNOT gates, and that a specific 17167×17167 matrix needs at least 51,535 CNOT gates — more t

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2026-08-01 05:20 UTC pith:PLF5MSKR

load-bearing objection First explicit CNOT-complexity family beating 3(n−1), with the asymptotic constant riding on Sergeev's theorem — internal logic clean, deserves refereeing. the 2 major comments →

arxiv 2607.22248 v1 pith:PLF5MSKR submitted 2026-07-24 quant-ph

Lower bounds for the CNOT-complexity of linear reversible operators

S{\o}ren Fuglede J{\o}rgensen This is my paper

classification quant-ph MSC 68Q1794B05 PACS 03.67.-a

keywords CNOT-complexitylinear reversible circuitsadditive complexityBCH codesparity-check matriceslower boundstransvectionsGL(n, F2)

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper addresses a concrete open problem: can anyone exhibit explicit families of linear reversible circuits (CNOT circuits) that require more than 3(n−1) gates, beating the cyclic permutations? It answers yes. The strategy is to take lower bounds that are easier to prove in the more permissive additive-complexity setting — where XOR gates may discard outputs and fan out freely — and lift them back to the reversible CNOT setting with a bounded loss. Applying this to an imported bound for block matrices built from BCH code parity-check matrices yields an explicit family whose CNOT-complexity is at least 4N − o(N), and a concrete 17,167-wire matrix whose minimal circuit provably needs at least 51,535 CNOT gates, exceeding the cyclic-permutation benchmark of 51,498.

Core claim

The central discovery is that hardness of non-reversible XOR computation transfers to reversible computation almost losslessly. Concretely, any n×n matrix A can be padded into an invertible matrix M of size N = n + s, where s is A's corank, so that d(M) ≥ L(A) − 1; a lower bound of cn on additive complexity thus becomes a lower bound of c/(1+γ) N on CNOT-complexity as long as the corank fraction stays small. For the block matrix A_U built from BCH-code parity checks, the paper adds a block-permutation matrix to force the rank up, bounding the corank by the number of columns, and combines the imported additive bound with a submatrix embedding to obtain d(M_N) ≥ 4N − o(N). Tracing the construc

What carries the argument

The load-bearing object is the block matrix M0 = [[U, I_a],[I_b, U^T]] formed by adding a block-permutation matrix to the block-diagonal matrix A_U = diag(U, U^T), where U is a parity-check matrix of a BCH code. The permutation addition preserves additive complexity up to an additive n penalty, while an isomorphism ker M0 ≅ ker(I_b + U^T U) bounds the corank by b, making the padding cost to invertibility negligible. Four lifting lemmas package the translation: submatrices inherit complexity, non-invertible matrices embed in invertible ones at small dimension cost, and adding permutations barely changes complexity. The driving force is an imported additive-complexity lower bound for BCH-based

Load-bearing premise

The entire construction leans on an imported result the paper does not reprove: that the BCH block matrices have additive complexity at least 5n − O(√n log n); if that bound failed, the constructed family would not exceed the old 3(n−1) barrier.

What would settle it

Run an optimal or near-optimal CNOT synthesis algorithm on the explicit 17,167-wire matrix M0 (the paper itself reports a roughly 40-million-gate circuit, far above the claimed bound): finding a circuit with fewer than 51,535 CNOT gates would refute the concrete claim. Alternatively, recompute the binary rank of I_784 + U^T U; a value below 784 would make M0 singular and void the instance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The cyclic permutations are dethroned: the first explicit family of linear reversible operators with CNOT-complexity above 3(n−1) is now known, asymptotically at 4n − o(n).
Any future improvement to additive-complexity lower bounds for code-based matrices automatically upgrades the reversible bound through the same lifting lemmas.
Because the hard instance is built from parity-check matrices of a well-known code, the lower bounds apply to synthesized circuits for encoding and syndrome extraction, where large gate counts are a practical liability.
The bound transfers to the model where qubit swaps are free, so the same 4n − o(n) statement holds under that cost measure.

Where Pith is reading between the lines

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

The 4/5 efficiency factor between the imported 5n additive bound and the 4n reversible bound is exactly the price paid to make the matrix invertible; a construction with zero corank loss would likely push the constant closer to 5.
The gap between the 51,535 lower bound and the roughly 4×10^7 upper bound reported for M0 suggests optimal synthesis tools might still find significantly better circuits, which would either confirm the gap or reveal slack in the lifting argument.
A quantitative version of the paper's hint that higher code distance yields harder instances — a function mapping code parameters to CNOT-complexity lower bounds — would give a practical tool for design-space exploration in error-correcting circuit synthesis.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

First explicit CNOT-complexity family beating 3(n−1), with the asymptotic constant riding on Sergeev's theorem — internal logic clean, deserves refereeing.

read the letter

Here's my take on the Jørgensen paper. The headline is simple: it delivers the first explicit family of reversible linear operators with CNOT-complexity 4n − o(n), breaking the 3(n−1) barrier that had stood as the best explicit lower bound. That is a genuine result, not a repackaging. The lifting lemmas (e.g., L(A+B) ≤ L(A)+L(B)+n, and the corank bound for the block matrix) are clean and do real work. The concrete 17167×17167 matrix with a lower bound of 51535 > 3(17167−1) is a nice touch; the rank computation is backed by code, and the arithmetic checks out.

The main soft spot is the one the stress-test flags: the entire asymptotic result rests on Sergeev's 2025 theorem giving L(A_U) ≥ 5n − O(√n log n) for the BCH-based construction. The paper does not reprove it, and after the lifting loss of n, you land only just above 3n. If Sergeev's hypotheses are not met by this particular U, the family would not beat the permutations. I read the conditions (row count n−m, column count m=O(√n log n), independence index ≥ s) and they look satisfied. So this is a dependency, not a flaw. It's the kind of dependency that editors should let a referee check by reading Sergeev, not by requiring reproduction.

Another minor point: the paper's family is only defined for a sparse set of N (those coming from the BCH parameters). That's fine for asymptotics, and the concrete instance compensates. The upper bound of ~40 million CNOTs from Patel–Markov–Hayes shows the lower bound is not tight, but that's expected and not a criticism.

The paper is honest: it states clearly what is new, credits Sergeev properly, and includes code for the computational check. No circularity, no hidden fitting. The presentation is compact to the point of terseness; some readers may want more intuition, but all the pieces are there.

Who should read this: anyone working on CNOT synthesis, linear reversible circuits, or lower bounds in quantum compilation. It deserves serious refereeing. I'd send it to a referee and expect acceptance after a careful check of the Sergeev dependency and the code.

Reading group: yes, I'd put it in a group focused on quantum circuit complexity.

Referee Report

2 major / 5 minor

Summary. The paper studies the CNOT-complexity of invertible matrices over F2, i.e., the minimum number of transvections needed to express a matrix in GL_n(F2). Its main contribution is a method for lifting lower bounds on additive complexity of non-reversible linear operators to lower bounds on CNOT-complexity, at the cost of a small loss. Using recent lower bounds of Sergeev [Ser25] for matrices built from BCH parity-check matrices, the author proves that an explicit family {M_N} satisfies d(M_N) ≥ 4N − o(N), asymptotically exceeding the previously best explicit lower bound 3(n−1) attained by cyclic permutations. The paper also constructs a concrete matrix M0 ∈ GL_17167(F2) with d(M0) ≥ 51535 > 3(17167−1). The argument proceeds through four lemmas (submatrix embedding, invertible extension, permutation perturbation, and a block decomposition with controlled corank) and two quoted theorems from [Ser25]. The internal derivations are compact and, conditional on the quoted results, correct.

Significance. Conditional on the cited results of Sergeev, the paper achieves a notable advance: it provides the first explicit family of linear reversible operators whose CNOT-complexity provably exceeds the 3(n−1) barrier of cyclic permutations, and it gives the first explicit matrix for which d(A) > 3(n−1) at a concrete size. The lifting technique (Lemmas 3.1–3.4) is simple, potentially reusable, and clearly presented. The paper also provides code for the rank computation used in Proposition 5.2, which is a good reproducibility practice. The main weakness is the heavy reliance on external lower bounds: the constant 5 in Theorem 4.1, and hence the final 4N, is entirely imported from [Ser25]. The paper does not reprove or even state the exact hypotheses of the cited corollary as applied to the shortened BCH matrix, so the applicability is not auditable from the manuscript alone.

major comments (2)

[§4, Theorem 4.1] The asymptotic result derives its constant 5 entirely from [Ser25, Cor. 1], which is quoted without stating the corollary's hypotheses. In the present construction, U is obtained by selecting only n−m rows from the full BCH matrix, rather than using all 2^p−1 nonzero field elements. The authors should state precisely the conditions under which [Ser25, Cor. 1] applies and verify them for this shortened U. In particular, it must be shown that the row-count reduction does not degrade the stated 5n bound. This is load-bearing: if the corollary only applies to the full matrix, the asymptotic family as parameterized may not beat 3(N−1).
[§5, Eq. (2) and Proposition 5.2] The numerical lower bound L(A_U) ≥ 68702 relies on Eq. (2), which is cited to [Ser25, Lem. 1 and Prop. 1], and on Theorem 5.1, which involves the reduced complexity L*, a quantity never defined in the manuscript. To make the concrete claim auditable, the authors should either state the exact lemmas/propositions used (or give a short derivation of Eq. (2)) and define L*, or give a self-contained proof of the numerical bound. Without this, the 51535 lower bound for M0 cannot be independently checked from the text.

minor comments (5)

[§3, Lemma 3.2] Lemma 3.2 is stated without proof. A one-sentence justification (extend a basis of the column space and row space) would make the paper self-contained.
[§5, Theorem 5.1] The formula in Theorem 5.1 is ambiguous as typeset: it should be written as L*(B) ≥ a + ((2k−2)/(2k+1)) a^{k/(k+1)} − m. The current text can be misread as ((a+2k−2)/(2k+1)) a^{k/(k+1)} − m.
[§5, Proposition 5.2] The rank computation rank(I_784 + U^T U) = 784 is stated as 'an exact computation' with only a GitHub link. Please describe the algorithm used (e.g., Gaussian elimination over F2) and consider including a short certificate or the code in the ancillary files.
[Abstract / Introduction] The abstract says no explicit family requiring a superlinear number of CNOT gates is known. The result here is linear (4n−o(n)) and does not resolve that open problem; the milestone is beating the constant 3 in front of n. The wording could be clarified to avoid a misleading impression.
[§5, Figure 1] Figure 1 is only alluded to as 'illustrated'; consider removing it or adding a textual description of the block structure, since the matrix is too large to be visually informative.

Circularity Check

0 steps flagged

No circularity: the lower bounds follow by reduction to an external theorem of Sergeev, with self-contained lifting lemmas and an explicit, code-verified construction.

full rationale

The derivation chain is not circular. The central bound d(M_N) ≥ 4N − o(N) is obtained by (i) importing L(A_U) ≥ 5n − O(√n log n) from [Ser25, Cor. 1] as an external, non-overlapping result; (ii) applying Lemma 3.3 to pass from A_U to M0 = A_U + P0 with a loss of n; (iii) bounding corank(M0) via Lemma 3.4; and (iv) embedding M0 into an invertible matrix with negligible dimension overhead. None of these steps defines its output in terms of its own input, and no parameter is fitted to a target value: the matrix M0 is constructed from a BCH parity-check matrix and then bounded below, and the concrete instance in Proposition 5.2 is a fixed matrix whose rank assertion is checked exactly over F2 and whose lower bound comes from an imported theorem. The external Sergeev bound is load-bearing and is not reproved in this paper, but that is an ordinary reliance on an independent cited result—not a self-citation and not a definitional shortcut. The internal inequalities and arithmetic are consistent, and the computational claim is backed by a code repository. Thus the paper contains no circular step by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central claim rests on two external theorems from [Ser25] (additive complexity lower bounds) and on standard coding theory facts. The hand-chosen parameters p=14, s=56, k=27 are arbitrary but needed to make the concrete instance's bound exceed 3(n−1). No invented entities or fitted constants.

free parameters (4)

p (extension degree) = 14
Hand-chosen for the concrete instance in §5; any p large enough would work in principle, 14 makes the matrix size 17167.
s (number of BCH roots) = 56
Chosen so that ind(U) ≥ 56 = 2k+2 with k=27, satisfying Theorem 5.1 hypothesis.
k (Theorem 5.1 exponent parameter) = 27
Hand-picked to make the final lower bound 51535 exceed 3(17167−1)=51498. The bound is sensitive to k.
asymptotic parameterization p=⌈log2 n⌉, s=⌈√n⌉ = n/a
Chosen so that m = ps = O(√n log n) = o(n), making the corank loss negligible in Theorem 4.2.

axioms (4)

domain assumption Sergeev's Theorem 4.1: L(A_U) ≥ 5n − O(√n log n) for the BCH-constructed A_U
Imported from [Ser25]; the entire constant-5 foundation comes from here. Located in §4, Theorem 4.1.
domain assumption Sergeev's Theorem 5.1: L*(B) ≥ a + (2k−2)/(2k+1) a^{k/(k+1)} − m under the stated row-weight and independence index conditions
Used in Proposition 5.2 to bound the concrete U. Cited as [Ser25, Thm. 2].
standard math Any set of s distinct nonzero α_i in F_{2^p} gives a full-rank Vandermonde matrix, so the expanded binary matrix has independence index at least s
Used in §4–5 to ensure ind(U) ≥ s; the authors sketch the proof in the paragraph before Theorem 4.1.
standard math Every invertible matrix over F2 is a product of transvections (Gaussian elimination)
Underpins the definition of CNOT-complexity; standard.

pith-pipeline@v1.3.0-alltime-deepseek · 7263 in / 19539 out tokens · 149351 ms · 2026-08-01T05:20:02.898265+00:00 · methodology

0 comments

read the original abstract

The CNOT-complexity of an invertible matrix over $\mathbb{F}_2$ is the minimum number of CNOT gates needed to synthesize the corresponding linear reversible operator. While the maximum CNOT-complexity over all $n \times n$ matrices is known to be $\Theta(n^2 / \log n)$, no explicit family of matrices requiring a superlinear number of CNOT gates is known, and until now the hardest explicitly known family has been the cyclic permutations, with CNOT-complexity $3(n-1)$. We show that lower bounds for the additive complexity of not-necessarily-reversible linear operators can be lifted to the reversible setting with only a small loss. As an application, we use this to describe an explicit family of matrices, constructed from parity-check matrices of error-correcting codes, with CNOT-complexity at least $4n - o(n)$, asymptotically surpassing the cyclic permutations. Moreover, this construction yields an explicit matrix $A \in \mathrm{GL}_{n}(\mathbb{F}_2)$, $n = 17167$, whose CNOT-complexity exceeds that of the cyclic permutation on $n$ symbols.

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Explicit Matrices over $\mathbb Z_2$ with CNOT and Row Complexity $4n-\mathrm{o}(n)$ and Local Logic Gates
quant-ph 2026-07 accept novelty 6.0

Explicit n imes n matrices over Z_2 require 4n−o(n) CNOT/row/2-local linear gates, and the same bound holds for the quantum complexity of the associated affine permutations.

Reference graph

Works this paper leans on

17 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

2013 , url =

Matthew Amy and Dmitri Maslov and Michele Mosca and Martin Roetteler , title =. 2013 , url =. doi:10.1109/TCAD.2013.2244643 , timestamp =

arXiv 2013
[2]

Quantum Science and Technology , abstract =

Amy, Matthew and Azimzadeh, Parsiad and Mosca, Michele , title =. Quantum Science and Technology , abstract =. 2018 , publisher =. doi:10.1088/2058-9565/aad8ca , url =

doi:10.1088/2058-9565/aad8ca 2018
[3]

Quantum Information Processing , volume =

Bataille, Marc , year =. Quantum Information Processing , volume =. doi:10.1007/s11128-022-03577-8 , url =

work page doi:10.1007/s11128-022-03577-8
[4]

Reversible Computation - 12th International Conference,

Timoth. Reversible Computation - 12th International Conference,. 2020 , url =. doi:10.1007/978-3-030-52482-1\_11 , timestamp =

work page doi:10.1007/978-3-030-52482-1 2020
[5]

2021 , issue_date =

De Brugi\`. 2021 , issue_date =. doi:10.1145/3474226 , journal =

work page doi:10.1145/3474226 2021
[6]

Minimum synthesis cost of CNOT circuits

Bu, Alan and Fan, Evan and Joo, Robert. Minimum synthesis cost of CNOT circuits. Quantum Information Processing. doi:10.1007/s11128-025-04831-5

work page doi:10.1007/s11128-025-04831-5
[7]

On exact sizes of minimal CNOT circuits

Christensen, Jens Emil and J rgensen, S ren Fuglede and Pavlogiannis, Andreas and van de Pol, Jaco. On exact sizes of minimal CNOT circuits. Lecture Notes in Computer Science. doi:10.1007/978-3-031-97063-4\_6

doi:10.1007/978-3-031-97063-4
[8]

2023 , volume=

Gheorghiu, Vlad and Huang, Jiaxin and Li, Sarah Meng and Mosca, Michele and Mukhopadhyay, Priyanka , journal=. 2023 , volume=. doi:10.1109/TCAD.2022.3213210 , url=

arXiv 2023
[9]

What is the complexity of this problem? , YEAR =

Timothy Gowers , HOWPUBLISHED =. What is the complexity of this problem? , YEAR =. https://mathoverflow.net/q/69873 , URL =
[10]

Complexity of linear Boolean operators

Jukna, Stasys and Sergeev, Igor. Complexity of linear Boolean operators. Found. Trends Theor. Comput. Sci. doi:10.1561/0400000063

doi:10.1561/0400000063
[11]

Proceedings of the 39th Annual Design Automation Conference , pages =

Iwama, Kazuo and Kambayashi, Yahiko and Yamashita, Shigeru , title =. Proceedings of the 39th Annual Design Automation Conference , pages =. 2002 , isbn =. doi:10.1145/513918.514026 , abstract =

arXiv 2002
[12]

Reversible Computation - 10th International Conference,

Giulia Meuli and Mathias Soeken and Giovanni De Micheli , title =. Reversible Computation - 10th International Conference,. 2018 , url =. doi:10.1007/978-3-319-99498-7\_12 , timestamp =

doi:10.1007/978-3-319-99498-7 2018
[13]

and Markov, Igor L

Patel, Ketan N. and Markov, Igor L. and Hayes, John P. , title =. Quantum Info. Comput. , pages =. 2008 , issue_date =

2008
[14]

2022 , pages=

Sarah Schneider and Lukas Burgholzer and Robert Wille , journal=. 2022 , pages=. doi:10.1145/3566097.3567929 , url=

arXiv 2022
[15]

Lower Bounds for Additive Complexity of Linear Operators and Bilinear Algorithms for Matrix and Polynomial Multiplication GF(2)

Sergeev, Igor S. Lower Bounds for Additive Complexity of Linear Operators and Bilinear Algorithms for Matrix and Polynomial Multiplication GF(2). Math. Notes. doi:10.1134/S0001434625605143 , url =

work page doi:10.1134/s0001434625605143
[16]

2024 , doi =

Irfansha Shaik and Jaco van de Pol , title =. 2024 , doi =

2024
[17]

Heuristic and optimal synthesis of CNOT and Clifford circuits

Webster, Mark and Koutsioumpas, Stergios and Browne, Dan E. Heuristic and optimal synthesis of CNOT and Clifford circuits. arXiv:2503.14660

Pith/arXiv arXiv

[1] [1]

2013 , url =

Matthew Amy and Dmitri Maslov and Michele Mosca and Martin Roetteler , title =. 2013 , url =. doi:10.1109/TCAD.2013.2244643 , timestamp =

arXiv 2013

[2] [2]

Quantum Science and Technology , abstract =

Amy, Matthew and Azimzadeh, Parsiad and Mosca, Michele , title =. Quantum Science and Technology , abstract =. 2018 , publisher =. doi:10.1088/2058-9565/aad8ca , url =

doi:10.1088/2058-9565/aad8ca 2018

[3] [3]

Quantum Information Processing , volume =

Bataille, Marc , year =. Quantum Information Processing , volume =. doi:10.1007/s11128-022-03577-8 , url =

work page doi:10.1007/s11128-022-03577-8

[4] [4]

Reversible Computation - 12th International Conference,

Timoth. Reversible Computation - 12th International Conference,. 2020 , url =. doi:10.1007/978-3-030-52482-1\_11 , timestamp =

work page doi:10.1007/978-3-030-52482-1 2020

[5] [5]

2021 , issue_date =

De Brugi\`. 2021 , issue_date =. doi:10.1145/3474226 , journal =

work page doi:10.1145/3474226 2021

[6] [6]

Minimum synthesis cost of CNOT circuits

Bu, Alan and Fan, Evan and Joo, Robert. Minimum synthesis cost of CNOT circuits. Quantum Information Processing. doi:10.1007/s11128-025-04831-5

work page doi:10.1007/s11128-025-04831-5

[7] [7]

On exact sizes of minimal CNOT circuits

Christensen, Jens Emil and J rgensen, S ren Fuglede and Pavlogiannis, Andreas and van de Pol, Jaco. On exact sizes of minimal CNOT circuits. Lecture Notes in Computer Science. doi:10.1007/978-3-031-97063-4\_6

doi:10.1007/978-3-031-97063-4

[8] [8]

2023 , volume=

Gheorghiu, Vlad and Huang, Jiaxin and Li, Sarah Meng and Mosca, Michele and Mukhopadhyay, Priyanka , journal=. 2023 , volume=. doi:10.1109/TCAD.2022.3213210 , url=

arXiv 2023

[9] [9]

What is the complexity of this problem? , YEAR =

Timothy Gowers , HOWPUBLISHED =. What is the complexity of this problem? , YEAR =. https://mathoverflow.net/q/69873 , URL =

[10] [10]

Complexity of linear Boolean operators

Jukna, Stasys and Sergeev, Igor. Complexity of linear Boolean operators. Found. Trends Theor. Comput. Sci. doi:10.1561/0400000063

doi:10.1561/0400000063

[11] [11]

Proceedings of the 39th Annual Design Automation Conference , pages =

Iwama, Kazuo and Kambayashi, Yahiko and Yamashita, Shigeru , title =. Proceedings of the 39th Annual Design Automation Conference , pages =. 2002 , isbn =. doi:10.1145/513918.514026 , abstract =

arXiv 2002

[12] [12]

Reversible Computation - 10th International Conference,

Giulia Meuli and Mathias Soeken and Giovanni De Micheli , title =. Reversible Computation - 10th International Conference,. 2018 , url =. doi:10.1007/978-3-319-99498-7\_12 , timestamp =

doi:10.1007/978-3-319-99498-7 2018

[13] [13]

and Markov, Igor L

Patel, Ketan N. and Markov, Igor L. and Hayes, John P. , title =. Quantum Info. Comput. , pages =. 2008 , issue_date =

2008

[14] [14]

2022 , pages=

Sarah Schneider and Lukas Burgholzer and Robert Wille , journal=. 2022 , pages=. doi:10.1145/3566097.3567929 , url=

arXiv 2022

[15] [15]

Lower Bounds for Additive Complexity of Linear Operators and Bilinear Algorithms for Matrix and Polynomial Multiplication GF(2)

Sergeev, Igor S. Lower Bounds for Additive Complexity of Linear Operators and Bilinear Algorithms for Matrix and Polynomial Multiplication GF(2). Math. Notes. doi:10.1134/S0001434625605143 , url =

work page doi:10.1134/s0001434625605143

[16] [16]

2024 , doi =

Irfansha Shaik and Jaco van de Pol , title =. 2024 , doi =

2024

[17] [17]

Heuristic and optimal synthesis of CNOT and Clifford circuits

Webster, Mark and Koutsioumpas, Stergios and Browne, Dan E. Heuristic and optimal synthesis of CNOT and Clifford circuits. arXiv:2503.14660

Pith/arXiv arXiv