Pith. sign in

REVIEW 1 major objections 6 minor 63 references

The paper claims that a net-free Solovay–Kitaev recursion is possible: with a good exponential basis in hand, each target unitary is compiled in time polynomial in the qudit dimension and polylogarithmic in 1/ε, and the formerly exponential

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

Replacing the depth-zero net search with an 'integerized trotterization' over a good exponential basis makes online synthesis poly(d, log 1/ε), moving the exponential net cost into a one-time preprocessing step.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection The main construction is sound and genuinely new; the reader's counterexample to Lemma 3.7 does not survive contact with the actual algorithm, but the proof of that lemma is sloppy and should be rewritten. the 1 major comments →

arxiv 2607.19874 v1 pith:IRPTC7EQ submitted 2026-07-22 cs.CG

Removing Online Exponential Net Search from Solovay-Kitaev

Henrique Ennes (DATASHAPE , UniCA , COATI) , Cl\'ement Maria (DATASHAPE) This is my paper
classification cs.CG MSC 68Q1281P6822E70 PACS 03.67.Lx
keywords Solovay–Kitaev algorithmgate synthesisunitary approximationtrotterizationgood exponential basisqudit dimensionLie group SU(d)net search
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 reading

The paper claims that the expensive coarse net search at the base of the Solovay–Kitaev recursion can be removed from the online phase. The replacement is a discrete, integerized trotterization step: the target unitary's logarithm is expanded in a 'good exponential basis' and approximated by integral powers of basis gates. For instruction sets that admit such a basis, each target is then synthesized in time polynomial in the qudit dimension d and polylogarithmic in 1/ε, after a one-time O(d^6) change-of-basis preprocessing. For arbitrary universal instruction sets, the exponential net cost is not eliminated but is moved into a one-time additive preprocessing term. If correct, this makes the algorithm's dimension dependence benign in practical settings where many unitaries are compiled against the same gate set.

Core claim

On the paper's own terms, the central discovery is that the depth-zero ε0-net query of the Solovay–Kitaev recursion can be replaced by a constructive local routine: integer trotterization along the geodesic from the identity to U. A 'good exponential basis'—a basis of exponentials exp(iH_j) whose logarithms are short and well-conditioned—guarantees that this routine meets the ε0 accuracy needed to start the recursion. With such a basis, Theorem 4.18 asserts online synthesis in O(d^6 + p d^4 log^{k_t}(1/ε)) with no net search. For arbitrary universal instruction sets, Corollary 4.29 first synthesizes a good basis in a preprocessing phase using the standard Solovay–Kitaev algorithm, absorbing

What carries the argument

The load-bearing objects are the good exponential basis and the integerized trotterization subroutine (Algorithm 4). The basis provides a short, well-conditioned set of Hamiltonian directions; the subroutine expands log(U)/n in that basis and rounds the real coefficients to integers, then builds U as n repetitions of the resulting product. The Lie-algebra product-formula expansion controls the product-formula error, and the minimum-gain parameter µ—the smallest operator norm attainable by a unit-ℓ∞ coefficient vector—controls the conditioning of the basis. The goodness condition makes the drift error and the product-formula error fit within ε0, thereby replacing an exponential-size precomput

Load-bearing premise

The load-bearing premise is that the greedy eigenvalue-balancing routine (Algorithm 3, Lemma 3.7) always keeps the running partial sum of eigenvalues inside [0, 2∥Z∥]; that invariant is asserted but is false for, e.g., eigenvalues (0.9, 0.9, 0.9, -1.0, -1.0, -0.7), and both the classic Solovay–Kitaev recursion and the new net-free version depend on it.

What would settle it

Run the balancing routine (Algorithm 3) on the diagonal Hermitian matrix diag(0.9, 0.9, 0.9, -1.0, -1.0, -0.7), whose operator norm is 1. The greedy stack rule pops 0.9, 0.9, -0.7, -1.0, and then -1.0 when the partial sum is 0.1, producing a final partial sum of -0.9 and violating the claimed 0 ≤ partial sum ≤ 2 invariant. A corrected routine that provably maintains the invariant for all inputs would revive the proof; without one, the stated theorems are not established as written.

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

If this is right

  • For any instruction set that contains or can efficiently build a good exponential basis, Algorithm 5 compiles a list of p target unitaries to error ε in O(d^6 + p d^4 log^{k_t}(1/ε)) total time, with no ε0-net search during synthesis.
  • For an arbitrary universal instruction set, the exponential net cost is confined to a one-time additive preprocessing step costing O(d^2(T0+d^3) log^{k_t} d), so compiling many targets amortizes the expensive base preparation.
  • The good-exponential-basis condition supplies a verifiable, real-RAM sufficient test for whether an instruction set is universal: compute logarithms, check that they form a basis of su(d), estimate the minimum gain µ, and verify the h_max bound.
  • Because only the depth-zero subroutine changes, established improvements to the Solovay–Kitaev recursion—inverse-free variants and tighter contraction exponents—can be carried over to this net-free version.
  • The synthesized circuits are represented as straight-line programs, and the working memory stays polynomial in d (plus log log(1/ε)) after preprocessing, rather than exponential.

Where Pith is reading between the lines

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

  • Higher-order splitting formulas should reduce the d^7-driven circuit-length blow-up by lowering the number of Trotter steps at the price of more exponentials per step; the paper's own error analysis suggests h_max ~ d^{-4} may be reachable, shortening circuits by a factor of d^3.
  • Scheduled integration, where integer coefficients are re-chosen at each step to cancel drift, could relax the restrictive h_max condition; the paper notes this may reduce to an approximate lattice problem at each step.
  • The net-free method trades online time for longer circuits—an extra d^{19/2} polylog(d) factor for arbitrary instruction sets—so in depth-sensitive applications, the classic Solovay–Kitaev algorithm may still be preferable despite its exponential net.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper proposes a modification of the Solovay-Kitaev (SK) algorithm that eliminates the online search over an ε0-net, which is the source of the exponential dependence on the qudit dimension d. It introduces the notion of a good exponential basis and a discrete trotterization base routine (Algorithm 4). Theorem 4.18 states that, given such a basis, Algorithm 5 approximates p target unitaries in time O(d^6 + p d^4 log^{k_t}(1/ε)) with no net search. For arbitrary universal instruction sets, Theorem 4.22 and Corollary 4.29 show that a good basis can be constructed in a one-time standard-SK preprocessing step, making the net-search cost T0 additive rather than multiplicative in p. The paper also discusses the conditioning parameter μ_N, the restrictiveness of h_max, and possible higher-order product-formula extensions.

Significance. If the main theorems are correct, this is a substantial contribution to the variable-dimension gate synthesis problem. It provides the first explicit construction (known to me) realizing the additive-cost form suggested by Kitaev-Shen-Vyalyi, and it does so with a clear separation of online and preprocessing costs. The good-exponential-basis criterion is a concrete, verifiable sufficient condition (Remark 4.19). The use of BCH-based trotterization to replace the depth-zero net is original and likely to stimulate further work. The paper is honest about limitations: the required h_max is very small (~d^-6 to d^-7), and for arbitrary instruction sets the exponential T0 remains, but only in preprocessing.

major comments (1)
  1. [§3, Lemma 3.7] The proof of Algorithm 3's invariant is not reliable as written. In the case w_{k-1} ≥ ∥Z∥, the text asserts the existence of a negative eigenvalue in (-w_{k-1}, 0); this may fail (e.g., Z=diag(1,1,-2) at w=2, remaining eigenvalue -2). It also does not justify that the eigenvalue actually popped from the N stack satisfies the needed lower bound. The lemma is nevertheless true: every remaining eigenvalue obeys |μ_j| ≤ ∥Z∥ ≤ w_{k-1}, so μ_j ≥ -w_{k-1}, and the invariant follows. The proof must be corrected (e.g., by adding this one-line observation). This is load-bearing because both Algorithm 1 and Algorithm 5 rely on the commutator decomposition.
minor comments (6)
  1. [§2] The paragraph beginning 'Many of the usual properties...' is duplicated immediately before Lemma 2.5.
  2. [§3, Algorithm 2] The loop 'for k=1 to d' uses the ket |k+1⟩ at k=d; this should be k=1 to d-1 (the proof in Lemma 3.6 uses k=0 to d-1).
  3. [§3, Algorithm 1 and §4, Algorithm 5] Line 5 writes 'log(U†Un−1)'; this should be 'log(U† U_{t-1})'.
  4. [§4.2, Theorem 4.22] The symbol h*_max is used both as the scalar defining H*_j = h*_max Λ_j and as the maximum norm max_j ||H*_j||. Since ||Λ_j||≤1 these are compatible only as an upper bound; it would be clearer to introduce a scalar h* and define h*_max accordingly.
  5. [§4.2, Corollary 4.29] The statement that the d^6 trotterization preprocessing is 'absorbed into the T0 SK cost' is only true because T0 is exponential in d (T0 = Ω(2^{c d^2}) in the intended regime). Please make this explicit, since for a hypothetical polynomial T0 the stated bound would need an additional d^6 term.
  6. [§2, Lemma 2.5] The uniform BCH constant C for M-term products is asserted without proof or reference. A citation for the M-term version (or a short derivation) would help.

Circularity Check

0 steps flagged

No significant circularity: the new net-free base case is an explicit constructive routine, and the net cost for arbitrary gate sets is honestly isolated as preprocessing.

full rationale

The central derivation chain is not circular. The paper introduces Algorithm 4 (NET FREE SEARCH), a discrete/integerized trotterization routine, and gives in Proposition 4.8 an explicit sufficient condition on the exponential basis for the routine to achieve error at most epsilon0. Definition 4.17 then defines a 'good exponential basis' by exactly that inequality (4.16), i.e. by explicit bounds on h_max and mu_N, not by any property that assumes Algorithm 5's success or the target theorem. Algorithm 5's correctness proof reduces directly to Proposition 4.8 plus the unchanged Solovay-Kitaev recursion from Theorem 3.11; there is no step in which an output quantity is equated to a fitted input. For arbitrary instruction sets, Corollary 4.29 uses the standard, external Solovay-Kitaev theorem to construct a good exponential basis from scaled Gell-Mann gates. This is an upfront reduction with cost T0 moved to a one-time additive preprocessing term, and the paper explicitly states that the exponential dependence on d^2 is not removed in that case. That is an honest isolation, not a reuse of the paper's own conclusion. The citations that carry mathematical weight ([22], [40], [18], and standard BCH references) are external, not self-citations; where the paper cites [40] for Algorithm 2's correctness, it immediately redoes the demonstration. There is no importation of a uniqueness theorem from the authors' prior work, and no ansatz is smuggled in via a self-citation. The paper's stated limitations (finite precision not analyzed, the net-free basis-construction conjecture, and the scheduling idea left to future work) are explicit and do not hide a circular input. The reader's concern about Lemma 3.7, if valid, would be a correctness flaw in the balancing subroutine, not a circularity; in any case, the skeptical analysis shows the alleged counterexample does not violate the invariant. Overall, no load-bearing step reduces by definition, by fitting, or by self-citation to the result being derived.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The paper introduces the concept of a 'good exponential basis' and the minimum-gain parameter μ_N, but these are mathematical definitions rather than new physical entities or extra degrees of freedom. The main unproven load-bearing assumptions are the uniform BCH constant and the correctness of the balancing subroutine; the free parameters are universal constants and a design scale, not fit parameters.

free parameters (4)
  • K (universal trotter constant) = unspecified; K ≥ 32√C (Remark 4.15)
    Introduced in Proposition 4.8 (eq. 4.11/4.14) to make the BCH remainder small enough; chosen by hand, not fitted to data.
  • C (BCH remainder constant) = unspecified universal constant
    Lemma 2.5 assumes a universal constant independent of d and M; the paper does not prove the independence and uses it throughout the error analysis.
  • δ0 (BCH convergence radius) = π > δ0 > 0 sufficiently small
    Assumed in Lemma 2.5 to guarantee BCH convergence; the paper notes δ0 ≤ log 2 but uses it as an unspecified constant.
  • h*_max (Gell-Mann scale) = Θ(d^{-7}) (eq. 4.21)
    Design choice for the scaled Gell-Mann basis that makes the constructed basis good; chosen by hand to satisfy eq. (4.16).
axioms (5)
  • standard math BCH formula with remainder constant independent of d and M (Lemma 2.5, Remark 2.8)
    Central to the product-formula error estimate in Proposition 4.8; the paper cites convergence but does not prove the uniform constant.
  • standard math Operator-norm bounds for exponentials (Lemma 2.9) and spectral theorem / principal branch log (Lemma 2.3)
    Used throughout the SK recursion and the trotterization error analysis.
  • domain assumption Real-RAM model and inverse-closed instruction sets
    Stated in the conventions; all complexity bounds ignore bit-precision, and inverses are assumed available.
  • domain assumption Universal instruction set whose words of length ≤ ℓ0 form an ε0-net
    Standard SK precondition imported from Theorem 3.11; needed for the preprocessing step that constructs a good basis.
  • domain assumption Existence/constructibility of a good exponential basis
    Theorem 4.18 assumes it as an input; Corollary 4.29 constructs it via SK, which still requires an exponential net in preprocessing.

reviewed 2026-08-01 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Removing Online Exponential Net Search from Solovay-Kitaev." pith.science (2026). https://pith.science/paper/IRPTC7EQ

@misc{pith2026260719874,
  author       = {Pith},
  title        = {Pith review of: Removing Online Exponential Net Search from Solovay-Kitaev},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRPTC7EQ}},
  note         = {Machine review of arXiv:2607.19874}
}
Share X Bluesky LinkedIn Reddit HN
abstract

The Solovay-Kitaev algorithm describes how to approximate, to arbitrary precision, a matrix in the special unitary group SU(d) using any fixed universal gate set. Although the algorithm scales as O(poly(log(1/$\epsilon$))), where $\epsilon$ is the maximum targeted approximation error, its running time depends exponentially on the qudit dimension d. This bad dependence can be traced to its explicit use of an $\epsilon$_0-net of size 2 $\Omega$(d^2) , which is queried O(poly(log(1/$\epsilon$))) times throughout the execution. For this reason, the standard Solovay-Kitaev theorem is usually stated for fixed d, with the base net and its lookup cost absorbed into the constants. We study the algorithmic problem in the variabledimension regime and show how to avoid searching an exponentially large precomputed net for each target unitary. In particular, we introduce the notion of a good exponential basis and show that such a basis can replace the usual depth-zero net-search routine. This yields a modification of the algorithm in which the use of an explicit net is fully moved to a preprocessing step. For instruction sets that already contain, or allow the efficient construction of, a good exponential basis, the resulting online synthesis algorithm is polynomial in d and polylogarithmic in 1/$\epsilon$. For arbitrary universal instruction sets, the exponential dependence on d^2 is not removed, but is isolated into a one-time additive preprocessing cost. Our technique uses differential-geometric methods to devise an integerized version of trotterization that replaces the depth-zero net query by a constructive local synthesis routine. The same framework also suggests possible extensions based on other discretized numerical integration schemes.

Figures

Figures reproduced from arXiv: 2607.19874 by Cl\'ement Maria (DATASHAPE), COATI), Henrique Ennes (DATASHAPE, UniCA.

Figure 1
Figure 1. Figure 1: The first three layers of a tree structure used to build a net out of four instruction gates, G1, G2, G† 1 , and G † 2 . Trivial relations such as G1G † 1 , G † 2G2, which equal the identity element I (already in the uppermost layer of the tree) have been omitted. practice, the number of calls is not as bad as it may look: assuming 0 < ε < ε0 and c 2 ε0 < 1 where c is a universal constant, the total depth … view at source ↗

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

63 extracted references · 2 linked inside Pith

  1. [1]

    Quantum computing, postselection, and probabilistic polynomial-time.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2063):3473–3482, 2005

    Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2063):3473–3482, 2005. [2]

    2063

  2. [2]

    Postbqp postscripts: A confession of mathematical errors.https://scottaaronson.blog/?p= 2072, 2014

    Scott Aaronson. Postbqp postscripts: A confession of mathematical errors.https://scottaaronson.blog/?p= 2072, 2014. Blog post, Shtetl-Optimized. [4]

    2072

  3. [3]

    The complexity zoo, 2005

    Scott Aaronson, Greg Kuperberg, and Christopher Granade. The complexity zoo, 2005. [2]

    2005

  4. [4]

    Quantum invariants of 3-manifolds and np vs# p.Quantum Information & Computation, 17(1-2):125–146, 2017

    Gorjan Alagic and Catharine Lo. Quantum invariants of 3-manifolds and np vs# p.Quantum Information & Computation, 17(1-2):125–146, 2017. [2]

    2017

  5. [5]

    Partitioning and geometric embedding of range spaces of finite vapnik-chervonenkis dimension

    Noga Alon, David Haussler, and Emo Welzl. Partitioning and geometric embedding of range spaces of finite vapnik-chervonenkis dimension. InProceedings of the third annual symposium on Computational geometry, pages 331–340, 1987. [2]

    1987

  6. [6]

    Chemical basis of trotter-suzuki errors in quantum chemistry simulation.Physical Review A, 91(2):022311, 2015

    Ryan Babbush, Jarrod McClean, Dave Wecker, Al´ an Aspuru-Guzik, and Nathan Wiebe. Chemical basis of trotter-suzuki errors in quantum chemistry simulation.Physical Review A, 91(2):022311, 2015. [13]

    2015

  7. [7]

    Dynamic programming.science, 153(3731):34–37, 1966

    Richard Bellman. Dynamic programming.science, 153(3731):34–37, 1966. [2]

    1966

  8. [8]

    The potential of multidimensional photonic computing

    Ivonne Bente, Shabnam Taheriniya, Francesco Lenzini, Frank Br¨ uckerhoff-Pl¨ uckelmann, Michael Kues, Harish Bhaskaran, C David Wright, and Wolfram Pernice. The potential of multidimensional photonic computing. Nature Reviews Physics, 7(8):439–450, 2025. [2]

    2025

  9. [9]

    Multidimensional binary search trees used for associative searching.Communications of the ACM, 18(9):509–517, 1975

    Jon Louis Bentley. Multidimensional binary search trees used for associative searching.Communications of the ACM, 18(9):509–517, 1975. [2]

    1975

  10. [10]

    Bloch vectors for qudits.Journal of Physics A: Mathematical and Theoretical, 41(23):235303, 2008

    Reinhold A Bertlmann and Philipp Krammer. Bloch vectors for qudits.Journal of Physics A: Mathematical and Theoretical, 41(23):235303, 2008. [20]

    2008

  11. [11]

    On the baker-campbell-hausdorff theorem: non- convergence and prolongation issues.Linear and Multilinear Algebra, 68(7):1310–1328, 2020

    Stefano Biagi, Andrea Bonfiglioli, and Marco Matone. On the baker-campbell-hausdorff theorem: non- convergence and prolongation issues.Linear and Multilinear Algebra, 68(7):1310–1328, 2020. [6]

    2020

  12. [12]

    Efficient universal quantum compilation: An inverse-free solovay- kitaev algorithm.arXiv preprint arXiv:2112.02040, 2021

    Adam Bouland and Tudor Giurgica-Tiron. Efficient universal quantum compilation: An inverse-free solovay- kitaev algorithm.arXiv preprint arXiv:2112.02040, 2021. [3, 4]

    Pith/arXiv arXiv 2021

  13. [13]

    A spectral gap theorem in su.(d).Journal of the European Mathematical Society (EMS Publishing), 14(5), 2012

    Jean Bourgain and Alex Gamburd. A spectral gap theorem in su.(d).Journal of the European Mathematical Society (EMS Publishing), 14(5), 2012. [19]

    2012

  14. [14]

    Near neighbor search in large metric spaces

    Sergey Brin. Near neighbor search in large metric spaces. InVLDB, volume 95, pages 574–584, 1995. [2]

    1995

  15. [15]

    Springer Science & Business Media, 2003

    Theodor Br¨ ocker and Tammo Tom Dieck.Representations of compact Lie groups, volume 98. Springer Science & Business Media, 2003. [4]

    2003

  16. [16]

    Trapped-ion quantum computing: Progress and challenges.Applied physics reviews, 6(2), 2019

    Colin D Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M Sage. Trapped-ion quantum computing: Progress and challenges.Applied physics reviews, 6(2), 2019. [2]

    2019

  17. [17]

    Toward the first quantum simu- lation with quantum speedup.Proceedings of the National Academy of Sciences, 115(38):9456–9461, 2018

    Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the first quantum simu- lation with quantum speedup.Proceedings of the National Academy of Sciences, 115(38):9456–9461, 2018. [13]

    2018

  18. [18]

    Theory of trotter error with commutator scaling.Physical Review X, 11(1):011020, 2021

    Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. Theory of trotter error with commutator scaling.Physical Review X, 11(1):011020, 2021. [13]

    2021

  19. [19]

    Chapman and hall/CRC, 2021

    Domenico d’Alessandro.Introduction to quantum control and dynamics. Chapman and hall/CRC, 2021. [14]

    2021

  20. [20]

    Exact and approximate unitary 2- designs and their application to fidelity estimation.Physical Review AAtomic, Molecular, and Optical Physics, 80(1):012304, 2009

    Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2- designs and their application to fidelity estimation.Physical Review AAtomic, Molecular, and Optical Physics, 80(1):012304, 2009. [19]

    2009

  21. [21]

    Topological quantum computation.Physics today, 59(7):32–38, 2006

    Sankar Das Sarma, Michael Freedman, and Chetan Nayak. Topological quantum computation.Physics today, 59(7):32–38, 2006. [2] 26 HENRIQUE ENNES AND CL ´EMENT MARIA

    2006

  22. [22]

    The solovay-kitaev algorithm.arXiv preprint quant-ph/0505030,

    Christopher M Dawson and Michael A Nielsen. The solovay-kitaev algorithm.arXiv preprint quant-ph/0505030,

    Pith/arXiv arXiv

  23. [23]

    Springer, 2008

    Mark De Berg, Otfried Cheong, Marc Van Kreveld, and Mark Overmars.Computational geometry: algorithms and applications. Springer, 2008. [2]

    2008

  24. [24]

    Universality in quantum computation.Proceedings of the Royal Society of London

    David Elieser Deutsch, Adriano Barenco, and Artur Ekert. Universality in quantum computation.Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 449(1937):669–677, 1995. [2]

    1937

  25. [25]

    Quantum control theory and applications: a survey.IET control theory & applications, 4(12):2651–2671, 2010

    Daoyi Dong and Ian R Petersen. Quantum control theory and applications: a survey.IET control theory & applications, 4(12):2651–2671, 2010. [14]

    2010

  26. [26]

    Springer, 2006

    Karin Erdmann and Mark J Wildon.Introduction to Lie algebras, volume 122. Springer, 2006. [4]

    2006

  27. [27]

    Topological quantum computation

    Michael Freedman, Alexei Kitaev, Michael Larsen, and Zhenghan Wang. Topological quantum computation. Bulletin of the American Mathematical Society, 40(1):31–38, 2003. [2]

    2003

  28. [28]

    Taylor & Francis, 2000

    Howard Georgi.Lie algebras in particle physics: from isospin to unified theories. Taylor & Francis, 2000. [20]

    2000

  29. [29]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. InProceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996. [2]

    1996

  30. [30]

    Quantum mechanics helps in searching for a needle in a haystack.Physical review letters, 79(2):325,

    Lov K Grover. Quantum mechanics helps in searching for a needle in a haystack.Physical review letters, 79(2):325,

  31. [31]

    Quantum computing with trapped ions.Physics reports, 469(4):155–203, 2008

    Hartmut H¨ affner, Christian F Roos, and Rainer Blatt. Quantum computing with trapped ions.Physics reports, 469(4):155–203, 2008. [2]

    2008

  32. [32]

    Lie groups, lie algebras, and representations

    Brian C Hall. Lie groups, lie algebras, and representations. InQuantum Theory for Mathematicians, pages 333–366. Springer, 2013. [4, 5]

    2013

  33. [33]

    Efficient discrete approximations of quantum gates

    Aram W Harrow, Benjamin Recht, and Isaac L Chuang. Efficient discrete approximations of quantum gates. Journal of Mathematical Physics, 43(9):4445–4451, 2002. [19]

    2002

  34. [34]

    SIAM, 2008

    Nicholas J Higham.Functions of matrices: theory and computation. SIAM, 2008. [6]

    2008

  35. [35]

    Cambridge university press, 2012

    Roger A Horn and Charles R Johnson.Matrix analysis. Cambridge university press, 2012. [15]

    2012

  36. [36]

    Native qudit entanglement in a trapped ion quantum processor

    Pavel Hrmo, Benjamin Wilhelm, Lukas Gerster, Martin W van Mourik, Marcus Huber, Rainer Blatt, Philipp Schindler, Thomas Monz, and Martin Ringbauer. Native qudit entanglement in a trapped ion quantum processor. Nature Communications, 14(1):2242, 2023. [2]

    2023

  37. [37]

    Approximate nearest neighbors: towards removing the curse of dimensionality

    Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. InProceedings of the thirtieth annual ACM symposium on Theory of computing, pages 604–613, 1998. [2]

    1998

  38. [38]

    The bloch vector for n-level systems.Physics Letters A, 314(5-6):339–349, 2003

    Gen Kimura. The bloch vector for n-level systems.Physics Letters A, 314(5-6):339–349, 2003. [20]

    2003

  39. [39]

    Quantum computations: algorithms and error correction.Russian Mathematical Surveys, 52(6):1191–1249, 1997

    A Yu Kitaev. Quantum computations: algorithms and error correction.Russian Mathematical Surveys, 52(6):1191–1249, 1997. [2]

    1997

  40. [40]

    Number 47

    Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi.Classical and quantum computation. Number 47. American Mathematical Soc., 2002. [2, 3, 10, 24]

    2002

  41. [41]

    How hard is it to approximate the jones polynomial?Theory OF Computing, 11(6):183–219,

    Greg Kuperberg. How hard is it to approximate the jones polynomial?Theory OF Computing, 11(6):183–219,

  42. [42]

    Breaking the cubic barrier in the solovay-kitaev algorithm.arXiv preprint arXiv:2306.13158,

    Greg Kuperberg. Breaking the cubic barrier in the solovay-kitaev algorithm.arXiv preprint arXiv:2306.13158,

    arXiv

  43. [43]

    Universal quantum simulators.Science, 273(5278):1073–1078, 1996

    Seth Lloyd. Universal quantum simulators.Science, 273(5278):1073–1078, 1996. [13]

    1996

  44. [44]

    SIAM, 2023

    Carl D Meyer.Matrix analysis and applied linear algebra. SIAM, 2023. [11]

    2023

  45. [45]

    Kluwer Academic Publishers, Boston, MA, 2002

    Daniele Micciancio and Shafi Goldwasser.Complexity of Lattice Problems: A Cryptographic Perspective. Kluwer Academic Publishers, Boston, MA, 2002. [25]

    2002

  46. [46]

    On the constant depth implementation of pauli exponentials.npj Quantum Information, 2026

    Ioana Moflic and Alexandru Paler. On the constant depth implementation of pauli exponentials.npj Quantum Information, 2026. [20]

    2026

  47. [47]

    Epsilon-approximations & epsilon-nets

    Nabil H Mustafa and Kasturi Varadarajan. Epsilon-approximations & epsilon-nets. InHandbook of Discrete and Computational Geometry, pages 1241–1267. Chapman and Hall/CRC, 2017. [2]

    2017

  48. [48]

    A geometric approach to quantum circuit lower bounds.Quantum Information and Computation, 6(3):213–262, 2006

    MA Nielsen. A geometric approach to quantum circuit lower bounds.Quantum Information and Computation, 6(3):213–262, 2006. [14]

    2006

  49. [49]

    Optimal control, geometry, and quantum computing.Physical Review AAtomic, Molecular, and Optical Physics, 73(6):062323, 2006

    Michael A Nielsen, Mark R Dowling, Mile Gu, and Andrew C Doherty. Optimal control, geometry, and quantum computing.Physical Review AAtomic, Molecular, and Optical Physics, 73(6):062323, 2006. [14]

    2006

  50. [50]

    Quantum computation as geometry

    Michael A Nielsen, Mark R Dowling, Mile Gu, and Andrew C Doherty. Quantum computation as geometry. Science, 311(5764):1133–1135, 2006. [14]

    2006

  51. [51]

    Five balltree construction algorithms

    Stephen M Omohundro. Five balltree construction algorithms. 1989. [2]

    1989

  52. [52]

    Epsilon-nets, unitary designs, and random quantum circuits.IEEE Transactions on Information Theory, 68(2):989–1015, 2021

    Micha l Oszmaniec, Adam Sawicki, and Micha l Horodecki. Epsilon-nets, unitary designs, and random quantum circuits.IEEE Transactions on Information Theory, 68(2):989–1015, 2021. [19]

    2021

  53. [53]

    Optimally generating su (2 n) using pauli strings.Physical Review Letters, 134(20):200601, 2025

    Isaac D Smith, Maxime Cautr` es, David T Stephen, and Hendrik Poulsen Nautrup. Optimally generating su (2 n) using pauli strings.Physical Review Letters, 134(20):200601, 2025. [20]

    2025

  54. [54]

    Topological quantum computationfrom basic concepts to first experiments

    Ady Stern and Netanel H Lindner. Topological quantum computationfrom basic concepts to first experiments. Science, 339(6124):1179–1184, 2013. [2] REMOVING ONLINE EXPONENTIAL NET SEARCH FROM SOLOV AY-KITAEV 27

    2013

  55. [55]

    On the construction and comparison of difference schemes.SIAM journal on numerical analysis, 5(3):506–517, 1968

    Gilbert Strang. On the construction and comparison of difference schemes.SIAM journal on numerical analysis, 5(3):506–517, 1968. [24]

    1968

  56. [56]

    General theory of fractal path integrals with applications to many-body theories and statistical physics.Journal of mathematical physics, 32(2):400–407, 1991

    Masuo Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics.Journal of mathematical physics, 32(2):400–407, 1991. [14]

    1991

  57. [57]

    Another NP-complete problem and the complexity of computing short vectors in a lattice

    Peter van Emde Boas. Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical Report Technical Report 81-04, Mathematisch Instituut, University of Amsterdam, 1981. [25]

    1981

  58. [58]

    Optimal hadamard gate count for clifford+ t synthesis of pauli rotations sequences.ACM Transactions on Quantum Computing, 5(1):1–29, 2024

    Vivien Vandaele, Simon Martiel, Simon Perdrix, and Christophe Vuillot. Optimal hadamard gate count for clifford+ t synthesis of pauli rotations sequences.ACM Transactions on Quantum Computing, 5(1):1–29, 2024. [20]

    2024

  59. [59]

    Photonic realization of qudit quantum computing.Photonic Quantum Technologies: Science and Applications, 2:651–674, 2023

    Yuchen Wang, Zixuan Hu, and Sabre Kais. Photonic realization of qudit quantum computing.Photonic Quantum Technologies: Science and Applications, 2:651–674, 2023. [2]

    2023

  60. [60]

    Qudits and high-dimensional quantum computing

    Yuchen Wang, Zixuan Hu, Barry C Sanders, and Sabre Kais. Qudits and high-dimensional quantum computing. Frontiers in Physics, 8:589504, 2020. [2]

    2020

  61. [61]

    Number 112

    Zhenghan Wang.Topological quantum computation. Number 112. American Mathematical Soc., 2010. [2]

    2010

  62. [62]

    Solving strongly correlated electron models on a quantum computer.Physical Review A, 92(6):062318, 2015

    Dave Wecker, Matthew B Hastings, Nathan Wiebe, Bryan K Clark, Chetan Nayak, and Matthias Troyer. Solving strongly correlated electron models on a quantum computer.Physical Review A, 92(6):062318, 2015. [13]

    2015

  63. [63]

    Springer, 2006

    Pavel Zezula, Giuseppe Amato, Vlastislav Dohnal, and Michal Batko.Similarity search the metric space approach. Springer, 2006. [2]

    2006

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.