Contributions to the Theory of Clifford-Cyclotomic Circuits

• URL: http://arxiv.org/abs/2508.14674v1
• Date: Wed, 20 Aug 2025 12:44:39 GMT
• Title: Contributions to the Theory of Clifford-Cyclotomic Circuits
• Authors: Linh Dinh, Neil J. Ross,
• Abstract summary: We make two contributions to the theory of Clifford-cyclotomic circuits.<n>We improve the existing synthesis algorithm by showing that, when $n=2k$ and $kgeq 4$, only $k-3$ ancillas are needed to synthesize a circuit for $U$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Let $n$ be a positive integer divisible by 8. The Clifford-cyclotomic gate set $\mathcal{G}_n$ consists of the Clifford gates, together with a $z$-rotation of order $n$. It is easy to show that, if a circuit over $\mathcal{G}_n$ represents a unitary matrix $U$, then the entries of $U$ must lie in $\mathcal{R}_n$, the smallest subring of $\mathbb{C}$ containing $1/2$ and $\mathrm{exp}(2\pi i/n)$. The converse implication, that every unitary $U$ with entries in $\mathcal{R}_n$ can be represented by a circuit over $\mathcal{G}_n$, is harder to show, but it was recently proved to be true when $n=2^k$. In that case, $k-2$ ancillas suffice to synthesize a circuit for $U$, which is known to be minimal for $k=3$, but not for larger values of $k$. In the present paper, we make two contributions to the theory of Clifford-cyclotomic circuits. Firstly, we improve the existing synthesis algorithm by showing that, when $n=2^k$ and $k\geq 4$, only $k-3$ ancillas are needed to synthesize a circuit for $U$, which is minimal for $k=4$. Secondly, we extend the existing synthesis algorithm to the case of $n=3\cdot 2^k$ with $k\geq 3$.

Related papers

• On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation [55.126702858312456]
  We show that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.<n>Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.
  arXiv  Detail & Related papers  (2026-01-06T18:34:07Z)
• Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates [0.0]
  We study ancilla-free approximation of single-qubit unitaries $Uin rm SU(2)$ by gate sequences over Clifford+$G$.<n>We prove three bounds on the minimum $G$-count required to achieve approximation error at most $varepsilon$.
  arXiv  Detail & Related papers  (2025-10-08T23:48:48Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• On Exact Sizes of Minimal CNOT Circuits [2.831145157553215]
  We consider circuits of CNOT gates, which are fundamental binary gates in reversible and quantum computing.<n>We develop a new approach for computing distances in $G_n$, allowing us to synthesize minimum circuits that were previously beyond reach.<n>We also confirm a conjecture that long cycle permutations lie at distance $3(n-1)$, for all $nleq 8$, extending the previous bound of $nleq 5$.
  arXiv  Detail & Related papers  (2025-03-03T12:20:48Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• LevAttention: Time, Space, and Streaming Efficient Algorithm for Heavy Attentions [54.54897832889028]
  We show that for any $K$, there is a universal set" $U subset [n]$ of size independent of $n$, such that for any $Q$ and any row $i$, the large attention scores $A_i,j$ in row $i$ of $A$ all have $jin U$. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well.
  arXiv  Detail & Related papers  (2024-10-07T19:47:13Z)
• Synthesis and Arithmetic of Single Qutrit Circuits [0.8192907805418581]
  We study qutrit circuits consisting of words over the Clifford$+D$ cyclotomic gate set.<n>The framework developed to formulate qutrit gate synthesis for Clifford$+D$ extends to qudits of arbitrary prime power.
  arXiv  Detail & Related papers  (2023-11-15T04:50:41Z)
• Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits [0.8411424745913132]
  We show that when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $mathcalG_n$. We moreover show that $log(n)-2$ ancillas are always sufficient to construct a circuit for $U$.
  arXiv  Detail & Related papers  (2023-11-13T20:46:51Z)
• On character table of Clifford groups [0.0]
  We construct the character table of the Clifford group $mathcalC_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.
  arXiv  Detail & Related papers  (2023-09-26T11:29:35Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
  We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
  arXiv  Detail & Related papers  (2020-02-09T03:39:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.