CNOT-Distance is NP-complete under all-to-all connectivity

• URL: http://arxiv.org/abs/2608.03825v1
• Date: Tue, 04 Aug 2026 15:34:26 GMT
• Title: CNOT-Distance is NP-complete under all-to-all connectivity
• Abstract summary: We show that $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity.<n>We prove that this problem is NP-complete.
• Score: 0.69670060449041
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given $A\in\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+τ(G)$, where $τ(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.

Related papers

• Near-Optimal Dimension Lower Bounds for Single-Vector Embeddings of Maximum Inner Product Similarity [75.14269295861845]
  For singleton queries, Chamfer becomes maximum inner product similarity (MAX-IP)<n>For every fixed $in(0,1)$, there are constants $A_,c_>0$.<n>Unit-sphere MAX-IP matrix is an exact two-valued affine image of the DNF pattern matrix with gap of at least $8$.
  arXiv  Detail & Related papers  (2026-07-22T17:27:20Z)
• Asymptotically Optimal Regret for Reinforcement Learning without Horizon Dependence [54.59847568544922]
  We study horizon-free regret for finite-horizon time-homogeneous Markov decision processes with $A$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$.<n>We prove a regret upper bound [tilde O(sqrtSAK+S8A3)] with failure probability $$, where $K$ is the number of episodes and $tilde O(sqrtSAK+S3K)$ hides $mathsfpoly
  arXiv  Detail & Related papers  (2026-07-22T07:42:19Z)
• Phase Transition in Convex Relaxations for Graph Alignment [6.515592049126884]
  We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices.<n>Our results precisely characterize that $|Xstar-star|_F2$ transitions from $o(n)$ for $= tildeo(n-1/2)$ to $(n)$ for $= tilde(n-1/2)$.
  arXiv  Detail & Related papers  (2026-06-14T03:47:45Z)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry [2.6140509675507384]
  We consider the problem of embedding the vertices of an $n$-vertex graph $G$ into the leaves of an $n$-leaf rooted binary tree $mathcalB$.<n>We numerically compare our congestion bounds on different families of graphs with regular structure (hypercubes and lattices), random graphs, and tensor network representations of quantum circuits.
  arXiv  Detail & Related papers  (2025-10-03T04:58:40Z)
• 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)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions [51.19236668224547]
  We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions. For tensor train decomposition, we give a bicriteria $(1 + eps)$-approximation algorithm with a small bicriteria rank and $O(q cdot nnz(A))$ running time. In addition, we extend our algorithm to tensor networks with arbitrary graphs.
  arXiv  Detail & Related papers  (2022-07-15T11:55:09Z)
• Solving Hamiltonian Cycle Problem using Quantum $\mathbb{Z}_2$ Lattice Gauge Theory [9.83302372715731]
  The Hamiltonian cycle (HC) problem in graph theory is a well-known NP-complete problem. We present an approach in terms of $mathbbZ$ lattice gauge theory defined on the lattice with the graph as its dual. For some random samples of small graphs, we demonstrate that the dependence of the average value of $g_c$ on $sqrtN_hc$, $N_hc$ being the number of HCs, and that of the average value of $frac1g_c$ on $N
  arXiv  Detail & Related papers  (2022-02-17T18:39:42Z)
• A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs [1.0435741631709405]
  An $soperatorname-t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects $s$ and $t$. In this work we describe a quantum algorithm for the minimum $soperatorname-t$ cut problem on undirected graphs.
  arXiv  Detail & Related papers  (2021-10-29T07:35:46Z)
• Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment [59.87663954467815]
  In graph-based binary learning, a subset of known labels $hatx_i$ are used to infer unknown labels. When restricting labels $x_i$ to binary values, the problem is NP-hard. We propose a fast projection-free method by solving a sequence of linear programs (LP) instead.
  arXiv  Detail & Related papers  (2021-06-03T07:22:48Z)
• 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.