An Algorithm for Reversible Logic Circuit Synthesis Based on Tensor Decomposition

• URL: http://arxiv.org/abs/2107.04298v4
• Date: Tue, 23 Jul 2024 14:01:28 GMT
• Title: An Algorithm for Reversible Logic Circuit Synthesis Based on Tensor Decomposition
• Authors: Hochang Lee, Kyung Chul Jeong, Daewan Han, Panjin Kim,
• Abstract summary: An algorithm for reversible logic synthesis is proposed. Map can be written as a tensor product of a rank-($2n-2$) tensor and the $2times 2$ identity matrix.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: An algorithm for reversible logic synthesis is proposed. The task is, for a given $n$-bit substitution map $P_n: \{0,1\}^n \rightarrow \{0,1\}^n$, to find a sequence of reversible logic gates that implements the map. The gate library adopted in this work consists of multiple-controlled Toffoli gates denoted by $C^m\!X$, where $m$ is the number of control bits that ranges from 0 to $n-1$. Controlled gates with large $m \,\,(>2)$ are then further decomposed into $C^0\!X$, $C^1\!X$, and $C^2\!X$ gates. A primary concern in designing the algorithm is to reduce the use of $C^2\!X$ gate (also known as Toffoli gate) which is known to be universal. The main idea is to view an $n$-bit substitution map as a rank-$2n$ tensor and to transform it such that the resulting map can be written as a tensor product of a rank-($2n-2$) tensor and the $2\times 2$ identity matrix. Let $\mathcal{P}_n$ be a set of all $n$-bit substitution maps. What we try to find is a size reduction map $\mathcal{A}_{\rm red}: \mathcal{P}_n \rightarrow \{P_n: P_n = P_{n-1} \otimes I_2\}$. %, where $I_m$ is the $m\times m$ identity matrix. One can see that the output $P_{n-1} \otimes I_2$ acts nontrivially on $n-1$ bits only, meaning that the map to be synthesized becomes $P_{n-1}$. The size reduction process is iteratively applied until it reaches tensor product of only $2 \times 2$ matrices.

Related papers

• 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)
• Efficient $1$-bit tensor approximations [1.104960878651584]
  Our algorithm yields efficient signed cut decompositions in $20$ lines of pseudocode. We approximate the weight matrices in the open textitMistral-7B-v0.1 Large Language Model to a $50%$ spatial compression.
  arXiv  Detail & Related papers  (2024-10-02T17:56:32Z)
• 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)
• Synthesis and Arithmetic of Single Qutrit Circuits [0.9208007322096532]
  We study single qutrit quantum circuits consisting of words over the Clifford+ $mathcalD$ gate set. We characterize classes of qutrit unit vectors $z$ with entries in $mathbbZ[xi, frac1chi]$.
  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)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates [0.0]
  We show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types are required to simulate it. The quantum circuit for implementing $mathcalU_cu(2otimes N)$ and $mathcalU_cd(Motimes N)$ are presented.
  arXiv  Detail & Related papers  (2022-09-11T06:24:24Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Coresets for Decision Trees of Signals [19.537354146654845]
  We provide the first algorithm that outputs such a $(k,varepsilon)$-coreset for emphevery such matrix $D$. This is by forging a link between decision trees from machine learning -- to partition trees in computational geometry.
  arXiv  Detail & Related papers  (2021-10-07T05:49:55Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

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.