arXiv subjects

Dmitry V. Zakablukov

Publications and source records attributed to Dmitry V. Zakablukov.

8 recordsLinked to original sources

arXiv2018

Synthesis methods for reversible circuits consisting of NOT, CNOT and 2-CNOT gates (Ph.D. thesis)

In this paper, reversible circuits consisting of NOT, CNOT and 2-CNOT gates are studied. Several asymptotically optimal by the order of magnitude synthesis methods are described. Some circuit's complexity reduction approaches are considered. Implementation of discrete logarithm within a reversible circuit is discussed. The main conclusion in the paper is that the usage of additional inputs (additional memory) in reversible circuits almost always allow to reduce their complexity, depth and weight.

cs.ET↗

arXiv2018

On Synthesis of Reversible Circuits with Small Number of Additional Inputs Consisting of NOT, CNOT and 2-CNOT Gates

The paper discusses the gate complexity of reversible circuits with the small number of additional inputs consisting of NOT, CNOT and 2-CNOT gates. We study Shannon's gate complexity function $L(n, q)$ for a reversible circuit implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with $q \leqslant O(n^2)$ additional inputs. The general bound $L(n,q) \asymp n2^n \mathop / \log_2 n$ is proved for this case.

cs.CC↗

arXiv2017

General Upper Bounds for Gate Complexity and Depth of Reversible Circuits Consisting of NOT, CNOT and 2-CNOT Gates

The paper discusses the gate complexity and the depth of reversible circuits consisting of NOT, CNOT and 2-CNOT gates in the case, when the number of additional inputs is limited. We study Shannon's gate complexity function $L(n, q)$ and depth function $D(n, q)$ for a reversible circuit implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with $8n < q \lesssim n2^{n-o(n)}$ additional inputs. The general upper bounds $L(n,q) \lesssim 2^n + 8n2^n \mathop / (\log_2 (q-4n) - \log_2 n - 2)$ and $D(n,q) \lesssim 2^{n+1}(2,5 + \log_2 n - \log_2 (\log_2 (q - 4n) - \log_2 n - 2))$ are proved for this case.

cs.CC↗

arXiv2016

Application of Permutation Group Theory in Reversible Logic Synthesis

The paper discusses various applications of permutation group theory in the synthesis of reversible logic circuits consisting of Toffoli gates with negative control lines. An asymptotically optimal synthesis algorithm for circuits consisting of gates from the NCT library is described. An algorithm for gate complexity reduction, based on equivalent replacements of gates compositions, is introduced. A new approach for combining a group-theory-based synthesis algorithm with a Reed-Muller-spectra-based synthesis algorithm is described. Experimental results are presented to show that the proposed synthesis techniques allow a reduction in input lines count, gate complexity or quantum cost of reversible circuits for various benchmark functions.

cs.ET↗

arXiv2016

On Asymptotic Gate Complexity and Depth of Reversible Circuits With Additional Memory

The reversible logic can be used in various research areas, e.g. quantum computation, cryptography and signal processing. In the paper we study reversible logic circuits with additional inputs, which consist of NOT, CNOT and C\textsuperscript{2}NOT gates. We consider a set $F(n,q)$ of all transformations $\mathbb B^n \to \mathbb B^n$ that can be realized by reversible circuits with $(n+q)$ inputs. An analogue of Lupanov's method for the synthesis of reversible logic circuits with additional inputs is described. We prove upper asymptotic bounds for the Shannon gate complexity function $L(n,q)$ and the depth function $D(n,q)$ in case of $q > 0$: $L(n,q_0) \lesssim 2^n$ if $q_0 \sim n 2^{n-o(n)}$ and $D(n,q_1) \lesssim 3n$ if $q_1 \sim 2^n$.

cs.CC↗

arXiv2016

On Asymptotic Gate Complexity and Depth of Reversible Circuits Without Additional Memory

Reversible computation is one of the most promising emerging technologies of the future. The usage of reversible circuits in computing devices can lead to a significantly lower power consumption. In this paper we study reversible logic circuits consisting of NOT, CNOT and 2-CNOT gates. We introduce a set $F(n,q)$ of all transformations $\mathbb Z_2^n \to \mathbb Z_2^n$ that can be implemented by reversible circuits with $(n+q)$ inputs. We define the Shannon gate complexity function $L(n,q)$ and the depth function $D(n,q)$ as functions of $n$ and the number of additional inputs $q$. First, we prove general lower bounds for functions $L(n,q)$ and $D(n,q)$. Second, we introduce a new group theory based synthesis algorithm, which can produce a circuit $\mathfrak S$ without additional inputs and with the gate complexity $L(\mathfrak S) \leq 3n 2^{n+4}(1+o(1)) \mathop / \log_2 n$. Using these bounds, we state that almost every reversible circuit with no additional inputs, consisting of NOT, CNOT and 2-CNOT gates, implements a transformation from $F(n,0)$ with the gate complexity $L(n,0) \asymp n 2^n \mathop / \log_2 n$ and with the depth $D(n,0) \geq 2^n(1-o(1)) \mathop / (3\log_2 n)$.

cs.ET↗

arXiv2016

Asymptotic bounds of depth for a reversible circuit consisting of NOT, CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuit consisting of NOT, CNOT and 2-CNOT gates. Reversible circuit depth function $D(n, q)$ for a circuit implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ is introduced as a function of $n$ and the number of additional inputs $q$. It is proved that for the case of implementing a permutation from $A(\mathbb Z_2^n)$ with a reversible circuit having no additional inputs the depth is bounded as $D(n, 0) \gtrsim 2^n / (3\log_2 n)$. It is proved that for the case of implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with a reversible circuit having $q_0 \sim 2^n$ additional inputs the depth is bounded as $D(n, q_0) \lesssim 3n$.

cs.ET↗

arXiv2016

On Gate Complexity of Reversible Circuits Consisting of NOT, CNOT and 2-CNOT Gates

The paper discusses the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The Shannon gate complexity function $L(n, q)$ for a reversible circuit, implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$, is defined as a function of $n$ and the number of additional inputs $q$. The general lower bound $L(n,q) \geq \frac{2^n(n-2)}{3\log_2(n+q)} - \frac{n}{3}$ for the gate complexity of a reversible circuit is proved. An upper bound $L(n,0) \leqslant 3n2^{n+4}(1+o(1)) \mathop / \log_2n$ for the gate complexity of a reversible circuit without additional inputs is proved. An upper bound $L(n,q_0) \lesssim 2^n$ for the gate complexity of a reversible circuit with $q_0 \sim n2^{n-o(n)}$ additional inputs is proved.

cs.ET↗