Lower bound on the number of Toffoli gates in a classical reversible circuit through quantum information concepts

In this paper we present a method for minimizing reversible quantum circuits using the Quantum Operator Form (QOF); a new representation of quantum circuit and of quantum-realized reversible circuits based on the CNOT, CV and CV$^\dagger$ quantum gates. The proposed form is a quantum extension to the well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the usage of different quantum gates. Therefore QOF permits minimization of quantum circuits by using prope...

This article is a short review on the concept of information. We show the strong relation between Information Theory and Physics, beginning by the concept of bit and its representation with classical physical systems, and then going to the concept of quantum bit (the so-called ``qubit'') and exposing some differences and similarities. This paper is intended to be read by non-specialists and undergraduate students of Computer Science, Mathematics and Physics, with knowledge of...

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 ...

This article is an attempt to generalize the classical theory of reversible computing, principally developed by Bennet [IBM J. Res. Develop., 17(1973)] and by Fredkin and Toffoli [Internat. J. Theoret. Phys., 21(1982)], to the quantum case. This is a fundamental step towards the construction of a quantum computer because a time efficient quantum computation is a reversible physical process. The paper is organized as follows. The first section reviews the classical theory of r...

We show a significant reduction of the number of quantum operations and the improvement of the circuit depth for the realization of the Toffoli gate by using qudits. This is done by establishing a general relation between the dimensionality of qudits and their topology of connections for a scalable multi-qudit processor, where higher qudit levels are used for substituting ancillas. The suggested model is of importance for the realization of quantum algorithms and as a method ...

The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its CNOT-optimality. We study three-qubit circuits which contain less than six CNOT gates and implemen...

Recently, various quantum computing and communication tasks have been implemented using IBM's superconductivity-based quantum computers which are available on the cloud. Here, we show that the circuits used in most of those works were not optimized and the use of the optimized circuits can considerably improve the possibility of observing unique features of quantum mechanics. Specifically, a systematic procedure is used here to obtain optimized circuits (circuits having reduc...

Among the cost metrics characterizing a quantum circuit, the $T$-count stands out as one of the most crucial as its minimization is particularly important in various areas of quantum computation such as fault-tolerant quantum computing and quantum circuit simulation. In this work, we contribute to the $T$-count reduction problem by proposing efficient $T$-count optimizers with low execution times. In particular, we greatly improve the complexity of TODD, an algorithm currentl...

This document describes a family of quantum circuits which load classical data into a quantum state. When loading $N$ classical bits, the result quantum state is of order $\log_2(N)$ qubits. Furthermore the gate depth of the data loading circuit is of order $\log_2(N)$. Limitations to the efficiency of the data loading process such as the Holevo bound are discussed. Methods to improve the efficiency of the data loading procedure such as combining classical compression techniq...

In this paper we consider circuit synthesis for n-wire linear reversible circuits using the C-NOT gate library. These circuits are an important class of reversible circuits with applications to quantum computation. Previous algorithms, based on Gaussian elimination and LU-decomposition, yield circuits with O(n^2) gates in the worst-case. However, an information theoretic bound suggests that it may be possible to reduce this to as few as O(n^2/log n) gates. We present an alg...