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

Reversible logic has applications in various research areas including low-power design and quantum computation. In this paper, a rule-based optimization approach for reversible circuits is proposed which uses both negative and positive control Toffoli gates during the optimization. To this end, a set of rules for removing NOT gates and optimizing sub-circuits with common-target gates are proposed. To evaluate the proposed approach, the best-reported synthesized circuits and t...

This paper presents novel techniques for the synthesis of reversible networks of Toffoli gates, as well as improvements to previous methods. Gate count and technology oriented cost metrics are used. Our synthesis techniques are independent of the cost metrics. Two new iterative synthesis procedure employing Reed-Muller spectra are introduced and shown to complement earlier synthesis approaches. The template simplification suggested in earlier work is enhanced through introduc...

This paper discusses a restriction of quantum theory, in which very complex states would be excluded. The toy theory is phrased in the language of the circuit model for quantum computing, its key ingredient being a limitation on the number of interactions that \textit{each} qubit may undergo. As long as one stays in the circuit model, the toy theory is consistent and may even match what we shall be ever able to do in a controlled laboratory experiment. The direct extension of...

Most of the work on implementing arithmetic on a quantum computer has borrowed from results in classical reversible computing (e.g. [VBE95], [BBF02], [DKR04]). These quantum networks are inherently classical, as they can be implemented with only the Toffoli gate. Draper [D00] has proposed an inherently "quantum" network for addition based on the quantum Fourier transform. His approach has the advantage that it requires no carry qubits (the previous approaches required O(n) ca...

We present the first exact quantum adder with sublinear depth and no ancilla qubits. Our construction is based on classical reversible logic only and employs low-depth implementations for the CNOT ladder operator and the Toffoli ladder operator, two key components to perform ripple-carry addition. Namely, we demonstrate that any ladder of $n$ CNOT gates can be replaced by a CNOT-circuit with $O(\log n)$ depth, while maintaining a linear number of gates. We then generalize thi...

The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented in detail. Throughout there is an emphasis on the physical as well as the abstract aspects of computation and the interplay between them. This report is presented as a Master's thesis at the department of Computer Science and Engineering at G{\"...

Quantum addition circuits are considered being of two types: 1) Toffolli-adder circuits which use only classical reversible gates (CNOT and Toffoli), and 2) QFT-adder circuits based on the quantum Fourier transformation. We present the first systematic translation of the QFT-addition circuit into a Toffoli-based adder. This result shows that QFT-addition has fundamentally the same fault-tolerance cost (e.g. T-count) as the most cost-efficient Toffoli-adder: instead of using a...

In this paper, we have implemented and designed a sorting network for reversible logic circuits synthesis in terms of n*n Toffoli gates. The algorithm presented in this paper constructs a Toffoli Network based on swapping bit strings. Reduction rules are then applied by simple template matching and removing useless gates from the network. Random selection of bit strings and reduction of control inputs are used to minimize both the number of gates and gate width. The method pr...

A historical review is given of the emergence of the idea of the quantum logic gate from the theory of reversible Boolean gates. I highlight the quantum XOR or controlled NOT as the fundamental two-bit gate for quantum computation. This gate plays a central role in networks for quantum error correction.

This paper considers the realizability of quantum gates from the perspective of information complexity. Since the gate is a physical device that must be controlled classically, it is subject to random error. We define the complexity of gate operation in terms of the difference between the entropy of the variables associated with initial and final states of the computation. We argue that the gate operations are irreversible if there is a difference in the accuracy associated w...