There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The organization of behavior: A neuropsychological theory

The Personal Interview

What is modal logic

IEEE transactions on computer-aided design of integrated circuits and systems : a publication of the IEEE Circuits and Systems Society

This paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits. The results we established are twofold. The first shows that ternary Swap, ternary Not and ternary Toffoli gates are universal for the realization of arbitrary $n\times n$ ternary quantum switching networks without ancilla bits. The second result proves that all $n\times n$ quantum ternary networks can be generated by Not, Controlled-Not, Multiply-Two, and Toffoli gates. Our approach is constructive. key words: ternary quantum logic synthesis, quantum circuit optimization, group theory.

/pdf/realizing-ternary-quantum-switching-networks-without-ancilla-1umhvzw7x7.pdf

Realizing Ternary Quantum Switching Networks without Ancilla Bits

Galois field sum of products (GFSOP) has been found to be very promising for reversible/quantum implementation of multiple-valued logic. In this paper, we show ten quaternary Galois field expansions, using which quaternary Galois field decision diagrams (QGFDD) can be constructed. Flattening of the QGFDD generates quaternary GFSOP (QGFSOP). These QGFSOP can be implemented as cascade of quaternary 1-qudit gates and multi-qudit Feynman and Toffoli gates. We also show the realization of quaternary Feynman and Toffoli gates using liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. Besides the quaternary functions, this approach can also be used for synthesis of encoded binary functions by grouping 2-bits together into quaternary value. For this purpose, we show binary-to-quaternary encoder and quaternary-to- binary decoder circuits using quaternary 1-quidit gates and 2-qudit Muthukrishnan-Stroud gates.

GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits.

The concept of a mixed-radix multiple-valued input exclusive sum of products (MRESP) is presented, and some possible circuit realizations for the concept are discussed. The algorithm starts from a Boolean function and generates an approximate MRESP form and the appropriate multioutput circuit. Such circuits can have smaller complexity than the EXOR forms with mixed polarity, the PLAs with decoders, and the networks with two-variable function generators. They are also easily testable. >

http://ieeexplore.ieee.org/iel4/245/1530/00037793.pdf

Minimization of multiple-valued input multi-output mixed-radix exclusive sums of products for incompletely specified Boolean functions

The paper discusses the evolutionary computation approach to the problem of optimal synthesis of Quantum and Reversible Logic circuits. Our approach uses standard Genetic Algorithm (GA) and its relative power as compared to previous approaches comes from the encoding and the formulation of the cost and fitness functions for quantum circuits synthesis. We analyze new operators and their role in synthesis and optimization processes. Cost and fitness functions for Reversible Circuit synthesis are introduced as well as local optimizing transformations. It is also shown that our approach can be used alternatively for synthesis of either reversible or quantum circuits without a major change in the algorithm. Results are illustrated on synthesized Margolus, Toffoli, Fredkin and other gates and Entanglement Circuits. This is for the first time that several variants of these gates have been automatically synthesized from quantum primitives.

Evolutionary approach to quantum and reversible circuits synthesis

This paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits. The results we established are twofold. The first shows that ternary Swap, ternary NOT and ternary Toffoli gates are universal for the realization of arbitrary n × n ternary quantum switching networks without ancilla bits. The second result proves that all n×n quantum ternary networks can be generated by NOT, Controlled-NOT, Multiply-Two and Toffoli gates. Our approach is constructive.

/pdf/realizing-ternary-quantum-switching-networks-without-ancilla-54hxrxryvv.pdf

Marek Perkowski

Papers

Realizing Ternary Quantum Switching Networks without Ancilla Bits

GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits.

Minimization of multiple-valued input multi-output mixed-radix exclusive sums of products for incompletely specified Boolean functions

Evolutionary approach to quantum and reversible circuits synthesis

Realizing ternary quantum switching networks without ancilla bits