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arXiv:quant-ph/0509192 · Journal of Physics A Mathematical and General

Realizing ternary quantum switching networks without ancilla bits

Guowu Yang, Xiaoyu Song B, Marek A. Perkowski, Jinzhao Wu

Published Oct 19, 2005• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/38/44/006

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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×nn\times nn×n ternary quantum switching networks without ancilla bits. The second result proves that all n×nn\times nn×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.
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Reversible quantum circuits are a necessary subclass of quantum computation and its realization is required for any quantum computer to be universal. This paper investigates how to synthesis of arbitrary ternary non-reversible logic circuits by adding inputs with constant value and garbage outputs. Group theory has been also used to solve the synthesis of reversible logic circuits. Our algorithm uses the SNT (ternary Swap gate, ternary NOT gate, ternary Toffoli gate) library, by reducing the ternary non-reversible logic circuit synthesis problem to group theory representation. The main result shows the relationship of ternary non-reversible logic circuits and the reversible circuits. The realization approach is constructive and can be further used to develop software for synthesis of arbitrary d-level circuits. This result is significantly different from the binary non-reversible logic circuits.
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This paper investigates the synthesis of quantum networks built to realize hybrid switching circuits in the absence of ancilla qudits. We prove that all hybrid reversible circuits can be constructed by hybrid Not and Multiple-Controlled-Not gates. We also prove that any hybrid reversible circuit with only 1 or 2 binary qudits and arbitrary number of other qudits, can be constructed by hybrid Not and Controlled-Not gates. We present two construction-based algorithms to synthesize hybrid reversible circuits without ancilla qudits. The algorithms use hybrid Not and Multiple-Controlled-Not gates or hybrid Not and `1'-Controlled-Not gates, which are exponentially lower than breadth-first search based synthesis algorithms with respect to the input number.
The designs using ternary logic exploit its logarithmic reduction in the number of qudits compared with the binary circuits. In this paper, we propose quantum ternary multiplication gate. We term it as QTMG. Then we present the symbol and the realization of QTMG. Researchers will be able to use this gate as well as its symbol and realizations in their future studies. We also present a new realization of ternary Toffoli gate in specific state. Moreover, in this paper, we propose 1-qutrit multiplier circuit. The symbol and the realization of the proposed 1-qutrit multiplier circuit are also provided. Afterward, we proposed ternary partial products generation circuit (TPPG) and summation network circuit in order to design quantum ternary 2-qutrit multiplier circuit. Overall, the proposed design of QTMG in this paper is suggested for the first time. In addition, the proposed realization of ternary Toffoli gate, TPPG, summation network and 2-qutrit multiplier circuits are compared with the existing designs and the improvements are reported. The proposed gate and circuits are realized using macro-level ternary gates which are built on the top of the ion-trap realizable 1-qutrit gates and 2-qutrit Muthukrishnan–Stroud gates.
Abstract. This paper presents an important result addressing a fundamental question in synthesizing binary reversible logic circuits for quantum computation. We show that any even-reversible-circuit of n (n>3) qubits can be realized using NOT gate and Toffoli gate (‘2’-Controlled-Not gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by (n+ ⌊ n 3 ⌋)(3×22n−3 −2 n+2)and4n(n+ ⌊ n 3 ⌋)2n, respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is 10 3 n2 · 2 n. Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities. 1