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IEEE Congress on Evolutionary Computation (CEC)

Synthesis of ternary non-reversible logic circuits

Xiaoyu Li, Guowu Yang, Desheng Zheng

Published Jul 1, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.1109/cec.2010.5586551

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Abstract — This paper investigates a universal approach of synthesizing arbitrary ternary logic circuits in quantum computation based on the truth table technology. It takes into account of the relationship of classical logic and quantum logic circuits. By adding inputs with constant value and garbage outputs, the classical non-reversible logic can be transformed into reversible logic. Combined with group theory, it provides an algorithm using the ternary Swap gate, ternary NOT gate and ternary Toffoli gate library. Simultaneously, the main result shows that the numbers of qutrits we use are minimal compared to other methods. We also illustrate with two examples to test our approach. Index Terms — ternary quantum computation, reversible logic, circuit synthesis, group theory I.
Reversible circuits play an important role in quantum computation. The realization of reversible quantum circuits is required for any quantum computer to be universal. This paper1studies the realization and synthesis problem of reversible circuits. Using group theory, we present two sets of new 3-bit reversible logic gates, which included g1g2g3and nine complex gates, respectively. It is shown that any 3-bit reversible logic circuit is realizable by cascading NOT and Feynman gates, and at most one instance of our proposed gates. Given any n-bit reversible function, there are N distinct input patterns different from their corresponding outputs, where N ≤ 2n, and the other (2n- N) input patterns will be the same as their outputs. According to a synthesis example, it demonstrates that the main results are constructive and practical than others.
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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
The problem of synthesis and optimization of reversible and quantum circuits have drawn the attention of researchers for the last two decades due to increasing interest in quantum computing. Although lot of works have been done on the synthesis of binary reversible circuits, very less works have been reported on the synthesis of ternary reversible circuits. Ternary circuits have lower cost of implementation as compared to their binary counterparts. However, the synthesis approaches that exist for ternary reversible circuits either use too many circuit lines (qutrits) or too many gates. Only one prior work has discussed the problem of generating cost-optimal ternary reversible circuits, but for a very restrictive gate library, which limits the approach to a specific subset of ternary reversible functions and often the solution becomes sub-optimal due to the imposed restrictions. The present paper overcomes that restriction, and uses multiple control ternary Toffoli gates with all possible ternary target operations as the gate library. This gate library is functionally complete and can be used to synthesize any arbitrary function. The proposed SAT-based synthesis approach provides low cost solutions in terms of the number of gates for any arbitrary ternary reversible function. Experimental results on various randomly generated permutations as well as standard ternary benchmarks establish this claim. The results can be used as template for other synthesis approaches by observing how far they deviate from the optimal solutions.