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arXiv:1304.5804

Tight Bounds on the Synthesis of 3-bit Reversible Circuits: NFT Library

Ahmed Younes

Published Apr 21, 2013• 3 citations
arXivHTMLPDF

Abstract

The reversible circuit synthesis problem can be reduced to permutation group. This allows Schreier-Sims Algorithm for the strong generating set-finding problem to be used to find tight bounds on the synthesis of 3-bit reversible circuits using the NFT library. The tight bounds include the maximum and minimum length of 3-bit reversible circuits, the maximum and minimum cost of 3-bit reversible circuits. The analysis shows better results than that found in the literature for the lower bound of the cost. The analysis also shows that there are 1960 universal reversible sub-libraries from the main NFT library.

DOI
10.48550/arxiv.1304.5804

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The reversible circuit synthesis problem can be reduced to permutation group. This allows Schreier-Sims Algorithm for the strong generating set-finding problem to be used to find tight bounds on the synthesis of 3-bit reversible circuits using the NFT library. The tight bounds include the maximum and minimum length of 3-bit reversible circuits, the maximum and minimum cost of 3-bit reversible circuits. The analysis shows better results than that found in the literature for the lower bound of the cost. The analysis also shows that there are 1960 universal reversible sub-libraries from the main NFT library.
This article presents an algorithm which can quickly find the exact minimum solution to almost all of 4-bit reversible functions. We assume minimization of quantum cost (MQC). This algorithm is designed in the most memory-efficient way, or it will quickly run out of memory. Therefore, we construct the shortest coding of permutations, the topological compression and flexible data structures for the memory savings. First, hash tables are used for all 8-gate 4-bit circuits with the minimization of gate count (MGC) by using the GT library (with NOT, CNOT, Toffoli and Toffoli-4 gates). Second, we merge and split the hash tables, thus generating a single longer hash table for high-performance. Third, we synthesize these circuits with MQC by using the GTP library (with GT, Peres, and Inverted Peres gates) based on the hash table. Finally, according to the comparison of the QC of circuits, the algorithm can quickly converge for any 4-bit reversible circuit with MQC. By synthesizing all benchmark functions, in comparison with Szyprowski and Kerntopf [2011], the running time and QC are reduced up to 99.95% and 18.2%, respectively.
In quantum computing, the synthesis of reversible circuits is an important topic. Reversible circuit synthesis is particularly challenging because the complexity grows as the number of bits increases. To date, many reversible circuit synthesis algorithms have been proposed, but most are unable to find the optima within an acceptable time. Because traditional methods only consider partial interests, the resulting cost would be more gates. This paper proposes a novel method, called Bound Oriented Algorithm, which has the ability to find the optimal solution with a high hit rate, one greater than 75% on average. Moreover, with the prediction of the optima by bound, it can reduce excess calculation to further improve efficiency. In addition, a special library containing only Toffoli gates is used, which simplifies algorithm design and is more easily converted to a common library. The experiment result shows that the proposed method performs better than other methods in terms of solution quality and time cost.
Owing to the exponential nature of the memory and run-time complexity, many methods can only synthesize 3-bit reversible circuits and cannot synthesize 4-bit reversible circuits well. We mainly absorb the ideas of our 3-bit synthesis algorithms based on hash table and present the efficient algorithms which can construct almost all optimal 4-bit reversible logic circuits with many types of gates and at mini-length cost based on constructing the shortest coding and the specific topological compression; thus, the lossless compression ratio of the space of n -bit circuits reaches near 2×n! . This paper presents the first work to create all 3120218828 optimal 4-bit reversible circuits with up to 8 gates for the CNT (Controlled-NOT gate, NOT gate, and Toffoli gate) library, and it can quickly achieve 16 steps through specific cascading created circuits.
Two questions have been addressed here: how many logic levels are necessary to synthesise a specific instance of a reversible circuit, and how much better to have large gate libraries, and what gates should be required in libraries. Group theory is applied to 3-bit reversible gate synthesis to create a library useful in hierarchical design. It has been shown that arbitrary 3-bit reversible circuit can be synthesised with four-logic levels, using this new gate library. The respective universal library for the four-level synthesis is constructed and optimised it on the level of nuclear magnetic resonance pulses. A very fast algorithm to synthesise arbitrary 3-bit reversible function to gates from our library is also presented. The algorithm demonstrates dramatic speed benefit and results in a maximum of four-level circuit for arbitrary 3-bit reversible function. The gates are optimised on the level of pulses to decrease their cost and allow for objective comparison with standard CNOT, NOT, Toffoli gates (CNT) circuits. This library guarantees a four-level circuit for any 3-qubit reversible function and is also intended to be used in a hierarchical design of larger circuits.
The synthesis and optimization of quantum circuits are essential for the construction of quantum computers. This paper proposes two methods to reduce the quantum cost of 3-bit reversible circuits. The first method utilizes basic building blocks of gate pairs using different Toffoli decompositions. These gate pairs are used to reconstruct the quantum circuits where further optimization rules will be applied to synthesize the optimized circuit. The second method suggests using a new universal library, which provides better quantum cost when compared with previous work in both cost015 and cost115 metrics; this proposed new universal library “Negative NCT” uses gates that operate on the target qubit only when the control qubit’s state is zero. A combination of the proposed basic building blocks of pairs of gates and the proposed Negative NCT library is used in this work for synthesis and optimization, where the Negative NCT library showed better quantum cost after optimization compared with the NCT library despite having the same circuit size. The reversible circuits over three bits form a permutation group of size 40,320 (23!), which is a subset of the symmetric group, where the NCT library is considered as the generators of the permutation group.