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Group theory based synthesis of binary reversible circuits

Guowu Yang, William N. N. Hung, Marek A. Perkowski

Published Apr 7, 2009• 5 citations
OpenAlex

Abstract

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

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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
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.
This paper presents a constructive synthesis algorithm for any n-qubit reversible function. Given any n-qubit reversible function, there are N distinct input patterns different from their corresponding outputs, where N les 2n, and the other (2n- N) input patterns will be the same as their outputs. We show that this circuit can be synthesized by at most 2nldrN '(n - 1)'-CNOT gates and 4n2ldr N NOT gates. The time complexity of our algorithm has asymptotic upper bound O(n ldr 4n). The space complexity of our synthesis algorithm is also O(n ldr 2n). The computational complexity of our synthesis algorithm is exponentially lower than the complexity of breadth-first search based synthesis algorithm.
Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics, and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT, and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting the distribution of circuit sizes. We also study canonical circuit decompositions where gates of the same kind are grouped together. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.
With increase in emphasis on low-power design and quantum computation, research in reversible logic synthesis has gained great attention. Various methods of synthesis exist in the literature, like exact synthesis, heuristic approaches, methods based on group theory, and those based on higher level representation of functions like Binary Decision Diagram (BDD). In this paper a group theory based synthesis approach for reversible circuits is presented, using both positive-control and negative-control Toffoli gates. The method uses group theoretic rules for factoring permutation cycles for synthesis. The use of both positive and negative control Toffoli gates results in reduction of total number of gates required as compared to previous methods based on group theory. Several worked out examples illustrate the advantage of the proposed approach.
Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.
Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.