Group Theory Based Synthesis of Binary Reversible Circuits (original) (raw)
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Bi-Directional Synthesis of 4-Bit Reversible Circuits
The Computer Journal, 2007
Reversible circuits play an important role in quantum computing, which is one of the most promising emerging technologies. In this paper, we investigate the problem of optimally synthesizing 4-bit reversible circuits. We present an enhanced bi-directional synthesis approach. Owing to the exponential nature of the memory and run-time complexity, all existing methods can only perform four steps for the Controlled-Not gate NOT gate, and Peres gate library. Our novel method can achieve 12 steps. As a result, we augment the number of circuits that can optimally be synthesized by over 5 3 10 6 times. We synthesized 1000 random 4-bit reversible circuits. The statistical analysis result supports our estimation. The quantum cost of our result is also better than the quantum cost of other approaches. The promising experimental results demonstrate the effectiveness of our approach.
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Iwls, 2003
In this paper we consider circuit synthesis for n-wire linear reversible circuits using the C-NOT gate library. These circuits are an important class of reversible circuits with applications to quantum computation. Previous algorithms, based on Gaussian elimination and LU-decomposition, yield circuits with O n 2 gates in the worst-case. However, an information theoretic bound suggests that it may be possible to reduce this to as few as O n 2 / log n gates. We present an algorithm that is optimal up to a multiplicative constant, as well as Θ(log n) times faster than previous methods. While our results are primarily asymptotic, simulation results show that even for relatively small n our algorithm is faster and yields more efficient circuits than the standard method. Generically our algorithm can be interpreted as a matrix decomposition algorithm, yielding an asymptotically efficient decomposition of a binary matrix into a product of elementary matrices.
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This dissertation describes the development of automated synthesis algorithms that construct reversible quantum circuits for reversible functions with large number of variables. Specifically, the research area is focused on reversible, permutative and fully specified binary and ternary specifications and the applicability of the resulting circuit to the physical limitations of existing quantum technologies. Automated synthesis of arbitrary reversible specifications is an NP hard, multiobjective optimization problem, where 1) the amount of time and computational resources required to synthesize the specification, 2) the number of primitive quantum gates in the resulting circuit (quantum cost), and 3) the number of ancillary qubits (variables added to hold intermediate calculations) are all minimized while 4) the number of variables is maximized. Some of the existing algorithms in the literature ignored tenth of my PhD. I close by thanking my friends Naveed and
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A rotation-based synthesis framework for reversible logic is proposed. We develop a canonical representation based on binary decision diagrams and introduce operators to manipulate the developed representation model. Furthermore, a recursive functional bi-decomposition approach is proposed to automatically synthesize a given function. While Boolean reversible logic is particularly addressed, our framework constructs intermediate quantum states that may be in superposition, hence we combine techniques from reversible Boolean logic and quantum computation. The proposed approach results in quadratic gate count for multiple-control Toffoli gates without ancillae, linear depth for quantum carry-ripple adder, and quasilinear size for quantum multiplexer.
Realization and synthesis of reversible functions
Theoretical Computer Science, 2011
Reversible circuits play an important role in quantum computing. This paper studies the realization problem of reversible circuits. For any n-bit reversible function, we present a constructive synthesis algorithm. Given any n-bit reversible function, there are N distinct input patterns different from their corresponding outputs, where N ≤ 2 n , and the other (2 n − N) input patterns will be the same as their outputs. We show that this circuit can be synthesized by at most 2n · N '(n − 1)'-CNOT gates and 4n 2 · N NOT gates. The time and space complexities of the algorithm are Ω(n · 4 n ) and Ω(n · 2 n ), respectively. The computational complexity of our synthesis algorithm is exponentially lower than that of breadth-first search based synthesis algorithms.
Fast synthesis of exact minimal reversible circuits using group theory
Proceedings of the ASP-DAC 2005. Asia and South Pacific Design Automation Conference, 2005., 2005
We present fast algorithms to synthesize exact minimal reversible circuits for various types of gates and costs. By reducing reversible logic synthesis problems to group theory problems, we use the powerful algebraic software GAP to solve such problems. Our algorithms are not only able to minimize for arbitrary cost functions of gates, but also orders of magnitude faster than the existing approaches to reversible logic synthesis. In addition, we show that the Peres gate is a better choice than the standard Toffoli gate in libraries of universal reversible gates.
Bi-Direction Synthesis for Reversible Circuits
IEEE Computer Society Annual Symposium on VLSI: New Frontiers in VLSI Design (ISVLSI'05), 2005
Quantum computing is one of the most promising emerging technologies of the future. Reversible circuits are an important class of Quantum circuits. In this paper, we investigate the problem of optimally synthesizing fourqubit reversible circuits. We present an enhanced bidirectional synthesis approach. Due to the superexponential increase on the memory requirement, all the existing methods can only perform four steps for the CNP (Control-Not gate, NOT gate, and Peres gate) library. Our novel method can achieve 12 steps. As a result, we augment the number of circuits that can be optimally synthesized by over 5*10 6 times. Moreover, our approach is faster than the existing approaches by orders of magnitude. The promising experimental results demonstrate the effectiveness of our approach.
An extension of transformation-based reversible and quantum circuit synthesis
2016 IEEE International Symposium on Circuits and Systems (ISCAS), 2016
Transformation-based synthesis is a well established systematic approach to determine a circuit implementation from a reversible function specification. Due to the inherent bidirectionality of reversible circuits the basic method can be applied in a bidirectional manner. In the approaches to date, gates are added either to the input side or the output side of the circuit on each iteration. In this paper, we introduce a new variation where gates may be added at both ends during a single iteration when this is advantageous to reducing the cost of the circuit. Experimental results show the advantage of the new approach over previous transformation-based synthesis methods and that the additional computation is justified by the possibility of improved circuit costs.