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107
Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis
- IEEE Trans. on CAD of Integrated Circuits and Systems
, 2006
"... Abstract—This paper proposes an approach to optimally synthesize quantum circuits by symbolic reachability analysis, where the primary inputs and outputs are basis binary and the internal signals can be nonbinary in a multiple-valued domain. The authors present an optimal synthesis method to minimiz ..."
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Cited by 51 (5 self)
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Abstract—This paper proposes an approach to optimally synthesize quantum circuits by symbolic reachability analysis, where the primary inputs and outputs are basis binary and the internal signals can be nonbinary in a multiple-valued domain. The authors present an optimal synthesis method to minimize quantum cost and some speedup methods with nonoptimal quantum cost. The methods here are applicable to small reversible functions. Unlike previous works that use permutative reversible gates, a lower level library that includes nonpermutative quantum gates is used here. The proposed approach obtains the minimum cost quantum circuits for Miller gate, half adder, and full adder, which are better than previous results. This cost is minimum for any circuit using the set of quantum gates in this paper, where the control qubit of 2-qubit gates is always basis binary. In addition, the minimum quantum cost in the same manner for Fredkin, Peres, and Toffoli gates is proven. The method can also find the best conversion from an irreversible function to a reversible circuit as a byproduct of the generality of its formulation, thus synthesizing in principle arbitrary multi-output Boolean functions with quantum gate library. This paper constitutes the first successful experience of applying formal methods and satisfiability to quantum logic synthesis. Index Terms—Formal verification, logic synthesis, model checking, quantum computing, reversible logic, satisfiability. I.
BDD-based synthesis of reversible logic for large functions
- in Design Automation Conf., 2009
"... Reversible logic is the basis for several emerging technologies such as quantum computing, optical computing, or DNA computing and has further applications in domains like low-power design and nan-otechnologies. However, current methods for the synthesis of re-versible logic are limited, i.e. they a ..."
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Cited by 46 (28 self)
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Reversible logic is the basis for several emerging technologies such as quantum computing, optical computing, or DNA computing and has further applications in domains like low-power design and nan-otechnologies. However, current methods for the synthesis of re-versible logic are limited, i.e. they are applicable to relatively small functions only. In this paper, we propose a synthesis approach, that can cope with Boolean functions containing more than a hundred of variables. We present a technique to derive reversible circuits for a function given by a Binary Decision Diagram (BDD). The cir-cuit is obtained using an algorithm with linear worst case behavior regarding run-time and space requirements. Furthermore, the size of the resulting circuit is bounded by the BDD size. This allows to transfer theoretical results known from BDDs to reversible cir-cuits. Experiments show better results (with respect to the circuit cost) and a significantly better scalability in comparison to previous synthesis approaches.
Toffoli network synthesis with templates
- IEEE Trans. on CAD of Integrated Circuits and Systems
, 2005
"... Abstract—Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. ..."
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Cited by 41 (9 self)
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Abstract—Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the above synthesis approach. We present a basic method and, based on that, a bidirectional synthesis algorithm which produces a network of Toffoli gates realizing a given reversible specification. An asymptotically optimal modification of the basic synthesis algorithm employing generalized mEXOR gates is also presented. Transformations are then applied using template matching. The basis for a template is a network of gates that realizes the identity function. If a sequence of gates in the synthesized network matches a sequence comprised of more than half the gates in a template, then a transformation using the remaining gates in the template can be applied resulting in a reduction in the gate count for the synthesized network. All templates with up to six gates are described in this paper. Experimental results including an exhaustive examination of all 3-variable reversible functions and a collection of benchmark problems are presented. The paper concludes with suggestions for further research. Index Terms—Logic synthesis, quantum computing, reversible logic. I.
Quantum circuit simplification using templates
- in Proc
, 2005
"... Optimal synthesis of quantum circuits is intractable and heuristic methods must be employed. Templates are a gen-eral approach to reversible and quantum circuit simplifi-cation. In this paper, we consider the use of templates to simplify a quantum circuit initially found by other means. We present a ..."
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Cited by 33 (10 self)
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Optimal synthesis of quantum circuits is intractable and heuristic methods must be employed. Templates are a gen-eral approach to reversible and quantum circuit simplifi-cation. In this paper, we consider the use of templates to simplify a quantum circuit initially found by other means. We present and analyze templates in the general case, and then provide particular details for circuits composed of NOT, CNOT and controlled-sqrt-of-NOT gates. We intro-duce templates for this set of gates and apply them to sim-plify both known quantum realizations of Toffoli gates and circuits found by earlier heuristic Fredkin and Toffoli gate synthesis algorithms. While the number of templates is quite small, the reduction in quantum cost is often significant. 1.
Synthesis of reversible logic
- In Design, Automation and Test in Europe
, 2004
"... Abstract — A function is reversible if each input vector produces a unique output vector. Reversible functions find applications in low power design, quantum computing, and nanotechnology. Logic synthesis for reversible circuits differs substantially from traditional logic synthesis. In this paper, ..."
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Cited by 20 (0 self)
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Abstract — A function is reversible if each input vector produces a unique output vector. Reversible functions find applications in low power design, quantum computing, and nanotechnology. Logic synthesis for reversible circuits differs substantially from traditional logic synthesis. In this paper, we present the first practical synthesis algorithm and tool for reversible functions with a large number of inputs. It uses positive-polarity Reed-Muller decomposition at each stage to synthesize the function as a network of Toffoli gates. The heuristic uses a priority queue based search tree and explores candidate factors at each stage in order of attractiveness. The algorithm produces near-optimal results for the examples discussed in the literature. The key contribution of the work is that the heuristic finds very good solutions for reversible functions with a large number of inputs. I.
Synthesis of Reversible Sequential Elements*
"... Abstract – To construct a reversible sequential circuit, reversible sequential elements are required. This work presents novel designs of reversible sequential elements such as D latch, JK latch, and T latch. Based on these reversible latches, we also construct the designs of the corresponding flip- ..."
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Cited by 19 (0 self)
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Abstract – To construct a reversible sequential circuit, reversible sequential elements are required. This work presents novel designs of reversible sequential elements such as D latch, JK latch, and T latch. Based on these reversible latches, we also construct the designs of the corresponding flip-flops. Comparing with previous work, the implementation cost of our new designs, including the number of gates and the number of garbage outputs is considerably reduced. I.
Comparison of the cost metrics for reversible and quantum logic synthesis
, 2005
"... A breadth-first search method for determining optimal 3-line circuits composed of quantum NOT, CNOT, controlled-V and controlled-V + (NCV) gates is introduced. Results are presented for simple gate count and for technology motivated cost metrics. The optimal NCV circuits are also compared to NCV cir ..."
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Cited by 19 (5 self)
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A breadth-first search method for determining optimal 3-line circuits composed of quantum NOT, CNOT, controlled-V and controlled-V + (NCV) gates is introduced. Results are presented for simple gate count and for technology motivated cost metrics. The optimal NCV circuits are also compared to NCV circuits derived from optimal NOT, CNOT and Toffoli (NCT) gate circuits. The work presented here provides basic results and motivation for continued study of the direct synthesis of NCV circuits, and establishes relations between function realizations in different circuit cost metrics. 1
Synthesis of reversible circuits with minimal lines for large functions
- in ASP Design Automation Conf., 2012
"... Abstract — Reversible circuits are an emerging technology where all computations are performed in an invertible manner. Motivated by their promising applications, e.g. in the domain of quantum computation or in the low-power design, the synthesis of such circuits has been intensely studied. However, ..."
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Cited by 18 (15 self)
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Abstract — Reversible circuits are an emerging technology where all computations are performed in an invertible manner. Motivated by their promising applications, e.g. in the domain of quantum computation or in the low-power design, the synthesis of such circuits has been intensely studied. However, how to auto-matically realize reversible circuits with the minimal number of lines for large functions is an open research problem. In this paper, we propose a new synthesis approach which relies on concepts that are complementary to existing ones. While “con-ventional ” function representations have been applied for synthe-sis so far (such as truth tables, ESOPs, BDDs), we exploit Quan-tum Multiple-valued Decision Diagrams (QMDDs) for this pur-pose. An algorithm is presented that performs transformations on this data-structure eventually leading to the desired circuit. Experimental results show the novelty of the proposed approach through enabling automatic synthesis of large reversible functions with the minimal number of circuit lines. Furthermore, the quan-tum cost of the resulting circuits is reduced by 50 % on average compared to an existing state-of-the-art synthesis method. I.
Simplification of Toffoli Networks via Templates
- In Symposium on Integrated Circuits and System Design
, 2003
"... Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This pap ..."
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Cited by 18 (4 self)
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Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the second step. Transformations are accomplished via template matching. The basis for a template is a network with Ñ gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation reducing the gate count can be applied. All templates for Ñ � � are described in this paper. 1
Synthesis of Fredkin–Toffoli Reversible Networks
- IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS
, 2005
"... Reversible logic has applications in quantum computing, low
power CMOS, nanotechnology, optical computing, and DNA computing.
The most common reversible gates are the Toffoli gate and the Fredkin gate.
We present a method that synthesizes a network with these gates in two steps. First, our synthesis ..."
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Cited by 17 (0 self)
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Reversible logic has applications in quantum computing, low
power CMOS, nanotechnology, optical computing, and DNA computing.
The most common reversible gates are the Toffoli gate and the Fredkin gate.
We present a method that synthesizes a network with these gates in two steps. First, our synthesis algorithm finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply transformations that reduce the number of gates in the network. Transformations
are accomplished via template matching. The basis for a template is a network with gates that realizes the identity function. If a sequence of gates in the network to be reduced matches a sequence of gates comprising more than half of a template, then a transformation that reduces the gate count can be applied.We have synthesized all three input, three output reversible functions and here compare our results to the optimal results. We also present the results of applying our synthesis tool to obtain networks
for a number of benchmark functions.
