We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as controlled-square-root-of-not (i.e., controlled-V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimization problem to a group permutation problem. The transformation enables us to utilize group theory to exploit the properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we find all reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates.
We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as controlled-square-root-of-not (i.e., controlled-V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimization problem to a group permutation problem. The transformation enables us to utilize group theory to exploit the properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we find all reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates.
Because of recent nano-technological advances, nano-structured systems have become highly ordered, making it quantum computing schemas possible. We propose an approach to optimally synthesise quantum circuits from non-permutative quantum gates such as controlled-square-root-of-not (i.e., controlled-V). Our approach reduces the synthesis problem to multiple-valued optimisation and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimisation problem to a permutable representation. The transformation enables us to use group theory to exploit the symmetric properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we found all reversible circuits with quantum costs of 4, 5, 6, etc., and give another algorithm to realise these reversible circuits with quantum gates. The approach can be used for both binary permutative deterministic circuits and probabilistic circuits such as controlled random-number generators and hidden Markov models.
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.
Since non-permutative quantum gates have more complex rules than permutative quantum gates, it is very hard to synthesize quantum logic circuits using non-permutative quantum gates, such as controlled-square-root-of-NOT gates (CV∕CV+ gates). In the efficient synthesis algorithm, direct use of quantum non-permutative gates should be avoided. Rather, the key method is to use quantum gates to create new permutative quantum gates that then replace non-permutative quantum gates. This method assumes the library of quantum gate primitives are constructed so as to have the lowest possible quantum cost. In this paper, we first propose some new CV∕CV+-like gates, i.e. controlled-kth-root-of-NOT gates where k = 2,4,8,…, and give all corresponding matrixes. Further, we also present a novel generic method to quickly and directly construct this new optimal quantum logic gate library using CNOT and these non-permutative quantum gates. Our method introduces new means to find permutative quantum gates with lower quantum cost.
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.
Owing to the exponential nature of the memory or run-time complexity,many existing methods can only synthesize 3-qubit logic circuits using quantum logic gate library,however,a few can optimally synthesize 3-qubit logic circuits for quantum NOT,CNOT,Controlled-V and Controlled-V+(NCV)gates,the key approach reduces the NCV quantum circuit synthesis problem to four-valued logic synthesis.This paper proposes using NCV gates to create a new quantum logic gate library,which is exactly the same as NCV gate library in the synthesis of all optimal 3-qubit circuits,thus it also reduces the four-valued logic synthesis to easily solved two-valued logic synthesis.We present a 3-qubit efficient synthesis algorithms based on perfect hash function,which can quickly construct all optimal 3-qubit circuits——the average speed that synthesizes circuits with minimum cost is nearly 127 times faster than that of the best result of Maslov 2007.
Reversible quantum logic plays an important role in quantum computing. In this paper, we propose an approach to optimally synthesize quantum circuits by symbolic reachability analysis where the primary inputs are purely binary. we use symbolic reachability analysis, a technique most commonly used in model checking (a way of formal verification), to synthesize the optimum quantum circuits. We present an exact synthesis method with optimal quantum cost and a speedup method with non-optimal quantum cost. Both our methods guarantee the synthesizeability of all reversible circuits. Unlike previous works which use permutative reversible gates, we use a lower level library which includes non-permutative quantum gates. For the first time, problems in quantum logic synthesis have been reduced to those of multiple-valued logic synthesis thus reducing the search space and algorithm complexity. We synthesized quantum circuits for gate, half-adder, full-adder, etc. with the smallest cost.. Our approach obtains the minimum cost quantum circuits for Miller's gate, half-adder, and full-adder, which are better than previous results. In addition, we prove the minimum quantum cost (using our elementary quantum gates) for Fredkin, Peres, and Toffoli gates. Our work constitutes the first successful experience of applying satisfiability with formal methods to quantum logic synthesis.