Minimization of Quantum Circuits using Quantum Operator Forms

In this paper we present a method for minimizing reversible quantum circuits using the Quantum Operator Form (QOF); a new representation of quantum circuit and of quantum-realized reversible circuits based on the CNOT, CV and CV$^\dagger$ quantum gates. The proposed form is a quantum extension to the well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the usage of different quantum gates. Therefore QOF permits minimization of quantum circuits by using properties of different gates than only the multi-control Toffoli gates. We introduce a set of minimization rules and a pseudo-algorithm that can be used to design circuits with the CNOT, CV and CV$^\dagger$ quantum gates. We show how the QOF can be used to minimize reversible quantum circuits and how the rules allow to obtain exact realizations using the above mentioned quantum gates.Comment: 11 pages, 14 figures, Proceedings of the ULSI Workshop 2012 (@ISMVL 2012

Synthesis of Reversible Circuits from a Subset of Muthukrishnan-Stroud Quantum Realizable Multi-Valued Gates

We present a new type of quantum realizable reversible cascade. Next we present a new algorithm to synthesize arbitrary single-output ternary functions using these reversible cascades. The cascades use “Generalized Multi-Valued Gates” introduced here, which extend the concept of Generalized Ternary Gates introduced previously. While there were 216 GTGs, a total of 12 ternary gates of the new type are sufficient to realize arbitrary ternary functions. (The count can be further reduced to 5 gates, three 2-qubit and two 1-qubit). Such gates are realizable in quantum ion trap devices. For some functions, the algorithm requires fewer gates than results previously published [1, 5, 8, 14]. In addition, the algorithm also does conversion from arbitrary ternary logic to reversible logic at the cost of relatively small garbage. The algorithm is implemented here in ternary logic, but generalization to arbitrary radix is both straightforward and sees a reduction in growth of cost as the radix is increased

USING HOMING, SYNCHRONIZING AND DISTINGUISHING INPUT SEQUENCES FOR THE ANALYSIS OF REVERSIBLE FINITE STATE MACHINES

A digital device is called reversible if it realizes a reversible mapping, i.e., the one for which there exist a unique inverse. The field of reversible computing is devoted to studying all aspects of using and designing reversible devices. During last 15 years this field has been developing very intensively due to its applications in quantumcomputing, nanotechnology and reducing power consumption of digital devices. We present an analysis of the Reversible Finite State Machines (RFSM) with respect to three well known sequences used in the testability analysis of the classical Finite State Machines (FSM). The homing, distinguishing and synchronizing sequences areapplied to two types of reversible FSMs: the converging FSM (CRFSM) and the nonconverging FSM (NCRFSM) and the effect is studied and analyzed. We show that while only certain classical FSMs possess all three sequences, CRFSMs and NCRFSMs have properties allowing to directly determine what type of sequences these machines possess

Regularity and Symmetry as a Base for Efficient Realization of Reversible Logic Circuits

We introduce a Reversible Programmable Gate Array (RPGA) based on regular structure to realize binary functions in reversible logic. This structure, called a 2 * 2 Net Structure, allows for more efficient realization of symmetric functions than the methods shown by previous authors. In addition, it realizes many non-symmetric functions even without variable repetition. Our synthesis method to RPGAs allows to realize arbitrary symmetric function in a completely regular structure of reversible gates with smaller “garbage” than the previously presented papers. Because every Boolean function is symmetrizable by repeating input variables, our method is applicable to arbitrary multi-input, multi-output Boolean functions and realizes such arbitrary function in a circuit with a relatively small number of garbage gate outputs. The method can be also used in classical logic. Its advantages in terms of numbers of gates and inputs/outputs are especially seen for symmetric or incompletely specified functions with many outputs

A Hierarchical Approach to Computer-Aided Design of Quantum Circuits

A new approach to synthesis of permutation class of quantum logic circuits has been proposed in this paper. This approach produces better results than the previous approaches based on classical reversible logic and can be easier tuned to any particular quantum technology such as nuclear magnetic resonance (NMR). First we synthesize a library of permutation (pseudobinary) gates using a Computer-Aided-Design approach that links evolutionary and combinatorics approaches with human experience and creativity. Next the circuit is designed using these gates and standard 1*1 and 2*2 quantum gates and finally the optimizing tautological transforms are applied to the circuit, producing a sequence of quantum operations being close to operations practically realizable. These hierarchical stages can be compared to standard gate library design, generic logic synthesis and technology mapping stages of classical CAD systems, respectively. We use an informed genetic algorithm to evolve arbitrary quantum circuit specified by a (target) unitary matrix, specific encoding that reduces the time of calculating the resultant unitary matrices of chromosomes, and an evolutionary algorithm specialized to permutation circuits specified by truth tables. We outline interactive CAD approach in which the designer is a part of feedback loop in evolutionary program and the search is not for circuits of known specifications, but for any gates with high processing power and small cost for given constraints. In contrast to previous approaches, our methodology allows synthesis of both: small quantum circuits of arbitrary type (gates), and permutation class circuits that are well realizable in particular technology

Function-driven Linearly Independent Expansions of Boolean Functions and Their Application to Synthesis of Reversible Circuits

The paper presents a family of new expansions of Boolean functions called Function-driven Linearly Independent (fLI) expansions. On the basis of this expansion a new kind of a canonical representation of Boolean functions is constructed: Function-driven Linearly Independent Binary Decision Diagrams (fLIBDDs). They generalize both Function-driven Shannon Binary Decision Diagrams (fShBDDs) and Linearly Independent Binary Decision Diagram (LIBDDs). The diagrams introduced in the paper, can provide significantly smaller representations of Boolean functions than standard Ordered Binary Decision Diagrams (OBDDs), Ordered Functional Decision Diagrams (OFDDs) and Ordered (Pseudo-) Kronecker Functional Decision Diagrams (OKFDDs) and can be applied to synthesis of reversible circuits

New generalizations of Shannon decomposition

ABSTRACT: New types of functional decompositions are presented which we call function-driven decompositions. Namely, it is shown in the paper that in the well-known Shannon formula it is possible to replace a variable by an arbitrary function having the property of self-duality with respect to a variable of the decomposed function. In this way a generalization of recently studied linear decompositions (used for constructing Linearly Transformed Binary Decision Diagrams) is obtained. Further extensions of functiondriven decompositions are also mentioned. These extensions can be defined by combining the functiondriven decompositions with other known decompositions, e.g. Reed-Muller (Davio) decompositions. It is also shown how to extend function-driven decompositions to multiple-valued functions. 1