Boolean Functions and Quantum Gates

Yorick Hardy, Willi-Hans Steeb

Yorick Hardy, Willi-Hans Steeb

We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.

Taghreed Ahmed, Ahmed Younes, Islam Elkabani

Abstract The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller (PPRM) expansion can be synthesized using Multiple-Control Toffoli (MCT) gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a PPRM expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the MCT gates into the NCV gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term d term , then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose MCT circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in the literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.

Anupam Chattopadhyay, Anubhab Baksi

Quantum computing necessitates the design of circuits via reversible logic gates. Efficient reversible circuit can be constructed by achieving low ancilla count, reducing logical depth and lowering Quantum costs. Generalized Peres gates have recently been realized with very low Quantum Cost (QC) by utilizing Quantum rotation gates. This is utilized in recent literature for efficient reversible circuit constructions for symmetric Boolean functions. In this paper, we extend this line of construction further by demonstrating efficient realization of adder circuits. In particular, we revisit the adder construction of Vedral, Barenco and Eckert to show that improvement of gate count and QC is achievable by exploiting a construction based only on Peres gates. We also report improved constructions of symmetric Boolean functions by following an approach recently proposed in the context of Boolean function complexity analysis.

Александр Алексеевич Горбацевич, Н. М. Шубин