The Design and Optimization of Quantum Circuits using the Palindrome Transform

The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2n×2n unitary matrices into efficient circuits of (n−1)-controlled single-qubit and (n−1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of selfinverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an enumerative algorithm for minimal (n−1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n − 1)-controlled single-qubit and (n − 1)-controlled-NOT gates.