Smaller Circuits for Arbitrary n-qubit Diagonal Computations
Stephen S. Bullock, Igor L. Markov · CERN Document Server (European Organization for Nuclear Research) · 2003
Several known algorithms for synthesizing quantum circuits in terms of elementary gates reduce arbitrary computations to diagonal. Circuits for n-qubit diagonal computations can be constructed using one (n-1)-controlled one-qubit diagonal computation and one inverter per pair of diagonal elements, not unlike the construction of classical AND-OR-NOT circuits based on the lines of a given truth table of a one-output Boolean function. More economical quantum circuits for diagonal computations are known in special cases. We propose a construction for combinational quantum circuits without ancilla qubits that allows one to implement arbitrary n-qubit diagonal computations exactly. Compared to known constructions, our circuits are an order of magnitude smaller and asymptotically smaller if no ancilla can be used. Rather than synthesize a pair of diagonal values at a time, our technique seeks tensor-product decompositions with 2x2 diagonal matrices and is applied recursively. In this process, we use 2^{n-1}-1 sub-circuits that perform diagonal computations. Each contains a diagonal rotation of the last qubit surrounded by two CNOT-chains, and can be viewed as an ``XOR-controlled'' rotation -- a new type of composite gate.