3 Smaller Circuits for Arbitrary n - qubit Diagonal Computations ∗

Several known algorithms for synthesizing quantum circuits in terms of elementary gates reduce arbitrary computations to diagonal [1, 2]. Circuits for n-qubit diagonal computations can be constructed using one (n − 1)-controlled one-qubit diagonal computation [3] 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 [5, 2] 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 2 × 2 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.