Elementary gates for quantum computation
A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, H. Weinfurter
TL;DR
This paper shows that quantum computation is universal when allowing all single-qubit gates together with a two-qubit CNOT-like gate, and it develops a suite of constructive methods to realize more complex multi-qubit unitaries from these elementary gates. It systematically derives exact and approximate gate-construction techniques, proving linear, quadratic, and near-linear resource bounds for simulating n-bit wedge gates and discussing phase-congruence variants that can improve efficiency. Through a combination of SU(2) decompositions and grey-code inspired sequences, the authors establish scalable approaches to synthesize general U(2^n) operations, with explicit bounds and limitations. They also contrast exact universal constructions with practical approximations and discuss Reck-type decompositions to place the results in the broader context of quantum circuit design. Overall, the work provides both foundational universal gate results and practical gate-count analyses essential for building scalable quantum networks.
Abstract
We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values $(x,y)$ to $(x,x \oplus y)$) is universal in the sense that all unitary operations on arbitrarily many bits $n$ (U($2^n$)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for $n$-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary $n$-bit unitary operations.