An arbitrary twoqubit computation In 23 elementary gates or less

Quantum circuits currently constitute a dominant model for quantum computation 14. Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates 1, 9 as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worstcase upper bound of 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previously known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm 4 has a worst case of 61 gates of which 18 may be CNOTs. Our techniques rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of a two-qubit gate with respect to the "magic basis" of phase-shifted Bell states 11, 12. We extend this decomposition in terms of elementary gates for quantum computation.