Minimum Construction of Two-Qubit Quantum Operations

Optimal construction of quantum operations is a fundamental problem in the realization of quantum computation. We here introduce a newly discovered quantum gate, $B$, that can implement any arbitrary two-qubit quantum operation with minimal number of both two- and single-qubit gates. We show this by giving an analytic circuit that implements a generic nonlocal two-qubit operation from just two applications of the $B$ gate. Realization of the $B$ gate is illustrated with an example of charge-coupled superconducting qubits for which the $B$ gate is seen to be generated in shorter time than the CNOT gate.

Figure 1

Tetrahedron $O{A}_1{A}_2{A}_3$ contains all the local equivalence classes of nonlocal gates [9], where $O\left(\right[0,0,0\left]\right)$ and $A_1\left(\right[\pi ,0,0\left]\right)$ both correspond to local gates, $L\left(\right[\frac{\pi }{2},0,0\left]\right)$ to the CNOT gate, $A_2\left(\right[\frac{\pi }{2},\frac{\pi }{2},0\left]\right)$ to the double-CNOT gate, $A_3\left(\right[\frac{\pi }{2},\frac{\pi }{2},\frac{\pi }{2}\left]\right)$ to the SWAP gate, and $B\left(\right[\frac{\pi }{2},\frac{\pi }{4},0\left]\right)$ to the new gate we introduce in this Letter. From the Cartan decomposition on su(4), any two-qubit unitary operation $U\in \mathrm{S}\mathrm{U}\left(4\right)$ can be written as $U=k_{1}A{k}_2=k_1{e}^{c_1(i/2){\sigma }_x^1{\sigma }_x^2}{e}^{c_2(i/2){\sigma }_y^1{\sigma }_y^2}{e}^{c_3(i/2){\sigma }_z^1{\sigma }_z^2}{k}_2$, where ${\sigma }_{\alpha }^1{\sigma }_{\alpha }^2={\sigma }_{\alpha }\otimes {\sigma }_{\alpha },{\sigma }_{\alpha }$ are the Pauli matrices, and $k_1,k_2\in \mathrm{S}\mathrm{U}\left(2\right)\otimes \mathrm{S}\mathrm{U}\left(2\right)$ are local gates [9, 10, 11]. Since the local gates are fully ac­cessible, it is evident that we need to construct a circuit for only the nonlocal block $A$ in terms of the available entangling gates (or Hamiltonians).

Figure 2

Trajectory of inductively coupled Josephson junction qubits that generates gate $B$ from the Hamiltonian $H_J$. Shown here is the minimum time solution, which is obtained for the Hamiltonian parameters $E_L=1$, $\alpha =1.1436$ with minimal time duration $t=1.5014$. The trajectory is confined to the $c_1,c_2$ basal plane of the tetrahedron $O{A}_1{A}_2{A}_3$ of Fig. 1.