Figure 1

Level diagram of a SQUID with the three lowest levels ∣0⟩, ∣1⟩, and ∣2⟩ forming a $\Lambda$-type structure.Reuse & Permissions

Figure 2

Illustration for the change of the level structure (reduced) of three SQUIDs (1, 2, 3) during a three-qubit controlled phase gate performance. In (a), (b), (c), and (d), figures from left to right represent the level structures for SQUIDs 1, 2, and 3, respectively; the nonidentical level spacings of the SQUIDs could be caused by nonuniform device parameters. $g_1$, $g_2$, and $g_3$ are the resonant coupling constants between the resonator mode and the $\mid 0⟩\leftrightarrow \mid 2⟩$ transition of SQUIDs 1, 2, and 3, respectively. The difference for $g_1$, $g_2$, and $g_3$ is due to device parameter nonuniformity or nonexact placement of each SQUID. ${\nu }_{\mu w}$ is the frequency of the applied microwave pulse while ${\nu }_{12}$ is the $\mid 1⟩\leftrightarrow \mid 2⟩$ transition frequency for SQUID 1. In (a), the level spacings of SQUID 1 is set to be much different from that of SQUIDs 2 and 3, such that SQUIDs 2 and 3 are decoupled from the applied pulse. The transition between any two levels linked by a dashed line is far off resonant with the resonator mode.Reuse & Permissions

Figure 3

Relationship between an $n$-qubit controlled-NOT gate and an $n$-qubit controlled phase gate. For the circuit on the left side, the element denoted by ⊕ corresponds to a controlled-NOT gate (with $n-1$ controls on the filled circles); if the $n-1$ controls are all in the state ∣1⟩, then the state at ⊕ is bit flipped. On the other hand, for the circuit on the right side, the element $Z$ represents a Pauli rotation ${\sigma }_z$, i.e., a phase flip operation (with $n-1$ controls on the filled circles); namely, if the $n-1$ control qubits are all in the state ∣1⟩, then the state ∣1⟩ at $Z$ is phase flipped as $\mid 1⟩\to -\mid 1⟩$ while nothing happens to the state ∣0⟩ at $Z$. In addition, the element containing $H$ corresponds to a Hadamard transformation described by $\mid 0⟩\to (1∕\sqrt{2})(\mid 0⟩+\mid 1⟩)$ while $\mid 1⟩\to (1∕\sqrt{2})(\mid 0⟩-\mid 1⟩)$.Reuse & Permissions

Figure 4

Sketch of the setup for three SQUIDs (1, 2, 3) and a standing-wave quasi-one-dimensional CPW resonator (not drawn to scale). Each SQUID is placed in the plane of the resonator between the two lateral ground planes (i.e., the $x-y$ plane) and at an antinode of the ${\mathbf{B}}_r$ field. The two curved lines represent the standing-wave ${\mathbf{B}}_r$ field, which is in the $z$ direction.Reuse & Permissions