Tunable coupling between three qubits as a building block for a superconducting quantum computer

Large-scale quantum computers will consist of many interacting qubits. In this paper we expand the two flux qubit coupling scheme first devised by P. Plourde et al. [Phys. Rev. B 70, 140501 (2004)] and realized by T. Hime et al. [Science 314, 1427 (2006)] to a three-qubit, two-coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the interaction strength between qubits changes in terms of the couplers’ dimensions. We explore two cases: the “on-state” where the interaction energy between two nearest-neighbor qubits is high, and the “off-state” where it is turned off. In both situations we study the undesirable crosstalk with the third qubit. Finally, we use the grape algorithm to find efficient pulse sequences for two-qubit gates subject to our calculated physical constraints on the coupling strength.

Figure 1

Two different arrangements of flux qubits 1, 2, 3 and couplers A, B. $a$ represents the arm length of the couplers, and $w$ their width. We assume that the wires of couplers A and B overlap at most in two places around qubit 2. (a) Straight-line shape. (b) Capital “L” shape.

Figure 2

The three flux qubits labeled 1, 2, and 3, and two DC-SQUIDs labeled A and B, form a three-qubit, two-coupler system. ${\gamma }_{n,i}$ is the phase difference across the $i$th junction in a coupler $n$. ${\Phi }_n$ and $J_n$ are the applied flux and circulating current of coupler $n$, respectively. $I_0$ is the critical current of the Josephson junctions used in the coupling DC-SQUIDs—for simplicity we assume that all the junctions, and hence their critical currents, are the same.

Figure 3

Interaction energies $K_{12}$ (horizontal axis), $K_{23}$ (vertical axis), and $K_{13}$ (out-of-page direction) for an L-shaped geometry with couplers’ arm width of $w=2\mu$m and arm length $a=100\mu$m. In the calculations we take $I_1=I_2=I_3=0.46\mu$A and $I_0=0.11\mu$A.

Figure 4

The size of the crosstalk interaction $K_{13}$ as a function of $K_{12}$ in the line-shaped geometry scenario where the coupling $K_{23}$ has been taken to be zero. In the calculations we take $I_1=I_2=I_3=0.46\mu$A and $I_0=0.11\mu$A.

Figure 5

The size of the crosstalk interaction $K_{13}$ as a function of $K_{12}$ in the L-shaped geometry scenario where the coupling $K_{23}$ has been taken to be zero. In the calculations we take $I_1=I_2=I_3=0.46\mu$A and $I_0=0.11\mu$A.

Figure 6

A plot consisting of values for the crosstalk $K_{13}$, while keeping $K_{12}=K_{23}\approx 0$. The horizontal axis represents different arm lengths of the coupling DC-SQUIDs. We find that the magnitude of $K_{13}$ is substantially smaller in the case of the straight-line geometry, but even in the L-shaped scenario dies off quickly. In the calculations we take $I_1=I_2=I_3=0.46\mu$A, $I_0=0.11\mu$A.

Figure 7

A pulse sequence for a $\sqrt{\text{iSWAP}}$ gate between qubits 1 and 2. Since the system coupling is “natural” for this gate, no single-qubit rotations are needed. The pulses lead to the desired gate with 99.9% fidelity. In the calculations we take ${\Delta }_1/h=5.0\text{GHz}$, ${\Delta }_2/h=5.4\text{GHz,}$ and ${\Delta }_3/h=5.8\text{GHz}$.

Figure 8

A pulse sequence for a CNOT gate between qubits 1 and 2. Single-qubit rotations are applied to all three qubits. For timely convergence of the grape algorithm, we modulate both ${\sigma }_x$ and ${\sigma }_y$ quadratures on the third qubit. Furthermore, we allow all three interaction energies to vary. The pulses lead to the desired gate with 99.9% fidelity. In the calculations we take ${\Delta }_1/h={\Delta }_2/h={\Delta }_3/h=5.0\text{GHz}$.