Protected gates for superconducting qubits

We analyze the accuracy of quantum phase gates acting on “0-$\pi$ qubits” in superconducting circuits, where the gates are protected against thermal and Hamiltonian noise by continuous-variable quantum error-correcting codes. The gates are executed by turning on and off a tunable Josephson coupling between an $LC$ oscillator and a qubit or pair of qubits; assuming perfect qubits, we show that the gate errors are exponentially small when the oscillator's impedance $\sqrt{L/C}$ is large compared to $\hslash /4e^2\approx 1\mathrm{k}\Omega$. The protected gates are not computationally universal by themselves, but a scheme for universal fault-tolerant quantum computation can be constructed by combining them with unprotected noisy operations. We validate our analytic arguments with numerical simulations.

Figure 1

The 0-$\pi$ qubit. The energy $E\left(\theta \right)$ of a superconducting circuit is a periodic function with period $\pi$ of the phase difference $\theta$ between its two leads, aside from exponentially small corrections. The two basis states $\left\{\right|0\rangle ,|1\rangle \}$ of the qubit, localized near the minima of the energy at $\theta =0$ and $\theta =\pi$, respectively, are nearly degenerate.

Figure 2

Two-rung superconducting circuit underlying the 0-$\pi$ qubit. If $\sqrt{L/C}$ is large, $C_1$ is large compared to $C$, and $JC$ is not too large, then the circuit's energy is a function of the combination of phases $({\theta }_2+{\theta }_4)-({\theta }_1+{\theta }_3)$, aside from corrections that are exponentially small in $\sqrt{L/C}$.

Figure 3

The circuit for the 0-$\pi$ qubit is obtained from the circuit in Fig. 2 by twisting one of the rungs and connecting the leads, thus identifying ${\theta }_2$ with ${\theta }_4$ and ${\theta }_1$ with ${\theta }_3$. In addition, another large capacitance is added to further suppress tunneling events that change ${\theta }_2-{\theta }_1$ by $\pi$.

Figure 4

A phase gate can be applied to a qubit by coupling it to a Josephson junction, but the gate is not protected against pulse errors and other noise sources.

Figure 5

A protected phase gate is executed by coupling a qubit (or a pair of qubits connected in series) to a “superquantum” $LC$ circuit with $\sqrt{L/C}\gg 1$.

Figure 6

An effective Josephson junction can be tuned by adjusting the flux $(\eta /2\pi ){\Phi }_0$ inserted in a circuit containing two identical junctions.

Figure 7

The profile of the tunable Josephson coupling $J\left(t\right)$ in the execution of the protected phase gate.

Figure 8

Coupling the qubit to the oscillator prepares a grid state in $\phi$ space, a superposition of narrowly peaked functions governed by a broad envelope function. The peaks occur where $\phi$ is an even multiple of $\pi$ if the qubit's state is $|0\rangle$, and where $\phi$ is an odd multiple of $\pi$ if the qubit's state is $|1\rangle$.

Figure 9

Ideal code words of the continuous variable code. The $\overline{Z}=\pm 1$ eigenstates $|0_C\rangle ,|1_C\rangle$, expressed in $\phi$ space, are uniform superpositions of position eigenstates with $\phi$ an even or odd multiple of $\pi$, respectively. The $\overline{X}=\pm 1$ eigenstates $|{+}_C\rangle ,|{-}_C\rangle$, expressed in $Q$ space, are uniform superpositions of momentum eigenstates with $Q$ an even or odd integer, respectively.

Figure 10

Approximate code words of the continuous variable code. The code word $|0_C\rangle$, expressed in $\phi$ space, is a superposition of Gaussian peaks, each of width $\Delta$, governed by a broad Gaussian envelope with width ${\kappa }^{-1}$. The code word $|{+}_C\rangle$, expressed in $Q$ space, is a superposition of Gaussian peaks, each of width $\kappa$, governed by a broad Gaussian envelope with width ${\Delta }^{-1}$.

Figure 11

The estimated gate error $|{\eta }_{\varepsilon }|$ (on a log scale) as a function of the rotation error $\varepsilon$, for ${\kappa }^{-2}=40$.

Figure 12

The estimated gate error $|{\eta }_{\varepsilon }|$ (on a log scale) as a function of the rotation error $\varepsilon$, for ${\kappa }^{-2}=80$.

Figure 13

Blue diamonds show on a logarithmic scale the numerically computed diamond-norm deviation from the ideal phase gate, for ${\kappa }^{-2}=\sqrt{L/C}=80$, ${\Delta }^{-2}=\sqrt{J_{0}C}=8$, and ${\tau }_J/C=80$, as a function of the overrotation parameter $\varepsilon$. The solid green line is the analytic prediction from Eq. (86) for the overrotation error alone. The discrepancy for small $\varepsilon$ arises from corrections due to diabatic transitions and squeezing.

Figure 14

The minimum diamond-norm deviation of the protected phase gate from the ideal gate, as a function of the initial oscillator eigenstate $n$, for $\sqrt{L/C}=80$ and $\sqrt{J_{0}C}=8$. Results from numerical simulations are shown in blue, and the analytic prediction Eq. (125) is shown in red. The discrepancy arises from corrections due to diabatic effects and squeezing, which are neglected in the derivation of Eq. (125).

Figure 15

The minimum diamond-norm deviation of the protected phase gate from the ideal gate, as a function of the oscillator's anharmonicity parameter $\alpha =\lambda L\sqrt{L/C}$. Here, as in Fig. 13, $\sqrt{L/C}=80$ and $\sqrt{J_{0}C}=8$.

Figure 16

Logical cnot gate acting on two blocks of the repetition code, shown here for code length $n=3$. Ancilla qubits are prepared in the $X=1$ eigenstate $|+\rangle$, interact via cphase gates with the data qubits, then are measured in the $X$ basis. (These preparations, gates, and measurements are repeated several times, and the result is determined by a majority vote; the measurement repetition is not shown.) Finally, each qubit in the two input blocks is measured in the $X$ basis, the results are decoded by a majority vote in each block, and logical Pauli errors in the output blocks are inferred from the results.

Figure 17

Adiabatic switch. For $\sqrt{L/C}\gg 1$, the effective Josephson energy $J_{\mathrm{eff}}$ of the switch depends sensitively on circuit parameters.

Figure 18

The inverse effective capacitance of a Josephson junction as a function of $JC$.

Figure 19

The effective Josephson parameter of the adiabatic switch as a function of $JC$ for $\sqrt{L/C}=40$.