Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit

We present a circuit design composed of two Josephson junctions coupled by a nonreciprocal element, the gyrator, whose ground space is doubly degenerate. The ground states are approximate code words of the Gottesman-Kitaev-Preskill code. We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron in a crystal, confined to a two-dimensional plane, and subjected to a strong, homogeneous magnetic field. We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction. We also propose realistic design parameters for an experimental realization, and we describe possible protocols to perform logical one- and two-qubit gates, state preparation, and readout.

Popular Summary

Building a universal quantum computer is a challenging task because of the inherent fragility of quantum bits. To deal with this problem, quantum error-correcting codes have been designed that could encode quantum information in a reliable way. Such codes typically rely on active protocols that require additional hardware and repeated measurements to detect errors, as well as further qubit operations to correct for these errors. Here, we design a circuit that implements a passive, hardware-efficient quantum error-correcting code that does not suffer from these issues.

Our design comprises a gyrator—a nonreciprocal two-port device that couples current on one port to voltage on the other—connected to two superconducting devices (Josephson junctions). The qubit is encoded into the twofold degenerate ground space of the system. This encoding waives the demand of active error detection and stabilization: The qubit is inherently protected against common types of noise. We also discuss possible ideas for logical single- and two-qubit gates, state preparation, and readout.

This work proposes a realistic circuit design that can be used for passive quantum error correction and can pave the way toward the realization of a fault-tolerant quantum computer.

Figure 1

Proposed hardware implementation of the GKP code. The circuit consists of two Josephson junctions coupled by a gyrator, highlighted in red.

Figure 2

Low-energy spectrum of the Hamiltonian in Eq. (21) as a function of the inverse flux ratio $q/p$ for a fixed value of $V/\hslash {\omega }_c=0.4$. The initially flat Landau levels at $E_n/\hslash {\omega }_c=n+1/2$ (obtained for $V=0$) are split into subbands dependent on $\mathit{k}$.

Figure 3

Hofstadter’s butterfly obtained by plotting the spectrum of the effective lowest Landau-level Hamiltonian $H_{\mathrm{LLL}}/V_0$, defined in Eq. (24), as function of the inverse magnetic flux ratio ${\mathrm{\Phi }}_0/\mathrm{\Phi }=q/p$. The two-dimensional GKP code space corresponds to the states of minimal energy at $p/q=1/2$. This point is marked by a red star in the figure.

Figure 4

Lowest ten eigenenergies of the Hamiltonian in Eq. (28) as functions of the confinement strength $\hslash {\omega }_0/V$ for $p/q=1/2$ and $V/\hslash {\omega }_c=0.4$. The gray dashed vertical line marks the value of $\hslash {\omega }_0/V=0.8$ that is experimentally relevant for our circuit proposal; see Sec. 6. Inset: Energy gap $\mathrm{\Delta }E=E_1-E_0$ between the two lowest eigenstates for the same values of $p/q$ and $V/\hslash {\omega }_c$. For $V/\hslash {\omega }_0\gg 1$, the energy gap decreases exponentially with $V/\hslash {\omega }_0$.

Figure 5

Expectation value of the LLL projector in the ground state of the Hamiltonian in Eq. (28) for different values of $V/\hslash {\omega }_c$ as function of $V/\hslash {\omega }_0$. As expected, the LLL projection becomes more accurate for smaller values of the energy ratios $V/\hslash {\omega }_c$ and $\hslash {\omega }_0/V$. We mark with a green circle the values of $V/\hslash {\omega }_c=0.4$ and $\hslash {\omega }_0/V=0.8$ that are experimentally relevant for our circuit proposal; see Sec. 6. For these values, we find $⟨{\psi }_0|{\mathrm{\Pi }}_{\mathrm{LLL}}|{\psi }_0⟩=0.981$.

Figure 6

Numerically obtained lowest-energy eigenfunctions ${\psi }_{H\pm }(X)$ of the effective Hamiltonian in Eq. (30) for $\hslash {\omega }_0^2/{\omega }_c{V}_0=0.05$, which corresponds to $\mathrm{\Delta }=0.25$; see Eq. (33). The quasidegenerate eigenfunctions are even and odd under Fourier transforming, respectively, and are well approximated by the analytic expressions (dashed light and dark gray lines, respectively) in Eq. (31).

Figure 7

Approximate GKP code words. We show with solid lines the linear combinations of the numerically obtained lowest-energy eigenfunctions ${\psi }_{H\pm }(X)$ of Eq. (30) that resemble the approximate GKP states ${\psi }_{0,1}(X)$; see Eq. (31). The dashed light and dark gray lines show the analytical results in Eq. (32), respectively. The figure is obtained by using the same parameters as in Fig. 6, i.e., $\hslash {\omega }_0^2/{\omega }_c{V}_0=0.05$, which corresponds to $\mathrm{\Delta }=0.25$. The widths of the individual peaks and that of the total envelope are the inverse of each other.

Figure 8

Absolute values squared of the wave functions (a) ${\mathrm{\Psi }}_{H+}(x_1,x_2)$ and (b) ${\mathrm{\Psi }}_{H-}(x_1,x_2)$. These wave functions approximate the low-energy eigenstates of the two-dimensional Hamiltonian in Eq. (21) and are constructed to be even and odd, respectively, under a $\pi /2$ rotation in the $x_1{x}_2$ plane. Note that the wave functions in the plot are not normalized. We remark that $|{\mathrm{\Psi }}_{H\pm }(x_1,x_2){|}^2$ are the Husimi $Q$ quasiprobability functions associated with the eigenstates ${\psi }_{H\pm }(X)$ [defined in Eq. (41)] of the projected Hamiltonian in Eq. (26).

Figure 9

Absolute values squared of the quasidegenerate ground state wave functions (a) ${\mathrm{\Psi }}_{H+}(x_1,x_2)$ and (b) ${\mathrm{\Psi }}_{H-}(x_1,x_2)$ of the Hamiltonian in Eq. (28), which includes a parabolic confinement potential. These functions are obtained by combining Eqs. (31) and (43) and are even and odd, respectively, under a $\pi /2$ rotation in the $x_1{x}_2$ plane. The wave functions here are normalized and are obtained by using $\mathrm{\Delta }=0.25$.

Figure 10

Circuit design implementing the Hamiltonian in Eq. (46), which approximates the GKP Hamiltonian.

Figure 11

Circuit implementation of the ideal logical $\overline{Z}$ gate with a current source. Assuming the system starts in the code subspace, the current source is turned on and kept constant at a value $I_1$ for a time $t_Z$ given in Eq. (54). A current source on the opposite port of the gyrator would instead implement a logical $\overline{X}$ gate.

Figure 12

Low-energy spectrum of the circuit shown in Fig. 10 as a function of the external flux ${\phi }_1^{\mathrm{ext}}=2\pi {\mathrm{\Phi }}_1^{\mathrm{ext}}/{\mathrm{\Phi }}_{0,s}$ for the parameters in Table 2 and with the other external fluxes set to zero. For these values, $|{\psi }_{H-}⟩$ is the second excited state. One can clearly notice the sweet spots when ${\phi }_1^{\mathrm{ext}}$ is an integer multiple of $\pi$.