Creation of quantum error correcting codes in the ultrastrong coupling regime

We propose to construct large quantum graph codes by means of superconducting circuits working at the ultrastrong coupling regime. In this physical scenario, we are able to create a cluster state between any pair of qubits within a fraction of a nanosecond. To exemplify our proposal, creation of the five-qubit and Steane codes is numerically simulated. We also provide optimal operating conditions with which the graph codes can be realized with state-of-the-art superconducting technologies.

Figure 1

(a) Schematic of a flux qubit, denoted by the Josephson junctions 1, 2, and 3. By varying the frustration parameter $f_3$, attained by an applied magnetic flux passing through the loop composed of the ${\mathrm{JJ}}_4$ and ${\mathrm{JJ}}_5$, the coupling between the qubit and the resonator can be tuned at will. This is a crucial aspect of our superconducting qubits design in order to realize cluster states in an ultrafast time scale. (b) A five-qubit cluster state. Each black bond represents the pairwise cluster state generation mechanism $U_{\mathcal{CZ}}^{ij}$ between $i\mathrm{th}$ and $j\mathrm{th}$ physical qubits (green circles) that are initially prepared in the $|+\rangle =\left(\right|g\rangle +|e\rangle )/\sqrt{2}$ state. (c) An array of five USC qubits embedded in a resonator to obtain the five-qubit quantum error correcting code.

Figure 2

Coupling coefficients (a) $c_x$ and (b) $c_z$ in Eq. (2) as a function $\alpha$, that is the size of junction ${\mathrm{JJ}}_3$, and the frustration parameter $f_1={\varphi }_1/{\Phi }_0$. In this simulation we have considered $E_J/h=221$ GHz and the capacitive energy of junctions as $E_C=E_J/32$. (c) Fidelity of achieving the desired controlled-phase gate $U_{\mathcal{CZ}}$, in the presence of nonzero transversal coupling strength $c_x$, while the red solid line, the black solid line, the dot-dashed line, the dashed line, and the dotted line represent the case when the resonator field is a thermal state at 15 mK, a vacuum state, a coherent state with amplitude $\gamma =0.25$, a coherent state with $\gamma =0.5$, and a coherent state with $\gamma =1$, respectively. (d) The enlarged figure of (c) shows the red and black solid lines overlap each other, indicating that the vacuum and the thermal states behave the same when considering two bosonic modes.

Figure 3

A ten-qubit cluster state emerging from the Steane code after appropriate projective measurements on the physical qubits $q_8,q_9$, and $q_{10}$ (orange circles). Each black bond represents the pairwise cluster state generation mechanism $U_{\mathcal{CZ}}^{ij}$ between the $i\mathrm{th}$ and $j\mathrm{th}$ physical qubits (green circles) that are initially prepared in the $|+\rangle =\left(\right|g\rangle +|e\rangle )/\sqrt{2}$ state.

Figure 4

Fidelity between the desired Jaynes-Cummings ground state $|{\psi }_{\text{JC}}\rangle =|gg\rangle \otimes |00\rangle$ and the instantaneous state $\left|\psi \right(t)\rangle$ during an adiabatic switch-off process, provided that an initial state of the evolution is $|{\psi }_G\rangle$, ground state of the quantum Rabi model. In another words, $\mathcal{F}=|\langle {\psi }_{\text{JC}}|\psi \left(t\right)\rangle {|}^2$ is plotted against $g/\omega$.

Figure 5

Monte Carlo simulation results. Fidelity of (a) the five-qubit code, where the average fidelity value is taken over 5000 runs, and (b) the Steane code, where the average fidelity value is taken over 1000 runs, is plotted against single-qubit gate error probability $p_1$ and two-qubit gate error probability $p_2$.

Figure 6

(a) A coplanar waveguide resonator (CWR) of length $L$ interrupted by $N$ identical and uniformly distributed flux qubits. (b) Lumped-element circuit model for a portion of the CWR, encompassing flux qubit $F^j$. (c) Schematic of the flux qubit. Numbers 1–6 with a cross sign label Josephson junctions, arrows refer to voltage drop between any two nodes, and ${\Phi }_{1,2,3}$ are external magnetic fluxes passing through each loop. Here, $\varphi$ means a node variable and $\tilde{\varphi }$ means a branch variable.