Demonstration of Weight-Four Parity Measurements in the Surface Code Architecture

We present parity measurements on a five-qubit lattice with connectivity amenable to the surface code quantum error correction architecture. Using all-microwave controls of superconducting qubits coupled via resonators, we encode the parities of four data qubit states in either the $X$ or the $Z$ basis. Given the connectivity of the lattice, we perform a full characterization of the static $Z$ interactions within the set of five qubits, as well as dynamical $Z$ interactions brought along by single- and two-qubit microwave drives. The parity measurements are significantly improved by modifying the microwave two-qubit gates to dynamically remove nonideal $Z$ errors.

Figure 1

(a) False-colored picture of a seven-qubit lattice. The five specific qubits used for the plaquette experiment are highlighted and labeled as data qubits ($D_i$, $i\in [1,4]$) and syndrome qubit $S_1$. (b) Cartoon representation of a plaquette. The arrows represent the cross-resonance two-qubit gate directions between the syndrome qubit and the four data qubits, with the convention of pointing from the control to the target qubit. The frequencies of the two quantum bus resonators bus 1 and bus 2 are measured to be ${\omega }_{B1}/2\pi =6.562\mathrm{GHz}$ and ${\omega }_{B2}/2\pi =6.810\mathrm{GHz}$, respectively. (c) The $ZZZZ$- and $XXXX$-parity measurement quantum circuits.

Figure 2

Two-qubit gate (${\mathrm{CR}}_2$) randomized benchmarking results with an (a) ${\mathrm{ECR}}_{\text{two}\text{−}\text{pulse}}$ Clifford generator and (b) an ${\mathrm{ECR}}_{\text{four}\text{−}\text{pulse}}$ Clifford generator using the cross-resonance interaction with $D_2$ as the control and $S_1$ as the target qubit. The inset of the figures show the different ECR gate pulse sequence decompositions. We show the decay of the ground state population of the target qubit. Data for the control qubit give the same RB figure within our confidence interval.

Figure 3

(a) Pulse sequence for the CR Ramsey experiments. The CR resonance tone spans the entirety of the Ramsey delay time and is applied with the same amplitude used in the parity-check operations. Time domain (b) and Fourier transform (c) of Ramsey interferometry on $D_3$ with ${\mathrm{CR}}_4$ on (triangles) and off (circles) for the case where all qubits other than $D_3$ and the target $S_1$ are kept at the ground state. The CR Ramsey experiment along with equivalent ones for all eight $D_1$, $D_2$, and $D_4$ states translate into the $IIIZ$ and $ZIIZ$ strengths shown in Table 2 for ${\mathrm{CR}}_4$. We attribute the slower oscillations in the no CR Ramsey experiment, around 30 kHz, to charge noise affecting $D_3$.

Figure 4

(a) Histogram of ground (triangles up) and excited (triangles down) state calibrations and the $XXXX$-parity check result (circles) when the data qubits ($D_1{D}_2{D}_3{D}_4$) are prepared into the $|--+-⟩$ state. The dashed line is the threshold used to determine the state of the syndrome qubit, obtained from calibration pulses, where the assignment fidelity is calculated to be 97.4%. In this parity-check experiment, 79.4% of the results lie above the threshold line [$P(1)$]. (b) Complete gate sequence used to run the $XXXX$-parity checks. The pulses that prepare the data qubits in each basis state are not shown. The vertical lines represent each of the cross-resonance pulses from which our four-pulse cnot primitives are constructed, with squares placed on the target qubit in each case.