Logical operations with a dynamical qubit in Floquet-Bacon-Shor code

TL;DR

The Floquet-Bacon-Shor code is experimentally implemented on a superconducting quantum processor, generating an error-detected logical Bell state with a fidelity of 75.9% and highlighting the potential of Floquet codes for resource-efficient FT quantum computation.

Abstract

Quantum error correction (QEC) protects quantum systems against inevitable noises and control inaccuracies, providing a pathway towards fault-tolerant (FT) quantum computation. Stabilizer codes, including surface code and color code, have long been the focus of research and have seen significant experimental progress in recent years. Recently proposed time-dynamical QEC, including Floquet codes and generalized time-dynamical code implementations, opens up new opportunities for FT quantum computation. By employing a periodic schedule of low-weight parity checks, Floquet codes can generate additional dynamical logical qubits, offering enhanced error correction capabilities and potentially higher code performance. Here, we experimentally implement the Floquet-Bacon-Shor code on a superconducting quantum processor. We encode a dynamical logical qubit within a $3\times 3$ lattice of data qubits, alongside a conventional static logical qubit. We demonstrate FT encoding and measurement of the two-qubit logical states, and stabilize these states using repeated error detection. We showcase universal single-qubit logical gates on the dynamical qubit. Furthermore, by implementing a logical CNOT gate, we entangle the dynamical and static logical qubits, generating an error-detected logical Bell state with a fidelity of 75.9\%. Our results highlight the potential of Floquet codes for resource-efficient FT quantum computation.

Figures (4)

• Figure 1: Implementation of the FBS code. (a) Schematic of the quantum device for the FBS code, consisting of 9 data qubits (white circles labeled as $D_i$, where i = 1 to 9) and 12 ancilla qubits (orange circles for $Z$ ancillas and blue circles for $X$ ancillas). The four central qubits are alternately used as $X$ and $Z$ ancillas in different stabilizer rounds. Extra ancilla qubits (grey circles) are used for circuit compression and the construction of logical CNOT gate. Each pair of qubits is connected by a tunable coupler (H shape). (b) Illustration of stabilizer measurements for the BS code (left) and the FBS code (right). The FBS code features period-4 stabilizer measurements, during which the logical operators $X_{\rm d}$ (purple lines) and $Z_{\rm d}$ (green lines) for the dynamical qubit evolve across different rounds. In contrast, the logical operators $X_{\rm s}$ and $Z_{\rm s}$ for the static qubit remain fixed and are identical in both codes. Weight-2 Pauli measurements are used to check the parity of the corresponding data qubits in the $X$ (blue) and $Z$ (orange) bases. (c) Cumulative distributions of simultaneous errors for single-qubit gates, CZ gates, ancilla qubit readout, and data qubit idle (with dynamical decoupling) during ancilla readout are depicted. The median error rates are 0.07%, 0.97%, 1.58%, and 1.43%, respectively, as indicated by the vertical lines.
• Figure 2: Encoding, stabilization, and measurement of two-qubit logical states in the FBS code. (a) Simplified circuit for encoding, stabilizing and measuring two-qubit logical states. All physical qubits are initialized to the ground state before the circuit. Period-4 stabilizer measurements are alternately performed following the encoding circuit. (b) Encoding circuits for logical Pauli states $|+,0\rangle$ and $|1,0\rangle$. BS stabilizer measurements are included in the latter case, with the dynamical qubit state post-selected to be $|0\rangle$. The notation $D_{147}$ denotes the three data qubits $D_1$, $D_4$, and $D_7$, and so on. The CNOT symbol denotes transversal CNOT gates between data and ancilla qubit groups, subject to connectivity constraints. (c) Example logical measurement circuits after stabilizer $S_Z^{\rm D}$: FT $ZZ$ measurement (left) and nFT $XZ$ measurement (right). Logical operators are indicated within the dashed-line boxes. Measurement outcomes of data qubits are used to calculate the values of the $X$ (blue) and $Z$ (orange) stabilizers, enabling error detection. (d) LQST fidelities of all 36 two-qubit Pauli states. The 16 states $\{|0\rangle,|1\rangle,|+\rangle,|-\rangle\}^{\otimes 2}$ are fault-tolerantly prepared and exhibit higher fidelity, as shown on the left side (white background). The blue portions indicate the raw fidelities (i.e., without using syndrome data), while the green portions highlight the fidelity improvement by error detection (i.e., with syndrome data post-selected). (e) Experimentally measured (filled symbols) and simulated (open symbols) expectation values of $X_{\rm s}Z_{\rm d}$ operator versus stabilizer round, for the encoded states $|-,1\rangle$ (left) and $|-,0\rangle$ (right). The solid and dashed lines represent exponential fits. Each stabilizer round lasts 920 ns, comprising 720 ns for ancilla readout and 200 ns for single-qubit and two-qubit gates. The cyan and green colors correspond to the raw data and error-detected data, respectively. The dashed grey lines indicate the average error rates of physical two-qubit states. Due to the exponential decrease in the retained data during error detection, the analysis is limited to 16 rounds, with a total of 100,000 experimental shots. The insets show the error detection probabilities for the weight-6 stabilizers over 24 measurement rounds, with the final round calculated from the logical measurements.
• Figure 3: Logical gates on the dynamical qubit. (a) FT Pauli gates on the dynamical qubit. Transversal Pauli gates are applied to the data qubits and inserted between stabilizers $S_Z^{\rm B}$ and $S_X^{\rm C}$, as shown at the top. The final state is characterized using LQST. The state infidelities with error detection after applying the Pauli gates on different encoded states are shown below the circuit. For each dynamical qubit state, the infidelity is averaged over 4 static qubit states {$|0\rangle$, $|1\rangle$, $|+\rangle$, $|-\rangle$}. Error bars correspond to 95% confidence intervals. (b) Logical rotation gates on the dynamical qubit around the $Z_{\rm d}$-axis (left) and $X_{\rm d}$-axis (right). The measurement expectation values of the operators $X_{\rm d}$ (cyan), $Y_{\rm d}$ (green) and $Z_{\rm d}$ (orange) are shown as functions of rotation angles $\varphi$ and $\theta$. The gate circuits (shown at the top) for rotations around the $Z_{\rm d}$-axis ($X_{\rm d}$-axis) are inserted between stabilizers $S_Z^{\rm B}$ ($S_X^{\rm C}$) and $S_X^{\rm C}$ ($S_Z^{\rm D}$). The filled (open) symbols represent data with (without) error detection in the four stabilizer rounds and the final logical measurement. The dashed curves correspond to trigonometric fits.
• Figure 4: Logical CNOT gate between the dynamical and static qubits. (a) The CNOT circuit (top) is inserted between stabilizers $S_Z^{\rm B}$ and $S_X^{\rm C}$ in the overall circuit (bottom). For Bell state generation, the logical qubits are initialized to $|+,0\rangle$ during the encoding process. For LQPT, 16 linearly independent logical cardinal states: $\{|0\rangle, |1\rangle, |-\rangle, |{-}i\rangle \}^{\otimes 2}$ are encoded. (b) Error-detected logical density matrix of the generated logical Bell state, obtained with four stabilizer rounds. The logical state fidelity is 75.9% with error detection and 38.8% without error detection. The wireframes indicate the ideal values. (c) Extracted LPTM $R_{\rm exp}$ of the CNOT gate, with a process fidelity $F_{\rm p} = 80.2\%$.