Fault-tolerant preparation of arbitrary logical states in the cat code

Abstract

Preparing high-fidelity logical states is a central challenge in fault-tolerant quantum computing, yet existing approaches struggle to balance control complexity against resource overhead. Here, we present a complete framework for the fault-tolerant preparation of arbitrary logical states encoded in the four-legged cat code. This framework is engineered to suppress the dominant incoherent errors, including excitation decay and dephasing in both the bosonic mode and the ancilla via error detection. Numerical simulations with experimentally realistic parameters on a 3D superconducting cavity platform yield logical infidelities on the order of $10^{-4}$. A scaling analysis confirms that the logical error rate grows nearly quadratically with the physical error rate, confirming that all first-order errors are fully suppressed. Our protocol is compatible with current hardware and is scalable to multiple bosonic modes, providing a resource-efficient foundation for magic state preparation and higher-level concatenated quantum error correction.

Figures (4)

• Figure 1: Arbitrary state preparation with error-detectable quantum operation. (a) Initialization of a target state $\xi_0\ket{0}+\xi_1\ket{1}$ through a noisy evolution, where the infidelity $\overline{f}$ of the final state is proportional to the physical error rate $p$. (b) Initialization of a logical target state $\xi_0\ket{0_L}+\xi_1\ket{1_L}$ through a noisy error-detectable operation, where the first order error can be suppressed through post-selection. (c) Schematic of the 3D superconducting cavities (circles) platform that can realize error-detectable operations. The three chips at the bottom correspond to ancillary qubits coupled to individual cavities, while the top two chips represent ancillas coupled to both adjacent cavities.
• Figure 2: Error-detectable Pauli-$X_\mathrm{c}$ measurement. (a) Circuit for the unambiguous quantum state discrimination of a coherent state component $|\beta\rangle$, employing a selective bit-flip gate $S$. (b) Phase-space schematic of the state discrimination for the $|-\alpha\rangle$ component of the cat code using the circuit in (a). When the ancilla is measured in $|f\rangle$, the measurement terminates, yielding an accurate conclusive result. However, an outcome of $|e\rangle$ indicates an error occurred, leading to the discard of the state and termination. A $|g\rangle$ outcome implies either that the system did not collapse into the target component or that it suffered from error, thus yielding an inconclusive result requiring subsequent measurements. (c) Implementation of the Pauli-$X_\mathrm{c}$ measurement $\mathcal{M}_X$ via sequential measurement of the four coherent state components of the cat code. The sequence terminates with an $|f\rangle$ outcome. If all four stages yield $|g\rangle$, the measurement fails and state is discarded. (d) Implementation of $H_\mathrm{c}$ gate through the error-detectable $Z_\mathrm{c}(\theta)$, $ZZ_\mathrm{c}(\theta)$, and $\mathcal{M}_X$.
• Figure 3: Fault-tolerant preparation of the logical state $(\ket{0_\mathrm{c}}\ket{0_\mathrm{c}}+{e}^(i\pi/4)\ket{1_\mathrm{c}}\ket{1_\mathrm{c}})/\sqrt{2}$. (a) The corresponding quantum circuit for the preparation of the logical state, composed of the elementary gates shown in Fig. \ref{['Fig:UniversalCircuit']}. Here, $\mathcal{M}_\mathrm{P}$ is the parity measurement post-selecting the even parity states, and $\mathcal{M}_\mathrm{X}$ is the Pauli-X measurement of the cat state. (b) Logical infidelity (main panel) and success probability (inset) for the preparation of the logical state as a function of the amplitude $\alpha$. (c) Logical infidelity (main panel) and success probability (inset) as a function of the physical error rate factor at $\alpha=2.6$. The blue dots represent the result from the new protocol, while the pink line is a baseline result of a single cavity and a single ancillary qubit initialized with SNAP gate. The black dashed line is a power-law fit to the blue points, given by $(1.6+1.8s^{2.12})\times10^{-4}$. The fitted scaling is greater than 1, which confirms the suppression of first-order noise. The parameters of the simulation are shown in the Supplemental Material.
• Figure 4: Fault-tolerant preparation of the logical state $\ket{\psi_\mathrm{L}(\theta,\phi)}=\mathrm{cos}\theta\ket{0_\mathrm{c}}\ket{0_\mathrm{c}}+ie^{i\phi}\mathrm{sin}\theta\ket{1_\mathrm{c}}\ket{1_\mathrm{c}}$. (a) The quantum circuit constructed from the elementary operations shown in Eq. (\ref{['Equ:UniversalGate']}). (b) Logical infidelity (main panel) and success probability (inset) for the logical state $\ket{\psi_\mathrm{L}(\pi/6,0)}$ as a function of the cat-code amplitude $\alpha$. (c) Logical infidelity (main panel) and success probability (inset) for the logical state $\ket{\psi_\mathrm{L}(\theta,0)}$ as a function of the parameter $\theta$, simulated at the amplitude $\alpha=2.6$. Other parameters of the simulation are shown in the Supplemental Material.