A fault-tolerant encoding for qubit-controlled collective spins

Abstract

Quantum error correction (QEC) is indispensable for scalable quantum computing, but implementing it with minimal hardware overhead remains a central challenge. Large spin systems with collective degrees of freedom offer a promising route to reducing the control complexity of qubit architectures while retaining a large Hilbert space for fault-tolerant encoding. However, existing proposals for logical gates and QEC in spin ensembles generally rely on inefficient higher-order interactions. Here we introduce spin-N-Cat codes, which encode logical qubits in superpositions of spin-coherent states and generalize bosonic Cat codes to the modular subspaces of permutationally symmetric spin ensembles. The code corrects collective and individual dephasing, excitation, and decay errors. We also present an efficient physical realization in central-spin systems, such as a quantum dot, where encoding, decoding, and a universal, fault-tolerant, and bias-preserving gate set are implemented using only first-order interactions. Numerical simulations demonstrate high logical fidelity under dephasing and excitation-decay noise, independent of noise bias, and that full QEC cycles are feasible with realistic microscopic parameters. For the large collective spins available in quantum dots, this translates into a substantial extension of coherence time. Our results establish spin-N-Cat codes as a scalable, hardware-efficient approach to QEC in spin-based quantum architectures.

Figures (9)

• Figure 1: (a) Left: Wigner distribution of the code states on the generalized Bloch sphere. These states are superpositions of three coherent states on the equator, separated by an angle of $\frac{2\pi}{3}$. Right: Their probability distribution in the collective basis lies in the $M \bmod{3} = 0$ subspace. (b) Effect of a dephasing-type error ($I_z$). The Wigner distribution broadens, but the logical states remain non-overlapping. (c) Effect of an excitation or decay error ($I_\pm$), which shifts the $M \bmod{3}$ value -- an error syndrome.
• Figure 2: (a) Physical system: the ensemble interacts with a single auxiliary spin, enabling nonlinearity and readout. (b) Available gates: unconditional qubit rotations $R_i(\theta)$, conditional rotations $R_{\{x,y,z\}}(\theta, M)$, ensemble rotations $\Theta(\theta)$, and flip-flop transitions $\Sigma(M, t)$. (c) The logical states $\ket{0}_L$ and $\ket{1}_L$ consist of spin-coherent components at different azimuthal angles. After a $\pi/2$ rotation about the $y$-axis, these states are mapped to different latitudes on the Bloch sphere, leading to distinct projections in $I_z$. (d) Probability distributions in the $I_z$-basis before and after a dephasing error. The rotated logical states have minimal overlap and remain distinguishable even under noise, enabling fault-tolerant conditional control.
• Figure 3: Sequence of operations for encoding and decoding the state of the electron into the codespace of the 6-Cat code. The joint electron–ensemble state is represented by a generalized Fresnel diagram: position indicates the collective state of the ensemble, color indicates the electron state (blue for ground, red for excited). Black circles mark states that begin with the electron in the ground state and, as required, end in the logical state $\ket{0}_L$. In step 3, we use $\theta = 2\arccos\!\left(\sqrt{\tfrac{2}{3}}\right)$.
• Figure 4: Fault-tolerant logical gates for the 6-Cat code. In all circuits, the top line represents the ensemble (logical qubit) and the bottom line the central spin (electron). (a) CNOT with ensemble as control. (b) CNOT with electron as control. (c) Logical Hadamard, implemented from a CNOT (ensemble control), free evolution under $A_c S_z I_z$, a corrective single-qubit unitary $U$, and a second CNOT. (d) Logical phase gate $P(\theta)$, realised by entangling the electron and ensemble, accumulating a controlled phase during free evolution, and mapping the phase back to the ensemble. All gates act identically on the codespace and all correctable error spaces, preserving $M \bmod \tfrac{N}{2}$ and thus the noise bias.
• Figure 5: Error Correction for the 6-Cat code. (a) Correction of collective dephasing errors. (I) The 6-Cat encoding is first mapped onto a 2-Cat encoding along the z-axis. Dephasing errors are now observable as reduced polarization of the coherent states. (II) A directional pumping process, implemented through alternating flip-flop and electron reset gates, restores the polarization and corrects dephasing errors. (b) Alternative method for correcting dephasing errors: The fault-tolerant CNOT gate transfers the information to the electron qubit, which is then used to encode the information onto a new ensemble. (c) Correction of decay or pumping errors: Selective flip-flop transitions bring the $M \bmod 3 = 1, 2$ subspaces back into the codespace, restoring the encoded information. (d) Collective vs. Local Errors: A local error alters the total angular momentum $\mathbf{I}$, but since the gate $\Sigma$ operates independently of $\mathbf{I}$, it corrects both collective and local errors. While a local error changes the total angular momentum of the codespace, all other properties remain unaffected.