Spin-Cat Qubit with Biased Noise in an Optical Tweezer Array

Abstract

Bias-tailored quantum error correcting codes (QECCs) offer a higher error threshold than standard QECCs and have the potential to achieve lower logical errors with less space overhead. The spin-cat qubit, encoded in a large nuclear spin-$F$ system, is a promising candidate for bias-tailored QECCs. Yet its feasibility is hindered by the difficulty of performing fast covariant SU(2) rotation with arbitrary rotation angles for nuclear spins and by a lack of noise characterization for gate operations in neutral atom platforms. Here we demonstrate single-qubit controls of ${}^{173}\mathrm{Yb}$ spin-cat qubits with nuclear spin $I=5/2$ in an optical tweezer array. We implement a covariant SU(2) rotation and non-linear rotations by optical beams and achieve an averaged single-Clifford gate fidelity of $0.961_{-5}^{+5}$. The measurement of the coherence time and spin relaxation time shows that the idling error becomes increasingly biased toward dephasing errors as the magnitude of the encoded sublevel $|m_F|$ increases. Furthermore, we benchmark the noise bias of rank-preserving gates on spin-cat qubits, demonstrating a finite bias of $18_{-11}^{+132}$, in contrast to the case of the two-level system in ${}^{171}\mathrm{Yb}$, which shows no bias within the experimental uncertainty. Our work demonstrates the feasibility of spin-cat qubits for realizing bias-tailored QECCs, paving the way for achieving hardware-efficient quantum error correction.

Figures (11)

• Figure 1: Overview of spin-cat state controls in an optical tweezer array.(a) Schematic illustration of the control beam geometries. The qubit is encoded in the nuclear-spin stretched states of the ${}^{173}\mathrm{Yb}$ atom ground state as $\ket{0}_{sc}=\ket{-5/2}$ and $\ket{1}_{sc}=\ket{+5/2}$. This encoding scheme suppresses hopping errors by leveraging the redundant states between the encoded qubit states. The spin-cat states $\ket{\pm}_{5/2}$ are generated as superpositions of the stretched states. We control single atoms trapped in an optical tweezer array by using a single-beam Raman technique. QB1 and QB2 are used for spin-cat state preparation and covariant SU(2) rotation, respectively, with different laser detunings. (b) Wigner function representation of the coherent manipulations for the spin-5/2 system. The spin-cat state rotations, denoted as $\hat{R}_x^{(cat)}(\pi/2)$, cyclically map the basis states: $\ket{0}_{sc}\rightarrow \ket{+}_{5/2}\rightarrow \ket{1}_{sc} \rightarrow \ket{-}_{5/2} \rightarrow \ket{0}_{sc}$. The central arrows indicate the application of the $\hat{X}$ gate and the $\hat{Z}$ gate.
• Figure 2: Spin-cat qubit manipulations. Time evolution of the $\ket{m_F}$ state population as a function of pulse duration for (a) a covariant SU(2) rotation, (b) a non-linear rotation, and (c) a $z$-axis rotation. All control lasers are applied in the horizontal plane with circular polarization. QB1 is utilized for the covariant SU(2) and $z$-axis rotations, and QB2 is used for the non-linear rotations. A bias magnetic field is applied orthogonally to the propagation axis of both QB1 and QB2 for the covariant SU(2) and non-linear rotations. In contrast, the magnetic field is aligned parallel to QB1 for the $z$-axis rotation. The simulated dynamics computed from the master equation (middle panels) show good agreement with the experimental data (top panels) for all rotations. The expectation value of the dynamics of the magnetization $\langle m_F \rangle$ (bottom panels) shows (a) a sinusoidal curve with a Rabi frequency of $2\pi\times 43.0\,\text{kHz}$, (b) a beat signal with five distinct frequencies, and (c) a sinusoidal curve with a Ramsey frequency of $2\pi\times 90.5\,\text{kHz}$. For the non-linear rotation, the spin-cat state is generated with a pulse duration of $85.1\,µs$. In the bottom panels, the solid red lines represent the simulated curves, and error bars represent $1\sigma$ confidence intervals.
• Figure 3: Clifford randomized benchmarking (CRB) for the spin-cat qubit.(a) Schematic illustration of coarse-grained (CG) measurement in the spin-cat encoding. For simplicity, we denote $\ket{^{1}S_{0}, m_F=k}$ as $\ket{k}$, where $k\in \{-5/2, -3/2, -1/2, +1/2, +3/2, +5/2\}$. CG levels are defined by the measured states: (Level 0) only the $\ket{0}_{sc}$ state; (Level 1) the $\ket{0}_{sc}$ and $\ket{-3/2}$ states; (Level 2) the $\ket{0}_{sc}$, $\ket{-3/2}$ and $\ket{-1/2}$ states. (b) Decay of the return probability after CRB circuits, measured using the CG measurements of level 0 (yellow triangle), level 1 (red square), and level 2 (blue circle). Solid curves are fits to the function $ap^m + b_i$, where $m$ is the circuit depth. For each level $i$, we use fixed noise floors $b_i$, which are determined by supplementary experiments on state-selective readout (see Appendix \ref{['method:SSR']}). (c) Averaged Clifford gate fidelities extracted from the CRB measurements. The fidelity improves as the CG level increases due to the redundancy in the qudit system. We obtain fidelities of $0.935_{-12}^{+10}$, $0.945_{-9}^{+8}$, and $0.961_{-5}^{+5}$ for level 0 (yellow), level 1 (red), and level 2 (blue), respectively. Shaded regions in (b) represent $1\sigma$-confidence intervals of the fit, and error bars in (b) and (c) represent $1\sigma$ confidence intervals.
• Figure 4: Coherence time $T_2^*$ and spin relaxation time $T_1$ measurements.(a) Ramsey oscillations of the spin-cat and kitten states at a short holding time. The spin-cat state $\ket{+}_{5/2}$ (blue, bottom) and kitten states $\ket{+}_{3/2}$ (red, middle), and $\ket{+}_{1/2}$ (green, top) are prepared, and their respective phase accumulations are measured. (b) Scaling of Ramsey frequency with the encoded sublevel $\abs{m_F}$. The Ramsey frequency extracted from (a) is proportional to the magnitude of the encoded sublevel $\abs{m_F}$, showing a slope of $1.651(6)~\text{kHz}$. (c) Decay of the Ramsey contrasts as a function of holding time $t$, where the $\ket{+}_{5/2}$ (blue, bottom), $\ket{+}_{3/2}$ (red, middle), and $\ket{+}_{1/2}$ (green, top) are initially prepared. The coherence time $T_2^*$ is extracted from a fitting function $\propto \exp\qty(-t/T_2^*)$, yielding $T_2^* = 94(10)\,\text{ms}$ for the spin-cat qubit. (d) Scaling of the coherence time $T_2^*$. The coherence time scales with $1/|m_F|$, with a coefficient of $251(21)~\mathrm{ms}$. (e) Scaling of the $T_1$ time. The $\ket{-1/2}$, $\ket{-3/2}$, and $\ket{-5/2}$ states are prepared, and the population of the $m_F>0$ states is measured after a varying holding time at a near-zero magnetic field. We observe that spin relaxation is suppressed for larger $\abs{m_F}$ encoding. Error bars in (a) represent $1\sigma$ confidence intervals, and shaded regions and error bars in (b-e) represent $1\sigma$-confidence intervals of the fit.
• Figure 5: Benchmarking noise bias characteristic for single-qubit gates.(a) Circuit diagrams for the $z$-basis and $x$-basis measurements in the noise-bias dihedral randomized benchmarking (DRB) protocol. (b) Detected non-dephasing errors (orange, right bars in each qubit) and dephasing errors (cyan, left bars in each qubit) extracted from DRB results. For the ${}^{173}\mathrm{Yb}$ atom, we perform the level 2 CG measurement in the last readout. (c) Noise bias comparison between a nuclear-pin qubit consisting of only two ground sublevels in $^{171}$Yb, and the spin-cat qubit with six Zeeman sublevels in $^{173}$Yb. The quantification of the noise bias, $\eta$, is extracted from the ratio of the dephasing error probability to the non-dephasing error probability. While the nuclear-spin qubit exhibits no noise bias ($\eta=0.8_{-0.3}^{+0.4}$), the spin-cat qubit has a significant and finite noise bias of $\eta=18_{-11}^{+132}$. Error bars represent $1\sigma$ confidence intervals.

Theorems & Definitions (2)

• Proposition 1
• proof