Leakage-protected idle operation of a triangular exchange-only spin qubit

Abstract

We characterize the coherence of a triangular exchange-only (EO) spin qubit operated at a leakage-protected idle (LPI) point. The triangular geometry enables independent control of all three pairwise exchange interactions, and the LPI condition occurs when these couplings are turned on simultaneously and tuned to equal strength. In this configuration, the exchange interaction induces an energy gap $E_g = 3J/2$ that suppresses leakage from the computational subspace while leaving the qubit state unaffected. We develop procedures to calibrate the LPI point and measure $E_g$, and use these to characterize the qubit dephasing time over a broad range of gap energies. While operating with large always-on exchange couplings exposes the qubit to charge noise, we find that $\tilde{T}_2^*$ still exceeds that of conventional exchange-only spin qubits for $E_g/h < 60$ MHz. The precise control of simultaneous, all-to-all connected exchange demonstrated here presents a natural path towards improving the performance of EO qubits and also enables novel qubit encodings.

Figures (5)

• Figure 1: Simultaneous exchange coupling of three spins. (a) SEM image of a device similar to the one measured. (b) An isolated pairwise exchange interaction generates a qubit rotation about one of the axes $\hat{n}$, $\hat{m}$, or $\hat{z}$, whereas simultaneously activating multiple interactions generates a rotation about some vector sum of these axes. (c) Eigenergies of Eq. \ref{['eq:hamiltonian']} plotted as a function of $J_{1,2}/h$ with $J_{1,3}/h$ = $J_{2,3}/h=100$ MHz and $B = 0$. The orange curve corresponds to the $S\mathbin{=}3/2$ leakage quadruplet and the blue curves correspond to the two degenerate $S\mathbin{=}1/2$ doublets used to encode the qubit. With equal exchange couplings ($J_{1,2} = J_{1,3} = J_{2,3} = J$; denoted by the black vertical line), there is no qubit rotation, but an energy gap $E_g = 3J/2$ is induced between the $S\mathbin{=}1/2$ and $S\mathbin{=}3/2$ subspaces. The energy gap suppresses leakage from local magnetic field fluctuations, yielding an LPI point.
• Figure 2: Locating the LPI point. A 20 ns 3-$J$ pulse is interleaved within a 1-$J$ RB sequence (top diagram). One of the 3-$J$ virtual exchange gate voltages is held fixed, while the other two are swept. The red circle marks the location of the LPI point where $J=J_{1,2}=J_{1,3}=J_{2,3}\approx h\times 200$ MHz. Inset: Simulation assuming an exponential dependence of exchange energies on gate voltages. Data are distorted relative to simulation due to a complex cross-coupled dependence of exchange energies on gate voltages, but the LPI feature (a closed disc of high $P_{| 0 \rangle}$) is robust to this distortion.
• Figure 3: EO qubit leakage spectroscopy. Inset: EO qubit eigenenergies plotted as a function of $B$ with $J_{1,2}=J_{1,3}=J_{2,3}=J$. The green circles identify points where the $S\mathbin{=}3/2$ leakage subspace intersects with the $S\mathbin{=}1/2$ qubit subspace. When $B$ is positive (negative), the qubit is preferentially initialized in the $m_S=+1/2$ ($m_S=-1/2$) gauge indicated by the darker shaded blue lines. Thus, only two level crossings are observed in practice (larger pair of green circles). Main plot: Leakage probability $P_L$ measured as a function of $B$. The two peaks correspond to the level crossings circled in the inset. The solid line is a fit to a double Gaussian from which we estimate $E_g/h=3.6\pm0.2$ MHz. The gap in data near $B=0$ is due to the ambient magnetic field (see Supplementary Information).
• Figure 4: Free evolution at the LPI point. (a) Free evolution at the LPI point with $E_g\approx h\times4.5$ MHz. $P_{| 0 \rangle}$ and $P_{| + \rangle}$ measure the probability that the qubit, initialized in the subscripted state, remains unperturbed as a function of time. $P_{| 1 \rangle}$ and $P_L$ measure the probabilities that the qubit, initialized in $| 0 \rangle$, is found to subsequently occupy either $| 1 \rangle$ or a leaked state, respectively. At long times, $P_{| 0 \rangle,| 1 \rangle}\rightarrow 0.5$ --- consistent with a mixture over the two-dimensional qubit subspace. Inset: $P_L$ measured at 1 ms as a function of $E_g$, with the black line a power law fit that scales approximately as $1/E_g^2$. (b) A similar set of experiments, but now performed at the conventional (exchange-off) idle point. Here, $P_L\rightarrow0.5$, $P_{| 0 \rangle,| 1 \rangle}\rightarrow 0.25$ consistent with a fully mixed state over the full eight-dimensional Hilbert space. (c) Dephasing time ($\tilde{T}_2^*$) measured at the LPI point as a function of $E_g$. The red diamond corresponds to the data in (a) and the purple line marks the $\tilde{T}_2^*$ derived from (b). Initially, $\tilde{T}_2^*$ increases as a function of $E_g$ due to suppression of leakage, but then (at $E_g/h\approx 4.5$ MHz) begins to decrease due to the mounting effects of charge noise. The black lines show power law fits (with the exponent a fitted parameter) to the low- and high-$E_g$ behavior. All data was taken at $B=0$.
• Figure S1: Magnetic field calibration. (a) Eigenenergies of the three-electron system as a function of applied magnetic field strength with a fixed 1-$J$ exchange energy $J_{1,3}=J$ ($J_{1,2}=J_{2,3}=0$). (b) Measured leakage population as a function of magnet coil current while applying a fixed 1-$J$ exchange energy $J_{1,3}=J$. The two peaks correspond to the green circles in (a) where the $| 1, 3/2, \pm 3/2 \rangle$ leakage states intersect with the $S_{1/2}$ computational subspace. (c) Measured exchange oscillations used to extract the value of $J$ (in this case, $\approx$ 23 MHz as shown in the inset). (d) Using the value of $J$ measured in (c) and the location of the peaks measured in (b), we can calculate the magnetic field at the position of the dots as a function of the applied coil current (blue circles). The red curve is a fit to this data that takes into account a potential ambient magnetic field component perpendicular to the direction of the applied magnetic field.