Four-state discrimination for a pair of spin qubits via gate reflectometry

Abstract

Single-electron spin qubits defined in quantum dots are used as building blocks of a semiconductor-based quantum computer. Readout in a scaled-up version of such a quantum computer is expected to rely on the Pauli Spin Blockade (PSB) mechanism. A desired functionality of PSB readout is that it reveals two bits of information on the two spin qubits that are involved in the process, such that the four computational basis states can be discriminated. In this work, we propose and quantitatively analyze an experimental procedure, based on gate reflectometry, which enables this four-state discrimination in a single measurement. We provide an intuitive recipe to maximize the contrast between the quantum capacitances of the four basis states. Focusing on silicon double quantum dots equipped with a micromagnet, we quantify how amplifier noise and phonon-mediated relaxation influence readout fidelity. Our results highlight a realistic opportunity to mitigate the overhead of readout ancilla qubits in a spin-based quantum computer.

Figures (6)

• Figure 1: Four-state discrimination with a single quantum capacitance measurement. (a) Schematics of the double quantum dot (DQD) in the presence of an inhomogeneous magnetic field. Energy spectrum (b) of the DQD, and quantum capacitances (c) of the energy eigenstates, as functions of detuning, for $t_0/h = 3.35$ GHz. Internal magnetic fields $\textbf{B}_L$, $\textbf{B}_R$ in the two dots are different, and emerge as combinations of the homogeneous external magnetic field $B_\mathrm{ext}$ and the inhomogeneous micromagnet field. Magnetic-field parameters, related to $\textbf{B}_L$ and $\textbf{B}_R$ via Eqs. \ref{['eq:Bs']}, \ref{['eq:Ba']}, \ref{['eq:Bsa']}: $B_s = 171.8$ mT, $B_{a\parallel} = -2.5$ mT, and $B_{a\perp} = 9.8$ mT. (d) Quantum capacitance contrast $\Delta C$ (gray), and four-state assignment fidelity $\mathcal{F}$ (blue), the latter computed with amplifier noise strength of $\sigma_C = 1$ fF. Gray dashed line in panels b, c, d indicate the maximum point of the capacitance contrast $\Delta C(\varepsilon)$.
• Figure 2: Four-state discrimination and readout error in the presence of amplifier noise. Colored curve $P_{r(j)}$ is the probability density function of the quantum capacitance value inferred from a reflectance measurement performed on state $j$, in the presence of amplifier noise. Vertical dashed lines are separators used for maximum-likelihood inference, separating the intervals defined in Eq. \ref{['eq:intervals']}. Means of the probability density functions correspond to the capacitances in Fig. \ref{['fig:setup']}(c) along the dashed vertical line: $C_{r(j)}=\{-15.6, 0, 9.0, 17.7\}$ fF. Amplifier noise strength: $\sigma_C = 2.5$ fF. The readout error is determined by the overlaps between the neighboring Gaussians.
• Figure 3: Maximizing the capacitance contrast for a fixed magnetic-field configuration. Energy levels (a,c,e) and quantum capacitances (b,d,f) corresponding to the four lowest-energy states, as functions of detuning $\varepsilon$, for different values of tunneling energy $t_0$. Magnetic-field parameters: $(B_{a\parallel},B_{a\perp },B_s) = (-2.5, 5, 171.8)$ mT. (a,b) Misaligned lower and upper singlet-triplet anticrossings, at $t_0/h = 3.72$ GHz. (c,d) Aligned anticrossings, at $t_0/h = t_\mathrm{align}/h = 3.37$ GHz. (e,f) Slightly misaligned anticrossings, at $t_0/h = 3.41$ GHz, which maximizes the capacitance contrast at the detuning value marked by the dashed vertical line.
• Figure 4: Maximizing the capacitance contrast in the detuning-tunneling parameter plane. (a,b) Detuning dependence of the quantum capacitances of the four states to be discriminated, for magnetic-field parameters $(B_{a\parallel},B_{a\perp}, B_s) = (-2.5, 3.0, 171.8)$ mT. Tunneling energy: (a) $t_0/h = 3.399$ GHz and (b) $t_0/h =3.354$ GHz, corresponding to $t_0 = t_\mathrm{align} \pm \delta t$, respectively. Solid: exact numerical result. Dashed: perturbative result (e.g., Eq. \ref{['eq:C1_smallBperp']}). (c) Detuning and tunneling dependence of the quantum capacitance contrast $\Delta C$, for the same magnetic-field parameters. Exact, numerical results are shown. Panel (a) [(b)] corresponds to the upper [lower] horizontal cut of panel (c) denoted by the thin dashed line. Horizontal thick dashed line indicates $t_\mathrm{align}/h = 3.377$ GHz. Diagonal thick dashed line is the line containing $(\varepsilon,t_0) = (0,t_\mathrm{align})$ and the perturbatively determined location of optimal capacitance contrast $(\varepsilon,t_0) = (\varepsilon_\mathrm{opt},t_\mathrm{align}+\delta t_\mathrm{opt})$ (see Eqs. \ref{['eq:optimaldeltat']},\ref{['eq:optimaldeltaepsilon']}). (d,e,f) Analogous to (a,b,c), with fourfold increased $B_{a\perp}$, i.e., $(B_{a\parallel},B_{a\perp}, B_s) = (-2.5, 12.0, 171.8)$ mT. Tunneling energy: (d) $t_0/h = 3.47$ GHz, (e) $t_0/h = 3.29$ GHz. Increasing $B_{a\perp}$ from (d) to (f) reduces the number of optimal readout positions (capacitance contrast maxima) from 4 to 2.
• Figure 5: Assignment infidelity in the presence of phonon-mediated relaxation and amplifier noise. (a) Red (left y axis): Average assignment infidelity $1-\mathcal{F}$ as function of detuning $\varepsilon$, along the diagonal dashed line of Fig. \ref{['fig:cq-perturbation-density']}(f), at an amplifier noise level $c = 3.136\, \mathrm{fF} \sqrt{\mu \mathrm{s}}$ (see Eq. \ref{['eq:sigmactmeas']}). Blue dashed (right y axis): measurement time $t_\textrm{meas}$, set as the minimum decay time at 50 mK via Eq. \ref{['eq:tmeas']}. Gray solid line (y axis not shown): capacitance contrast $\Delta C$. Two infidelity dips appear, correlated with the two capacitance-contrast peaks. Difference of the depths of the two infidelity dips is explained by the difference of the relaxation rates. (b) Phonon-mediated downhill relaxation times among the four computational basis states at zero temperature. The $4\to 1$ relaxation paths are denoted by arrows; the slow (fast) path requires spin flip (no spin flip). (c) Two-electron DQD energy spectrum. The common x axis of the three panels corresponds to the detuning $\varepsilon$ along the diagonal dashed line of Fig. \ref{['fig:cq-perturbation-density']}(f).