Pauli Blockade in Silicon Quantum Dots with Spin-Orbit Control

Quantum computation relies on accurate measurements of qubits not only for reading the output of the calculation, but also to perform error correction. Most proposed scalable silicon architectures utilize Pauli blockade of triplet states for spin-to-charge conversion. In recent experiments there have been instances when instead of conventional triplet blockade readout, Pauli blockade is sustained only between parallel spin configurations, with $|T_0⟩$ relaxing quickly to the singlet state and leaving $|T_{+}⟩$ and $|T_{-}⟩$ states blockaded—which we call parity readout. Both types of blockade can be used for readout in quantum computing, but it is crucial to maximize the fidelity and understand in which regime the system operates. We devise and perform an experiment in which the crossover between parity and singlet-triplet readout can be identified by investigating the underlying physics of the $|T_0⟩$ relaxation rate. This rate is tunable over 4 orders of magnitude by controlling the Zeeman energy difference between the dots induced by spin-orbit coupling, which in turn depends on the direction of the applied magnetic field. We suggest a theoretical model incorporating charge noise and relaxation effects that explains quantitatively our results. Investigating the model both analytically and numerically, we identify strategies to obtain on demand either singlet-triplet or parity readout consistently across large arrays of dots. We also discuss how parity readout can be used to perform full two-qubit state tomography and its impact on quantum error-detection schemes in large-scale silicon quantum computers.

Popular Summary

Qubits, the building blocks of quantum computers, are of current interest as we approach the long-term goal of scaling up quantum processors. With the use of silicon, common in everyday classical computers, scaling up to millions of qubits will benefit greatly by leveraging well-established fabrication technologies used to commercially manufacture integrated circuits. The readout of these qubits is integral for the operation of quantum computers. In this work, the spins of two electrons are studied, with a single qubit being defined by each spin. Harnessing the Pauli exclusion principle, the state of this pair of spins can be determined, but in a noisy environment, this measurement method has intricacies. Here, we develop a theory describing how parameters in the system can affect these spins, determining the readout outcome.

This article focuses on work performed in silicon metal-oxide-semiconductor devices that can form quantum dots where single electron qubits reside. The work includes an experimental study of a device that can transition between two readout schemes (ways in which the spin state can be determined) that originate from the Pauli spin blockade - which are commonly referred to as singlet-triplet readout, and parity readout. This article shows how to vary the blockade characteristics to transition between the readout methods. This is explored theoretically, followed by examples of how the readout schemes are viable for scalable quantum computing.

Our work gives insight into how spin qubits can be scaled up. The parity readout technique is shown theoretically to be as useful for readout as the more traditional singlet-triplet readout. This means that we have more options in our quantum toolbox to adopt when considering scaling up silicon spin qubit systems.

Figure 1

Pauli spin blockade and latched readout. (a) False-colored scanning electron microscope (SEM) image of a device nominally identical to the one used in the experiments. The dashed line shows the location of the cross section of the device represented at the bottom of the image. (b) The charge stability diagram showing the electron occupancy in the device. The valence electrons in each dot are shown by the numbers highlighted in yellow; the actual electron occupancy is shown by the numbers highlighted in blue. The arrows show the different loading and unloading stages of the experiment and the symbols represent the different stages of the experiment. The circle represents the initialization point, which is beyond the region of the stability diagram shown here (the arrow pointing out of the figure represents this). (c) A diagram representing the different stages of the experimental process and how the Pauli blockade allows for spin readout. The decay of $|T_0⟩$ and $|T_{-}⟩$ triplets is shown in (d) as a function of wait time $\tau$ at the blockade region near the (1,1)-(0,2) transition. We normalize the probabilities of the decay of the states according to their initialization fidelity. (e) Energy diagram showing the Zeeman energy difference coupling the $|T_0(1,1)⟩$ and $|S(1,1)⟩$ states. The decay of $|S(1,1)⟩$ into $|S(0,2)⟩$ is shown by the arrow between the two states.

Figure 2

Correlation between Zeeman energy difference and the rate of lifting the triplet blockade. (a) The rate of lifting the triplet blockade (red squares) extracted from decay plots similar to Fig. 1 and is plotted as a function of in-plane magnetic field angle for $B=600$ mT. (b) The difference in Zeeman energies plotted as a function of in-plane magnetic field angle, experimentally determined with two different methods, ESR (blue squares) and $S-T_0$ oscillations (green squares). (c) The energy diagram of a double-dot system as a function of detuning, showing schematically the initialization and control strategy for each of the methods presented in (b) to find the Zeeman energies. The arrow shows the fast, diabatic plunge into (1,1), which initializes into $|S(1,1)⟩$ and allows for observation of $S-T_0$ oscillations. The two frequencies $f_{\mathrm{ESR}1}$ and $f_{\mathrm{ESR}2}$ in (c) are shown as dips in (d), which are the individual resonance frequencies of the electron spins in each dot. (e) The $S-T_0$ oscillations as a function of dwell time and in-plane magnetic field angle.

Figure 3

Numerical model for the blockade lifting. (a) For an initial state $|\psi (t=0)⟩=|T_0(1,1)⟩$, the probability as a function of time $p(t)=|⟨T_0(1,1)|\psi (t)⟩{|}^2$ is calculated numerically using Eqs. (A1), (A2), and (A4), including dephasing processes only. The blockade rate ${\mathrm{\Gamma }}_{\mathrm{blockade}}$ can be extracted from the decay (red dashed line). Comparison between all three plots shows that the decay rate is not monotonically dependent on the dephasing time $T_2^{\mathrm{charge}}$. (b) The extracted decay rates ${\mathrm{\Gamma }}_{\mathrm{blockade}}$ as a function of charge relaxation and dephasing calculated from the simulation for four values of $\mathrm{\Delta }{E}_Z$ (blue). The values of ${\mathrm{\Gamma }}_{\mathrm{blockade}}$ that correspond to $\mathrm{\Delta }{E}_Z$ from the experimental data (black dashed lines) are then used to extract the relaxation and dephasing times. These are $T_2^{\mathrm{charge}}=0.2$ ns and $T_1^{\mathrm{charge}}=0.15\mu \mathrm{s}$. For small values of $T_2^{\mathrm{charge}}$ and $T_1^{\mathrm{charge}}$, the blockade lifting is halted [37]. (c) The comparison of ${\mathrm{\Gamma }}_{\mathrm{blockade}}$ and $\mathrm{\Delta }{E}_Z$ for the experiment (black dots) against the numerical (squares) and analytical (solid line) theoretical decay processes adopting $T_2^{\mathrm{charge}}=0.2$ ns (red) or $T_1^{\mathrm{charge}}=0.15\mu \mathrm{s}$ (green). The blue stars indicate the sampled points used to match the relaxation and dephasing times in (b).

Figure 4

Validity limits of the analytical perturbative expression. Comparing ${\mathrm{\Gamma }}_{\mathrm{blockade}}$ obtained by the analytical expression in Eq. (A13) to the numerical calculations for different values of $\mathrm{\Delta }{E}_Z/J$, where the inset shows this on a larger range. The analytical approximation is shown to be valid for ratios $\mathrm{\Delta }{E}_Z/J$ between ${10}^{-2}$ and 2. The upper limit is inherent to the perturbative nature of the analysis, while the lower limit relates to the regime where the nonunitary evolution can no longer be treated by the conventional SW method and must be incorporated explicitly in the analysis.