Bell-state tomography in a silicon many-electron artificial molecule

An error-corrected quantum processor will require millions of qubits, accentuating the advantage of nanoscale devices with small footprints, such as silicon quantum dots. However, as for every device with nanoscale dimensions, disorder at the atomic level is detrimental to quantum dot uniformity. Here we investigate two spin qubits confined in a silicon double quantum dot artificial molecule. Each quantum dot has a robust shell structure and, when operated at an occupancy of 5 or 13 electrons, has single spin-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document}12 valence electron in its p- or d-orbital, respectively. These higher electron occupancies screen static electric fields arising from atomic-level disorder. The larger multielectron wavefunctions also enable significant overlap between neighbouring qubit electrons, while making space for an interstitial exchange-gate electrode. We implement a universal gate set using the magnetic field gradient of a micromagnet for electrically driven single qubit gates, and a gate-voltage-controlled inter-dot barrier to perform two-qubit gates by pulsed exchange coupling. We use this gate set to demonstrate a Bell state preparation between multielectron qubits with fidelity 90.3%, confirmed by two-qubit state tomography using spin parity measurements.

From the single dot shell structure [1], one can try to predict which double dot occupations will lead to a single spin ½ qubit in each dot. But in order to confirm that the spin structure of the double dot can be extrapolated from single dot results, we obtain the spin ordering of the dots performing magnetospectroscopy. Traditionally, magnetospectroscopy is performed studying the shifts of chemical potentials of each dot as a function of the externally applied magnetic field. This assumes that the quantum dot is in diffusive equilibrium with a reservoir (same chemical potential). Such reservoir is assumed to be spinless, such that its chemical potential does not shift with magnetic field and the absolute shift in dot chemical potential with magnetic field can be assessed. In our system, the two dots are in equilibrium with each other, but all transitions conserve the total number of electrons in the double dot system (isolated double dot) -there is no reference reservoir, as shown in Supplementary Figure 1a. Therefore, only relative Zeeman shifts are observed.
The hypothetical field dependencies, assuming that the shell structure from Ref. Note that the lever arm we extracted from the slope in Supplementary Figure 1c is the sum of lever arm from Q1 and Q2, approximately α Q1 + α Q2 = 0.53 eV/V. Differences in lever arm α Q1 − α Q2 cannot be obtained from this method.

SUPPLEMENTARY NOTE 2: ADIABATIC INVERSION AND QUBIT OPERATION POINTS
In order to achieve single qubit EDSR control fidelities exceeding 99 %, compliant with the demands for quantum error correction in the surface code architecture, we must adjust the inter-dot detuning and J gate voltage such that we achieve the most efficient Rabi drive for both Q1 and Q2.
We perform an adiabatic spin inversion experiment by sweeping the microwave frequency applied to the EDSR gate electrode (in our case the Co magnet) at fixed power, such that when each of the qubit resonance frequencies f ESR is found, that spin is flipped with an efficiency given by the comparison between the sweeping speed and the Rabi frequency (limited by the spin relaxation time) [2]. This is observed as an increase in the probability of measuring an odd parity readout after preparing the even initial state |↓↓⟩, with an example shown in Figure 3a us to determine the resonance frequencies, as well as the region of high qubit fidelity, as a function of detuning and J gate voltage.
The colour scale in Figure 1d shows the extracted adiabatic inversion probability of each qubit at various detuning and J gate voltages. We interpolated these probabilities and plotted them again in Supplementary Figure 2a and b. At first glance, we notice that P odd is symmetric along the axis of detuning ε = 37 mV, implying that detuning the dots in either direction has the same effect on dot shape and spin behaviour.
The strategy to quickly calibrate the ideal operation points is to choose a few potential operation points on the 2D map where P odd shows a high adiabatic inversion probability, and measure the Rabi oscillation frequency at a fixed microwave power. We then choose the highest Rabi frequency point that meets some constrains. Firstly, for individual addressability by frequency modulation, the ESR frequency f ESR of both qubits should be at least 10 MHz apart, which means ∆V J < 20 mV or > 100 mV in Figure 3a. Also, we would like to minimise the exchange coupling during single qubit operation, which is achieved for ∆V J < −20 mV, setting J < 1 MHz as observed from Figure 2h. As a result, we are generally limited to the bottom half of the 2D map in Supplementary Figure 2a  Note that the operation point chosen for Q2 is not the one with the absolute maximum Rabi frequency, as we also would like to minimise gate voltage fluctuation when ramping between Q1 and Q2 logic gate operations. We observe a significant influence of ramping range on the final outcome of the Bell state preparation, but a thorough evaluation of this source of error is not warranted, since this relates to instrument limitations.

SUPPLEMENTARY NOTE 3: EXCHANGE OSCILLATION, COHERENCE AND Q FACTORS OF INTERACTING SPINS
The oscillations observed from Ramsey-like experiments in the main text Figure 2c, d are due to difference in precession frequency of the qubits in the period between π 2 -pulses. The difference in frequencies arises from both Stark shift, which is in the order of 10 MHz in our experiments, and exchange coupling J, between 100 kHz and 10 MHz. As a result, the total Ramsey frequency will be dominated by Stark shift, making the J-coupling effect difficult to observe without a high resolution scan of precession time. Therefore, we adjust the phase of the second π 2 -pulse to match a rotating frame of reference which is not the same as the qubit Q1 precession frequency f Q1 , but instead it is offset by a value f ref chosen to reduce the impact of the Stark shift to the oscillation observed in experiment. This reference frequency is adjusted ad hoc between different experiments in order to facilitate the extraction of the exchange coupling effect.
In the left panel of Figure 2, where the quantum dots are detuned, f ref is set to 10.5 MHz throughout the experiment. However, for direct J gate controlled CZ, the oscillation frequency varies across a range of 20 MHz, as shown in Figure  2f. In order to capture the oscillation data efficiently, we assign various f ref for each ∆V J targeting a shift of approximately −1 MHz from the CZ frequency f CZ (which could differ depending on whether the control spin is up or down).
In a qubit rotating frame, positive and negative phase accumulation will result in the same Ramsey oscillation if only a single measurement projection is taken. To determine the sign of ESR frequency shift, we repeat every Ramsey experiment with additional phase shift on the second π 2 pulse, in order to extract X,−X,Y,−Y projections of the qubit. Note that all four measurements are taken in a interleaved fashion to minimise the impact of quasi-static noise.

SUPPLEMENTARY NOTE 4: MEASUREMENT FEEDBACK
Low frequency noise is a major limitation for high fidelity operation of qubits in MOS devices [3]. An efficient approach to mitigate high amplitude noise that occurs in a sub-Hz scale is to recalibrate the most critical qubit control parameters periodically.
There are 10 parameters that require feedback throughout the experiments due to the intricate way by which the qubit operations are defined with different gate configurations targeting the optimisation of each qubit. These parameters are the SET Coulomb peak alignment, the readout level set by the dot gate, both qubit ESR frequencies, a total of five relative phases acquired when pulsing between operating points, and the exchange coupling controlled by the J gate. The SET feedback is used to maintain its high sensitivity during charge transition, while read level feedback is to ensure the readout is done within a Pauli spin blockade region for parity readout. SET and readout level feedbacks are performed with first order corrections, with a predefined target SET current. SET top gate voltage V ST and read level voltage (controlled via V G1 ) are updated based upon the difference between measured current and target current.
We adopt the ESR frequency tracking protocol from Ref. 3 in order to follow the resonance frequency jumps due to quasi-static noise such as hyperfine coupling with residual 29 Si nuclear spin in the silicon wafer, as well as low frequency electrical noise. We perform checks of each of the two resonance frequencies shown in Figure 3a independently every 10 measurement data points. If the spin rotation is unsuccessful at the assumed resonance frequency, we recalibrate the frequency with a series of Ramsey experiments.
In Figure 3a, the ESR frequency shift ∆f ESR is taken as 0 MHz at the microwave driving frequency that matches the resonance frequency of Q2 at voltage ∆V J = −70 mV, which is the operating point for Q2. At all the other operation points where ∆f ESR is non-zero, a phase will accumulate due to variations in precession frequency. Since our Clifford set requires 3 operation voltages, each with two phases for Q1 and Q2 to track, excluding the reference frequency f ESR = f Q2 , that results in 5 phase accumulations to recalibrate.
Although phase accumulation can be calculated by the extracted ESR frequency (∆f ESR ) and gate time t g , i.e. ϕ = ∆f ESR × t g , such method assumes an instantaneous step from one gate voltage to another, which in reality is limited by the 80 MHz bandwidth of the measurement cable, meaning during the ramp both qubits spend a nonnegligible amount of time in an intermediate voltage state, accumulating phases that are non-trivial to calculate, especially when the Stark shift is highly non-linear as seen in Figure 3a. Moreover, it is unclear whether the low frequency noise will affect the overall shape of the gate dependency of the resonant frequencies.
In quantum computing, all operations can be performed by a sequence of gates taken from a primitive gate set. The processing unit is fully calibrated if all the primitive gates are calibrated individually. Supplementary Table 1 shows the pulse sequences required to extract each of the 5 phases accumulated, each associate with certain qubit and primitive gates.  Figure 3a. Element at column Qn row ∆VJ corresponds to pulse sequence required to extract phase accumulated in qubit n when inter-dot barrier gate voltage is at ∆VJ. Rn represents a π 2 rotation around R-axis on qubit n, with R ∈ {X, Y}, while In means identity gate with ∆VJ equals to the voltage where single qubit operation is performed for qubit n.
Phase calibration is performed every ten measurements, after the ESR frequencies are updated. In each calibration, the corresponding pulse sequence from Supplementary Table 1 is applied with various phases ϕ for the last π 2 pulse with respect to the other pulses. The results are then fitted with a function P odd = A cos(2π(ϕ − ϕ ′ ))+b, where A and b are fitting constants related to the oscillation visibility and dark counts, while ϕ ′ is the phase accumulated from the target gate. Since this protocol may rely on multiple primitive gates in a sequence, the phase associated with each gate in Supplementary Table 1 has to be calibrated following a certain order , to ensure the phase extracted corresponds to one particular primitive gate only. These phases ϕ ′ will be used for compensation of unwanted accumulated phases as we apply the corresponding Clifford gates in the experiment.

SUPPLEMENTARY NOTE 5: EXCHANGE COUPLING FEEDBACK
The exchange coupling J may fluctuate between experiments due to low frequency electrical noise, which can be compensated by monitoring and recalibrating the CZ gate operation with a feedback protocol. The sampling rate of the arbitrary waveform generator (AWG) and microwave IQ modulation used here, 8 ns and 10 ns respectively, limit our gate operation times to the least common multiple of these two, τ CZ = 40 ns, or any multiples of that. This means that updating the CZ exchange time τ CZ is not accurate enough for high fidelity operation. Instead, we update the inter-dot barrier gate voltage V J , which compensates the change in J while leaving τ CZ unchanged.
The initial calibration method is as follows: two CZ identical sequences are performed, each one with an opposite control qubit state (spin down or up). We vary the readout projection angles ϕ and fit the parity readout probability to a sinusoidal wave similar to the case of the phase feedback, which we use to extract the phase offset ϕ ′ fit . The difference in phase accumulated in the control spin down and up cases are due to the composition of an exchange coupling from the CZ operation and from the extra X2 2 gate necessary for the control spin up calibration step. The latter can be compensated by re-scaling ϕ ′ fit to 0 at low exchange coupling regime. This experiment is repeated with various exchange gate voltages ∆V J , as shown in Supplementary Figure 3a  Upon choosing the desired value of J with the associated ∆V J , which should correspond to a δϕ ′ fit = π phase difference between the two initial states, a feedback protocol can be implemented to recalibrate J periodically. The feedback protocol is similar to the initial calibration mentioned above, but optimised for speed by focusing on a smaller range of ∆V J , and the exponential fit used in Supplementary Figure 3c is replaced with a linear fit. With that, the value of ∆V J is updated using the fit in order to maintain the same exchange coupling strength J.
This exchange coupling feedback is performed after ten measurements, immediately after the phase calibration step. Note that the pulse sequence used in Supplementary Figure 3a is identical to the one in Supplementary Table 1. Therefore, the X1−CZ−X1 sequence is omitted from the phase calibration stage, but extracted from the subsequent exchange coupling feedback stage.
Supplementary Figure 4 is an example of a Bell state tomography experiment, with all ten feedback loops active, and the variation of the respective parameters over 40 minutes of laboratory time. The parameters that are calibrated only every ten measurements have larger gaps between data points.

SUPPLEMENTARY NOTE 6: TWO QUBIT TOMOGRAPHY WITH PARITY READOUT
A two qubit density matrix is a 4 × 4 matrix spanning a 4 2 − 1 = 15 dimensional space and requires 15 linearly independent projection measurements. Ref. 4 gives a detailed explanation on how to perform two-qubit state tomography using parity readout. Supplementary Table 2 lists the gate operation sequences adopted here for each of the 15 projection measurements, using a combination of primitive gates described in the main text.  In order to accurately estimate the fidelity of the control steps in preparing a Bell state, some post-processing techniques are applied to the outcome of the measured odd parity probability P odd corresponding to the 15 projections from Supplementary Table 2.
Firstly, we factor in the errors associated with state initialisation and measurement (SPAM error), by renormalising the parity readout probability of the two qubits for ZZ readout.
Next, we reconstruct the density matrix from the measurement data. Let E υ be the measurement outcome projector, ρ be density matrix, p υ be the measurement probability, where υ = 1...30 (notice that measurements of the projector P M N , where M, N ∈ {I, X, Y, Z}, produce not only probability p M N , but also p −M N = 1−p M N , so that 15 projections yield 30 probabilities). We define a matrix A as where ⃗ E † υ stands for the vectorised form of the projection E υ . Similarly, all elements of ρ can also be vectorised. This yields the relation: With matrix A constructed from our choice of measurement projection, and ⃗ p from measurement data. We then perform a (pseudo) linear inversion to estimate the density matrixρ.
Since the matrix computed numerically by linear inversion can be an unphysical state for a qubit (leading to a measured matrix ⃗ p that does not have the properties of a density matrix), a maximum likelihood technique is used to numerically estimate the density matrix [5] under several constrains. A legitimate qubit density matrix must be