Electron cascade for distant spin readout

The spin of a single electron in a semiconductor quantum dot provides a well-controlled and long-lived qubit implementation. The electron charge in turn allows control of the position of individual electrons in a quantum dot array, and enables charge sensors to probe the charge configuration. Here we show that the Coulomb repulsion allows an initial charge transition to induce subsequent charge transitions, inducing a cascade of electron hops, like toppling dominoes. A cascade can transmit information along a quantum dot array over a distance that extends by far the effect of the direct Coulomb repulsion. We demonstrate that a cascade of electrons can be combined with Pauli spin blockade to read out distant spins and show results with potential for high fidelity using a remote charge sensor in a quadruple quantum dot device. We implement and analyse several operating modes for cascades and analyse their scaling behaviour. We also discuss the application of cascade-based spin readout to densely-packed two-dimensional quantum dot arrays with charge sensors placed at the periphery. The high connectivity of such arrays greatly improves the capabilities of quantum dot systems for quantum computation and simulation.


Supplementary Note 1. SIGNAL IN CASCADE CSD
The signal for Fig. 2b in the main text was taken with the fourth dot and the sensor tuned such that all charge occupations result in clearly distinguishable signals. The signal for (1101) is higher than that for (0101), because the rightmost dot was close to the Fermi level for (1101), and thus only partially occupied. The signal for (1200) is higher than for (0200), because the signal for these occupations is from the high-voltage flank of a sensing dot Coulomb peak. The other relative signals are as intuitively expected, namely adding charges and bringing charges closer to the sensor both result in a reduced sensor signal.

Supplementary Note 2. OPERATING WINDOW FOR CASCADE READOUT
In order to get insight in the size of the operating window for PSB, CPSB and inter-dot CPSB (iCPSB), we start from the single-band Fermi-Hubbard Hamiltonian for the quantum dot array [1]: where i is the single-particle energy offset, n i = c † i c i is the dot occupation, and c ( †) i is the annihilation (creation) operator, U i is the on-site Coulomb repulsion, and V ij the inter-site Coulomb repulsion. For simplicity, we assume in what follows homogeneous Coulomb repulsion, thus U i = U , V i,i+1 = V , V i,i+2 = V , and V i,i+3 = V , and neglect tunnel coupling. The shifts of charge transition lines due to capacitive couplings, and the tuning of dot potentials for each of the different readout schemes, namely PSB, CPSB and iCPSB, are obtained by solving sets of constraints. Note that, because the interaction strength decays with distance, the size of the operating window for a charge transition only depends on the occupation of nearby dots. This implies that the sizes of the operating windows remain constant as the length of the cascade is extended.

Supplementary Note 3. SINGLE-SHOT HISTOGRAM
The single-shot histograms are modeled with [4] N with N tot the total number of single-shot repetitions, P S the average singlet probability over al single-shot outcomes, w bin the bin width, and n S and n T the probability density distribution for the singlet and triplet states respectively. The probability density distributions are modeled by noise-broadened Gaussians as and where the second term accounts for relaxation during the integration time, where µ S and µ T are the means and σ the standard deviation of the gaussians for the singlet and triplet density distributions respectively. T 1 is the triplet-singlet relaxation time obtained from an exponential fit to the averaged data as shown in the insets of Fig. 3, and t int is the integration time.
From the probability density distributions we obtain the uncorrected readout fidelities as with V T the signal threshold, and F avg = 1 2 (F S + F T ). These uncorrected readout fidelities include errors due to residual overlap of the histograms and relaxation during the integration time. We afterwards correct the readout fidelity for errors due to relaxation during the arming time, and errors due to excitation during the integration and arming time as described in the main text.

Supplementary Note 4. INTER-DOT CASCADE PAULI SPIN BLOCKADE
An alternative implementation for cascade-based readout in a quadruple dot is shown in Supplementary Fig. 1. The additional electron moves from the third dot to the fourth dot, thus the cascade involves an inter-dot transition. The signal is from the left flank of a Coulomb peak of the sensing dot. The signal changes of the two charge transitions now add up, thus a singlet state corresponds, as with Pauli spin blockade, to the peak at lower signal and a triplet state corresponds to the peak at higher signal.

Supplementary Note 5. THEORY ON CASCADE SPEED AND SUCCESS PROBABILITY
In order to assess the scalability of the cascade-based readout, we analyse the speed and adiabaticity of the movement of charges in the cascade. The speed of the cascade is important since spin measurement must be faster than spin relaxation for achieving high-fidelity spin readout. Furthermore, spin readout must be faster than spin decoherence (with dynamical decoupling) for achieving fault-tolerance using feedback in quantum error correction. The adiabaticity with respect to charge is important when the Zeeman splitting is different between quantum dots. For different Zeeman splitting, the uncertainty in the electron position results in a phase error. On the left and the right, the electron in dot 3 or 4 stays in place when an electron moves from dot 1 to dot 2. In the center, an electron moves from dot 3 to dot 4 when the electron on dot 1 is pushed to dot 2. This corresponds to an inter-dot cascade effect. c Ladder diagram corresponding to the readout point R , illustrating the tuning of the dot potentials for the cascade Pauli spin blockade with inter-dot transition. Note that µ2,S(0210) is drawn below µ1(1110), but for the cascade it could also be above. d Histograms and fits of 10,000 single-shot measurements for iCPSB readout. The integration time is tint = 1.5 µs. Red and green solid lines correspond to the respectively triplet and singlet probability distributions, obtained from the fit to the histogram [4,5]. For iCPSB readout the singlet corresponds to charge occupation (0201) and the triplet to (1110). The inset shows the signal averaged over the single-shots and an exponential fit, with T1 = (75.0 ± 0.2) µs.

A. Co-tunnel cascade Pauli spin blockade
When the cascade is operated such that E(S(0201)), E(1100) > E(1101), then the cascade occurs via a co-tunnel process, and the cascade can be operated adiabatically. For a quantum dot array with length four and when the cascade involves a dot-reservoir transition, the relevant charge states are (1101), (0200), (1100) and (0201). The Hamiltonian in this basis is with t c,12 = √ 2t c,12 , and t c,4R the tunnel coupling between the rightmost dot and the right reservoir. Rewrite the Hamiltonian as with = − U + V + V + V , and δ = δ − U + V − 2V , where = 12 + 4 , and δ = 12 − 4 , with 12 = − 1 + 2 . By diagonalising this Hamiltonian, with approximation | | |V −V ± δ|, the co-tunnel coupling between the eigenstates that are predominantly (1101) and (0200)

B. Co-tunnel inter-dot cascade Pauli spin blockade
The analysis for a cascade involving co-tunnelling and only inter-dot transitions is very similar as for the co-tunnel cascade with a dot-reservoir transition. For cascade with an inter-dot transition, the relevant charge states are (1110), (0201), (1101) and (0210). The Hamiltonian in this basis is Rewrite the Hamiltonian as with = − U + 2V − V and δ = δ − U − V + V + V , where = 12 + 34 , and δ = 12 − 34 with ij = − i + j . By diagonalising this Hamiltonian, with approximation | | |V − 2V + V ± δ|, the co-tunnel coupling between the eigenstates that are predominantly (1110) and (0201) is t co = tc,12tc,34 ∆+

C. Controlled propagation
The cascade can be implemented such that the propagation is controlled by a sequence of gate voltages. As example, we consider the cascade Pauli spin blockade as described in the main text. First, conventional PSB is performed with µ 4 (0201) < 0. The electron on the fourth dot remains there. Next, gate voltages are changed such that µ 4 (1101) < 0 < µ 4 (0201). Then the cascade will propagate and the electron on the fourth dot will move to the reservoir. A similar scheme can be designed for inter-dot cascade PSB. For a longer cascade path, which involves more than two charge transitions, the propagation could be controlled at each transition. The motivation for controlled propagation becomes clear in the next subsection.

D. Longer cascade
We now discuss how the total duration of the cascade scales with the length of the cascade path for three different scenarios.
First we consider a cascade where all the charges are displaced in one single co-tunnel process. This involves N simultaneous tunnel events that are each energetically forbidden, but where the final state is lower in energy than the initial state. Then, for a chain with length 2N (with every other site occupied, except for the first two sites where the PSB mechanism is implemented), and homogeneous tunnel coupling, t c,ij = t c , and when the cascade involves only inter-dot transitions between neighbouring pairs, the co-tunnel coupling is [7] where for simplicity we only included inter-site Coulomb repulsion between nearest-neighbour sites. Charge adiabaticity will require increasingly slower gate voltage changes, because t c < V , thus t co decreases exponentially with increasing cascade length. When the adiabaticity condition is not met, the cascade can get stuck along the way. Next, for the sequential tunneling regime, thus with E(11LL . . . L) > E(02LL . . . L) > E(02RL . . . L) > . . . > E(02RR . . . R), with L = 10 and R = 01, the expected duration, assuming homogeneous tunnel rates, Γ, for the individual transitions is [8] τ ∼ N Γ .
For the sequential regime, charge adiabaticity need not be preserved. Charge tunnelling is here a stochastic process and the duration only scales linearly with the length. Note that there is an intermediate regime, which does not fully rely on co-tunnelling, but is also not completely sequential. Theory on this regime is beyond the scope of this work. Finally, both the charge adiabaticity and speed can be largely maintained in a cascade with controlled propagation. The total duration of the cascade increases linearly with the cascade length, similar to the sequential case, but now uncertainties in the timing of the charge movement can be suppressed, which is important when the Zeeman splittings are not homogeneous along the path.
Alternatively, co-tunnel, sequential, and cascades with controlled propagation could be combined, such that different parts of the cascade have different character.

E. Scaling of success probability
Here we consider the probability density function for the total cascade duration and its scaling with cascade length in the context of charge getting stuck. The probability density function for the cascade duration with N +1 transitions, which each have decay rate Γ, is given by the Erlang distribution [9] P N (t) = Γ N +1 t N N ! e −Γt .
The probability for the cascade to take longer than time t is with τ = Γt. We will show that the probability for a cascade with N + 1 transitions to take longer than time τ N decreases for increasing N given τ N > N N −1 τ N −1 , with τ i the time used to obtain the probability for the cascade of length i + 1. From the constraint on τ N it follows that τ N ≥ N τ 1 , thus we consider τ N = N τ 1 , because if the scaling holds for this τ N , then it will certainly hold for τ N > N τ 1 . The derivative of the probability with respect to N is Thus the probability decreases as function of cascade length, which shows that the time for a cascade to complete with a given probability scales sublinearly with respect to the cascade length. Alternatively formulated, the probability for a charge to get stuck and interrupt the cascade thus scales sublinearly with the cascade length.