Single hole spin relaxation probed by fast single-shot latched charge sensing

Hole spins have recently emerged as attractive candidates for solid-state qubits for quantum computing. Their state can be manipulated electrically by taking advantage of the strong spin-orbit interaction (SOI). Crucially, these systems promise longer spin coherence lifetimes owing to their weak interactions with nuclear spins as compared to electron spin qubits. Here we measure the spin relaxation time T1 of a single hole in a GaAs gated lateral double quantum dot device. We propose a protocol converting the spin state into long-lived charge configurations by the SOI-assisted spin-flip tunneling between dots. By interrogating the system with a charge detector we extract the magnetic-field dependence of T1 ∝ B−5 for fields larger than B = 0.5 T, suggesting the phonon-assisted Dresselhaus SOI as the relaxation channel. This coupling limits the measured values of T1 from ~400 ns at B = 1.5 T up to ~60 μs at B = 0.5 T. Research on spin qubit systems for use in quantum computational devices has recently focused on the use of hole spins rather than the conventional single electron spins. The authors report the spin relaxation time of a single hole by developing a novel spin-sensitive charge-latching technique using a GaAs gated double quantum dot device.

C oherence of solid-state spins is of crucial importance in the context of their utilization in quantum computation and communication [1][2][3] . This property is quantified by the spin relaxation time T 1 and the decoherence time T Ã 2 3 . While both of them are essentially material parameters, T Ã 2 can be extended dynamically by refocusing techniques 4 . On the other hand, 1/T 1 measures the spin relaxation rate from an arbitrary linear superposition to the ground state, by which the quantum information is lost irreversibly. Extending T 1 is thus essential. Here, solid-state holes offer an advantage, as they interact much weaker than the electrons with the nuclei in the crystal lattice 5 .
Two mechanisms of the spin relaxation process in quantum dots have been identified 3,6 . The first involves flip-flop interactions with nuclear spins of the crystal lattice and is active at magnetic fields B up to several mT. The resulting T 1 was measured to be 10-100 ns at very small fields for electrons in gated GaAs quantum dots 3 , rising to~70 μs at B = 100 mT 7 . It can be radically enhanced by moving to 28 Si samples where nuclear spins are absent 8 . However, GaAs hole spins have a competitive edge, since their hyperfine interaction strength in bulk was predicted 9-12 and measured [13][14][15] to be an order of magnitude weaker than that of the electrons. This suppression translates into improved values of T 1 , reaching 1 ms for holes in InGaAs samples 16,17 .
In the second mechanism, dominant at higher magnetic fields, the spin relaxation occurs via the phonon-mediated spin-orbit interaction (SOI). Crossover into this regime occurs when the spin Zeeman energy exceeds the hyperfine interaction strength 18 and is marked by a large increase of T 1 3 . In Si/SiGe dots, T 1 was reported to reach~3 s at B = 1-2 T 19,20 ; in GaAs systems at similar fields T 1~1 s 3,21 , whereas in InGaAs T 1~2 0 ms at B = 2 T 22 and T 1~1 ms at B = 5 T 23 . The trend follows the strength of the SOI, which is weaker in centrosymmetric materials such as Si, and strongest in III-V In-based systems. The importance of the phonon component is revealed in the dependence of T 1 on the magnetic field, as the increasing spin Zeeman energy tracks the increasing phonon density of states. The dependence of T 1 ∝ B −5 was predicted theoretically 24,25 and confirmed experimentally in GaAs quantum dots 3,22,23,[26][27][28] , revealing values of T 11 00 μs at B~10 T. For hole spins, theory also predicts a decrease of T 1 with increasing magnetic field, with T 1 ∝ B −5 for the Dresselhaus SOI, or T 1 ∝ B −9 for the Rashba SOI 29,30 . Moreover, the exact functional relationship depends on the structural properties of the dots, such as size, thickness, shape, and strain, characteristic for the strong SOI experienced by holes in solids [31][32][33] . However, there has been little systematic experimental analysis of T 1 in p-type quantum dots at higher magnetic fields. Measurements in Ge/Si samples suggest T 1 of hundreds of microseconds 34 to submicrosecond values 35 at B~1 T. In the Ge hut wire system, values of tens to a hundred microseconds were recorded for magnetic fields from 0.5 T to 1.5 T 36 . T 1 times of several microseconds in hole Si Complementary Metal Oxide Semiconductor (CMOS) devices 37 , and several nanoseconds in gated GaAs samples at similar fields 38 have been reported. Two-axis coherent control of the hole spin has been demonstrated both in the Ge hut sample 39 and in the Si dots 40 . With the exception of the Si device, however, all measurements mentioned above involved systems with several holes.
Here we report on the detailed experimental study of the single-hole T 1 as a function of the magnetic field. We show that, in the range of B = 0.5-1.5 T, T 1 changes from 60 μs down to 400 ns along the power law of T 1 ∝ B −5 . This result, in agreement with theory 29,30 , reveals the dominance of the Dresselhaus SOI in spin relaxation. Although our values of T 1 are lower than those for electrons in GaAs at similar fields, extrapolation to very small magnetic fields is consistent with previous optical measurements 16,17 , showing holes to be superior (T 1 potentially in tens of milliseconds at B~100 mT) owing to the reduced hyperfine interactions. We also report on the development of a technique to read out a single-hole spin qubit in one quantum dot by converting the spin state into a latched charge state in a secondary dot. Charge-latching techniques involving spin-to-charge mapping have been applied to a variety of qubits [41][42][43][44] . In the gated electronic double dot, the latching technique proposed by our group 42 involved the mapping of the spin of a singlet-triplet qubit onto the charge states differing by one electron charge, rather than by the charge distribution of different spin states. The technique relies on a long-lived (latched) nature of the ground and excited charge states allowing for a high-contrast charge sensor readout. The latched technique was used to demonstrate single-shot high-fidelity spin measurements 44,45 and to extend the sensitivity of the spin detection at large magnetic-field gradients 46 . The spin-to-charge conversion allows measuring T 1 by fitting the leakage current in the Pauli-blockaded state to a theoretical formula describing a relaxation cascade in the electron 3,19,27,47 or hole systems 34,35,37,38 . However, this technique cannot be used directly for our hole GaAs dot because of the strong spin-flip tunneling due to the SOI 48,49 , resulting in the suppression of the spin blockade. Another approach utilizes the spin-selective time-resolved tunneling to leads 20,23,26,28 ; however, the charge detection in our system is not sufficiently fast for high-fidelity sub-microsecond measurements. Instead, we propose a method of projective measurement of a single-hole spin utilizing fast resonant tunneling between dissimilar spin states in adjacent dots. This spin-flip tunneling is much weaker in electronic GaAs quantum dots but was recently reported to be strong in hole quantum dots in the single-hole 49 and two-hole regime 48 . We convert short-lived spin states into long-lived metastable charge states utilizing spin-selective, spin-flip resonant tunneling of a single hole between the dots of the lateral double dot. This enables single-shot measurements of spin states of a relaxing hole with timescales much faster than could be achieved by nonlatching methods.

Results
Double-dot gated device confining a single hole. Our lateral double quantum dot (DQD) was defined by electrostatic gates on the surface of an undoped GaAs/AlGaAs heterostructure [48][49][50] . Figure 1a shows the gate layout of the device, with the two QD potential minima denoted as 1 and 2, respectively. The star denotes the DC-biased quantum point contact (QPC) utilized to measure the charge configurations of the system. The device is operated in the single-hole regime, i.e., with gate voltages V L and V R chosen to correspond to the regions of stability of the (N L , N R ) = (1, 0), (0, 1), or (0,0), as shown in Fig. 1b (N i denotes the occupation of the ith dot). The magnetic field B = [0, 0, B] is applied in the direction perpendicular to the sample surface. Owing to a strong hole confinement in the growth direction of the system, the heavy and light hole subbands are separated by a substantial gap (~10 meV), resulting in a nearly pure heavy-hole character of the confined carriers. This is evidenced by a strong gfactor anisotropy, leading to a nearly zero effective g factor for the in-plane magnetic field, with g* ≈ 1.35 for the out-of-plane field 48 . However, the spin-orbit interaction is sufficiently strong to enable a spin-flip tunneling channel, with the characteristic matrix element being of the same order as that characterizing the usual spin-conserving tunneling process 48,49 .
Measurement protocol. We extract T 1 by probing the relaxation process of the excited state |L⇑〉 into the ground state |L⇓〉 of the charge configuration (1,0), where ⇑ (⇓) denotes the hole spin up (down). The magnetic field opens the Zeeman gap E Z = gμ B B between the two spin states, with μ B being the Bohr magneton and the effective hole g factor g = 1.35 in our system 48,49 . Since in our system the in-plane effective g factor is close to zero 48,49 , our protocol is only sensitive to the out-of-plane component of the magnetic field. We map the spin state of the left dot onto the long-lived charge state of the right dot, which plays the role of the memory register. Its state can then be analyzed using the charge sensor. The use of the right dot as a long-lived, latched memory register is enabled by isolating it from the right lead, i.e., raising the right tunnel barrier to prevent the direct tunneling between the (0, 0) and (0, 1) charge states. The tunneling rate Γ R between the right dot and the right lead can be measured in a transport measurement 49 , and in our system it was~2 Hz. On the other hand, the left dot is strongly coupled to the left lead (the tunneling rate Γ L~1 00 MHz). Moreover, the interdot barrier is set to enable the resonant tunnel coupling of order of~0.2 μeV 49 , giving the tunneling rate of~50 MHz. The rate of the inelastic interdot tunneling, occurring when the right and left dot levels are off-resonance, is estimated at Γ C~0 .5 MHz. By this arrangement, the charge state of the right dot is long-lived, while the left dot can be quickly emptied to the left lead by raising its energy above the Fermi level of the lead.
The left-dot spin states |L⇑〉 and |L⇓〉 are mapped respectively onto the charge states (0, 1) and (0, 0) of the latching memory register using the single-shot preparation and measurement protocol outlined in Fig. 2. The protocol involves applying a fivestep voltage pulse ΔV L (t) to the left gate along the profile shown in Fig. 2a. This pulse transfers the system among three points, T, R, and M, in the charging diagram of Fig. 2b, corresponding to the three phases of the sequence: transfer, relaxation, and measurement, respectively. The evolution of the state of the system at each voltage step is illustrated with diagrams c-j of Fig. 2.
The protocol begins with emptying the DQD, which requires clearing the latched memory. To this end, we visit the point R, in which both spin states of the left dot are lower in energy than those of the right dot, as depicted in Fig. 2c. By maintaining this alignment for 10 μs (i.e., long compared to the inelastic transfer rate Γ C ), we let the system relax into the (1, 0) charge configuration. Next, we pulse rapidly to the point M, placing the left-dot states at energy higher than those of the right dot or the Fermi level of the left lead, as shown in Fig. 2d. Owing to the asymmetry of the barriers, the hole is promptly ejected into the left lead and the system is cast in the charge configuration (0, 0).
In the third step, we pulse back to the point R, as shown in Fig. 2e and h. Here, a single-hole tunnel from the left lead into the left dot, occupying the |L⇑〉 state with the probability P L⇑ (t = 0), and the |L⇓〉 state with the probability P L⇓ (t = 0). This arrangement is maintained over time T R , during which the spin relaxation takes place. At the end of this step, the probability , it decays exponentially with the rate defined by T 1 . For further discussion, we consider two limiting cases of the spin state: P L⇑ (t = 0) = 0 (Fig. 2e) and P L⇑ (t = 0) = 1 (Fig. 2h).
The fourth step is the essence of our spin readout protocol. The system is positioned at the point T of the charging diagram, in which the spin-up state |L⇑〉 is resonant with the spin-down state of the right dot (the memory register), as shown in Fig. 2f, i. These two states are coupled by the spin-orbit tunneling process, in which the charge is resonantly transferred between the dots with a spin flip 48,49 . No tunneling occurs if the state of the left dot is |L⇓〉, as shown in Fig. 2f. However, if the left-dot state has a finite spin-up component, i.e., P L⇑ (T R ) > 0, this component will tunnel resonantly into the right dot (Fig. 2i). As already mentioned, the rate of this resonant tunneling process is~50 MHz. We maintain our system in this configuration for T T = 100 ns. After this time the spin-up charge probability density P L⇑ (T R ) is split into the density P L⇑ (T R + T T ) remaining on the state |L ⇑ 〉 and the density P R⇓ (T R + T T ) transferred onto the orbital |R⇓〉 of the right dot. The probability of occupation of the |L⇓〉 orbital, including any relaxation taking place during the transfer time, is now P L⇓ (T R + T T ). We note that the possible small differences of the value of gfactor from one dot to another do not influence this spin-to-charge mapping process, as they can be compensated for by an appropriate choice of the gate voltage in this phase.
Having mapped the hole spin onto the charge configuration, we now proceed to the latching and readout step. As shown in Fig. 2g, j, we pulse the system to the point M, positioning the leftdot orbitals above the Fermi energy of the left lead. This ejects any charge residing in the left dot, while the charge state of the right dot remains latched. Figure 2g, j show, respectively, the states of the latched memory register in the limiting cases of P L⇑ = 0 and P L⇑ = 1. The former case corresponds to the memory state of (0, 0), while the latter corresponds to the state (0, 1), and these two charge configurations are distinguished by the QPC charge detector with high fidelity: for the state (0, 0) [(0, 1)], we expect a high [low] QPC current I QPC . In reality, the measurement by the charge detector collapses the hole state, and detects the configuration (0, 1) with the probability P 01 = P R⇓ (T R + T T ). As we can see, the detection of this memory state is equivalent to the measurement of a finite spin-up component of the hole state at the end of the relaxation step. We note that the readout step takes place in the region of the charging diagram in which the groundstate configuration is (0, 0) (as indicated by the position of point M in the stability diagram in Fig. 2b). This makes the latched In 2g and j, this situation is reflected by the fact that the two states of the right dot are above the Fermi energy of the left lead. However, the readout can also be performed if the M point were in the (0, 1) stability region. This would occur if in panels g and j the lowest-energy level of the right dot was below the Fermi energy level of the left lead. In such case, the filled memory register would constitute the ground state of the system, whereas the empty memory register would be in a latched excited state. Thus, the choice of the readout position should not have any influence on the outcome of our protocol.

Discussion
Since the transfer stage is the key element of our protocol, we examine the overall performance of the measurement as a function of the voltage ΔV L at that stage. This voltage translates into an energy detuning ε between dot levels with the same spin. Figure 2f, i shows the system at ε = −E Z . In this study, we choose B = 1 T and map out all possible alignments of levels of the left and right dot, from ε < −E Z to ε > E Z , at the transfer stage. In this mapping, we preserve the overall shape of the pulse as in Fig. 2a, but move the three points, R, T, and M, along the line R-M toward the (0, 0) region (Fig. 2b). Figure 3a shows the memory state P 01 averaged over 1000 measurements for two different times T R = 100 ns (blue downward triangles) and 100 μs (red upward triangles) of the relaxation stage. We see a clear dependence of P 01 on the time T R , indicating that we are indeed sensitive to the evolution of the hole state caused by spin relaxation. However, the change ΔP 01 of that probability, obtained as the difference between the two traces in Fig. 3a, strongly depends on the detuning ε, as shown in Fig. 3b. In the following, we focus on three alignments of levels, corresponding to ε = −E Z , 0, and +E Z , and depicted schematically in Fig. 3c-e, respectively. In the first case, the spin-up level of the left dot is in resonance with the spindown level of the right dot, as also shown in Fig. 2f, i. As a result, only the spin-up component |L⇑〉 of the hole in the left dot can transfer into the right dot, while the component |L⇓〉 is lower in energy and therefore is blocked. Due to spin relaxation, the spin density P L⇑ (T R ) decreases as the relaxation time T R increases, resulting in a decrease of the probability P 01 , hence in a positive value of ΔP 01 . We find that at this detuning our protocol operates with maximal sensitivity. On the other hand, at the detuning ε = 0 (Fig. 3d) the same-spin levels of the dots are on resonance and therefore the charge transfer process is not spin selective. As a result, here ΔP 01 ≈ 0 (Fig. 3b), i.e, our protocol executed with ε = 0 is insensitive to the spin relaxation processes. Lastly, at the detuning ε = E Z (Fig. 3e), we again observe a resonant spin-selective alignment of states, in this case |L⇓〉 and |R⇑〉. Here, one would expect that our protocol should detect the opposite spin component |L⇓〉 of the left dot. This is indeed the case, as ΔP 01 < 0 close to that detuning, reflecting the relaxation induced increase of occupation of |L⇓〉 with time T R . However, at this alignment, the sensitivity is reduced. This is because during the pulse rise time the system passes across configurations ε = −E Z and ε = 0 undergoing, respectively, the spin-selective and spinnon-selective Landau-Zener tunneling process (see Supplementary Note 1). Also, at this alignment, multiple, non-resonant (inelastic) charge transfer channels are active and influence the state of the memory register. At detunings ε > E Z , the probability P 01 approaches unity for both relaxation times T R due to the details of the pulsing sequence, as discussed in the Supplementary Note 1. Next, we evaluate the fidelity of our spin measurement protocol at its maximum sensitivity. We define the fidelity of detection of the spin-up state F ⇑ in terms of the conditional probability P((0, 1)|L⇑), i.e., probability that the memory register holds a hole on condition that the left-dot state was |L⇑〉. Similarly, the fidelity of detection of the spin-down state is F ⇓ = P((0, 0)|L⇓). Based on the redistribution of probability densities after the transfer phase, we expect F ⇓ ≈ 1; however, a similarly high F ⇑ is unlikely due to the fact that the |L⇑〉 state is mapped onto a distributed charge configuration. We confirm these expectations by a Bayesian analysis of our results (see Supplementary Notes 2 and 3), giving F ⇓ = 0.99 and F ⇑ = 0.52, and yielding the maximum visibility V = F ⇑ + F ⇓ − 1 = 0.51. We find that the portion |L⇑〉 of the hole state after the relaxation phase is mapped in the transfer phase onto a roughly equal charge distribution in both dots, i.e., P L⇑ (T R + T T ) ≈ P R⇓ (T R + T T ). The fidelity F ⇑ can be improved by replacing the constant voltage ΔV L in the transfer phase of the protocol by a voltage pulse shaped to give P L⇑ (T R + T T ) ≈ 0, e.g., with techniques similar to the adiabatic passage 49,[51][52][53][54][55] . In the present version of the protocol, we optimized the duration of the transfer pulse for signal strength and range of magnetic fields. A longer transfer time reduces the upper limit of the magnetic field because it becomes comparable to T 1 .
A shorter transfer time, on the other hand, becomes comparable to the interdot resonance tunneling time (~20 ns).
In the unoptimized form, the protocol is expected to represent correctly the evolution of the hole state at the end of the relaxation phase as a function of T R , provided that the relaxation time is the only parameter being adjusted. In Fig. 4a, we show single-shot measurements of the QPC current I QPC at B = 1 T as a function of T R revealing the state of the memory register. For short T R , we find indications of both filled memory state (0, 1) (low QPC current), corresponding to the detection of the hole spin up, and empty memory state (0, 0) (high QPC current), corresponding to the detection of the hole spin down. On the other hand, for sufficiently long T R , we find the memory register only in the empty state. In Fig. 4b, we show I QPC obtained by Gaussian averaging of 1000 measurements for each T R , showing a clear exponential profile. We reveal this profile by fitting to the relationship with I 0 and I 0 + I 1 corresponding to the filled and empty memory register, respectively. The fit shown with the solid black line is obtained with T 1 = 2.8 μs for B = 1 T. In Fig. 4c, we show the dependence of the extracted relaxation time T 1 on the magnetic field. It follows the power law T 1 = A ⋅ B −N with A = 2.7 ± 0.3 μs and N = 4.4 ± 0.3 (solid green line). Theoretical predictions of this dependence indicate two very different exponents, depending on the nature of the SOI 29,30 . The Dresselhaus SOI, arising as a result of the lack of inversion symmetry of the crystal lattice, is expected to give N = 5, while the Rashba SOI, brought about by the asymmetry of the sample geometry, should result in N = 9. The red-dashed line in Fig. 4c shows the fit with N = 5, indicating clearly that the spin relaxation in our system is dominated by the phonon-assisted Dresselhaus SOI. The theoretical exponent N = 5 is the lowest-order estimate only. Our results exhibit a deviation toward somewhat shorter values of T 1 , particularly at lower magnetic fields. This is consistent with the recent optical measurements in self-assembled dots 56 , where the dependence of T 1 on the magnetic field is more complex than a power law, and at low fields the theory overestimates the T 1 times as compared with the experimental values. This is likely due to the structural details of the quantum dot, which become less relevant at higher fields, when the cyclotron radius is smaller than the dot diameter and the system becomes more isotropic. We note that the values of T 1 , ranging from~400 ns at B = 1.5 T tõ 60 μs at B = 0.5 T, are much longer than those measured to date in hole GaAs samples (e.g.~300 ns at B = 0.5 T in ref. 38 ), but shorter by an order of magnitude than those in Ge-based 34,35 or Si-based devices 37 . This is most likely due to the fact that centrosymmetric lattices such as Si do not exhibit the Dresselhaus SOI, leading to the dominance of the Rashba SOI in the spin relaxation mechanism. Also, our values of T 1 are about four orders of magnitude shorter than those in electronic GaAs dots at equivalent magnetic fields. Again, this is a consequence of much weaker SOI in the electronic systems 18,57-59 and a strongly elongated shape of our dot 48 , which is known to enhance the spin-orbit coupling 21 . However, extrapolation of our values of T 1 to very low magnetic fields promises superior lifetime of the hole spin over that of the electron owing to the much weaker hyperfine interactions. Indeed, the record value of T 1 = 57 s reported in GaAs devices confining electrons 60 is limited by the isotropic hyperfine interaction between the electron spin and that of the nuclei, and depends on the magnetic field as B −3 . The hole exponent of N~5 offers a more promising low-field scaling, and to date neither theoretical calculations 29,30 , nor optical measurements 16,17,56 suggest any higher-order mechanisms leading to its deterioration.
Moreover, the hyperfine interaction in GaAs (as in all A 3 B 5 materials) cannot be removed by isotopic purification. In our system, the hyperfine-mediated relaxation is not relevant, and the phonon-mediated spin-orbit relaxation is the dominant mechanism. The Dresselhaus SOI in A 3 B 5 materials is unavoidable. However, we propose that the hole T 1 could be further extended by engineering the phonon density of states to eliminate the phonon modes mediating the spin-flip transitions. Preliminary work toward that end has already been demonstrated for electronic dots in nanowires 61 .

Methods
Sample details. The experimental study was performed on a DQD fabricated from an undoped GaAs/Al x Ga 1−x As (x = 50%) heterostructure employing lateral splitgate technology [48][49][50] . A suitable DQD potential profile was defined by the deposited lateral Ti/Au gates. Holes were generated by a global gate deposited above the structure (not shown in Fig. 1a) separated by a 110-nm-thick Al 2 O 3 dielectric layer grown by an atomic layer deposition technique. Left and right plunger gates, labeled as L and R, respectively, were used to tune the hole potentials individually in each dot, while the central gate was used to adjust the interdot tunneling barrier. The sample was cooled down in a dilution refrigerator at the nominal lattice temperature of 60 mK. The effective hole temperature, measured from the temperature dependence of the width at half maximum of a Coulomb blockade peak, was~100 mK.  pairs running from the room temperature top Fisher connector to a 1-K pot, followed by 60-cm-long, 0.5-mm-diameter thermo-coaxes (SS outer, Ni-Cr inner conductor, MnO dielectric) from the 1-K pot to the mixing chamber. A homemade loom of copper wires was used from the mixing chamber to the sample carrier to simultaneously cool the sample positioned in vacuum in the center of the 18-T superconducting solenoid. For high-frequency lines, we used 0.085″-diameter silver-plated Cu-Be semi-rigid coaxial cables with Anritsu V101 series connectors running from the room flange to the 1-K pot. Anritsu 20 dB attenuators were installed and thermo-anchored at the 1-K pot. Superconducting Nb coaxial cables were used from the 1-K pot to the mixing chamber to minimize the thermo-load. Modified Anritsu V251 bias-tees were installed and thermo-anchored at the mixing chamber followed by Ø 0.085″ copper semi-rigid coaxial cables down to the sample socket. In principle, the described radio-frequency wiring allows to deliver pulses with a sub-nanosecond rise time 62 . However, in the current experiments, we were limited by the sample design. In particular, in order to generate holes, our DQD sample had a relatively large global gate with increased parasitic capacitance to the ground that limited the bandwidth in current experiments to 100 MHz, or 10 ns rise time for pulses. The charge state of the DQD was probed using a QPC charge sensor with a constant source-drain bias voltage of 100 μV. The measurement cycles were synchronized to the power line cycle by using the line trigger of the HP-3458A digital multimeter used to record the current readings. We independently exported the line trigger from our Tektronix TDS6154C oscilloscope and used it to trigger the arbitrary waveform generator. The rising edge of each power line cycle triggered both the pulse sequence and the reading. The charge state was measured by integrating the current I QPC over one full period of the cycle: 1/60 Hz = 16.6(6) ms. Our setup could be modified to perform measurements in shorter cycles, however not reaching the sub-microsecond range.
Range of magnetic fields. Our measurements were performed in the magneticfield range from~0.4 T to~1.6 T. The lower limit of the magnetic field is set by the condition of spin-selective resonant tunneling. For high protocol fidelity, the Zeeman gap between the left-dot levels has to be larger than the width of the resonance peaks shown in Fig. 3a. In our system, this occurs at B ≥ 0.4 T, which corresponds to the Zeeman energy E Z ≥ 30 μeV. The upper limit of the magnetic field is defined by the value at which the spin relaxation time T 1 drops below the tunneling time constants set in our experiment, i.e., the length of the transfer step of 100 ns plus the minimum initialization time of 100 ns. At B~1.5 T, these two times are of the same order as T 1~4 00 ns. However, both magnetic-field limits are more of technical than fundamental nature and can be extended by tuning the system parameters.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.