Sensor dot readout
Readout of the exchange-only qubits is performed through PSB41, in which an electron either tunnels or remains confined, depending on its spin state. Nearby sensing dots, which function as single electron transistors (SET), are capacitively coupled to the quantum dot electrons and the conductance of SET will change depending on whether a charge tunnels or not. In this set-up, the change in the SET conductance is detected using a heterojunction-bipolar-transistor-based cryogenic amplifier42. The amplifier and SET are embedded in a lock-in detection circuit in which signals of around 50 μV rms at 1 MHz are biasing one ohmic of the SET and the other ohmic is connected to the input of a dual-stage, AC-coupled voltage amplifier. Readout performance in this first-generation cryoamplifier is limited by noise, voltage division and bandwidth set by the 100-kΩ shunt resistors.
Device fabrication
All measurements are performed on an Intel Tunnel Falls device that was fabricated using state-of-the-art high-volume manufacturing techniques. The quantum dots are electrostatically confined in a two-dimensional electron gas (2DEG) defined by a Si/SiGe heterostructure. To protect electrons trapped in the quantum dots from incurring noise owing to surrounding nuclear spins, we use 28Si, isotopically enriched to 800-ppm 29Si remnants, to grow the quantum well. Although valley splittings were not measured on this particular device, we extract a range of 50–140 μeV for the one-electron valley splitting on devices from the same 300-mm wafer. More details on device fabrication can be found in ref. 7 and an extensive discussion on device uniformity is presented in ref. 1.
Device tuning
A 2DEG is accumulated under gate electrodes P 1 –P 3 and P 10 –P 12 to facilitate access of the middle quantum dots to an electron reservoir. We tune quantum dots QD 4 –QD 9 to the (1,3,1,1,3,1) electron occupation (charge stability diagrams are presented in Extended Data Fig. 2a,f,k,p,u). The two exchange-only qubits Q 1 and Q 2 are encoded on dots QD 4 –QD 6 and QD 7 –QD 9 , respectively. QD 5 and QD 8 are populated with three electrons to increase the PSB readout window size that would otherwise be limited by valley splitting43; this makes readout more robust against charge drift and miscalibration. PSB readout is established on dot pairs QD 4 –QD 5 and QD 8 –QD 9 to minimize crosstalk owing to capacitive coupling of quantum dots in the quantum dot array. Extended Data Fig. 5 demonstrates the necessity of a two-lattice-site separation between quantum dot pairs used for PSB in the scenario in which both quantum dot pairs are pulsed simultaneously for readout. Charge sensing is performed using two sensing quantum dots (SD 5 for Q 1 and SD 8 for Q 2 ) that host many electrons; these are placed on the opposite side of the central screening gate. With this readout arrangement, we find that the sensitivity of SD 5 to QD 8 –QD 9 (performing PSB for Q 2 ) and of SD 8 to QD 4 –QD 5 (performing PSB for Q 1 ) is small enough to not be a concern.
Definition of exchange-only qubits
To define an exchange-only qubit, three spins in three quantum dots are needed13,14. The computational states of exchange-only qubits are defined as |0⟩ = |S⟩|↓⟩ (|0⟩ = |S⟩|↑⟩) and \(| 1\rangle =\sqrt{\frac{2}{3}}| {T}_{+}\rangle | \downarrow \rangle -\sqrt{\frac{1}{3}}| {T}_{0}\rangle | \uparrow \rangle \) (\(| 1\rangle =\sqrt{\frac{2}{3}}| {T}_{+}\rangle | \uparrow \rangle -\sqrt{\frac{1}{3}}| {T}_{0}\rangle | \downarrow \rangle \)), in which \(| S\rangle =\sqrt{\frac{1}{2}}(| \uparrow \downarrow \rangle -| \downarrow \uparrow \rangle )\), \(| {T}_{0}\rangle =\sqrt{\frac{1}{2}}(| \uparrow \downarrow \rangle +| \downarrow \uparrow \rangle )\) and \({| T\rangle }_{+}=| \uparrow \uparrow \rangle \). The other possible states are leakage states outside the computational subspace, which have the same readout signature as the |1⟩ state. For qubit manipulation, the quantum dot pair with which PSB readout is performed defines the z axis of the Bloch sphere of the qubits and rotations around this J z axis can be performed by turning on the exchange interaction between these quantum dots. The second axis of control is given by J n , positioned at 120° from the J z axis and can be controlled by turning on the exchange interaction between one of the quantum dots involved in PSB and the gauge spin (third quantum dot). We initialize the qubits using a charge-locking PSB readout sequence, recording the state of the system before qubit manipulation. Although this method is not scalable to larger systems as the fraction of accepted records is only 1/4N for N qubits, it allows reducing the set-up complexity. An alternative initialization approach uses deterministic initialization by means of a reservoir44. As we cannot discern the computational |1⟩ state from leakage states, only records indicating the computational |0⟩ state on both qubits can be used for data analysis. Although an external magnetic field is not strictly required for exchange-only qubit operation, as the gauge spin does not need to be initialized in a specific state, we apply a small 1-mT magnetic field to suppress nuclear spin dynamics25. For more details on the exchange-only qubit encoding, see refs. 13,14 and the Supplementary information.
Readout considerations for multi-exchange-only-qubit systems
Exchange-only qubits are encoded in a decoherence-free (or noiseless) subsystem of three spins in three quantum dots13,14. In the tuning chosen here, the state of Q 1 (Q 2 ) can be determined by performiang PSB readout on the QD 4 –QD 5 (QD 8 –QD 9 ) pair. Neither of the exchange-only qubit computational states are ground states or eigenstates of the three-spin system, which implies that the qubits can be brought into leakage states through initialization, excitation, relaxation or dephasing. In particular, the computational states will be mixed with the leakage states during idle operations on the timescale of the singlet lifetime \({T}_{2}^{* }\) (Extended Data Fig. 2). Experimentally, this implies that idle time, even for the initialized |0⟩ state, needs to be avoided. Similarly, any idle time introduced before readout will lead to information loss on the \({T}_{2}^{* }\) timescale, rather than the single-spin relaxation time scale T 1 relevant for LD qubits. In turn, this means that we cannot simply perform sequential readout of both qubits as the second qubit would decay, unless (1) the readout integration time is much shorter than \({T}_{2}^{* }\) (typically 2–3 μs in 800-ppm 28Si but the readout integration time used in this experiment is 18 μs per PSB readout) or (2) we introduce dynamical decoupling to extend the lifetime of the idle qubit45 or (3) we turn on a sufficiently large exchange such that the PSB pair Hamiltonian eigenbasis corresponds to the qubit readout basis46 or (4) we involve ancillary quantum dots and swap the PSB quantum dot pairs away from each other such that both can be read in parallel with individual sensors. In our device, the two neighbouring sensors share an ohmic contact and part of the accumulated 2DEG, both with substantial series resistance. This couples the two SET drain currents, which in turn prohibits the fully parallel readout of neighbouring qubits, even though they have their own sensors. We need to either implement one of the solutions listed above or find new techniques that would allow us to recover the complete two-qubit information.
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