Operating two exchange-only qubits in parallel

Abstract

Semiconductors are among the most promising platforms to implement large-scale quantum computers, as advanced manufacturing techniques allow fabrication of large quantum dot arrays¹. Various qubit encodings can be used to store and manipulate quantum information on these quantum dot arrays. Regardless of qubit encoding, precise control over the exchange interaction between electrons confined in quantum dots in the array is critical. Furthermore, it is necessary to execute high-fidelity quantum operations concurrently to make full use of the limited coherence of individual qubits. Here we demonstrate the parallel operation of two exchange-only qubits, consisting of six quantum dots in a linear arrangement. Using randomized benchmarking (RB) techniques, we show that issuing pulses on the five barrier gates to modulate exchange interactions in a maximally parallel way maintains the quality of qubit control relative to sequential operation. The techniques developed to perform parallel exchange pulses can be readily adapted to other quantum-dot-based encodings. Moreover, we show the first, to our knowledge, experimental demonstrations of an iSWAP gate for exchange-only qubits and of a charge-locking Pauli spin blockade (PSB) readout method. The results are validated using cross-entropy benchmarking (XEB)², a technique useful for performance characterization of larger quantum computing systems; here it is used for the first time on a quantum system based on semiconductor technology.

Parallel operation of two exchange-only qubits consisting of six quantum dots arranged linearly is shown to be achievable and maintains qubit control quality compared with sequential operation, with potential for use in scaled quantum computing.

Main

Parallelizing operations is one of the key enablers for quantum computation with many qubits, regardless of the technological platform. It minimizes qubit idle time, which leads to decoherence and information loss. Although parallel operations are important for running near-term algorithms within the coherence time of the device, they become crucial for effective error correction to enable fault-tolerant computation^3–6. It is therefore necessary to establish experimental procedures for parallelization of qubit operations and evaluate how much parallelization is possible to guide workload development and future device design.

Among various quantum computing technologies, spin qubits encoded on quantum dots are emerging as a promising platform through their inherent compatibility with silicon foundries^7–11. In these devices, either electrons or holes are trapped in voltage-electrode-defined quantum dot potentials and the quantum information is encoded on either a single¹² or several spins^13–16. The choice of qubit encoding determines the type of operations necessary to execute single-qubit or two-qubit operations. For example, the Loss–DiVincenzo (LD) qubit encodes the quantum information onto the spin of a single electron; it relies on microwave (MW) pulses resonant with the Larmor precession frequency of the spin (typically 1–20 GHz) to perform single-qubit gates^17,18. Parallelization can then be achieved by applying two or more (for example, frequency multiplexed) MW pulses to the sample, for which special care needs to be taken to avoid crosstalk effects¹⁹. Two-qubit gates with LD qubits are typically performed by dynamically turning on exchange coupling through voltage pulsing on barrier gates between two quantum dots^20–23. So far, these operations have only been executed individually.

Another prominent qubit encoding using quantum dots are exchange-only qubits^13,14, in which the quantum information is encoded on the collective spins of three electrons. The key appeal of this qubit type is that the single-qubit and two-qubit gates are composed only of voltage pulses that dynamically turn on exchange couplings J, eliminating the need for MW pulses. This simplifies set-up, fabrication, control and calibration requirements, especially for scalable two-dimensional arrays. Recently, high-fidelity single-qubit gates²⁴ and universal logic using two exchange-only qubits²⁵ have both been demonstrated by applying sequential exchange pulses. Operating exchange-only qubits in parallel, however, requires simultaneous control of exchange pulses, which has only been explored without estimates of the impact on gate fidelity^26,27.

In this work, we demonstrate the feasibility of simultaneous exchange control using a 12 quantum dot Tunnel Falls sample⁷ (Fig. 1a) (see Methods). We compare the system performance for sequential versus simultaneous pulsing in single-qubit and two-qubit RB measurements²⁸. We experimentally demonstrate a new iSWAP gate implementation for exchange-only qubits and show two-qubit gates with reduced overall duration owing to parallelization of exchange pulses. Last, we implement XEB^2,29 using sequential as well as simultaneous pulsing.

Fig. 1. Device and experimental pulse sequences.

a, Typical pulse sequence used in this work (left) and the Tunnel Falls device (right). The scanning electron microscope image of a device similar to that used in this work indicates the position of quantum dots (denoted by ‘QD’) used to form qubits (blue) and charge sensors (yellow), as well as regions of accumulated two-dimensional electron gas (Acc., green). The exchange-only qubits Q₁ and Q₂ are encoded on the six quantum dots (dashed lines). Voltage pulses applied sequentially (orange box) or in parallel (blue box) to the electrodes dynamically control the exchange coupling strength between quantum dots, allowing for exchange-only qubit operations. b, Charge-locking readout sequence used in all measurements presented in this paper (time axis not to scale). First, we ramp QD₄–QD₅ to the PSB readout window in 21.84 ns (1) and subsequently allow a 3.64 ns projection time before pulsing Q₂ to its PSB readout window (2), before reducing the tunnel coupling between quantum dots involved in readout (3, optional). We repeat these steps for QD₈–QD₉ in (4), (5) and (6). Then, the signal for QD₄–QD₅ is integrated for 18 μs (7). Next we integrate the QD₈–QD₉ signal for 18 μs (8), before ramping both qubits to their manipulation position in 21.84 ns each in (9) and (10). c, Quantum state tomography of the |00⟩ state. We initialize by post-selection using the PSB readout sequence and measure a fidelity of 86.4 ± 1.4%.

We operate the central six quantum dots (hereafter denoted by ‘QD’) of the 12 quantum dot array⁷ to encode two exchange-only qubits Q₁ and Q₂ on dots QD₄–QD₆ and QD₇–QD₉, respectively (Fig. 1a; see Methods). Characterization of the double quantum dot used in the experiment can be found in Extended Data Fig. 2. To perform full two-qubit state extraction and to overcome the spin lifetime limitation of sequential readout with exchange-only qubits, we introduce charge-locking PSB readout (Fig. 1b; see Methods), building on the frozen PSB technique³⁰, which we tune with a separation of at least two quantum dots to avoid crosstalk (Extended Data Fig. 5). In the experiments presented here, the readout sequence is also used as a means of initialization through post-selection³¹. We perform all exchange pulses with the same pulse duration of t_p = 10.92 ns, followed by a buffer time of t_b = 10.92 ns. Furthermore, we use exponential overshoot pulse pre-distortions to compensate for extended wiring, limiting our signal path bandwidth, which was instrumental to achieving high-fidelity control (see Methods). The impact of pulse pre-distortions can be visualized with buffer time spectroscopy measurements (Extended Data Fig. 3). We perform state tomography on the |00⟩ state to verify functionality of the system (Fig. 1c) and compute a fidelity of 86.4 ± 1.4%.

Extended Data Fig. 2. Double quantum dot characterization.

a,f,k,p,u, Charge stability diagrams for QD₄–QD₅ in (1,3) charge configuration (a), for QD₅–QD₆ in (3,1) (f), for QD₆–QD₇ in (1,1) (k), for QD₇–QD₈ in (1,3) (p) and for QD₈–QD₉ in (3,1) (u). b,g,l,q,v, Exchange fingerprints for QD₄–QD₅ with four 10.92-ns pulses (b), for QD₅–QD₆ with four 10.92-ns pulses (g), for QD₆–QD₇ with eight 10.92-ns pulses (l), for QD₇–QD₈ with eight 10.92-ns pulses (q) and for QD₈–QD₉ with eight 10.92-ns pulses (v). c,h,m,r,w, Calibrated exchange coupling tunability for QD₄–QD₅ (c), for QD₅–QD₆ (h), for QD₆–QD₇ (m), for QD₇–QD₈ (r) and for QD₈–QD₉ (w). d,i,n,s,x, Measurement of exchange oscillations for QD₄–QD₅ (d), for QD₅–QD₆ (i), for QD₆–QD₇ (n), for QD₇–QD₈ (s) and QD₈–QD₉ (x). e,j,o,t,y, Singlet lifetime measurement for QD₄–QD₅ (e), for QD₅–QD₆ (j), for QD₆–QD₇ (o), for QD₇–QD₈ (t) and for QD₈–QD₉ (y).

Extended Data Fig. 5. Crosstalk in charge-locking PSB readout.

a, Two-exchange-only qubit arrangement in which PSB readout is performed using QD₅–QD₆ for Q₁ and QD₈–QD₉ for Q₂. b,c, We perform charge-locking readout as described in the main text for Q₁ and Q₂, with the PSB readout arrangement as visualized in panel a. We scan the Q₂ detuning pulse magnitude, whereas readout parameters for Q₁ remain constant. At each Q₂ detuning value, we record the PSB readout result for both Q₁ and Q₂. As we do not modify Q₁ readout parameters, we expect to see a variable signal signifying a singlet for high results and a triplet for low results. Instead, we see that, at the point at which Q₂ crosses a charge transition from (3,1) to (4,0) (panel c), the Q₁ readout is pushed out of calibration and into a (4,0) charge state. This is because of capacitive coupling between quantum dots participating in Q₁ and Q₂ readout. Similar effects have been used for cascade readout in ref. ⁵¹. d, Two-exchange-only qubit arrangement in which PSB readout is performed using QD₄–QD₅ for Q₁ and QD₈–QD₉ for Q₂. e,f, With two dot separation between PSB readout pairs, Q₁ readout remains calibrated, regardless of the Q₂ detuning scan or Q₂ charge transition.

Extended Data Fig. 3. Buffer time spectroscopy.

a, We apply a train of 12 consecutive exchange pulses. On the x axis, we vary the targeted angular evolution for each exchange pulse between 0 and 2π. On the y axis, we vary the buffer time t_b between the exchange pulses. Ideally, we expect to see oscillations with 12 periods, one per pulse, regardless of the buffer time. However, in the presence of finite rise and fall times, we observe more oscillations at short buffer times, indicating that the overlap of consecutive pulses invalidates our calibrations. b, Buffer spectroscopy measurement after introducing pulse pre-distortions that reduce pulse overlap. The plot is visibly more consistent with expectations. Therefore, buffer times as short as 7.28 ns can be used in the experiment.

Crosstalk in semiconductor qubit devices

Semiconductor qubit devices experience signal crosstalk through (1) capacitive coupling of electrodes to nearby dots affecting their charge state and tunnel couplings, as well as (2) on-chip signal path routing and (3) off-chip signal routing. To enable efficient control of the system, the sum of these crosstalk contributions needs to be compensated. The capacitive coupling crosstalk is partially corrected in a process called virtualization^32,33, allowing for orthogonal control over the electrochemical potentials of the quantum dots (see Methods). This reduces control complexity and ensures that the quantum dots retain their intended electron population during operation. In a typical spin qubit experiment, the barrier–barrier elements are left unpopulated, as all operations requiring voltage pulsing are performed sequentially. In this case, the exponential relation of the exchange coupling to the barrier electrode voltage suppresses the effects of the existing crosstalk on idle exchange axes. In this regime, a large voltage change would be needed to substantially affect the exchange coupling from its off value and thereby limit performance. So far, barrier–barrier virtualization was realized primarily for tuning purposes^27,34,35 but was not implemented for high-fidelity parallel exchange pulsing, which requires precise calibrations.

We evaluate the impact of simultaneous pulses on the exchange between the QD₇–QD₈ pair (barrier electrode B₈) by measuring an exchange fingerprint²⁰ (see Methods) directly (Fig. 2a) and compare it with an exchange fingerprint with a simultaneous pulse of voltage v′B₆ on barrier electrode B₆ (Fig. 2b). By adding the v′ prefix to the voltage label, we highlight that the virtualization is incomplete and does not yet contain barrier–barrier correction factors. We note four changes to the fingerprint: (1) a shift of the barrier voltage axis; (2) a shift of the detuning axis; (3) minor changes to the fingerprint shape; and (4) a change in exchange tunability. The difference in the two fingerprints is visualized in Fig. 2c, in which linecuts of the fingerprints for three different detunings are shown. The individual exchange pulses show less exchange for the same barrier pulse voltage as compared with simultaneous pulses. This can be understood using the diagram of the potential landscape (Fig. 2d). The uncompensated, simultaneous exchange pulses on electrodes B₆ and B₈ lower each others’ potential barriers further (Fig. 2d, orange curve) than the desired values (Fig. 2d, black curve), therefore also increasing the resulting exchange coupling. Furthermore, when pulsing barrier electrode B_N, an increase in voltage vB_N not only reduces the potential-barrier height between QD_N−1 and QD_N but also attracts the quantum dots towards each other, further enhancing their effective coupling³⁵ (Fig. 2d). The orange curve in Fig. 2d illustrates this effect for the case of simultaneous pulses applied to electrodes B₆ and B₈, attracting QD₅–QD₆ and QD₇–QD₈ towards each other. This dot movement in turn reduces the exchange coupling between QD_N−2 and QD_N−1 as well as QD_N and QD_N+1.

Fig. 2. Simultaneous exchange pulsing.

a, Exchange fingerprint for the double dot QD₇–QD₈, measured with eight consecutive exchange pulses of 10.92 ns, each followed by a 10.92-ns pulse buffer. b, Exchange fingerprint for the double dot QD₇–QD₈, measured in the same manner as a, while also simultaneously applying a voltage v′B₆ on the B₆ electrode to emulate an exchange pulse between QD₅–QD₆. c, Linecuts along the dashed lines of the fingerprints in a and b, illustrating the change in the exchange coupling owing to simultaneous pulsing. d, Qualitative diagram of potential landscape during simultaneous exchange pulses on electrodes B₆ and B₈ under different barrier–barrier compensation schemes: ideal compensation (black), no compensation (orange) and next-nearest-barrier compensation (blue). Using next-nearest-barrier compensation, the impact of simultaneous pulses on the potential landscape, and therefore the resulting exchange couplings, is suppressed. e, Change in the linecut of the QD₇–QD₈ fingerprint at  −10 mV detuning as a function of the pulse amplitude applied to B₆ without barrier–barrier compensation. f, Same measurement as in e but with barrier–barrier compensation enabled. The B₆ amplitude dependence of the fingerprint linecut is suppressed.

Next-nearest-neighbour barrier–barrier compensation

Intuitively, all exchange couplings, including nearest-neighbour elements, would be compensated in the process of barrier–barrier virtualization³⁴. This approach would implement ideal compensation and make the system seem insensitive to crosstalk from nearby electrodes (Fig. 2d, black curve). In the case of nearest-neighbour barrier–barrier virtualization, we apply a positive voltage pulse to barrier electrodes B_N−1 and B_N+1 to counteract any dot movement effects and keep the tunnel coupling to neighbouring dots unchanged. However, by disallowing shifts in the quantum dot position and removing the enhancement of the tunnel coupling from that effect, we reduce the exchange coupling tunability dJ_N/dvB_N for all barrier electrodes, which is undesirable.

We note that, in both single-exchange-only and two-exchange-only qubit gates, exchange pulses are never issued on two neighbouring barrier electrodes. As a result, only the crosstalk contributions from next-nearest neighbours need to be managed (Fig. 2). In fact, pulsing neighbouring barrier electrodes simultaneously would correspond to having an electron being involved in two exchanges at the same time³⁵, leading to different operations, that is, a J_x rotation for exchange-only qubits³⁶ or Toffoli-like gates for LD qubits³⁷. Because there is no situation in which we turn on neighbouring exchange couplings simultaneously, the substantial impact of an exchange pulse issued on B_N−1 on the effective neighbouring barrier voltage vB_N only results in a small change of the resulting exchange J_N, as dJ_N/dvB_N is small in the regime of low vB_N. We find it acceptable for the tunnel coupling between dots QD_N−2–QD_N−1 and QD_N–QD_N+1 to change while coupling QD_N−1 and QD_N but we ensure that the couplings between QD_N−3–QD_N−2 and QD_N+1–QD_N+2 remain unchanged. We illustrate this in the blue curve of Fig. 2d for simultaneous exchange pulses applied to barrier electrodes B₆ and B₈. We apply further barrier pulsing to B₅, B₇ and B₉ to implement the next-nearest compensation and correct for crosstalk during simultaneous operation. Detailed description of the barrier–barrier virtualization method can be found in the Methods. The resulting insensitivity in barrier calibration to parallel pulsing is visualized in Extended Data Fig. 6.

Extended Data Fig. 6. Barrier–barrier crosstalk measurements.

Panels a, f and l show horizontal lines for which the total angular evolution under the extra barrier pulse swept on the y axis leads the exchange-only qubit to leave its computational subspace.

It is worth highlighting the regularity shown by the extracted cross-capacitance matrix (Extended Data Table 1), in which we observe repeated elements for plunger–plunger, plunger–barrier and barrier–barrier elements. This regularity indicates a degree of lateral translational symmetry, confirming good sample uniformity and the benefits of the industrial manufacturing processes^1,7. The observed uniformity reduces the calibration time necessary for virtualization and allows values from a small section (or simulations) to be used as initial guess for the rest of the cross-capacitance matrix or to be directly transferred to lithographically identical devices.

Extended Data Table 1.

Virtual matrix used in the experiment

We highlight the newly introduced next-nearest barrier–barrier elements that allowed for parallel operation in bold.

Individual vs. simultaneous single-qubit operation

We measure the individual qubit performance using the blind RB protocol²⁴ (see Methods). To avoid changes in the control scheme, both qubits are initialized but only one qubit is operated at any time. We apply a random sequence of N Clifford gates, including a final recovery gate that returns the qubit state to either |0⟩ or |1⟩. The return probability decay as a function of increasing sequence length is fitted and Clifford gate fidelities of 99.84 ± 0.02% (including 0.08 ± 0.02% leakage error) for Q₁ (Fig. 3b) and 99.41 ± 0.03% (including 0.13 ± 0.02% leakage error) for Q₂ (Fig. 3d) are extracted.

Fig. 3. Individual versus simultaneous single-qubit gates.

a,b, Blind RB of Q₁ with both qubits initialized. We measure an average Clifford fidelity of 99.84 ± 0.02%, including 0.08 ± 0.02% leakage error. The error bar is calculated as a standard deviation to the mean on five measurement repetitions. c,d, Blind RB of Q₂, measured similarly to Q₁. We measure 99.41 ± 0.03% average Clifford gate fidelity, including 0.13 ± 0.02% leakage error. e, Simultaneous blind RB of Q₁ and Q₂. The two qubits perform different Clifford sequences with the same number of gates at the same time. We measure 99.77 ± 0.02% (including 0.11 ± 0.04% leakage error) and 99.36 ± 0.03% (including 0.16 ± 0.03% leakage error) average Clifford gate fidelity for Q₁ (f) and Q₂ (g), respectively. This constitutes a 0.05–0.07% reduction in fidelity compared with individual blind RB for both qubits.

Next we examine the simultaneously driven single-qubit performance. We generate several sequences of independent, random Clifford gates with increasing length for the individual qubits. We initialize both qubits to their |00⟩ state and then execute the respective sequences simultaneously. As single-qubit Clifford gates vary in length, it is possible that one qubit finishes manipulation earlier. It would then idle until the other qubit completes its random Clifford sequence. As each of the qubits consists of three quantum dots and two exchange axes, simultaneous pulses will be issued on the various combinations of exchange axes: J_z–J_z (most distant), J_z–J_n, J_n–J_z, J_n–J_n (next-nearest gates) or individually J_n or J_z on qubits Q₁–Q₂. This scrutinizes the average single-qubit performance in the presence of simultaneous pulsing. We measure that the Clifford gate fidelity for Q₁ is 99.77 ± 0.02% (including 0.11 ± 0.04% leakage error) and for Q₂ is 99.36 ± 0.03% (including 0.16 ± 0.03% leakage error) (Fig. 3f,g), constituting a 0.05–0.07% fidelity reduction compared with individual pulsing. This is an encouraging result as it confirms that our techniques limit the impact of crosstalk and enable simultaneous control of neighbouring exchange-only qubits, an important functionality for efficient quantum computation.

Two-qubit gates with simultaneous exchange pulses

Parallelizing exchange pulses not only benefits single-qubit control of exchange-only qubits but also has important implications for the two-qubit gate design. A long-standing issue for exchange-only qubits is the complexity of the two-qubit gates, which consist of substantially longer exchange pulse sequences than the single-qubit gate sequences. This implies that the two-qubit gate duration, being substantial in comparison with typical coherence times, limits the operation fidelity. We note that the exchange pulse sequences for CNOT, iSWAP and SWAP gates contain many commuting exchange pulses. If exchange pulses commute, they can be executed at the same time, thereby reducing the 23-pulse-long CNOT sequence to 15 time steps, the 28 pulse-long iSWAP sequence to 17 time steps and 9 pulse-long SWAP sequence to five time steps. This constitutes up to an approximately 40% reduction in the two-qubit gate duration. Shorter two-qubit gates are beneficial, as they reduce error originating from decoherence. This is true for both qubits involved in the two-qubit gate but also reduces the errors accumulated by idle qubits waiting for the two-qubit gate completion.

We characterize the performance of our two-exchange-only qubit system with two-qubit RB. As with single-qubit RB, we execute random gate sequences of length N − 1 using sequential exchange pulses, followed by a recovery gate to |00⟩. The gates in the random sequence are sampled from the two-qubit Clifford gate set, containing 11,520 different operations (see Methods). Next to the SWAP and CNOT two-qubit gates, we make explicit use of the available iSWAP gate (Extended Data Fig. 7) in the two-qubit Clifford gate decomposition, bringing the average number of exchange pulses to 32.3 per Clifford gate. We also make use of mirroring gates to adjust the configuration of the two exchange-only qubits to shorten two-qubit gate sequences and to issue most exchange pulses on an exchange axis with a high quality factor (see Methods). By fitting the decay of the |00⟩ recovery probability, we extract a two-qubit Clifford gate fidelity of 96.25 ± 0.07% (Fig. 4a,b). We note that this version of two-qubit RB could be improved to better account for leakage (see Extended Data Fig. 4; similar to blind single-qubit RB), as the decay to each of the two-qubit basis states is different (see Methods).

Extended Data Fig. 7. Validation of the phase imparted by an iSWAP gate.

a–c, Quantum circuits executed in this experiment. a, Prepare Q₁ in the superposition state $\mid {\mathrm{Q}}_1{\mathrm{Q}}_2⟩=\frac{1}{\sqrt{2}}\left(\mid 0⟩+\mid 1⟩\right)\mid 0⟩$. b, Prepare Q₁ in a superposition state and apply a SWAP gate, bringing the system to $\mid {\mathrm{Q}}_1{\mathrm{Q}}_2⟩=\mid 0⟩\left(\frac{1}{\sqrt{2}}\left(\mid 0⟩+\mid 1⟩\right)\right)$. c, Prepare Q₁ in a superposition state and apply an iSWAP gate, bringing the system to $\mid {\mathrm{Q}}_1{\mathrm{Q}}_2⟩=\mid 0⟩\left(\frac{1}{\sqrt{2}}\mid 0⟩+\frac{i}{\sqrt{2}}\mid 1⟩\right)$. d–f, Quantum state tomography results for the circuits in a–c. We note that, as well as swapping the superposition state between qubits, the iSWAP gate imparts the expected i phase.

Fig. 4. Sequential versus parallelized two-qubit gates.

a, Components of sequential two-qubit Clifford gate sequences. We execute single-qubit gates sequentially and two-qubit gates are executed as sequential exchange pulses. Colours indicate exchange pulses with a π-rotation (green, spin-swap), a $\frac{3}{2}\pi$rotation (yellow) and a $\frac{1}{2}\pi$-rotation (blue). Orange indicates custom rotation angles necessary for specific gate implementations (CNOT → 2.007 rad, 4.689 rad, 4.736 rad, 1.134 rad; iSWAP → 0.162 rad, 4.550 rad). b, Two-qubit RB yields a Clifford gate fidelity of 96.25 ± 0.07%. c, Interleaved RB of CNOT, iSWAP and SWAP gates and standard two-qubit RB as reference (measured at the same time as interleaved RB for best accuracy). We measure the fidelity of the CNOT gate to be 97.55 ± 0.47%, iSWAP 97.43 ± 0.43% and SWAP 99.03 ± 0.33%. d, Components of two-qubit Clifford sequences with parallel pulsing. We apply single-qubit gates simultaneously to both qubits and maximally parallelize exchange pulses in two-qubit gates. e, Two-qubit RB using parallel pulsing. We measure a two-qubit Clifford fidelity of 95.80 ± 0.08%. f, Parallel interleaved RB of CNOT, iSWAP and SWAP gates and standard two-qubit RB as reference. We measure that the fidelity of the parallelized CNOT gate is 96.70 ± 0.57%, parallelized iSWAP is 96.15 ± 0.57% and parallelized SWAP is 98.48 ± 0.55%. The error bars are estimated as a standard deviation on the mean value of five measurement repetitions.

Extended Data Fig. 4. Two-qubit RB with recovery to all computational basis states.

a, |00⟩ state. b, |10⟩ state. c, |01⟩ state. d, |11⟩ state. The return probability for all two-qubit basis states is recorded. These data can be used for the development of blind two-qubit RB.

To characterize the two-qubit gates themselves, we implement interleaved RB. Here the gate to be characterized, for example, the CNOT gate, is inserted between all of the two-qubit Clifford gates of a random gate sequence³⁸. The recovery gate is calculated by taking into account the interleaved gate. We fit the decay of the |00⟩ recovery probability for the reference curve (Fig. 4c, blue) and compare it with the recovery probability decay when gates are interleaved (Fig. 4c, orange, CNOT; red, SWAP; green, iSWAP). We measure that the fidelity of the CNOT gate is 97.55 ± 0.47%, iSWAP is 97.43 ± 0.43% and SWAP is 99.03 ± 0.33%. We note the trend of increased error per gate with increased number of exchange pulses per gate.

We perform Monte Carlo simulations (see Methods) to evaluate the impact of magnetic noise and charge noise on our two-qubit gates. We estimate $T_2^{*}$ to contribute about 1.03% to the CNOT gate error (SWAP ≈ 0.17%, iSWAP ≈ 1.22%) and  about 0.48% is because of charge noise (SWAP ≈ 0.14%, iSWAP ≈ 0.56%). In our simulations, we use N_osc rather than a single-frequency quality factor Q of the exchange oscillations, as the experimental data show a change in the exchange oscillation frequency owing to slow settling of the voltage on the gate electrode (Extended Data Fig. 1). The remaining error of  about 0.94% is composed of calibration errors (including J(V_B) deviating from the fitted function), non-Markovian errors (such as pulse overlaps) and other unaccounted-for effects (SWAP ≈ 0.66%, iSWAP ≈ 0.79%).

Extended Data Fig. 1. Exchange quality factor versus N_osc.

a, Exchange oscillations measured for QD₈–QD₉. b–e, Zoom-in on the exchange oscillations with sinusoidal fits to extract the momentary frequency. We note that the frequency increases with increasing pulse length, which can be correlated with the final settling of the barrier electrode voltage. For qubit control, only the total angular evolution during the exchange pulse is relevant. Overlap with transients from previously issued pulses can be accounted for with further calibrations such as pulse-domain pre-distortions. It is, however, difficult to reliably extract quality factors for exchange oscillations at a fixed frequency. f, When fitting the data with a decaying sinusoid, we extract a quality factor of 21.5 at 49.856 MHz. Ignoring the steadily increasing frequency, we also fit the envelope of the decaying oscillation to extract N_osc of 33.7 (ref. ²⁵).

Next we implement two-qubit gates with parallel exchange pulses. The two-qubit Clifford gates are performed by executing the single-qubit gates on both qubits in parallel (see Methods) and we use the reduced-length two-qubit gates with up to three exchange pulses at the same time, as shown in Fig. 4d. With this parallelization, the average number of exchange pulse time steps is 20.3 per Clifford gate, a reduction of  approximately 37% in comparison with the sequential pulse execution. Using two-qubit RB, we measure an average fidelity of 95.80 ± 0.08% per Clifford gate (Fig. 4e). We then use interleaved RB to extract the fidelities of the parallelized gates to be 96.70 ± 0.57% for parallelized CNOT, 96.15 ± 0.57% for parallelized iSWAP and 98.48 ± 0.55% for parallelized SWAP. For the parallelized CNOT gate, we estimate error contributions of  approximately 0.48% from nuclear spins (SWAP ≈ 0.05%, iSWAP ≈ 0.43%) and almost unchanged contributions from charge noise of  about 0.45% (SWAP ≈ 0.13%, iSWAP ≈ 0.56%). Other effects such as non-Markovian contributions and calibration errors account for  about 2.37% (SWAP ≈ 1.34%, iSWAP ≈ 2.86%). A comparison of sequential and parallelized two-qubit gate fidelities is provided in Extended Data Table 2. The gate error from nuclear spins typically scales as ${\left(t_{\mathrm{gate}}/T_2^{*}\right)}^2$, in which t_gate is the gate time²⁵. We therefore note a substantial decrease by more than 60% in the nuclear spin error contribution for parallelized two-qubit gates, as compared with sequential operation. Although the error contribution from nuclear spins has decreased, the calibration precision of parallel pulses is not sufficient to observe an overall improvement in two-qubit gate fidelity. This behaviour differs from the case of single-qubit gate fidelities, in which simultaneous and parallel performance was comparable up to about 0.05–0.07%. We attribute this to a miscalibration of the coupling between the barrier electrode B₇ (controlling the exchange between the two qubits) to the barrier electrodes B₅ and B₉. Specifically, we see a change in the fingerprint shape owing to simultaneous pulsing that is larger than in other exchange pairs. We attribute this to the asymmetric tuning of the dots neighbouring the reservoir, for which we intentionally reduce the tunnel coupling to the reservoir ($t_{\mathrm{c}}^{\mathrm{res}}$) to isolate the quantum dot array. Dot-to-dot tunnel coupling is much larger than $t_{\mathrm{c}}^{\mathrm{res}}$. This problem can be alleviated with a more consistent tuning, operating on dots inside a larger quantum dot array or by introducing quadratic components to the crosstalk compensation.

Extended Data Table 2.

Two-qubit RB error budget

Comparison of the two-qubit gate performance between sequential and parallel pulsing, including estimated error contributions from magnetic noise, charge noise and other effects.

Cross-entropy benchmarking

As a final experiment, we implement XEB² for both sequential (Fig. 5a,b) and parallel exchange pulsing (Fig. 5c,d). We choose to use XEB as a characterization tool as it produces an easy-to-track overall performance metric that can be used for systems consisting of up to several dozens of qubits. Each XEB experiment consists of several cycles, which in turn consist of random single-qubit gates or exchange pulses, followed by an entangling two-qubit gate. When running the parallel version of the experiment, the single-qubit gates are applied simultaneously in each cycle and the parallel version of the two-qubit gate is issued. The resulting figure of merit is obtained by comparing the measured bit-string distribution for 100 random circuits against their expected bit-string distribution as determined by simulations³⁹. We measure a per-cycle XEB fidelity of 96.8% and 97.0% for the sequential and simultaneous operating modes, respectively, which are matched within the error bars. We find that the extracted error rate per cycle is comparable with those from two-qubit RB experiments. This is not surprising, as the composition of two-qubit Clifford gates closely resembles the base structure of the random quantum circuits used in XEB (see Supplementary information for more details).

Fig. 5. Two-qubit XEB.

a,b, Experimental implementation using sequential exchange pulsing. For single-qubit gates, we use randomly selected J_n and J_z axis rotations. We measure a per-cycle fidelity of 96.8 ± 0.4%. c,d, Experimental implementation using parallel exchange pulsing. The J_n and J_z rotations are issued in parallel and the maximally parallel version of the CNOT gate is used. We measure a per-cycle fidelity of 97.0 ± 0.4%.

Conclusions

The results presented in this paper demonstrate that it is possible to perform high-quality, simultaneous exchange pulses on next-nearest electrodes, allowing for maximum parallelization of quantum circuits in exchange-only qubits. An important enabling factor for this experiment is the low defect density of the studied device, which is manufactured on a state-of-the-art 300-mm semiconductor process line. Still, we find that certain combinations of parallel exchange pulses on the studied device cannot be fully compensated only by linear virtualization of the corresponding barrier electrodes owing to the resulting change in fingerprint shape. Quadratic corrections to the crosstalk compensation⁴⁰ will mitigate this issue. We show an experimental implementation of an exchange-only iSWAP gate and identify a way to schedule parallelized two-qubit Clifford gates. We use RB to show that the resulting overall quality of parallel exchange pulses is sufficient to match the performance of sequentially issued single-qubit gates. We execute parallelized two-qubit gates with only a 0.55% to 1.28% reduction in fidelity as compared with sequential two-qubit gates, while reducing the gate duration by about 40%. A detailed breakdown of resulting error budget shows the need for further refinement of pulse control and highlights the importance of improving material characteristics by using higher-purity isotopes and reducing charge noise. Furthermore, we adapt XEB as a technique to characterize the overall performance of a semiconductor-based quantum system.

Methods

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 P₁–P₃ and P₁₀–P₁₂ to facilitate access of the middle quantum dots to an electron reservoir. We tune quantum dots QD₄–QD₉ 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 Q₁ and Q₂ are encoded on dots QD₄–QD₆ and QD₇–QD₉, respectively. QD₅ and QD₈ 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 QD₄–QD₅ and QD₈–QD₉ 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 (SD₅ for Q₁ and SD₈ for Q₂) 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 SD₅ to QD₈–QD₉ (performing PSB for Q₂) and of SD₈ to QD₄–QD₅ (performing PSB for Q₁) 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 needed^13,14. The computational states of exchange-only qubits are defined as |0⟩ = |S⟩|↓⟩ (|0⟩ = |S⟩|↑⟩) and $\mid 1⟩=\sqrt{\frac{2}{3}}\mid T_{+}⟩\mid \downarrow ⟩-\sqrt{\frac{1}{3}}\mid T_0⟩\mid \uparrow ⟩$ ($\mid 1⟩=\sqrt{\frac{2}{3}}\mid T_{+}⟩\mid \uparrow ⟩-\sqrt{\frac{1}{3}}\mid T_0⟩\mid \downarrow ⟩$), in which $\mid S⟩=\sqrt{\frac{1}{2}}\left(\mid \uparrow \downarrow ⟩-\mid \downarrow \uparrow ⟩\right)$, $\mid T_0⟩=\sqrt{\frac{1}{2}}\left(\mid \uparrow \downarrow ⟩+\mid \downarrow \uparrow ⟩\right)$ and ${\mid T⟩}_{+}=\mid \uparrow \uparrow ⟩$. 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 1/4^N 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 dots^13,14. In the tuning chosen here, the state of Q₁ (Q₂) can be determined by performiang PSB readout on the QD₄–QD₅ (QD₈–QD₉) 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 T₁ 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.

Charge-locking readout

Charge-locking readout is a multi-qubit readout protocol in which we first pulse all participating quantum dot pairs to their respective PSB readout window and only then start signal integration. In this manner, we prevent information loss through decoherence or relaxation while qubits are idling at their respective manipulation points (see Methods). This could be supplemented by the frozen PSB technique³⁰ if the relaxation times at the PSB window are not substantially longer than the signal integration time (steps 3 and 6 in Fig 1b). The steps of the charge-locking PSB readout are described as follows. (1) The first quantum dot pair used for PSB is pulsed into the readout window using a 21.84-ns-long voltage ramp. (2) At this readout point, a 3.64-ns wait time allows the spins to either be projected to a singlet and move to the (0,4) charge state or be projected to a triplet and remain in the (1,3) charge state. (3) The tunnel coupling between QD₄ and QD₅ can be lowered to suppress relaxation rates of these charge states³⁰. (4) After projection of the first quantum dot pair, the second quantum dot pair is ramped to its readout window, (5) spins are projected to the singlet or triplet states and (6) the charge relaxation rates are suppressed. We find the charges to be locked in the readout window, with relaxation times substantially longer than the integration time of 18 μs required to obtain a signal of sufficient quality. Therefore, we consider steps (3) and (6) not strictly necessary in our experiment. Once both qubits have been projected, a voltage pulse is applied to the SD₅ plunger to shift its electrochemical potential to the flank of a Coulomb peak to allow for charge sensing. (7) Then, a pulsed readout stimulus signal is applied to the sensor that allows determination of the state of Q₁. (8) After the readout of Q₁ is completed, the SD₅ plunger is brought back into Coulomb blockade and the process is repeated for Q₂. We achieve a signal-to-noise ratio of 4.25 and 3.10 for readout of Q₁ and Q₂, respectively. (9,10) At the end of the sequence, both qubits are pulsed back to their manipulation point.

Virtualization

To efficiently control the array of quantum dots, we use standard virtualization techniques^32,33. The relative effect between the dedicated plunger electrode and the surrounding plunger and barrier electrodes on the chemical potential of a quantum dot are measured by tracking electron addition lines in charge stability diagrams as a function of the applied voltages. These values are collected in the cross-capacitance matrix, which in turn is inverted to perform a linear basis transformation into the virtual voltage space. These virtual voltages now only affect the chemical potential of the quantum dot they are supposed to control, leaving the chemical potentials of other dots unaffected.

The voltage virtualization is fine-tuned after the quantum dot array is brought into the correct charge configuration and is isolated from the environment (Extended Data Fig. 2a,f,k,p,u). However, at that point, charge addition lines are no longer visible and cannot be used to update the voltage virtualization. Therefore, the crosstalk matrix entries are adjusted in an iterative process in which the position of inter-dot charge transition lines is monitored as virtual voltages on surrounding electrodes are changed. Proper virtualization is achieved once the position of charge transition lines does not change with altering virtual voltages. This procedure can be used for plunger–plunger as well as for plunger–barrier crosstalk compensation; the technique for barrier–barrier crosstalk compensation is described in the main text.

Barrier–barrier crosstalk compensation

To compensate the barrier–barrier crosstalk, the centre linecut of Fig. 2a is tracked as a function of the simultaneous barrier pulse amplitude vB₆ (Fig. 2e). As vB₆ is increased, we observe an increase of the exchange between QD₇ and QD₈, in line with our observations in Fig. 2c. We note that the exchange pattern is linearly shifted as a function of vB₆, indicating that the linear crosstalk compensation framework is applicable. The value of the crosstalk element is given by the slope of the resulting line pattern. We enter it into the existing voltage virtualization framework that relies only on a linear basis transformation. Also, we include finer corrections to compensate any outstanding movement along the detuning axis by optimizing barrier–plunger elements. Furthermore, we note a slight fingerprint shape change that reduces the exchange tunability during simultaneous exchange pulses as compared with individual exchange pulses for some of the quantum dot pairs (most notably B₅ and B₇). We account for this by adding small nearest-neighbour barrier–barrier elements to the cross-capacitance matrix (Extended Data Table 1). After the calibration cycle, we check the dependence of exchange strength as a function of the simultaneous barrier pulse amplitude (Fig. 2f) in comparison with the crosstalk-uncompensated situation (Fig. 2e). Using the described corrections, we observe that the resulting exchange pattern is independent of the simultaneously applied exchange pulse.

Exchange pulse calibrations

Exchange-only qubits are operated by issuing well-calibrated voltage pulses to dynamically control the exchange interaction between the relevant quantum dots^13,14. We choose to keep the durations t_p and t_b of the resulting exchange pulses equal and constant throughout the experiment. The rotation angle produced by an exchange pulse is given as φ = t_pJ(V_B) and we control it with the voltage V_B issued to the electrode controlling the exchange interaction. To calibrate each exchange axis, a qubit containing this exchange axis is encoded and prepared in its |0⟩ state, exchange pulses on the axis in question are applied and the qubit is read out. To ensure adequate measurement contrast (for example, to calibrate the J_z axis), pre-rotations or excitation swaps making use of previously calibrated axes are applied. Then, for each axis, the following three steps are performed:

• Fingerprint calibration. The goal of the fingerprint calibration is to ensure that exchange pulses are insensitive to charge noise in the double-dot detuning direction. Therefore, the exchange strength is measured as a function of barrier and detuning voltages²⁰. Then the crosstalk compensation between the exchange barrier electrode and involved plunger electrodes is adjusted to keep the sweet spot along the exchange axis at a constant detuning voltage. Extended Data Fig. 2b,g,l,q,v shows fingerprints for the exchange axis used in the experiment. Although this calibration can be performed with a single exchange pulse, we make use of an error amplification scheme by applying eight repeated exchange pulses to increase the sensitivity of the calibration.

• Initial exchange calibration. Next we measure the exchange coupling magnitude in the function of the barrier gate voltage at the selected detuning voltage. For this, exchange pulses of increasing amplitudes are applied and the |1⟩ state return probability is measured. The resulting data can be fit to b + acos(J(V_B)t_p), in which J(V_B) = αexp(γV_B) and γ is the exchange tunability.

• Fine exchange calibration. With an initial knowledge of J(V_B), it is possible to issue pulses of a desired rotation angle on this axis. To amplify the calibration error, a train of N pulses with a fixed rotation angle is measured. As the angle is swept, the resulting oscillations can be used to extract a more precise relation of J(V_B), which can be described as $J\left(V_{\mathrm{B}}\right)=\alpha \exp \left(\gamma V_{\mathrm{B}}+\kappa V_{\mathrm{B}}^2\right)$, in which κ is a second-order correction to the exchange tunability. Further precision in the calibration can be gained by directly using a heuristic interpolation of J(V_B). The final calibration is presented in the Extended Data Fig. 2c,h,m,r,w.

Once the relation J(V_B) is established, pulses with rotation angle φ can be produced by issuing barrier pulses with amplitude

$$V_{\mathrm{B}}\left(\phi \right)=\frac{-\gamma +\sqrt{{\gamma }^2+4\kappa \text{ln}\left(\phi /\left(\alpha t_{\mathrm{p}}\right)\right)}}{2\kappa }\text{for}\kappa \ne 0,$$

and

$$V_{\mathrm{B}}\left(\phi \right)=\frac{1}{\gamma }\text{ln}\left(\frac{\phi }{t_{\mathrm{p}}\alpha }\right)\text{for}\kappa =0.$$

Furthermore, we ensure negligible error contributions from residual coupling $J_{\mathrm{off}}=J\left(V_{{\mathrm{B}}_{\mathrm{off}}}\right)$ by reducing the voltage $V_{{\mathrm{B}}_{\mathrm{off}}}$ on the electrode when the exchange coupling is not active. We need to ensure that this condition is met and that rotation angles of up to 2π are available within the dynamic range of the arbitrary waveform generator (AWG). We achieve this by pulsing negatively on the barrier while its coupling is not active to make full use of the available dynamic range of the AWG.

Pulse pre-distortions

To minimize the effects of decoherence caused by magnetic noise from the spin bath, it is beneficial to operate with as-fast-as-practical exchange pulses. However, as pulse time t_p and buffer time t_b are reduced, the qubit fidelity is limited by non-Markovian control errors resulting from overlap of subsequent pulses owing to filtering effects of the transmission lines used to deliver the pulses. More concretely, the finite bandwidth of both the control electronics as well as the transmission lines results in the ideally square pulses produced by the control electronics being distorted. Although a systematic distortion that only affects the current pulse can be accounted for by standard calibrations, finite pulse fall times of one pulse that affect the amplitude of the subsequent pulse create an effective system memory. This system memory cannot be accounted for by only using standard calibrations. Further effects leading to a system memory can include pulse reflections at impedance mismatches in the signal chain or the high-pass effect of the bias tees used. To counter these effects, we implement exponential undershoot/overshoot pulse pre-distortions. Properly pre-distorted pulses avoid pulse overlap and reduce the pulse rise and fall time^47,48.

We tackle the task of finding the appropriate pre-distortions in two steps. First, we introduce a buffer time spectroscopy measurement in which t_b is varied for a train of pulses with varying amplitudes (Extended Data Fig. 3). Tracking the voltage at which a particular overall rotation angle is achieved as a function of t_b gives a good approximation to the step response. On the basis of this dependence, exponential parameters are extracted to define the pre-distortion coefficients for short (about 8 ns) and medium (about 50 ns) timescales.

The pre-distortion corrections are applied in real time at the last output stage of the AWG. The effect of pre-distorted pulses can be assessed by taking another buffer time spectroscopy measurement for which now the approximate step response for these timescales is substantially flatter. However, these corrections still leave long pulse sequences of several thousand exchange pulses vulnerable to effects on longer timescales.

To further reduce the effect of non-Markovian errors on longer timescales, a second step to optimize the best pre-distortion parameters is performed. As the cost function of the optimization procedure, we choose the visibility of a sequence of random Clifford gates (difference between the recovered |1⟩ and |0⟩ states). To reduce the complexity of the procedure, the same pre-distortion parameters are applied on all control lines. We find that an exponential model with a timescale of about 500 ns describes the filter function of the transmission lines well. We choose to apply these corrections pulse by pulse by modifying the amplitude of the exchange pulses based on the history of exchange pulses on the respective channel rather than applying them sample by sample. This is why Extended Data Fig. 1 can show a time-dependent oscillation frequency but we can still measure high-fidelity exchange pulses over extended periods of time. Otherwise, this technique follows the same mathematical framework. This procedure can be further improved by designing a targeted error-amplifying sequence to act as the cost function for the optimization.

Exchange-only gate library

General single-qubit gates for exchange-only qubits need to be synthesized from positive angle rotations around the n and z axes^13,14. The precise decomposition of Clifford gates to n-exchange and z-exchange pulses can be found in ref. ²⁴. With this decomposition, a single-qubit Clifford gate needs on average 2.666 exchange pulses.

Two-qubit gates for exchange-only qubits are also composed only of exchange pulses. However, the precise pulse sequence, and therefore also the number of pulses and the duration of the gate, depend on the connectivity between the qubits as well as their configuration. The two-qubit gate library implementation of ref. ⁴⁹ is used to determine the appropriate sequence of pulses. The two-qubit gates supported in this library are CNOT, iSWAP and SWAP. Of the 11,520 two-qubit Clifford gates, 576 contain no two-qubit gate, 576 include a SWAP, 5,184 include a CNOT and 5,184 include an iSWAP gate, with an average of 32.3 exchange pulses.

To improve the efficiency of pulse execution on two qubits, exchange pulses can be applied in parallel. This can be done straightforwardly in the case of single-qubit gates on the respective qubits. For two-qubit gates, an algorithm that issues pulses ‘as soon as possible’ for the two-qubit gate sequence while avoiding scheduling pulses on neighbouring barriers is used. A general two-qubit Clifford gate can be decomposed into single-qubit Clifford gates acting on either qubit, followed by a two-qubit gate, followed again by single-qubit Clifford gates acting on either qubit. We implement the parallelization of a general two-qubit Clifford gate by parallelizing its respective parts.

Mirroring gates for exchange-only qubits

The flexibility of exchange-only qubits can further be enhanced by the use of mirroring gates. Which exchange axis is responsible for n rotations and which axis is responsible for z rotations—the configuration of the exchange-only qubit—is defined by the location of the quantum dot pair that performs PSB readout. This gives each exchange-only qubit two possible configurations. The configuration can be changed by choosing the other possible quantum dot pair to perform PSB readout or it can be changed in situ, by applying mirroring gates consisting of three exchange π-pulses around the n− , z− and n− axes. Here changing the quantum dot pairs performing readout is not possible but applying mirroring gates after initialization and before readout provide us with this extra flexibility. Choosing the optimal combination of configurations for the two qubits has two advantages. First, two-qubit gates with fewer pulses can become available⁴⁹, reducing overall two-qubit gate complexity. Second, because gates have an unequal distribution of pulses on the various exchange axes, fewer pulses can be scheduled on exchange axes with reduced quality factors, increasing average gate fidelity.

Randomized benchmarking

We choose to use RB as the primary tool to characterize the fidelities of single-qubit and two-qubit gates. For RB, a random sequence of N single-qubit or two-qubit Clifford gates is applied, followed by a recovery gate that returns the qubit(s) to a known state. The Clifford gates are decomposed into exchange pulses as described in the previous section. By interpolating the state recovery probability as a function of N, the average error per Clifford gate can be extracted. In more detail, for the single-qubit experiments, we use a technique called blind RB²⁴, in which each seed is run twice, recovered one time to |0⟩ and another time to |1⟩. Analysis of the resulting curves allows extraction of the leakage error per Clifford, which we find to be in line with expectations given pulse durations and abundance of nuclear spins (see the main text).

When performing parallelized RB, single-qubit gates are issued at the same time on the respective qubits and the parallelized versions of the two-qubit gates in the gate library are used. We also emphasize that the full two-qubit state readout is performed. This ensures that state leakage is properly accounted for, which can be challenging when only partial information of the two-qubit state is extracted (Extended Data Fig. 4). For single-qubit and two-qubit RB, we report the mean fidelity and error of the mean of five measurements that each are averaged over 20 random seeds. For interleaved two-qubit RB, we use five measurements with 25 random seeds each.

Monte Carlo simulation

To improve our understanding of the system, we estimate the impact of magnetic noise and charge noise with a Monte Carlo simulation by assuming quasi-static noise δB and δJ on both the Zeeman and exchange coupling, respectively. The two-qubit gate fidelity is then calculated as the average projection of the expected state with that of the final state for all computational states. The magnitude of δB is chosen to fit the average pure dephasing time ${\overline{T}}_2^{*}$, calculated as $\frac{1}{{\overline{T}}_2^{*}}=\sqrt{{\sum }_n^N{\left(\frac{1}{n{T}_2^{*}}\right)}^2}/N$ of the N quantum dot pairs involved, in which $\genfrac{}{}{0ex}{}{n}{}{T}_2^{*}$ is the singlet lifetime in the nth double dot (Extended Data Fig. 2e,j,o,t,y). An appropriate δJ is chosen to fit the number of resolvable exchange oscillations N_osc for each of the exchange axes (Extended Data Fig. 2d,i,n,s,x). We note that the evolution of the exchange coupling remains coherent well past the decay of the fixed frequency fit. The parameter N_osc is extracted by fitting a Gaussian decay to the oscillation envelope, providing a better estimate for the impact of charge noise as it is insensitive to small changes in the exchange oscillation frequency²⁵.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-025-09767-5.

Author contributions

M.T.M. performed measurements, with assistance from F.L., F.B., J.Z., M.J.C. and B.H. J.D.C., G.G.G., F.L. and M.T.M. developed the gate library. F.L. wrote the measurement software, with assistance from J.Z., R.F., F.B., M.T.M. T.M.M. and M.R. F.A.M. estimated the error contributions. G.G.G., F.A.M., S. Premaratne, A.R. and A.Y.M. developed the XEB implementation for exchange-only qubits. S.N., O.K.Z., A.N., P.L.B., J.D.R., D.K., T.F.W., E.J.C., J.C., R.S., R.O., B.P. and H.E.E. provided feedback used for the development of the fabrication process. F.L., F.B., N.K., R.F., R.W.M., L.P.O.I. and S. Pellerano developed the pre-distortion framework. L.F.L. designed the thermalization hardware. H.C.G., E.H., M.M.I., S.A. and R.P. fabricated the sample. M.T.M. and F.L. wrote the manuscript, with contributions from F.A.M., G.G.G., N.C.B., J.R., J.S.C. and input from all authors.

Peer review

Peer review information

Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

Data availability

The raw data and analysis that support the findings of this study are available in the Zenodo repository⁵⁰ (http://zenodo.org/doi/10.5281/zenodo.15090726).

Competing interests

The authors declare no competing interests.

Extended data

is available for this paper at 10.1038/s41586-025-09767-5.

Supplementary information

The online version contains supplementary material available at 10.1038/s41586-025-09767-5.