Main

To realize the promised benefits of quantum computing, large arrays of qubits will need to operate within densely packed cryogenic platforms, and the heating effects will eventually impose temperatures well above the millikelvin regime12,14,15,16,17,18. Spins in semiconductor quantum dots are rising candidates for this undertaking, thanks to their low error rates, long information hold time and industrial manufacturing compatibility2,3,6,22. Initial studies of spin qubit operation above 1 K have verified its feasibility, despite suffering from degraded state-preparation-and-measurement (SPAM) and gate fidelities15,16,17,18. Tackling these challenges requires combining previously unknown device designs and engineering techniques, in areas from initialization to control and readout.

In this work, we operate electron-spin qubits in silicon with SPAM and universal logic fidelities approaching the requirements for surface code error correction19,20,21,23. We enable deterministic two-qubit initialization in silicon above 1 K by an entropy-transferring algorithmic initialization protocol based on two-qubit logic and single-shot readout. The excellent high-temperature performance of semiconductor spin qubits underpins their potential for scalability and integration with classical control electronics. We elaborate on this presenting a complete error analysis in the two-qubit space and characterize every aspect of operation at different temperatures and external magnetic field B0 to open up further studies on error correction and performance of scaled-up systems.

Device and two-qubit operation

We conduct our study on a prototype two-qubit processor based on a silicon-metal-oxide-semiconductor (SiMOS) double quantum dot (Fig. 1a,b). Each qubit is encoded in the spin state of an unpaired electron24,25. The device incorporates multi-level aluminium gate-stacks26 fabricated on an isotopically enriched 28Si substrate with 50 ppm residual 29Si (ref. 27). The quantum dots are electrostatically defined in areas of around 80 nm2 underneath the plunger gates (P1, P2) at the Si/SiO2 interface. An exchange gate (J) controls the inter-dot separation and two-qubit exchange28,29,\({T}_{1}^{{\rm{PSB}}}\) nor charge readout, whereas above 1 K, these two limitations are present. T1-induced errors increase at a higher rate than charge readout errors and seem to be the dominating factor towards even higher temperatures. Overall, SPAM around 1 K is comparable to that at millikelvin temperatures and remains workable at least until 1.4 K.

Finally, we test repeated parity readout at T = 1 K. We apply machine learning to infer the parity readout errors and probabilities during SPAM and reconstruct the true initial state parity using the cumulative readout outcomes39. Figure 2e shows this protocol along with the results of our analysis. See the Methods for details on the machine learning approach. The SPAM fidelities are captured by Pinit and Pread, and the probabilities of state changes during each readout cycle are captured by Peven→odd and Podd→even. With the algorithmic |↓↓⟩ initialization and using 20 readout cycles (n = 20), we infer Pinit,even, Pinit,odd to be 99.34 ± 0.27%, 94.67 ± 0.73%, and Pread,even, Pread,odd to be 99.34 ± 0.08%, 96.15 ± 0.44% respectively. For the |↓↓⟩ initialization with n = 5, the reconstructed Pblockade increases from 99.2% to 99.3%, and for the |↑↓⟩ initialization with n = 12, Pblockade decreases from 5.8% to 5.1%. The full set of probabilities are detailed in the Supplementary Information.

Single-qubit performance

Relaxation time (T1) and dephasing time (T2) as well as the single-qubit control fidelities tend to be notable in silicon40,41,42, with single-qubit fidelities around 99% previously attained at T > 1 K (refs. 16,17). The improvements in device materials and design further improved the performance of individual qubits.

We first study T1 and T2 (Fig. 3a,b) in this device in the (1, 3) and (5, 3) charge states near the optimal B0. These regimes are expected to have similar relaxation mechanisms as (3, 3) (ref. 43), which is studied in ref. 16. However, the absence of a micromagnet in the present study affects some of the physical mechanisms of relaxation and decoherence. The dominating relaxation contributors—charge noise, Johnson noise, Orbach and Raman phonon scattering, and their coupling to the qubits—may change at different temperature ranges, giving rise to an intricate temperature dependence of T1 (ref. 15). Moreover, the parity readout convolutes the relaxation and thermal excitation of both qubits. Nevertheless, we note that the temperature dependence of T1 shown in Fig. 3a falls between T−2.0 and T−3.1 above 1 K. Moreover, the thermal equilibrium shifts from |↓↓⟩ when kBT ≪ hfqubit to a mixed state when kBT ≥ hfqubit. See Extended Data Fig. 4 for a more detailed T1 analysis. The temperature dependence of \({T}_{2}^{{\rm{Hahn}}}\) in different configurations falls between T−1 and T−1.1, whereas \({T}_{2}^{* }\) scales uniformly to T−0.2. The temperature scaling power of both T1 and T2 are lower than those in most of the previous results15,16,18. We expect that the more purified silicon and the absence of a micromagnet in this study affect some of the physical mechanisms of relaxation and decoherence. The bias of Z errors (dephasing noise) to X errors (depolarization noise) can be indicated by the T1/T2 ratio shown in Extended Data Fig. 5a.

Fig. 3: Single-qubit performance.
figure 3

a, Relaxation time, T1, of different states as a function of temperature from 0.14 K to 1.4 K in different charge configurations for Q1, Q2. b, Dephasing times, \({T}_{2}^{* }\) and \({T}_{2}^{{\rm{Hahn}}}\), as a function of temperature from 0.14 K to 1.4 K in different charge configurations for Q1. c, Single-qubit noise PSD of Q1 based on the CPMG protocol44,45 at different temperatures at B0 = 0.79 T. d, Single-qubit Clifford gate fidelity as a function of B0 from 50 mT to 1 T at T = 1.2 K in different charge configurations for Q1. e, Single-qubit Clifford gate fidelity as a function of temperature at B0 = 0.4 T with the one-electron configuration. f, Demonstration of a dressing protocol, the SMART protocol on Q1 (ref. 8) at B0 = 0.5 T and T = 1 K with fRabi = 1.44 MHz, showing periodical modulation optima. Error bars represent the 95% confidence level.

Full size image

We perform noise spectroscopy based on a Carr–Purcell–Meiboom–Gill (CPMG) protocol44,45, which uses a single qubit as a noise probe to determine the noise power spectral density (PSD) at different temperatures. As shown in Fig. 3c, the overall noise level rises with temperature within the detectable frequency range. In the white noise regime (<200 kHz), the noise power spectral density increases by an order of magnitude from 0.14 K to 1.2 K. We notice an increase in the apparent PSD at higher frequencies that we characterize as blue noise, possibly because of accumulated microwave pulse miscalibration, or an effect from the high-power driving9,46. See Extended Data Fig. 5e for the full set of power noise spectral density traces, and Extended Data Fig. 5f for another measurement of the microwave effect.

The optimized single-qubit Clifford fidelity in randomized benchmarking47 is up to 99.85 ± 0.01% (see Supplementary Information for data). Correction of crosstalk is crucial because of the relatively small difference in fESR in this device. See the Methods for crosstalk correction and the implementation of randomized benchmarking. As shown in Fig. 3d, we observe a fidelity reduction at low B0, limited by crosstalk (Extended Data Fig. 7c), or near an excited state degeneracy at high B0 in which dephasing is enhanced by spin–orbit coupling48. Even when kBT ≈ 7hfqubit, we still measure larger than 99% fidelities. The qubits are operable with distinguishable frequencies at B0 as low as 25 mT, at which point all operation protocols must be revisited4. See Extended Data Fig. 6 for the full study. These results suggest the possibility of ultralow B0 operation to markedly reduce the hardware and power cost49.

Furthermore, we extend the recent demonstration of a dressing protocol, the SMART protocol, for a single qubit from 0.1 K (refs. 4,5,8) to 1 K. See Extended Data Fig. 7e for the gate sequence. This demonstration substantiates the potential to continuously drive a large number of spin qubits with a global field above 1 K in future architectures.

Two-qubit performance

Two-qubit gate fidelities in silicon have recently reached the fault-tolerant requirements23,36,50,51,

Methods

Measurement setup

The full experimental setup is shown in Extended Data Fig. 1. The device is measured in a Bluefors XLD400 dilution refrigerator. The device is mounted on the cold finger. Within T = 1 K, elevation from the base temperature is achieved by switching on and tuning the heater near the sample. Temperatures above 1 K are attained by reducing the amount of He mixture in the circulation and consequently the cooling power. Temperature control becomes non-trivial above 1.2 K and nonviable above 1.5 K.

An external d.c. magnetic field is supplied by an American Magnetics AMI430 magnet. The magnetic field points in the [110] direction of the Si lattice. The d.c. voltages are supplied with Basel Precision Instruments SP927 LNHR DACs through d.c. lines with a bandwidth of 0–20 Hz. Dynamic voltage pulses are generated with a Quantum Machines OPX and combined with d.c. voltages by custom voltage combiners at the 50 K stage in the refrigerator. The OPX has a sampling time of 4 ns. The dynamic pulse lines in the fridge have a bandwidth of 0–50 MHz, which translates into a minimum rise time of 20 ns. Microwave pulses are synthesized using a Keysight PSG8267D Vector Signal Generator with the baseband I/Q and pulse modulation signals from the OPX. The modulated signal spans from 250 kHz to 44 GHz but is band-limited by the fridge line and the d.c. block.

The charge sensor comprises a single-island SET connected to a tank circuit for reflectometry measurement. The return signal is amplified by a Cosmic Microwave Technology CITFL1 LNA at the 4 K stage and a Mini-circuits ZX60-P33ULN+ LNA followed by two Mini-circuits ZFL-1000LN+ LNAs at room temperature. The Quantum Machines OPX generates the tones for the RFSET and digitizes and demodulates the signals after the amplification.

Device tune-up

We first load the electrons according to the map** of the double-dot charge configurations over a large range, using lock-in charge sensing measurement61 with the RFSET. The measurement can be done in the physical gate basis by swee** VP1 and VP2, as shown in Extended Data Fig. 2a, or in the virtual gate basis by swee** VP1 − VP2 and VJ, as shown in Fig. 1a. In the virtual gate basis, voltages of −0.32 VJ and −0.25 VJ are applied on P1 and P2 to compensate for the effect of pulsing J. During operation, each dot is loaded with an odd number of electrons, from which the unpaired electron carries the spin information. This is denoted as the (m + 1, n + 1) charge state in the charge maps, where m and n are even numbers.

The tune-up proceeds with locating the PSB region around the inter-dot charge transition, as indicated by the dashed square in Extended Data Fig. 2c,d. The initial PSB search involves loading a mixed spin state in (m + 1, n + 1), which has some probability of being even-parity (|↓↓⟩ or |↑↑⟩), and subsequently pulsing to a location near the inter-dot charge transition point. Single-shot charge readout is performed before and after reaching the location and the final readout signal is provided by subtracting the two signals. Except at ultra-low B0, the readout mechanism is dominated by parity readout because of the relatively large dEZ between the two qubits32. An even-parity spin state appears as blockaded in the PSB region, which translates to a lower radiofrequency signal compared with that from an unblockaded state. The averaged radiofrequency signal, therefore, indicates the probability of having an even-parity state across multiple shots.

The two-level behaviour in the PSB region is used to perform single-shot spin readout. The readout signal in each shot of the experiment is compared with a preset threshold that lies between the two levels, as we see in the readout histograms in Fig. 2b. We assign value 1 to a blockaded readout, and value 0 to an unblockaded readout. Finally, we average over all shots to obtain Pblockade for the statistics.

Extended Data Fig. 2f shows the ESR spectrum as a function of VJ, in which we identify two regimes. At VJ < 1.175 V, only two transitions pertaining to the driven rotation of the individual qubits are detected. Driven over time, these transitions correspond to the Rabi oscillations in Fig. 1e. At VJ > 1.175 V, in which the exchange energy is large, we see four transitions among the four two-qubit states corresponding to the controlled rotation operations38,50. The layout of the transitions, together with the background signal, shows the composition of the initialized qubit state. The traces in Fig. 2a are taken from these measurements at high VJ. A more scalable two-qubit operation is the electrically pulsed controlled phase operation (CZ)35,36. This is adopted in this work to construct the CZ gate (Extended Data Fig. 2g), or the DCZ gate in the main text.

Algorithmic initialization

When the qubit energy hfqubit is greater than the thermal energy kBT, electron-spin qubit initialization may rely on intrinsic polarization mechanisms such as spin-dependent tunnelling from a reservoir62,63,64, PSB17,18,49,65 or relaxation16,43,66. Higher-fidelity single-qubit state preparation can be achieved using initialization by measurement39,67 and conditional single-qubit pulses9,68. These approaches either partially rely on intrinsic polarization or require readout with a reservoir, which are incompatible with operation at elevated temperatures. In this work, we design a generic two-qubit algorithmic initialization protocol that works in conditions for which hfqubit is comparable to or less than kBT. The method is applicable to a large-scale qubit array, in which initialization and readout are performed pairwise16,49,65.

The algorithmic initialization protocol, as shown in Extended Data Fig. 3a, proceeds as follows:

  1. 1.

    Enter (m + 1, n + 1) to create two unpaired spins in the double-dot system.

  2. 2.

    This results in one of the |↓↓⟩, |↓↑⟩, |↑↓⟩ and |↑↑⟩ states. The probability of creating the ground state |↓↓⟩ decreases as the temperature increases, as the thermal energy becomes comparable or greater than the qubit exchange coupling and the Zeeman energies.

  3. 3.

    Ramp to the PSB region for parity readout and apply a filter that rejects odd-parity states. The parity readout preserves the even-parity states as long as it is performed faster than the spin relaxation time32.

    1. (a)

      If the state is unblockaded and thus determined as an odd-parity (|↓↑⟩, |↑↓⟩) or excited state, the initialization is restarted.

    2. (b)

      If the state is blockaded and thus determined as even-parity (|↓↓⟩, |↑↑⟩), the initialization proceeds to the next stage.

  4. 4.

    This results in either |↓↓⟩ or |↑↑⟩, with an increased probability of |↓↓⟩ from step 3. We calibrate the CZ gate at this stage, either from the exchange-induced splitting of the ESR transitions (Extended Data Fig. 2f) or from the CZ oscillations (Extended Data Fig. 2g).

  5. 5.

    A zero-CNOT (zCNOT) gate23 is performed to convert |↑↑⟩ into |↑↓⟩, leaving |↓↓⟩ unchanged. The construction of the zCNOT gate in this work is shown in Extended Data Fig. 2g.

  6. 6.

    Ramp to the PSB region for parity readout, and apply a filter that rejects odd-parity states.

    1. (a)

      If the state is unblockaded and thus determined as |↑↓⟩ or an excited state, the initialization is restarted.

    2. (b)

      If the state is blockaded and thus determined as |↓↓⟩, the initialization is determined to be completed.

  7. 7.

    The resulting state is purely |↓↓⟩.

The if conditions above are implemented using real-time logic in the FPGA.

The protocol can also be adapted to prepare any other state on the parity basis. |↑↓⟩ and |↓↑⟩ can be prepared from |↓↓⟩ with a microwave π pulse on Q1 and Q2. |↑↑⟩ can be prepared by replacing the zCNOT with CNOT in the algorithm.

We test the algorithmic initialization in a wide range of B0 from 1 T down to 25 mT. The results at different stages of the protocol are seen in Fig. 2a and Extended Data Fig. 3b. Stage I, the outcome of a 100-μs ramp into the operation point, has a mixture of |↓↓⟩, |↓↑⟩, |↑↓⟩ and |↑↑⟩ states and the measured ESR transitions are almost indistinguishable. After Stage II, the output is a mixture of |↓↓⟩ and |↑↑⟩ with the odd-parity states or excited states filtered out through PSB, which can be identified from the associated ESR transitions. Stage III converts the remnant |↑↑⟩ into |↑↓⟩, which is then filtered out through PSB because of the odd parity.

It is also important to assess the time cost for the algorithmic initialization, as it involves multiple control and readout iterations. The table in Extended Data Fig. 3b breaks down the time spent on control and readout. We see that the readout integration time tintegration dominates the time consumption. At B0 = 85 mT and T = 1 K, the full algorithmic initialization takes an average of around three iterations, which totals around 150 μs. Evaluating this in the context of different B0 and temperatures, we obtain the dependence shown in Extended Data Fig. 3c,d. At ultralow B0, for which a reduction in the control and readout fidelity is seen, Niteration decreases, possibly because the system deviates from the parity basis. Higher B0 provides a larger qubit energy, increasing the likelihood of obtaining a |↓↓⟩ state after the load ramp and reducing Niteration. Similarly, Niteration also increases with higher temperatures. At B0 above 1 T, the onset of excited state-level crossings enhances spin randomization after the load ramp, and thus more Niteration is required. We expect that Niteration may be reduced by incorporating corrective control based on measurement9,68 to accelerate the polarization towards the target state.

SPAM error analysis with repeated readout

A more comprehensive SPAM error analysis uses machine learning of the increased statistics from multiple measurements. The experimental sequence consists of initialization followed by repeated parity readout that results in a series of binary measurement outcomes m1, m2, …, mn, where mi ∈ {evenparity = 0, oddparity = 1}. This initialization-(measurement)n sequence is performed 1,000 shots.

A hidden Markov model (HMM) can describe this series of measurements formalism in which the true, but hidden, spin state s1, s2, …, sn follows the Markov chain and the measurement outcomes, mi, are probabilistically related to the underlying spin state. Three different tensors completely determine HMMs:

  1. 1.

    A start probability vector, Π, encoding the initializing probabilities in each spin state.

  2. 2.

    A transition probability matrix, A, encoding the probabilities of transiting between spin states during measurements.

  3. 3.

    A measurement probability matrix, Θ, encoding the probability of the measurement outcomes conditioned on the current hidden spin state.

To find the likely HMM model for a given set of data, we perform expectation maximization in which we maximize the marginal likelihood, which is dependent on the marginalized hidden spin state, such that

$$\begin{array}{l}L({\boldsymbol{\Pi }},{A},{\Theta }\,;{\bf{m}})\,:=\,p({\bf{m}}|{\boldsymbol{\Pi }},{A},{\Theta })\\ \,\,=\int \,p({\bf{s}},{\bf{m}}|{\boldsymbol{\Pi }},{A},{\Theta }){\rm{d}}{\bf{s}}.\end{array}$$
(1)

For HMM models, there exists the Baum–Welch algorithm that can perform this expectation maximization by an iterative update rule, without the need for backpropagation of gradients69.

We use the Cramer–Rao bound to quantify the level of uncertainty in these parameters when fitted by expectation maximization70. The Cramer–Rao bound states that if \({{\rm{est}}}_{{\boldsymbol{\theta }}}({\bf{m}})\) is an unbiased estimate of the parameters \({\boldsymbol{\theta }}:=({\boldsymbol{\Pi }},{A},{\Theta })\) given the data m, such as that produced by expectation maximization, then

$${{\rm{cov}}}_{{\boldsymbol{\theta }}}\left({{\rm{est}}}_{{\boldsymbol{\theta }}}({\bf{m}})\right)\ge I{\left({\boldsymbol{\theta }};{\bf{m}}\right)}^{-1},$$
(2)

where \(I{({\boldsymbol{\theta }};{\bf{y}})}_{ij}=-{\partial }^{2}\log L({\boldsymbol{\theta }}\,;{\bf{m}})/\partial {\theta }_{i}\partial {\theta }_{j}\), the Fisher information matrix. Therefore, we can obtain lower bounds on the uncertainty of each parameter from the diagonal elements of the inverse of the Fisher information matrix. We used the Forward–Backward algorithm to compute the marginal likelihood defined in equation (1) needed to compute the Fisher information matrix.

Finally, we use the Viterbi algorithm to compute the most likely set of true spin states that gave rise to the set of measurements given a set of model parameters69,71.

Crosstalk correction

The relatively small ΔEZ even at higher B0 requires cancellation of crosstalk between the two qubits, that is, the effect on the other qubit when one qubit is being driven. This can be addressed to the first order by considering the following aspects.

To cancel off-resonance driving, we enforce

$$\sqrt{\Delta {E}_{{\rm{Z}}}^{2}+{f}_{{\rm{Rabi}}}^{2}}=N{f}_{{\rm{Rabi}}},$$
(3)

where fRabi is the Rabi frequency of the target qubit, and N = 4, 8, 12, …. Consequently, each π/2 microwave pulse on the target qubit incurs a full 2πN off-resonance rotation on the ancilla qubit, as exemplified in Extended Data Fig. 7a. Failure to cancel the off-resonance driving can result in large errors under parity readout, as shown in Extended Data Fig. 7b. With N = 4, this cancellation criterion dictates the fastest Rabi possible and is therefore expected to limit the single-qubit gate fidelities, especially at low B0 where ΔEZ is small. The full set of fRabi used for single-qubit randomized benchmarking at different B0 is shown in Extended Data Fig. 7c. In this case, we can alternatively execute X(π/2) as a 3π/2 gate for faster driving at the cost of redundancy. We implemented this with the three- and five-electron qubit at 0.1 T, 1.2 K in Fig. 3d.

In two-qubit sequence runs, it is also necessary to correct AC Stark shift by an amount of

$$\frac{{f}_{{\rm{Rabi}}}^{2}}{2\Delta {E}_{{\rm{Z}}}},$$
(4)

apart from cancelling the off-resonance driving. Extended Data Fig. 7d measures the AC Stark shift on an ancilla qubit by preparing it on the equator, driving it off-resonantly and projecting the phase. Before the correction, the AC Stark shift is seen as the linear fringes that correspond to the phase accumulation given by equation (4).

We note that the above cancellation of crosstalk does not prevent it from incurring errors. The perturbation on the ancilla qubit induces decoherence. At ultra-low B0 at which ΔEZ becomes diminishing, higher-order crosstalk terms cannot be neglected, and the control of individual qubits becomes unmanageable. However, these problems are circumvented in the SMART control scheme, which addresses all the qubits simultaneously.

Randomized benchmarking

Single-qubit randomized benchmarking sequences for Fig. 3d–e are constructed from elementary π/2 gates [X(π/2), Z(π/2), −X(π/2), −Z(π/2)], π gates [X(π), Z(π)] and an I gate. Each Clifford gate contains one physical elementary gate on average, excluding the virtual Z(π/2) and Z(π) gates.

To optimize the single-qubit gate fidelity, we study different B0 (Fig. 3d) and tightly confine the qubits with low barrier gate voltages to reduce noise coupling. In single-qubit randomized benchmarking, the coherent driving decouples the qubit from noise to a certain extent72, and the random rotations of the qubit also have the effect of refocusing38,73. Here we optimize the microwave power and thus fRabi, such that the spins are driven quickly without excessive microwave-induced noise46.

Two-qubit randomized benchmarking sequences for Fig. 4b are constructed from single-qubit elementary π/2 gates [X1(π/2), Z1(π/2), X2(π/2), Z2(π/2)] for Q1 and Q2, and a two-qubit elementary gate DCZ. Each Clifford gate contains 1.8 single-qubit elementary gates and 1.5 two-qubit elementary gates on average. All gates are sequentially executed, which means Q1 idles while X2(π/2) or Z2(π/2) takes place, and the same for Q2. The generated random sequences are used in both randomized benchmarking and FBT. In the case of IRB, we incorporate an interleaved DCZ gate between adjacent Clifford gates. The experimental implementation and the analysis protocol are shown in Extended Data Fig. 9a,b, and the IRB results are shown in Extended Data Fig. 9c.

We then fit the randomized benchmarking decay curve to the formula38,41

$$a{{\rm{e}}}^{-{(bx)}^{c}}+d,$$
(5)

from which 1 − 0.5b gives the Clifford fidelity in single-qubit randomized benchmarking, and 1 − 0.75b gives the Clifford fidelity in two-qubit randomized benchmarking. The term c represents the decay exponent and reflects the error Markovianity; a is subjected to the readout fidelity and d is close to 0.5.

From the two-qubit IRB decays, we first obtain an IRB fidelity50 of 99.8 ± 0.2% at T = 0.1 K and 97.7 ± 1.5% at T = 1 K for the DCZ gate. This fidelity reflects the combined effect of dephasing during texchange and echoing in the DCZ gate and the results can be understood from the stronger temperature dependence of \({T}_{2}^{{\rm{Hahn}}}\) compared with that of \({T}_{2}^{{\rm{* }}}\). We also note the numerical instabilities in IRB fidelities, which result in large error bars.

Fast Bayesian tomography

FBT53 is an agile gate set process tomography protocol that can self-consistently reconstruct all gate set process matrices based on previous calibration. In principle, FBT learns and updates the model using the gate sequence information and its experimental outcome. In this work, we feed FBT with the variable-length two-qubit randomized benchmarking sequences and the corresponding experimental data. Clifford gates in the randomized benchmarking sequences are decomposed into their elementary gate implementation of X1(π/2), Z1(π/2), X2(π/2), Z2(π/2) and DCZ. The randomized benchmarking experiments at T = 0.1 K and T = 1 K run through 32,000 and 26,000 sequences, respectively, sufficient for FBT to reliably reconstruct the error channels. We feed the native parity readout results directly to FBT, without converting them to the standard two-qubit measurement basis.

To initiate the FBT analysis, we must bootstrap the model from educated guesses to help the analysis converge with a finite amount of experiments. Here, we do this by injecting guessed fidelity numbers as introduced in refs. 53,74. FBT models each noisy gate \(\widetilde{G}\) as the product of the noise channel \(\mathop{G}\limits^{ \sim }=\varLambda G\) and the ideal gate G, in which the noise channel Λ is linearized about I by expressing it as Λ = I + ε. Each update of the FBT analysis is essentially on the statistics of the noise channel residuals ε. Extended Data Fig. 9d shows the reconstructed Pauli transfer matrices of the DCZ gate. Supplementary Information shows the reconstructed noise channel residuals of the three physical elementary gates DCZ, X1(π/2), and X2(π/2) at T = 0.1 K and T = 1 K.

As FBT does not guarantee that the reconstructed channels are physical or flag any gauge ambiguity, we perform CPTP projection and gauge optimization over the entire gate set at the output stage.

FBT extracts DCZ fidelities of 99.15 ± 0.13% at T = 0.1 K and 98.92 ± 0.67% at T = 1 K. Here, a single-qubit gate on one qubit always leaves the other qubit idling, which considerably limits the single-qubit process fidelities (Supplementary Information) and consequently the Clifford fidelity in two-qubit randomized benchmarking, even at T = 0.1 K (Fig. 4b). However, the reduction in the Clifford fidelity from 0.1 K to 1 K mainly originates from the degradation of the DCZ gate, exhibiting a similar factor.

Error taxonomy with pyGSTi

When examining the fidelity results, we are also interested in understanding the dominant error sources behind the DCZ gate infidelity and their variation at different temperatures. FBT is a flexible and efficient gate set process tomography that enables us to extract gate errors from randomized sequence runs53,74. To categorize the gate errors, we perform post-processing of the tomography results obtained by FBT using tools for decomposing errors implemented in the pyGSTi package59,60.

Error taxonomy for FBT can be achieved by converting the noise channels Λ for each gate to their error generator \({\mathbb{L}}\) using the following relationship:

$$G=\varLambda {G}_{0}={{\rm{e}}}^{{\mathbb{L}}}{G}_{0},$$
(6)

where G is the estimated noisy gate, and G0 is the ideal gate.

Using the pyGSTi package59,60, we project \({\mathbb{L}}\) into the subspace of Hamiltonian and stochastic errors, extracting the coefficients of each elementary error generator. We perform this analysis on each of the gates [DCZ, X1(π/2) and X2(π/2)] for both temperatures of 0.1 K and 1 K. The coefficients of the elementary error generators are represented in the Pauli basis and presented in the Supplementary Information. The five largest components of the Hamiltonian and stochastic errors for the DCZ gate are shown in Fig. 4c.

We also estimate the generator or entanglement infidelity \(1-{{\mathcal{F}}}_{{\rm{ent}}}\) based on these error coefficients, given by60

$$1-{{\mathcal{F}}}_{{\rm{ent}}}\approx \sum _{P}{s}_{P}+\sum _{P}{h}_{P}^{2},$$
(7)

where the sum is performed over the extracted coefficients and P denotes non-identity Pauli elements. The approximation is validated by the domination of Hamiltonian errors over stochastic errors in magnitude. To obtain the average gate fidelities (\({{\mathcal{F}}}_{{\rm{avg}}}\)), which are the quantities quoted based on IRB and FBT measurements, it can be connected to \({{\mathcal{F}}}_{{\rm{ent}}}\) in the following way75:

$${{\mathcal{F}}}_{{\rm{avg}}}=\frac{d\cdot {{\mathcal{F}}}_{{\rm{ent}}}+1}{d+1},$$
(8)

where d is the dimension of the Hilbert space (4 for a two-qubit system). This means that generally stochastic errors contribute more to the gate infidelities, even in the case in which the magnitudes of the Hamiltonian errors are larger.