Abstract
Highfidelity control of quantum bits is paramount for the reliable execution of quantum algorithms and for achieving fault tolerance—the ability to correct errors faster than they occur^{1}. The central requirement for fault tolerance is expressed in terms of an error threshold. Whereas the actual threshold depends on many details, a common target is the approximately 1% error threshold of the wellknown surface code^{2,3}. Reaching twoqubit gate fidelities above 99% has been a longstanding major goal for semiconductor spin qubits. These qubits are promising for scaling, as they can leverage advanced semiconductor technology^{4}. Here we report a spinbased quantum processor in silicon with singlequbit and twoqubit gate fidelities, all of which are above 99.5%, extracted from gateset tomography. The average singlequbit gate fidelities remain above 99% when including crosstalk and idling errors on the neighbouring qubit. Using this highfidelity gate set, we execute the demanding task of calculating molecular groundstate energies using a variational quantum eigensolver algorithm^{5}. Having surpassed the 99% barrier for the twoqubit gate fidelity, semiconductor qubits are well positioned on the path to fault tolerance and to possible applications in the era of noisy intermediatescale quantum devices.
Main
Quantum computation involves the execution of a large number of elementary operations that take a qubit register through the steps of a quantum algorithm^{6}. A major challenge is to implement these operations with sufficient accuracy to arrive at a reliable outcome, even in the presence of decoherence and other error sources. The higher the accuracy, or fidelity, of the operations, the higher the likelihood that nearterm applications for quantum computers come within reach^{7}. Furthermore, for most presently known algorithms, the number of operations that must be concatenated will unavoidably lead to excessive accumulation of errors, and these errors must be removed using quantum error correction^{1}. Correcting quantum errors faster than they occur is possible when the error probability per operation is below a certain threshold, known as the faulttolerance threshold. For the widely considered surface code, for instance, the faulttolerance threshold is between 0.6% and 1%, under certain assumptions, albeit at the cost of a large redundancy in the number of physical qubits^{2,3}.
Among all the candidate platforms, electron spins in semiconductor quantum dots have advantages, such as their long coherence times^{8}, small footprint^{9}, the potential for scaling up^{10} and the compatibility with advanced semiconductor manufacturing technology^{4}. Singlequbit operations of spin qubits in quantum dots achieve fidelities of 99.9% (refs. ^{11,12}) but the twoqubit gate fidelities reported vary from 92% to 98% (refs. ^{13,14}). This has limited the twoqubit Bellstate fidelities to 94% (ref. ^{15}) and quantum algorithms implemented with spin qubits gave only coarsely accurate outcomes^{16,17}. Pushing the twoqubit gate fidelity well beyond 99% requires not only low chargenoise levels and the elimination of nuclear spins by isotopic enrichment but also careful Hamiltonian engineering.
In this paper, using a precisely engineered twoqubit interaction Hamiltonian, we report the demonstration of singlequbit and twoqubit gates with fidelities above 99.5%. We use gateset tomography (GST) not only to characterize the gates and to quantify the fidelity but also to improve the gate calibration. The highfidelity gates allow us to compute the dissociation energy of molecular hydrogen with a variational quantum eigensolver (VQE) algorithm, reaching an accuracy for the dissociation energy of around 20 mHa, limited by readout errors.
We use a gatedefined double quantum dot in an isotopically enriched ^{28}Si/SiGe heterostructure^{17} (Fig. 1a), with each dot occupied by a single electron (see Methods). The spin states of the electrons serve as qubits. The spin states are measured with the help of a sensing quantum dot (SQD), which is capacitively coupled to the qubit dots^{18}. A micromagnet on top of the device provides a magnetic field gradient enabling electricdipole spin resonance^{19} and separates the resonance frequencies of the qubits in the presence of an external magnetic field (~320 mT) to 11.993 GHz (Q_{1}) and 11.890 GHz (Q_{2}). Singlequbit X and Y gates are implemented by frequencymultiplexed microwave signals applied to gate MW and virtual Z gates are implemented by a phase update of the reference frame^{20}. The plunger gates (LP and RP) control the chemical potentials of the quantum dots.
The native twoqubit gate for spin qubits uses the exchange interaction^{21,22}, originating from the wavefunction overlap of electrons in neighbouring dots. This selectively shifts the energy of the antiparallel spin states and, thus, enables an electrically pulsed adiabatic conditional Z (CZ) gate^{8,16,23}. The barrier gate (B) controls the tunnel coupling between the dots, allowing the precise tuning of the exchange coupling from <100 kHz to 20 MHz. To minimize the sensitivity to charge noise, we activate the exchange coupling while avoiding a tilt in the doubledot potential^{24,25} (Fig. 1a). This symmetric condition can be determined accurately by decoupled adiabatic exchange pulses inside a Ramsey sequence (Fig. 1c, d). The tunnel barrier is controlled by simultaneously pulsing gate B and compensating LP and RP to avoid shifts in the electrochemical potentials^{24}, constituting a virtual barrier gate. The detuning between quantum dots is controlled by additional offsets to the LP and RP pulses in opposite directions. As the decoupling pulses remove additional singlequbit phase accumulation from electron movement in the magnetic field gradient, the spinup probability of Q_{1} results in a symmetric chevron pattern, with the symmetry point at the centre (Fig. 1d).
Among the various quantum benchmarking techniques, quantum process tomography (QPT) is designed to reconstruct all details in a target process^{6}. Owing to the susceptibility of QPT to state preparation and measurement (SPAM) errors, selfconsistent benchmarking techniques such as GST^{26} and alternative techniques such as randomized benchmarking^{27} have been developed. In contrast to randomized benchmarking, GST inherits the advantage of QPT in that it reports the detailed process, which allows us to isolate Hamiltonian errors from stochastic errors and to correct for such errors in the control signals (Extended Data Fig. 5). In addition, GST accounts for gatedependent errors. We benchmark the fidelities of a universal gate set using GST^{26,28} (Fig. 2a). The gate set we choose contains an idle gate (I), sequentially operated singlequbit π/2 rotations about the \(\hat{x}\) and \(\hat{y}\) axes for each qubit (\({{\rm{X}}}_{{{\rm{Q}}}_{1}}\), \({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\), \({{\rm{X}}}_{{{\rm{Q}}}_{2}}\) and \({{\rm{Y}}}_{{{\rm{Q}}}_{2}}\)) and a twoqubit controlledphase (CZ) gate. A total of 36 fiducial sequences containing \(\{{\rm{n}}{\rm{u}}{\rm{l}}{\rm{l}},{{{\rm{X}}}_{{{\rm{Q}}}_{i}}}^{n=1,2,3},{{{\rm{Y}}}_{{{\rm{Q}}}_{j}}}^{n=1,3}\}\) on each qubit, where null (unlike the idle gate) has no waiting time, are used to tomographically measure the twoqubit state. These fiducials are interleaved by germ sequences and their powers up to a sequence depth of 16. Germs are short sequences of gates taken from the universal gate set (see Methods). They are repetitively executed to amplify different types of gate errors in the gate set, such that SPAM errors can be isolated. GST allows using a maximumlikelihood estimator to compute completely positive and tracepreserving process matrices for each element of the gate set^{6}. The gate fidelity can be calculated by comparing the measured process using the Pauli transfer matrix (PTM), \({ {\mathcal M} }_{\exp }\), with the ideal PTM, \({ {\mathcal M} }_{{\rm{ideal}}}\), \({F}_{{\rm{gate}}}=({\rm{Tr}}({ {\mathcal M} }_{\exp }^{1}{ {\mathcal M} }_{{\rm{ideal}}})+d)/\)[d(d+1)], where d is the dimension of the Hilbert space. These process matrices provide a detailed error diagnosis of the gate set, allowing for efficient feedback calibration^{29} (Fig. 2a). Analysing the error generator \( {\mathcal L} =\,\log ({ {\mathcal M} }_{\exp }{ {\mathcal M} }_{{\rm{ideal}}}^{1})\) provides easy access to information. For example, coherent Hamiltonian errors can be isolated from incoherent stochastic errors and singlequbit errors can be isolated from each other and from twoqubit errors^{30}.
Figure 2b, c shows the reduced PTMs of \({{\rm{X}}}_{{{\rm{Q}}}_{1}}\) and \({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\) operations in the Q_{1} subspace and Fig. 2d shows the full PTM of \({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\) in twoqubit space (\({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\otimes {{\rm{I}}}_{{{\rm{Q}}}_{2}}\)) containing additional errors from decoherence and crosstalk on Q_{2} while operating Q_{1} (see Extended Data Figs. 1 and 2 for other PTMs) and from unintentional entanglement due to a residual exchange interaction. The average singlequbit gate fidelity is 99.72% in the singlequbit subspace (\({{\rm{X}}}_{{{\rm{Q}}}_{1}}\): 99.68%; \({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\): 99.73%; \({{\rm{X}}}_{{{\rm{Q}}}_{2}}\): 99.61%; \({{\rm{Y}}}_{{{\rm{Q}}}_{2}}\): 99.87%; see Extended Data Fig. 2 for all error bars). A metric that is rarely reported is the singlequbit gate fidelity in the full twoqubit space, here 99.16% on average (see Methods and Extended Data Fig. 1). These results highlight that singlequbit benchmarking is not sufficient to identify all errors occurring during singlequbit operations. By analysing the error generators, we find that errors from uncorrelated dephasing of the idling qubit dominate the drop in singlequbit gate fidelity when characterized in the twoqubit space. Coherent, microwaveinduced phase shifts—the main source of crosstalk errors—have been corrected by applying a compensating phase gate to the idling qubit (Extended Data Fig. 4). The elimination of idling errors and other crosstalk errors from the microwave drive, such as through heating effects, will be a crucial step to improve the quality of the singlequbit operations further.
For a highfidelity adiabatic CZ gate, precise control of the exchange coupling, J, between the two qubits is required. Specifically, in order to avoid unintended state transitions due to nonadiabatic dynamics, we must be able to carefully shape the envelope of J. We characterize J over a wide range using a Ramsey sequence interleaved by a virtual barrier pulse with incremental amplitude v_{B}. Figure 3a shows the measured frequency shift of each qubit as functions of the barrier pulse amplitude and the state of the other qubit. The exchange interaction is modelled to be exponentially dependent on the barrier pulse amplitude \(J({v}_{{\rm{B}}})\propto {{\rm{e}}}^{2\alpha {v}_{{\rm{B}}}}\) (refs. ^{31,32}). The micromagnetinduced singlequbit frequency shifts are approximated by linear functions within the voltage window of the CZ gate in the numerical simulations. By fitting the measured datasets simultaneously to theoretical models (see Methods), J can be extracted very precisely as the difference between the two conditional frequencies of each qubit^{16,33} (Fig. 3b). The barrier pulse \({v}_{{\rm{B}}}\propto \,\log ({A}_{{v}_{{\rm{B}}}}(1\,\cos (2{\rm{\pi }}t/{t}_{{\rm{g}}{\rm{a}}{\rm{t}}{\rm{e}}}))/2)\) (Fig. 3d) compensates the exponential dependence such that J ∝ (1 − cos(2πt/t_{gate})) follows a cosine window function, which ensures good adiabaticity^{34} (Fig. 3e). In addition, the virtual gates are calibrated such that the symmetric operation point is maintained for each barrier setting, minimizing the influence of charge noise via the doubledot detuning. The most relevant remaining noise sources include charge noise, affecting J through fluctuations in the virtual barrier gate δv_{B}, and fluctuating qubit frequencies \(\delta {f}_{{{\rm{Q}}}_{1}}\) and \(\delta {f}_{{{\rm{Q}}}_{2}}\) from charge noise entering through artificial spin–orbit coupling from the micromagnet and residual nuclear spin noise coupling through the hyperfine interaction. By analysing the decay of the Ramsey oscillations at each transition frequency, individual dephasing times \({T}_{2}^{\ast }\) can be extracted and, from there, also δv_{B}, \(\delta {f}_{{{\rm{Q}}}_{1}}\) and \(\delta {f}_{{{\rm{Q}}}_{2}}\) (Fig. 3c).
Figure 4a shows an example GST pulse sequence that contains twice in a row the germ \([{\rm{C}}{\rm{Z}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{\rm{C}}{\rm{Z}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}]\). The PTM of the CZ gate obtained from GST is shown in Fig. 4b. Using the detailed information from the error generator to finetune the calibration parameters, we can achieve a CZ fidelity of 99.65 ± 0.15% (Extended Data Figs. 4 and 5). Error bars included here and elsewhere are the 2σ ≈ 95% confidence intervals computed using the Hessian of the loglikelihood function^{35}. The CZ error generator reveals that, at this point, incoherent errors dominate. The virtual barrier gate technique used here efficiently suppresses crosstalk errors during twoqubit gates. Therefore, we expect the CZ fidelity to be mostly affected by dephasing errors of idling qubits in a larger space, which can be corrected for using decoupling pulses. From the obtained PTMs, we can numerically estimate Bellstate fidelities by multiplications of the PTMs necessary to construct the corresponding state, giving an estimate of 97.75%–98.42%, neglecting SPAM errors, for the four Bell states (Fig. 4c and Extended Data Fig. 3).
Next, we use the highfidelity gate set in the context of an actual application, in order to provide a quantitative benchmark for future work under realistic conditions. Specifically, we implement a VQE algorithm to compute the groundstate energy of molecular hydrogen (H_{2}) (Fig. 5a). In a VQE algorithm, a quantum processor is used to implement a classically inefficient subroutine (see Methods and Extended Data Fig. 6). The second quantized H_{2} Hamiltonian can be mapped onto two qubits under the Bravyi–Kitaev (BK) transformation H = h_{0}II + h_{1}ZI + h_{2}IZ + h_{3}ZZ + h_{4}XX + h_{5}YY. Here I, X, Y and Z are Pauli operators, for example, ZI is shorthand for Z ⊗ I, and the coefficients h_{0}–h_{5} are classically computable functions of the internuclear distance, R. Figure 5b shows the schematic of the VQE algorithm and its circuit implementation for a H_{2} molecule. The qubit is initialized in 01⟩, which represents double occupation of the lowest molecular orbital, corresponding to the Hartree–Fock (HF) ground state. A parameterized ansatz state is then prepared by considering single and double excitation, which, after the BK transformation, yields \(\psi (\theta )\rangle ={{\rm{e}}}^{{\rm{i}}\theta {\rm{XY}}}01\rangle \), with θ the parameter to variationally optimize. By performing partial tomography on the ansatz state with an initial guess θ_{0}, the expectation value of the Hamiltonian for ψ(θ_{0})⟩ can be calculated. A classical computer can efficiently compute the next guess θ_{1} as the new input for the quantum computer. This loop is iterated until the result converges. For a H_{2} molecule, there is only one parameter θ to optimize, thus, a scan of the entire parameter range of 2π with finite samples is sufficient to interpolate the smoothly changing measured expectation values. This emulates a real variational algorithm, where θ can be estimated to arbitrary precision by increasing the number of repetitions to suppress statistical fluctuations^{36}. Figure 5c shows the partial tomography result after normalization of the visibility window. The data demonstrate highquality phase control in the quantum circuits. The deviations in the oddparity expectation values indicate correlations in the readout of the two qubits^{37}. Figure 5d shows the energy curves of the H_{2} molecule from both theory^{38} and the VQE experiment. We observe a minimum energy at around 0.72 Å and an error of approximately 20 mHa at the theoretical bond length 0.7414 Å, mainly attributed to slow drift in the readout parameters. This accuracy matches the results obtained using superconducting and trapped ion qubits with comparable gate fidelities^{36,39}.
The twoqubit gate with fidelity above 99.5% and singlequbit gate fidelities in the twoqubit gate space above 99% on average place semiconductor spin qubit logic at the error threshold of the surface code. Recently, a twoqubit operation between nuclear spin qubits in silicon, mediated by an electron spin qubit, has been demonstrated to surpass 99% fidelity as well, further highlighting that semiconductor spin qubits offer precise twoqubit logic^{40}. Independent studies have shown spin qubit readout with a fidelity above 98% in only a few μs (ref. ^{41}), with further improvements underway^{42}. Combining highfidelity initialization, readout and control into a demonstration of fault tolerance poses several key challenges to be overcome. First, sufficiently large and reliable quantum dot arrays must be constructed, with good connectivity between the qubits. Second, the fidelities achieved in smallscale systems must be maintained across such larger systems, which will require reducing idling and crosstalk errors. The same advances will allow us to implement more sophisticated algorithms in the noisy intermediatescale quantum era, such as solving energies involving excited states of more complex molecules.
Methods
Measurement setup
The measurement setup and device are similar to those used in ref. ^{17}. We summarize a few key points and all the differences here. The gates LP, RP and B are connected to arbitrary waveform generators (AWGs, Tektronix 5014C) via coaxial cables. The position in the chargestability diagram of the quantum dots is controlled by voltage pulses applied to LP and RP. Linear combinations of the voltage pulses applied to B, LP and RP are used to control the exchange coupling between the two qubits at the symmetry point. The compensation coefficients are v_{LP}/v_{B} = −0.081 and v_{RP}/v_{B} = 0.104. A vector signal generator (VSG, Keysight E8267D) is connected to gate MW and sends frequencymultiplexed microwave bursts (not necessarily timemultiplexed) to implement electricdipole spin resonance (EDSR). The VSG has two I/Q input channels, receiving I/Q modulation pulses from two channels of an AWG. I/Q modulation is used to control the frequency, phase and length of the microwave bursts. The current signal of the sensing quantum dot is converted to a voltage signal and recorded by a digitizer card (Spectrum M4i.44), and then converted into 0 or 1 by comparing it to a threshold value.
Two differences between the present setup and that in ref. ^{17} are that (1) the programmable mechanical switch is configured such that gate MW is always connected to the VSG and not to the cryoCMOS control chip and (2) a second AWG of the same model is connected to gate B, with its clock synchronized to the first AWG.
Gate calibration
In the gate set used in this work, \(\{{\rm{I}},{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{\rm{C}}{\rm{Z}}\}\), the duration of the I gate and the CZ gate are set to 100 ns, and we calibrate and keep the amplitudes of the singlequbit drives fixed and in the linearresponse regime, where the Rabi frequency is linearly dependent on the driving amplitude. The envelopes of the singlequbit gates are shaped following a ‘Tukey’ window, as it allows adiabatic singlequbit gates with relatively small amplitudes, thus, avoiding the distortion caused by a nonlinear response. The general Tukey window of length t_{p} is given by
where r = 0.5 for our pulses. Apart from these fixed parameters, there are 11 free parameters that must be calibrated: singlequbit frequencies \({f}_{{{\rm{Q}}}_{1}}\) and \({f}_{{{\rm{Q}}}_{2}}\), burst lengths for singlequbit gates t_{XY1} and t_{XY2}, phase shifts caused by singlequbit gates on the addressed qubit itself ϕ_{11} and ϕ_{22}, phase shifts caused by singlequbit gates on the unaddressed ‘victim qubit’ ϕ_{12} and ϕ_{21} (ϕ_{12} is the phase shift on Q_{1} induced by a gate on Q_{2} and similar for ϕ_{21}), the peak amplitude of the CZ gate \({A}_{{v}_{{\rm{B}}}}\) and phase shifts caused by the gate voltage pulses used for the CZ gate on the qubits θ_{1} and θ_{2} (in addition, we absorb into θ_{1} and θ_{2} the 90° phase shifts needed to transform diag(1, i, i, 1) into diag(1, 1, 1, −1)).
For singlequbit gates, \({f}_{{{\rm{Q}}}_{1}}\) and \({f}_{{{\rm{Q}}}_{2}}\) are calibrated by standard Ramsey sequences, which are automatically executed every 2 h, at the beginning and in the middle (after 100 times the average of each sequence) of the GST experiment. The EDSR burst times t_{XY1} and t_{XY2} are initially calibrated by an AllXY calibration protocol^{43}. The phases ϕ_{11}, ϕ_{12}, ϕ_{21} and ϕ_{22} are initially calibrated by measuring the phase shift of the victim qubit (Q_{1} for ϕ_{11} and ϕ_{21}; Q_{2} for ϕ_{22} and ϕ_{12}) in a Ramsey sequence interleaved by a pair of \([{{\rm{X}}}_{{{\rm{Q}}}_{i}},{{\rm{X}}}_{{{\rm{Q}}}_{i}}]\) gates on the addressed qubit (Q_{1} for ϕ_{11} and ϕ_{12}; Q_{2} for ϕ_{22} and ϕ_{21}) (Extended Data Fig. 4).
The optimal pulse design presented in Fig. 3 gives a rough guidance of the pulse amplitude \({A}_{{v}_{{\rm{B}}}}\). In a more precise calibration of the CZ gate, an optional πrotation is applied to the control qubit (for example, Q_{1}) to prepare it into the 0⟩ or 1⟩ state, followed by a Ramsey sequence on the target qubit (Q_{2}) interleaved by an exchange pulse. The amplitude is precisely tuned to bring Q_{2} completely out of phase (by 180°) between the two measurements (Extended Data Fig. 4d, e). The phase θ_{2} is determined such that the phase of Q_{2} changes by zero (π) when Q_{1} is in the state 0⟩ (1⟩), corresponding to CZ = diag(1, 1, 1, −1) in the standard basis. The same measurement is then performed again with Q_{2} as the control qubit and Q_{1} as the target qubit to determine θ_{1} (ref. ^{16}).
In such a ‘conventional’ calibration procedure of the CZ gate, we notice that the two qubits experience different conditional phases (Extended Data Fig. 4). We believe that this effect is caused by offresonant driving from the optional πrotation on the control qubit. Similar effects can also affect the calibration of the phase crosstalk from singlequbit gates.
This motivates us to use the results from GST as feedback to adjust the gate parameters. The error generators not only describe the total errors of the gates but also distinguish Hamiltonian errors (coherent errors) from stochastic errors (incoherent errors). We use the information on seven different Hamiltonian errors (IX, IY, XI, YI, ZI, IZ and ZZ) of each gate to correct all 11 gate parameters (Extended Data Fig. 5), except \({f}_{{{\rm{Q}}}_{1}}\) and \({f}_{{{\rm{Q}}}_{2}}\), for which calibrations using standard Ramsey sequences are sufficient. For singlequbit gates, t_{XY1} and t_{XY2} are adjusted according to the IX, IY, XI and YI errors. The phases ϕ_{11}, ϕ_{12}, ϕ_{21} and ϕ_{22} are adjusted according to the ZI and IZ errors. For the CZ gate, θ_{1} and θ_{2} are adjusted according to the ZI and IZ errors, and \({A}_{{v}_{{\rm{B}}}}\) is adjusted according to the ZZ error. The adjusted gates are then used in a new GST experiment.
Theoretical model
In this section, we describe the theoretical model used for the fitting, the pulse optimization and the numerical simulations. The dynamics of two electron spins in the (1,1) charge configuration can be accurately described by an extended Heisenberg model^{21}
with \({{\bf{S}}}_{j}={({X}_{j},{Y}_{j},{Z}_{j})}^{{\rm{T}}}/2,\) where X_{j}, Y_{j} and Z_{j} are the singlequbit Pauli matrices acting on spin j = 1, 2, μ_{B} the Bohr’s magneton, g ≈ 2 the gfactor in silicon and h is Planck’s constant. The first and second terms describe the interaction of the electron spin in dot 1 and dot 2 with the magnetic fields \({{\bf{B}}}_{j}={({B}_{x,j},0,{B}_{z,j})}^{{\rm{T}}}\) originating from the externally applied field and the micromagnet. The transverse components B_{x,j} induce spinflips, thus, singlequbit gates if modulated resonantly via EDSR. For later convenience, we define the resonance frequencies by \(h{f}_{{Q}_{1}}=g{\mu }_{{\rm{B}}}{B}_{z,1}\) and \(h{f}_{{Q}_{2}}=g{\mu }_{{\rm{B}}}{B}_{z,2}\), and the energy difference between the qubits ΔE_{z} = gμ_{B}(B_{z,2} − B_{z,1}). The last term in the Hamiltonian of equation (2) describes the exchange interaction J between the spins in neighbouring dots. The exchange interaction originates from the overlap of the wave functions through virtual tunnelling events and is, in general, a nonlinear function of the applied barrier voltage v_{B}. We note that v_{B} determines the compensation pulses applied to LP and RP for virtual barrier control. We model J as an exponential function^{31,32}
where J_{res} ≈ 20–100 kHz is the residual exchange interaction during idle and singlequbit operations and α is the lever arm. In general, the magnetic fields \({{\bf{B}}}_{j}\) depend on the exact position of the electron. We include this in our model \({B}_{z,j}\to {B}_{z,j}({v}_{{\rm{B}}})={B}_{z,j}(0)+{\beta }_{j}{v}_{{\rm{B}}}^{\gamma },\) where β_{j} accounts for the impact of the barrier voltage on the resonance frequency of qubit j. The transition energies described in the main text are now given by diagonalizing the Hamiltonian from equation (2) and computing the energy difference between the eigenstates corresponding to the computational basis states {00⟩, 01⟩, 10⟩, 11⟩} (ref. ^{44}). We have
where \( {\mathcal E} (\xi \rangle )\) denotes the eigenenergy of eigenstate ξ ⟩ and 0⟩ = ↓⟩ is defined by the magnetic field direction.
In the presence of noise, qubits start to lose information. In silicon, charge noise and nuclear noise are the dominating sources of noise. In the absence of twoqubit coupling and correlated charge noise, both qubits decohere largely independently of each other, giving rise to a decoherence time set by the interaction with the nuclear spins and charge noise coupling to the qubit via intrinsic and artificial (via the inhomogeneous magnetic field) spin–orbit interaction. We describe this effect by \({f}_{{Q}_{1}}\to {f}_{{Q}_{1}}+\delta {f}_{{Q}_{1}}\) and \({f}_{{Q}_{2}}\to {f}_{{Q}_{2}}+\delta {f}_{{Q}_{2}}\), where \(\delta {f}_{{Q}_{1}}\) and \(\delta {f}_{{Q}_{2}}\) are the singlequbit frequency fluctuations. Charge noise can additionally affect both qubits via correlated frequency shifts and the exchange interaction through the barrier voltage, which we model as v_{B} → v_{B} + δv_{B}. In the presence of finite exchange coupling, one can define four distinct, pure dephasing times, each corresponding to the dephasing of a single qubit with the other qubit in a specific basis state. In a quasistatic approximation, the four dephasing times are then given by
Fitting qubit frequencies and dephasing times
The transition energies in equations (4)–(7) are fitted simultaneously to the measured results from the Ramsey experiment (see Fig. 3a). For the fitting, we use the NonLinearModelFit function from the software Mathematica with the least squares method. The best fits yield the following parameters: α = 12.1 ± 0.05 V^{−1}, β_{1} = −2.91 ± 0.11 MHz V^{−γ}, β_{2} = 67.2 ± 0.63 MHz V^{−γ}, γ = 1.20 ± 0.01 and J_{res} = 58.8 ± 1.8 kHz.
The dephasing times in equations (8)–(11) are fitted simultaneously to the measured results from the Ramsey experiment (see Fig. 3c) using the same method. The best fits yield the following parameters: δv_{B} = 0.40 ± 0.01 mV, \(\delta {f}_{{Q}_{1}}=11\pm 0.1{\rm{kHz}}\) and \(\delta {f}_{{Q}_{2}}=24\pm 0.7{\rm{kHz}}\).
Numerical simulations
For all numerical simulations, we solve the timedependent Schrödinger equation
and iteratively compute the unitary propagator according to
where \(\hbar =h/(2{\rm{\pi }})\) is the reduced Planck’s constant. Here H(t + Δt) is discretized into N segments of length Δt such that H(t) is constant in the time interval [t, t + Δt]. All simulations are performed in the rotating frame of the external magnetic field (B_{z,1} + B_{z,2})/2 and neglecting the counterrotating terms, making the socalled rotatingwave approximation. This allows us to choose Δt = 10 ps as a sufficiently small time step.
For the noise simulations, we included classical fluctuations of \({f}_{{Q}_{1}}\to {f}_{{Q}_{1}}+\delta {f}_{{Q}_{1}}\), \({f}_{{Q}_{2}}\to {f}_{{Q}_{2}}+\delta {f}_{{Q}_{2}}\) and v_{B} → v_{B} + δv_{B}. We assume the noise coupling to the resonance frequencies \(\delta {f}_{{Q}_{1}}\) and \(\delta {f}_{{Q}_{2}}\) to be quasistatic and assume 1/f noise for v_{B}, which we describe by its spectral density \({S}_{{v}_{{\rm{B}}}}(\omega )=\delta {v}_{{\rm{B}}}/\omega \), where ω is the angular frequency. To compute time traces of the fluctuation, we use the approach introduced in refs. ^{45,46} to generate timecorrelated time traces. The fluctuations are discretized into N segments with time Δt such that δv_{B}(t) is constant in the time interval [t, t + Δt), with the same Δt as above. Consequently, fluctuations that are faster than \({f}_{{\rm{\max }}}=\frac{1}{\Delta t}\) are truncated.
CZ gate
We realize a universal CZ = diag(1, 1, 1, −1) gate by adiabatically pulsing the exchange interaction using a carefully designed pulse shape. Starting from equation (2), the full dynamics can be projected on the oddparity space spanned by 01⟩ and 10⟩. The entangling exchange gate is reduced in this subspace to a global phase shift, thus, the goal is to minimize any dynamics inside the subspace. Introducing a new set of Pauli operators in this subspace σ_{x} = 01⟩⟨10 + 10⟩⟨01, σ_{y} = −i01⟩⟨10 + i10⟩⟨01 and σ_{z} = 01⟩⟨01 − 10⟩⟨10, we find
In order to investigate the adiabatic behaviour, it is convenient to switch into the adiabatic frame defined by \({U}_{{\rm{ad}}}={{\rm{e}}}^{\frac{{\rm{i}}}{2}{\tan }^{1}\left(\frac{hJ({v}_{{\rm{B}}}(t))}{\Delta {E}_{z}}\right){\sigma }_{y}}.\)The Hamiltonian accordingly transforms as
where the first term is unaffected and describes the global phase accumulation due to the exchange interaction, the second term describes the singlequbit phase accumulations and the last term, \(f(t)={h}^{2}\dot{J}/(4{\rm{\pi }}\Delta {E}_{z})\), describes the diabatic deviation proportional to the derivative of the exchange pulse. From equation (15) and equation (16), we assumed a constant ΔE_{z}(t) ≈ ΔE_{z} and hJ(t) ≪ ΔE_{z}. The transition probability from state ↑↓⟩ to ↓↑⟩ using a pulse of length t_{p} is then given by^{34}
From the first to the second line, we identify the integral by the (shorttimescale) Fourier transform, allowing us to describe the spinflip error probability by the energy spectral density S_{s} of the input signal f(t). Minimizing such errors is, therefore, identical to minimizing the energy spectral density of a pulse, a wellknown and solved problem from classical signal processing and statistics. Optimal shapes are commonly referred to as window functions W(t) due to their property of restricting the spectral resolution of signals. A highfidelity exchange pulse is consequently given by J(0) = J(t_{p}) and
while setting \(J(t)={A}_{{v}_{{\rm{B}}}}W(t){J}_{{\rm{res}}}\) (ref. ^{34}), with a scaling factor \({A}_{{v}_{{\rm{B}}}}\) that is to be determined. In this work, we have chosen the cosine window
from signal processing, which has a high spectral resolution. The amplitude \({A}_{{v}_{{\rm{B}}}}\) follows from condition equation (19). For a pulse length of t_{p} = 100 ns and a cosine pulse shape, we find \({A}_{{v}_{{\rm{B}}}}{J}_{{\rm{res}}}=10.06\,{\rm{MHz}}\). As explained in the main text, owing to the exponential voltageexchange relation, the target pulse shape for J(t) must be converted to a barrier gate pulse, following^{47}
Our numerical simulations predict an average gate infidelity 1 − F_{gate} < 10^{−6} without noise and 1 − F = 0.22 × 10^{−3} with the inclusion of noise through the fluctuations \(\delta {f}_{{Q}_{1}}\), \(\delta {f}_{{Q}_{2}}\) and δv_{B}, discussed in the previous section. The measured PTMs reveal much higher rates of incoherent errors, which we attribute to drifts in the barrier voltage on a timescale much longer than the timescale on which \(\delta {f}_{{Q}_{1}}\), \(\delta {f}_{{Q}_{2}}\) and δv_{B} were determined.
Gateset tomography analysis
We designed a GST experiment using the gate set \(\{{\rm{I}},{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{\rm{CZ}}\},\)where I is a 100ns idle gate, \({{\rm{X}}}_{{{\rm{Q}}}_{1}}\) (\({{\rm{Y}}}_{{{\rm{Q}}}_{1}}\)) and \({{\rm{X}}}_{{{\rm{Q}}}_{2}}\) (\({{\rm{Y}}}_{{{\rm{Q}}}_{2}}\)) are singlequbit π/2 gates with rotation axis \(\hat{x}\) (\(\hat{y}\)) on Q_{1} and Q_{2}, with durations of 150 ns and 200 ns, respectively, and CZ = diag(1, 1, 1, −1). A classic twoqubit GST experiment consists of a set of germs designed to amplify all types of error in the gate set when repeated and a set of 36 fiducials composed by the 11 elementary operations \(\{{\rm{null}},{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}},\) \({{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}\}\)required to carry out quantum process tomography of the germs^{48}. We use a set of 16 germs \(\{{\rm{I}},{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{\rm{CZ}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}\)\({{\rm{Y}}}_{{{\rm{Q}}}_{2}},{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{\rm{C}}{\rm{Z}},{{\rm{C}}{\rm{Z}}{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{2}},\) \({{\rm{C}}{\rm{Z}}{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{C}}{\rm{Z}}{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{X}}}_{{{\rm{Q}}}_{2}}{{\rm{X}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{1}}{{\rm{Y}}}_{{{\rm{Q}}}_{2}}\}\)(ref. ^{35}). Note that the null gate is the instruction for doing nothing in zero time, different from the idle gate. Simple errors such as errors in the rotation angle of a particular gate can be amplified by simply repeating the same gate. More complicated errors such as tilts in rotation axes can only be amplified by a combination of different gates. The germs and fiducials are then compiled into GST sequences, such that each sequence consists of two fiducials interleaved by a single germ or power of germs^{35} (as illustrated in Fig. 2a). The GST sequences are classified by their germ powers into lengths L = 1, 2, 4, 8, 16…, where a sequence of length n consists of n gates plus the fiducial gates. We note that the sequences used in GST are shorter than the sequences involved in other methods to selfconsistently estimate the gate performance, such as randomized benchmarking. As a result, GST suffers less from drift in qubit frequencies and readout windows induced by long sequences of microwave bursts.
After the execution of all sequences, a maximumlikelihood estimation is performed to estimate the process matrices of each gate in the gate set and the SPAM probabilities. We use the open source pyGSTi Python package^{49,50} to perform the maximumlikelihood estimation, as well as to design an optimized GST experiment by eliminating redundant circuits and to provide statistical error bars by computing all involved Hessians. The circuit optimization allows us to perform GST with a maximum sequence length L_{max} = 16 using 1,685 different sequences in total. The pyGSTi package quantifies the Markovianmodel violation of the experimental data, counting the number of standard deviations exceeding their expectation values under the χ^{2} hypothesis^{50}. This model violation is internally translated into a more accessible goodness ratio from 0 to 5, with 5 being the best^{49}, where we obtain a 4 out of 5 rating, indicating remarkably small deviations from expected results. The total number of standard deviations exceeding the expected results for each L, as well as the contribution of each sequence to this number, can be found in the pyGSTi report, along with the supporting data.
From the GST experiment, we have extracted the PTM \({ {\mathcal M} }_{\exp }\) describing each gate in our gate set \(\{{\rm{I}},{{\rm{X}}}_{{{\rm{Q}}}_{1}},{{\rm{Y}}}_{{{\rm{Q}}}_{1}},{{\rm{X}}}_{{{\rm{Q}}}_{2}},{{\rm{Y}}}_{{{\rm{Q}}}_{2}},{\rm{CZ}}\}\). The PTM is isomorphically related to the conventionally used χ matrix describing a quantum process. A completely positive, tracepreserving, twoqubit PTM has 240 parameters describing the process. To obtain insight into the errors of the gates in the experiment, we first compute the error in the PTM given by \(E={ {\mathcal M} }_{\exp }{ {\mathcal M} }_{{\rm{ideal}}}^{1}\), where we have adapted the convention to add the error after the ideal gate. The average gate fidelity is then conveniently given by
It is related to the entanglement fidelity via \(1{F}_{{\rm{ent}}}=\frac{d+1}{d}(1{F}_{{\rm{gate}}})\) (ref. ^{51}), where d is the dimension of the twoqubit Hilbert space. Although the PTM \( {\mathcal M} \) perfectly describes the errors, it is more intuitive to analyse the corresponding error generator \( {\mathcal L} =\,\log (E)\) of the process^{30}. The error generator \( {\mathcal L} \) relates to the error PTM E in a similar way as a Hamiltonian H relates to a unitary operation U = e^{−iH}. The error generator can be separated into several blocks. A full discussion about the error generator can be found in ref. ^{30}. In this work, we have used the error generator to distinguish the dynamics originating from coherent Hamiltonian errors, which can be corrected by adjusting gate parameters (see Extended Data Fig. 5), and from noisy/stochastic dynamics, which cannot be corrected easily. The coherent errors can be extracted by projecting \( {\mathcal L} \) onto the 4 × 4dimensional Hamiltonian space H. In the Hilbert–Schmidt space, the Hamiltonian projection is given by^{30}
where \({ {\mathcal L} }_{{\rm{\sup }}}\) is the error generator in Liouville superoperator form, P_{m} ∈ {I, X, Y, Z} are the extended Pauli matrices with m, n = 0, 1, 2, 3, 1_{d} is the ddimensional identity matrix and d = 4 is the dimension of the twoqubit Hilbert space. To improve the calibration of our gate set, we use the information of seven different Hamiltonian errors (IX, IY, XI, YI, ZI, IZ and ZZ). To estimate coherent Hamiltonian errors and incoherent stochastic errors, two new metrics are considered^{30}: the Jamiołkowski probability
which describes the amount of incoherent error in the process, and the Jamiołkowski amplitude
which approximately describes the amount of coherent Hamiltonian errors (Extended Data Table 1). Here \({\rho }_{{\rm{J}}}( {\mathcal L} )=( {\mathcal L} \otimes {1}_{{d}^{2}})[\varPsi \rangle \langle \Psi ]\) is the Jamiołkowski state and Ψ⟩ is a maximally entangling fourqubit state that originates from the relation of quantum processes to states in a Hilbert space twice the dimension via the Choi–Jamiołkowski isomorphism^{52}. For small errors, the average gate infidelity can be approximated by^{30}
For a comparison of the performance of the singlequbit gates with previous experiments reporting singlequbit gate fidelities, we compute the fidelities projected to the singlequbit space from the PTMs or the error generators. In Fig. 2 and Extended Data Fig. 2, singlequbit gate fidelities are estimated by projecting the PTMs onto the corresponding subspace. Let \({{\mathscr{P}}}_{j}\) be the projector on the subspace of qubit j, then the fidelity is given by
Error bars for the fidelity projected to the subspace are computed using standard error propagation of the confidence intervals of \({ {\mathcal M} }_{\exp }\) provided by the pyGSTi package. A more optimistic estimation for the fidelities in the singlequbit subspace is given by projecting the error generators instead of the PTMs.
Variational quantum eigensolver
We follow the approach of ref. ^{36} to using the VQE algorithm to compute the groundstate energy of molecular hydrogen, after mapping this state onto the state of two qubits. We include this information here for completeness. The Hamiltonian of a molecular system in atomic units is
where \({{\bf{R}}}_{i}\), M_{i} and Q_{i} are the position, mass and charge, respectively, of the ith nuclei and \({{\bf{r}}}_{j}\) is the position of the jth electron. The first two sums describe the kinetic energies of the nuclei and electrons, respectively. The last three sums describe the Coulomb repulsion between nuclei and electrons, nuclei and nuclei, and electrons and electrons, respectively. As we are primarily interested in the electronic structure of the molecule, and nuclear masses are a few orders of magnitude larger than the electron masses, the nuclei are treated as static point charges under the Born–Oppenheimer approximation. Consequently, the electronic Hamiltonian can be simplified to
Switching into the secondquantization representation, described by fermionic creation and annihilation operators, \({a}_{p}^{\dagger }\) and a_{q}, acting on a finite basis, the Hamiltonian becomes
where p, q, r and s label the corresponding basis states. The antisymmetry under exchange is retained through the anticommutation relation of the operators. The weights of the two sums are given by the integrals
where \({{\boldsymbol{\sigma }}}_{i}=({{\bf{r}}}_{i},{s}_{i})\) is a multiindex describing the position \({{\bf{r}}}_{i}\) and the spin s_{i} of electron i. Such a secondquantized molecular Hamiltonian can be mapped onto qubits using the Jordan–Wigner (JW) or the BK transformation^{5}. The JW transformation directly encodes the occupation number (0 or 1) of the ith spin orbital into the state (0⟩ or 1⟩) of the ith qubit. The number of qubits required after JW transformation is, thus, the same as the number of spin orbitals that are of interest. The BK transformation, on the other hand, encodes the information in both the occupation number and parities, whether there is an even or odd occupation in a subset of spin orbitals.
Taking molecular hydrogen in the HF basis as an example, we are interested in investigating the bonding (O_{1}↑⟩, O_{1}↓⟩) and the antibonding orbital state (O_{2}↑⟩, O_{2}↓⟩). The initial guess of the solution is the HF state in which both electrons occupy the O_{1}⟩ orbital. The JW transformation encodes the HF initial state as 1100⟩, representing \({N}_{{O}_{1}\downarrow }{N}_{{O}_{1}\uparrow }{N}_{{O}_{2}\downarrow }{N}_{{O}_{2}\uparrow }\rangle \) from left to right, where \({N}_{{O}_{i}S}\) is the occupation of the O_{i}S spin orbital with S = ↑, ↓. The BK transformation encodes the HF initial state as 1000⟩, where the first and the third qubits (counting from the right) encode the occupation number of the first and third spin orbitals (\({N}_{{O}_{1}\uparrow }=1\) and \({N}_{{O}_{2}\uparrow }=0\)), the second qubit encodes the parity of the first two spin orbitals (\(({N}_{{O}_{1}\uparrow }+{N}_{{O}_{1}\downarrow })\) mod 2 = 0) and the fourth qubit encodes the parity of all four spin orbitals (\(({N}_{{O}_{1}\uparrow }+{N}_{{O}_{1}\downarrow }+{N}_{{O}_{2}\uparrow }+{N}_{{O}_{2}\downarrow })\) mod 2 = 0). With the standard transformation rules for fermionic creation and annihilation operators, the system Hamiltonian becomes a fourqubit Hamiltonian
The subscripts are used to label the qubits. We see that, owing to the symmetry of the represented system in H_{BK}, qubit 2 and qubit 4 are never flipped, allowing us to reduce the dimension of the Hamiltonian to
where qubit 1 has been relabelled as qubit 2 and qubit 3 has been relabelled as qubit 1. The HF initial state is, therefore, reduced to 01⟩ and the Hamiltonian is rephrased to be consistent with the partial tomography expression in Fig. 5. This reduced representation requires only two qubits to simulate the hydrogen molecule. We emphasize that such a reduction of the BK Hamiltonian is not a special case for the H_{2} molecule but is connected to symmetry considerations to reduce the complexity of systems, in a scalable way.
VQE is a method to compute the groundstate energy of the Hamiltonian. The total energy can be directly calculated by measuring the expectation value of each Hamiltonian term. This can be done easily by partial quantum state tomography. All the expectation values are then added up with a set of weights (h_{0} through h_{5}). The weights are only functions of the internuclear separation (R) and can be computed efficiently by a classical computer. Here we use the OpenFermion Python package to compute these weights^{38}.
The main task of the quantum processor is, then, to encode the molecular spinorbital state into the qubits. The starting point is the HF initial state, which is believed to largely overlap with the actual ground state. In order to find the actual ground state, the initial state needs to be ‘parameterized’ into an ansatz to explore a subspace of all possible states. We apply the unitary coupled cluster (UCC) theory to the parameterized ansatz state, which is used to describe manybody systems and cannot be efficiently executed on a classical computer^{53}. The UCC operator has a format
with
representing single and double excitation of the electrons. The indices i and j label the occupied spin orbitals and m and n are the labels of the unoccupied spin orbitals. The vector θ is the set of all parameters to optimize. In the case of a H_{2} molecule, the UCC operator is transformed into a qubit operator as
where θ is a single parameter to variationally optimize.
Data availability
Data supporting this work are available at Zenodo, https://doi.org/10.5281/zenodo.5044450.
Code availability
The codes used for data acquisition and processing are from the open source Python packages QCoDeS (https://github.com/QCoDeS/Qcodes), QTT (https://github.com/QuTechDelft/qtt) and PycQED (https://github.com/DiCarloLabDelft/PycQED_py3). The codes used for the design and analysis of the gateset tomography experiment are from pyGSTi (https://github.com/pyGSTio/pyGSTi). The codes used for the design and analysis of the variational quantum eigensolver experiment are from OpenFermion (https://github.com/quantumlib/OpenFermion).
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Acknowledgements
We acknowledge discussions with P. Cerfontaine, C. BureauOxton, M. T. Madzik, A. Morello, J. Helsen, B. Terhal, M. Veldhorst and all the members of the spin qubit team, and technical assistance from O. Benningshof, M. Sarsby, R. Schouten and R. Vermeulen. This research was funded by the Dutch Ministry of Economic Affairs through the allowance for Top Consortia for Knowledge and Innovation (TKI) and the Army Research Office (ARO) under grant number W911NF1710274. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the ARO or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.
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X.X. performed the experiment, with help from N.S. and B.U. M.R. developed the theory model and analysed the data with X.X. N.S. fabricated the quantum dot device. A.S. and G.S. designed and grew the Si/SiGe heterostructure. X.X. and L.M.K.V. conceived the project. L.M.K.V. supervised the project. X.X., M.R. and L.M.K.V. wrote the manuscript, with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Twoqubit processes.
Average gate infidelities, process matrices (PTMs) and error generators of the six quantum gates in the chosen gate set. These results are analysed by the pyGSTi package using maximumlikelihood estimation.
Extended Data Fig. 2 Singlequbit processes.
Average gate infidelities and process matrices (PTMs) of the identity gates (idle gates) and singlequbit X/Y gates in the subspace of the individual qubits. The individual PTMs are calculated from the PTMs in the twoqubit space (see Methods).
Extended Data Fig. 3 Bell states predicted from the quantum processes.
Top panels show the real part of the reconstructed density matrices of the four Bell states \({\Psi }^{+}\rangle =(01\rangle +10\rangle )/\sqrt{2}\) (a), \({\Psi }^{}\rangle =(01\rangle 10\rangle )/\sqrt{2}\) (b), \({\Phi }^{+}\rangle =(00\rangle +11\rangle )/\sqrt{2}\) (c) and \({\Phi }^{}\rangle =(00\rangle 11\rangle )/\sqrt{2}\) (d). The colour code is the same as in Fig. 4. Bottom panels show the quantum circuit used to reconstruct the Bell states. \({{\rm{Z}}}_{{\rm{Q}}i}^{2}\) is a virtual πrotation around the \(\hat{z}\) axis on the ith qubit, which is executed by a phase update on the microwave reference clock of the qubit and, therefore, is errorfree. We numerically estimate the state fidelities to be 98.42% for the Ψ^{+}⟩ and Ψ^{−}⟩ states and 97.75% for the Φ^{+}⟩ and Φ^{−}⟩ states.
Extended Data Fig. 4 Initial gate calibrations.
a, Decomposition of singlequbit and twoqubit gates. After each microwave burst for singlequbit rotations, a corresponding phase correction is applied to each qubit. The CZ gate is implemented by a barrier voltage pulse applied to gate B (orange) and negative compensation pulses applied to gates LP (blue) and RP (red), with the same shape as the barrier pulse. Singlequbit phase corrections are then applied on each qubit to compensate the frequency detuning induced by electron movement in the magnetic field gradient. b, c, Calibration of phase corrections on Q_{1} induced by a singlequbit gate applied on Q_{2} (ϕ_{21}, b) and on Q_{1} (ϕ_{11}, c). A relative phase shift, 2ϕ_{21} (2ϕ_{11}), is determined by interleaving the target gate (a π/2 rotation) and its inverse (a −π/2 rotation) on Q_{2} (Q_{1}) in a Ramsey interference sequence. d, e, Calibration of phase corrections on each qubit after the CZ gate, using Q_{1} (d) and Q_{2} (e) as the control qubits, respectively. When the amplitude of the barrier pulse is perfectly calibrated, the two curves in each experiment should be out of phase by 180°. However, when the barrier pulse amplitude is calibrated such that one of the two experiments shows a 180° phase difference (d), the phase difference in the other calibration experiment always deviates by a few degrees. One possible explanation is that the optional πrotation applied to the control qubit induces a small, offresonance rotation on the other qubit, causing an additional phase on the target qubit to appear in the measurement due to the commutation relation of the Pauli operators.
Extended Data Fig. 5 Pulse optimization.
a, b, Full error generators for a CZ gate calibrated by conventional Ramsey sequences (a) and after improving the calibration using the information extracted from a (b), resulting in fidelities of 97.86% and 99.65%, respectively. c, d, Seven Hamiltonian errors (IX, IY, XI, YI, IZ, ZI and ZZ) extracted from the error generators shown in a (c) and b (d). Owing to the crosstalkinduced additional phases shown in Extended Data Fig. 4, errors IZ, ZI and ZZ occur systematically in conventional calibrations. Error bars indicate the 2σ confidence intervals computed using the Hessian of the loglikelihood function. e, f, Shapes of the barrier pulses (e) and their corresponding J envelopes (f) for a CZ gate before and after being corrected by GST. Since the Hamiltonian to generate a CZ gate is H = (II + IZ + ZI − ZZ)/2, the positive ZZ error shown in c is corrected by increasing the amplitude of the pulse. The IZ and ZI errors are corrected by decreasing the phase shifts θ_{1} and θ_{2} after the CZ gate. Hamiltonian errors in singlequbit gates are corrected similarly. The results presented in b and d are achieved in four loops of correction, with each loop correcting the parameters by approximately 70% of the measured deviation.
Extended Data Fig. 6 Workflow of the variational quantum eigensolver algorithm.
The qubit Hamiltonian is typically transformed from the molecular Hamiltonian by JW transformation or BK transformation by a classical processor (see Methods). A HF initial state is encoded into the qubit states according to JW or BK transformation and then transformed by the quantum processor into a parameterized ansatz state by considering single and double excitation in the molecule using the UCC theory. The expectation value of each individual Hamiltonian term is directly measured by partial state tomography. The expectation of the total energy is then calculated by the weighted sum of the individual expectations. The result is fed into a classical optimizer, which suggests a new parameterized ansatz state for the next run. This process is repeated until the expectation of the total energy converges.
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Xue, X., Russ, M., Samkharadze, N. et al. Quantum logic with spin qubits crossing the surface code threshold. Nature 601, 343–347 (2022). https://doi.org/10.1038/s4158602104273w
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DOI: https://doi.org/10.1038/s4158602104273w
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