Quantum logic with spin qubits crossing the surface code threshold – Nature

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 charge-stability 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 vLP/vB = −0.081 and vRP/vB = 0.104. A vector signal generator (VSG, Keysight E8267D) is connected to gate MW and sends frequency-multiplexed microwave bursts (not necessarily time-multiplexed) to implement electric-dipole 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 cryo-CMOS 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 single-qubit drives fixed and in the linear-response regime, where the Rabi frequency is linearly dependent on the driving amplitude. The envelopes of the single-qubit gates are shaped following a ‘Tukey’ window, as it allows adiabatic single-qubit gates with relatively small amplitudes, thus, avoiding the distortion caused by a nonlinear response. The general Tukey window of length tp is given by

$$W(t,r)={begin{array}{cc}frac{1}{2}left[1-,cos ,left(frac{2{rm{pi }}t}{r{t}_{{rm{p}}}}right)right] & 0le tle frac{r{t}_{{rm{p}}}}{2}\ 1 & frac{r{t}_{{rm{p}}}}{2} < t < {t}_{{rm{p}}}-frac{r{t}_{{rm{p}}}}{2}\ frac{1}{2}left[1-,cos ,left(frac{2{rm{pi }}({t}_{{rm{p}}}-t)}{r{t}_{{rm{p}}}}right)right] & {t}_{{rm{p}}}-frac{r{t}_{{rm{p}}}}{2}le tle {t}_{{rm{p}}}end{array},$$

(1)

where r = 0.5 for our pulses. Apart from these fixed parameters, there are 11 free parameters that must be calibrated: single-qubit frequencies ({f}_{{{rm{Q}}}_{1}}) and ({f}_{{{rm{Q}}}_{2}}), burst lengths for single-qubit gates tXY1 and tXY2, phase shifts caused by single-qubit gates on the addressed qubit itself ϕ11 and ϕ22, phase shifts caused by single-qubit gates on the unaddressed ‘victim qubit’ ϕ12 and ϕ21 (ϕ12 is the phase shift on Q1 induced by a gate on Q2 and similar for ϕ21), 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 θ1 and θ2 (in addition, we absorb into θ1 and θ2 the 90° phase shifts needed to transform diag(1, i, i, 1) into diag(1, 1, 1, −1)).

For single-qubit 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 tXY1 and tXY2 are initially calibrated by an AllXY calibration protocol43. The phases ϕ11, ϕ12, ϕ21 and ϕ22 are initially calibrated by measuring the phase shift of the victim qubit (Q1 for ϕ11 and ϕ21; Q2 for ϕ22 and ϕ12) in a Ramsey sequence interleaved by a pair of ([{{rm{X}}}_{{{rm{Q}}}_{i}},-{{rm{X}}}_{{{rm{Q}}}_{i}}]) gates on the addressed qubit (Q1 for ϕ11 and ϕ12; Q2 for ϕ22 and ϕ21) (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 π-rotation is applied to the control qubit (for example, Q1) to prepare it into the |0 or |1 state, followed by a Ramsey sequence on the target qubit (Q2) interleaved by an exchange pulse. The amplitude is precisely tuned to bring Q2 completely out of phase (by 180°) between the two measurements (Extended Data Fig. 4d, e). The phase θ2 is determined such that the phase of Q2 changes by zero (π) when Q1 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 Q2 as the control qubit and Q1 as the target qubit to determine θ1 (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 off-resonant driving from the optional π-rotation on the control qubit. Similar effects can also affect the calibration of the phase crosstalk from single-qubit 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 single-qubit gates, tXY1 and tXY2 are adjusted according to the IX, IY, XI and YI errors. The phases ϕ11, ϕ12, ϕ21 and ϕ22 are adjusted according to the ZI and IZ errors. For the CZ gate, θ1 and θ2 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 model21

$$H=g{mu }_{{rm{B}}}{{bf{B}}}_{1}cdot {{bf{S}}}_{1}+g{mu }_{{rm{B}}}{{bf{B}}}_{2}cdot {{bf{S}}}_{2}+hJ({{bf{S}}}_{1}cdot {{bf{S}}}_{2}-frac{1}{4}),$$

(2)

with ({{bf{S}}}_{j}={({X}_{j},{Y}_{j},{Z}_{j})}^{{rm{T}}}/2,) where Xj, Yj and Zj are the single-qubit Pauli matrices acting on spin j = 1, 2, μB the Bohr’s magneton, g ≈ 2 the g-factor 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 Bx,j induce spin-flips, thus, single-qubit 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 ΔEz = B(Bz,2 − Bz,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 vB. We note that vB determines the compensation pulses applied to LP and RP for virtual barrier control. We model J as an exponential function31,32

$$J({v}_{{rm{B}}})={J}_{{rm{res}}}{{rm{e}}}^{2alpha {v}_{{rm{B}}}},$$

(3)

where Jres ≈ 20–100 kHz is the residual exchange interaction during idle and single-qubit 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 βj 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

$$h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|0rangle )= {mathcal E} (|10rangle )- {mathcal E} (|00rangle ),$$

(4)

$$h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|1rangle )= {mathcal E} (|11rangle )- {mathcal E} (|01rangle ),$$

(5)

$$h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|0rangle )= {mathcal E} (|01rangle )- {mathcal E} (|00rangle ),$$

(6)

$$h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|1rangle )= {mathcal E} (|11rangle )- {mathcal E} (|10rangle ),$$

(7)

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 two-qubit 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  single-qubit  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 vB → vB + δvB. 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

$${T}_{2}^{ast }({{rm{Q}}}_{1}({{rm{Q}}}_{2}=|0rangle ))=frac{1}{sqrt{2}{rm{pi }}sqrt{{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|0rangle )))}{d{v}_{{rm{B}}}}right]}^{2}delta {v}_{{rm{B}}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|0rangle ))}{dh{f}_{{{rm{Q}}}_{1}}}right]}^{2}delta {f}_{{{rm{Q}}}_{1}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|0rangle ))}{dh{f}_{{{rm{Q}}}_{2}}}right]}^{2}delta {f}_{{{rm{Q}}}_{2}}^{2}}},$$

(8)

$${T}_{2}^{ast }({{rm{Q}}}_{1}({{rm{Q}}}_{2}=|1rangle ))=frac{1}{sqrt{2}{rm{pi }}sqrt{{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|1rangle )))}{d{v}_{{rm{B}}}}right]}^{2}delta {v}_{{rm{B}}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|1rangle ))}{dh{f}_{{{rm{Q}}}_{1}}}right]}^{2}delta {f}_{{{rm{Q}}}_{1}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{1}}({{rm{Q}}}_{2}=|1rangle ))}{dh{f}_{{{rm{Q}}}_{2}}}right]}^{2}delta {f}_{{{rm{Q}}}_{2}}^{2}}},$$

(9)

$${T}_{2}^{ast }({{rm{Q}}}_{2}({{rm{Q}}}_{1}=|0rangle ))=frac{1}{sqrt{2}{rm{pi }}sqrt{{left[frac{d(h{f}_{{rm{Q2}}}({{rm{Q}}}_{1}=|0rangle )))}{d{v}_{{rm{B}}}}right]}^{2}delta {v}_{{rm{B}}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|0rangle ))}{dh{f}_{{{rm{Q}}}_{1}}}right]}^{2}delta {f}_{{{rm{Q}}}_{1}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|0rangle ))}{dh{f}_{{{rm{Q}}}_{2}}}right]}^{2}delta {f}_{{{rm{Q}}}_{2}}^{2}}},$$

(10)

$${T}_{2}^{ast }({{rm{Q}}}_{2}({{rm{Q}}}_{1}=|1rangle ))=frac{1}{sqrt{2}{rm{pi }}sqrt{{left[frac{d(h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|1rangle )))}{d{v}_{{rm{B}}}}right]}^{2}delta {v}_{{rm{B}}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|1rangle ))}{dh{f}_{{{rm{Q}}}_{1}}}right]}^{2}delta {f}_{{{rm{Q}}}_{1}}^{2}+{left[frac{d(h{f}_{{{rm{Q}}}_{2}}({{rm{Q}}}_{1}=|1rangle ))}{dh{f}_{{{rm{Q}}}_{2}}}right]}^{2}delta {f}_{{{rm{Q}}}_{2}}^{2}}}.$$

(11)

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, β1 = −2.91 ± 0.11 MHz Vγ, β2 = 67.2 ± 0.63 MHz Vγ, γ = 1.20 ± 0.01 and Jres = 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: δvB = 0.40 ± 0.01 mV, (delta {f}_{{Q}_{1}}=11pm 0.1{rm{kHz}}) and (delta {f}_{{Q}_{2}}=24pm 0.7{rm{kHz}}).

Numerical simulations

For all numerical simulations, we solve the time-dependent Schrödinger equation

$${rm{i}}hbar frac{{rm{d}}}{{rm{d}}t}|{rm{psi }}
(12)

and iteratively compute the unitary propagator according to

$$U(t+Delta t)={{rm{e}}}^{-frac{{rm{i}}}{hbar }H(t+Delta t)}U
(14)

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}}}
(15)

$$approx frac{1}{2}(-hJ({v}_{{rm{B}}}
(16)

where the first term is unaffected and describes the global phase accumulation due to the exchange interaction, the second term describes the single-qubit phase accumulations and the last term, (f
(17)

$$propto {S}_{{rm{s}}}(f
(18)

From the first to the second line, we identify the integral by the (short-timescale) Fourier transform, allowing us to describe the spin-flip error probability by the energy spectral density Ss of the input signal f(t). Minimizing such errors is, therefore, identical to minimizing the energy spectral density of a pulse, a well-known 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 high-fidelity exchange pulse is consequently given by J(0) = J(tp) and

$${int }_{0}^{{t}_{{rm{p}}}}{rm{d}}tJ({v}_{{rm{B}}}
(19)

while setting (J
(20)