Hamiltonian Phase Error in Resonantly Driven CNOT Gate Above the Fault-Tolerant Threshold

Because of their long coherence time and compatibility with industrial foundry processes, electron spin qubits are a promising platform for scalable quantum processors. A full-fledged quantum computer will need quantum error correction, which requires high-fidelity quantum gates. Analyzing and mitigating the gate errors are useful to improve the gate fidelity. Here, we demonstrate a simple yet reliable calibration procedure for a high-fidelity controlled-rotation gate in an exchange-always-on Silicon quantum processor allowing operation above the fault-tolerance threshold of quantum error correction. We find that the fidelity of our uncalibrated controlled-rotation gate is limited by coherent errors in the form of controlled-phases and present a method to measure and correct these phase errors. We then verify the improvement in our gate fidelities by randomized benchmark and gate-set tomography protocols. Finally, we use our phase correction protocol to implement a virtual, high-fidelity controlled-phase gate.


Introduction
Spin qubits in solid state devices [1] are a promising platform for large-scale quantum computers.Universal control has recently been demonstrated in a six qubit device in Silicon [2], and a four qubit device in Germanium [3], marking a first step of scaling up spin qubit devices.Spin qubits in Silicon exhibit long coherence times [4,5], fast manipulation [6] and ability to operate at an elevated temperature [7,8,9].The compatibility with the already matured semiconductor industry processes allows mass fabrication of devices [10], integration with cryo-electronics [11], and opens the potential for high-performance integrated quantum circuits in the future [12].Quantum error correction, a critical feature of large-scale quantum computers, has also been demonstrated in a three-qubit device recently [13].These progresses make spin qubits a viable qubit platform for the future.
To implement large-scale quantum computers the ability to implement quantum error correction code is required.One of the most promising quantum error correction codes is the surface code [14].Typically, under certain assumptions of the error model, the surface code gives an error threshold of 1% [15].High fidelity single-qubit [4,16] and twoqubit gates [17,18,19] which satisfy this error threshold have been demonstrated with spin qubits in isotopically enriched Silicon.Among these results, the two-qubit gates are implemented as a controlled-phase (CZ) gate or controlled-rotation (CROT) gate.A highfidelity CZ gate requires fast and precise pulses to control the exchange coupling between two qubits.The CROT gate, on the other hand, can be implemented in a less demanding way by keeping the always-on exchange [17,20].In the exchange-always-on system, the CROT gate fidelity is reduced by a coherent off-resonant Hamiltonian phase error which has the form of a controlled-phase.This phase error must be mitigated to obtain highfidelity CROT gates above the fault-tolerant threshold.Previous work avoids this problem by shifting control microwave frequecy [17].The microwave frequency is adjusted by a feed-back loop to minimize this phase error.Here we systematically compensate the effect of these phase errors by shifting the phase of the applied microwave pulses.We measure the phase errors with a calibration sequence and compensate its effects.Our procedure to compensate these controlled-phase errors enables us to implement a CZ gate virtually, similar to a virtual single-qubit z-gate [21], thus without additional execution time in the quantum circuit.The ability to implement both high-fidelity CROT and a virtual CZ gate without complicated pulse engineering makes the exchange-always-on system interesting to study.Compared to a synthesized implementation using CZ gates [18,19], the CROT gate allows for a native, resonant CNOT logical gate with a fidelity above the fault tolerant threshold [17], which makes this gate relevant for future spin based quantum processors.
Here, we demonstrate a procedure to obtain a high-fidelity resonantly driven CROT gate in an exchange-always-on two-qubit system.We first present a systematic way to measure the accumulated phase error and then a method to compensate for these gate errors.We use randomized benchmarking (RB) protocol [22,23] to compare the gate fidelity with and without compensation.We then perform gate-set tomography (GST) [24] to obtain the details on the error processes of our quantum gates using experimental and simulated data.The experimental and simulation data results show good agreement, which proves the validity of the quantum gate model we use for simulation.Finally, we demonstrate the implementation of a virtual high-fidelity CZ gate using the compensation method and benchmark the performance of this virtual CZ gate with GST.

Device and controlled rotation gates
Fig. 1 (a) shows the device used for the experiment, which is a triple quantum dot device fabricated in isotopically purified silicon quantum well, the same device as used in Ref. [17].
A three-layer aluminum gate stack is deposited to fabricate the gate electrodes, which control the electric confinement potential of the quantum dots.A cobalt micro-magnet is deposited on the gate stack to achieve a gradient magnetic field.The gradient magnetic field allows individual addressing of the spins in the quantum dots and manipulating the spin qubit state by performing electric dipole spin resonance (EDSR) with gradient magnetic field generated by the micro-magnet.The device has a charge sensor quantum dot in the upper part of the device and an array of three quantum dots in the lower part.
We perform charge sensing with reflectometry [25] and accumulate an electron in the center (qubit Q 1 ) and right dot (qubit Q 2 ), while the leftmost dot is used as an extension of the left reservoir.Energy selective single-shot readout is used for qubit readout and initialization [26].As the exchange coupling between the two qubits is turned on, when the Zeeman energy difference δE z between two qubits is much larger than the exchange coupling J, where Ēz is the averaged Zeeman energy of the two qubits, δ Ẽz = J 2 + δE 2 z the effective Zeeman energy difference and B(t) the effective magnetic field induced by the EDSR.exciting one of the four transition frequencies, we implement the resonantly driven zerocontrolled rotations (ZCROT) and controlled rotations (CROT) [28].The ZCROT rotates the target qubit if the control qubit is in |↓ (0 state), and the CROT rotates the target qubit if the control qubit is in |↑ (1 state).The notation CROT ctrl, targ (ZCROT ctrl, targ ) indicates a rotation of the target qubit (targ = 1, 2) if the control qubit (ctrl = 1, 2) is in |↑ (|↓ ) state.Fig. 1 (d) shows the measured EDSR frequencies and δE z ∼ 310 MHz.We choose J = 18 MHz such that the system is in an optimal condition for high two-qubit gate fidelities [17].
The effective magnetic field induced by the EDSR has the form with f MW the microwave driving frequency and f R the Rabi frequency.To implement the CROT 12 gate, where Q 1 is the control qubit, and Q 2 is the target qubit, we choose a driving frequency that is resonant with the corresponding transition frequency, i.e., f MW = f 2,↑ .By substituting B(t) into the Hamiltonian given in Eq. ( 1), transforming to the rotating frame (see Methods 4.3) and neglecting far off-resonance terms using the rotating wave approximation (RWA), the Hamiltonian becomes [20,27] The upper-left 2-by-2 sub-block provides the desired controlled rotation, while the lowerright 2-by-2 sub-block introduces error to the gate.Choosing f R = J/ √ 15 cancels out the population transfer caused by the lower-right sub-block for the π and the half-π CROT, but two z-phases resulting from e ±2iπJt terms will be accumulated in this sub-block.This results in a controlled-phase error which accumulates in the |↓↓ and |↓↑ states [20,27].
We verify this source of error using GST experiments as we discuss later.We call these phase errors the off-resonant Hamiltonian phase errors since they are errors arising from the control Hamiltonian.We change the rotating frame by offsetting the microwave phase in the pulse sequence to account for the accumulated phase errors.This allows us to correct these off-resonant Hamiltonian phase errors.

Measuring the off-resonant Hamiltonian phase error
There We use the measured off-resonant Hamiltonian phase errors to compensate for the unwanted phases accumulated in the calibration sequences.Fig. 2 (e) shows the procedure we use to compensate for the phase errors in the CROT 12 sequence.We record all the phase errors at each step of the sequence.From these phase errors, we obtain the offset needed for each pulse to compensate for the effect of phase error, which is the phase accumulated on the states the pulse is acting on (see Methods 4.1).To compensate for the accumulated phase error, we subtract the accumulated phase from the microwave phase, which implements a virtual z-gate [21].These virtual z-gates change the rotating frame according to the accumulated phase errors such that the effects on the qubit gates caused by the phase errors are canceled.Fig. 2  • .We notice that there are still non-zero phase shifts in the two ZCROT sequences.This is also observed in simulation (see Methods 4.3 and Supplementary Fig. S3) from which we notice that this phase is not originating from the far off-resonant terms.This residual phase is only observed in sequences when both qubits are operated and its origin is not yet clear.A possible source for this residual phase could be a correlated gate error on Q 2 (Q 1 ) when applying gates acting on

Compensation of the off-resonant Hamiltonian phase error
Next, we extend the use of the compensation procedure to general pulse sequences.Fig. 3 (a) shows the phase compensation procedure for general pulse sequence U 1 , U 2 , ..., U N .
We keep a phase error table that records the phase errors accumulated on the four basis states and check the sequence pulse by pulse with this phase error table to obtain the phase offsets.Since the errors are time-idependent, we can compensate the phase errors pulse by pulse.For each applied pulse, we check the phase error which is accumulated on the states which the pulse acts on and offset the microwave phase correspondingly as done in the previous section.We then add the phase error accumulated by the pulse to the phase error table and move on to the next pulse.This procedure is performed in software before the execution of the physical pulses.We emphasize that this method can also be implemented in real-time, e.g., on an FPGA using a phase counter [2].
To evaluate the performance of our calibration, we compare the pulse fidelity with and without the compensation by performing a two-qubit RB experiment [17,20].We use 15 (59) different random sequences for the experiment without (with) the phase com-pensation protocol (see Methods 4.2).Fig. 3    a General off-resonant Hamiltonian phase error compensation procedure.We check the phase error table for phases accumulated in the relevant states for each pulse (1).We then subtract the offset from the microwave phase of the pulse (2).Finally, the phase error caused by the pulse is added to the phase error table and used to obtain the offsets for the following pulses To get a more detailed report on the performance of the quantum gates, we conduct a GST experiment [24,29] (see also Methods 4.4).Here the GST experiment generates the two-qubit Pauli transformation matrix (PTM) of the implemented quantum gates, which describes how Pauli matrices are transformed under the quantum gate.The experimentally obtained PTM is then compared to the ideal PTM to get the error generator which gives more specific gate error processes.By writing the error generator into a linear combination of terms representing different error processes, we can interpret the errors of our gates more intuitively [30].S1.
After the phase compensation protocol, the CNOT 12 gate still has an infidelity of ∼ 0.5 %.The precision of our GST result does not allow us to make a definite statement on whether the source of this infidelity is coherent Hamiltonian errors or incoherent stochastic errors.The correlated gate error mentioned in Sec.2.2 could be an indication that there are still Hamiltonian errors in our gate.Further investigations are required to determine whether the CNOT 12 fidelity can be increased by compensating this error.

Virtual CZ Gate
Finally, we demonstrate the implementation of a virtual CZ gate with the compensation procedure, shown in Fig. 4 (a).In the compensation procedure, we use the phase error table to obtain the microwave phase offset for each pulse in the sequence U 1 , U 2 , ..., U N .
When a π phase is added to the |↑↑ row, the following CROT 12 pulses will acquire an additional π phase in the offset while the offsets of ZCROT The graph shows the dominant stochastic error generator component ZX for the and virtual CZ 12 .We implement the identity gate by idling both qubits for the duration of a π/2 gate time (62 ns), and X 1 by applying a CROT 21 and a ZCROT 21 sequentially.Error bars represent the 1σ standard deviation from the mean.
One advantage of virtually implementing quantum gates is that the gate time is reduced to zero such that the qubits are not affected by dephasing.Fig. 4 (c) shows the ZX stochastic error component for I ⊗ I, X 1 = X ⊗ I and virtual CZ 12 .We use the identity gate I ⊗ I which idles both qubits for 62 ns to emulate the dephasing in a physical CZ gate.We find experimentally that ZX is the dominant component for these gates (see become more complicated.While further investigations are required, we anticipate that the procedure discussed here can also be used to calibrate the controlled rotations in a three-qubit exchange-always-on system.
In summary, we demonstrated a systematic way to calibrate high-fidelity CROT gates in an exchange-always-on two-qubit system.We present a calibration procedure to compensate for the Hamiltonian off-resonant phase errors in our CROT gates, allowing us to achieve universal single and two-qubit gate fidelities above the fault-tolerant threshold of 99%.Finally, we implement a high-fidelity virtual CZ gate using our phase error compensation protocol.
Using a similar argument, we can write the phase shifts measured into a linear combination of the off-resonant Hamiltonian phase errors Solving these four equations gives the explicit form of the off-resonant Hamiltonian phase errors.The relation is With the obtained off-resonant Hamiltonian phase errors we write down a table of phase errors accumulated on each basis states before the pulse is applied.We calculate the offsets needed by

Two-Qubit Randomized Benchmarking
We choose sequence lengths L = (1,8,16,23,31) in our two-qubit randomized benchmarking experiment.For each length we use 59 (15) different random sequences for the experiment with (without) the phase compensation protocol.We combined three datasets measured over a time span of one week, demonstrating the stability of our qubits.For each sequence, the probability is obtained by averaging 150 single-shot measurements.
The gates in each sequence are randomly chosen from the two-qubit Clifford group, which contains 11520 elements [37].We use a computer search to find the combinations of primitive gates (see Supplementary Fig. S2) to construct all two-qubit Clifford group elements [17,20].At the end of the sequence, we search recovery gate, which projects the state into the target state |↑↑ or |↓↓ .This results in two sequence fidelities F ↑↑ (n) and F ↓↓ (n).
We fit the difference between two sequences F (n) = F ↑↑ (n) − F ↓↓ (n) with the formula

CROT Simulation
We start with the Hamiltonian given in Eq. ( 1) and transform the Hamiltonian into the rotating frame using with R = diag(e −2iπEzt , e −iπ(−δ Ẽz−J)t , e −iπ(δ Ẽz−J)t , e 2iπEzt ).The Hamiltonian in the rotating frame is then with the effective EDSR magnetic field B(t) = f R e −2iπf MW t+iφ .We substitute this magnetic field into the rotating frame Hamiltonian and calculate the propagator.If the RWA is used, we set the elements in the far off-resonant terms to zero before calculating the propagator.
We choose J = 16 MHz and f R = J/ √ 15 4.13 MHz, which results in a π/2 gate time T π/2 60.5 ns.We compute the unitary propagator with this Hamiltonian by with T the time-ordering operator.By choosing the driving frequency f MW , we select which ZCROT or CROT is implemented.Changing the microwave phase φ changes the rotation angle of the pulse.This unitary is then used for simulating the implemented pulses.

Gate-set Tomography
For the GST experiments, depending on the implemented gates in the system, a different target gate set is chosen.This target gate set is then used to compose a preparation and measurement gate set and a set of germ sequences.The preparation-and measurementfiducial gates are used to make tomographic measurements.These fiducial gates must be able to prepare and measure an information-complete set of states.The germ sequence in between is chosen from the germ set, which is amplificationally-complete [24] and therefore capable of amplifying all possible errors that can occur during the gate operation.
We perform GST with the python package pyGSTi [29].We use the default gate set provided by the pyGSTi package and the fiducial pair reduction function to reduce the number of sequences required.The CNOT 12 gate-set contains {I, The identity gate I is implemented by idling both qubits for a time of T π/2 = 62 ns.X 1,2 (Y 1,2 ) are π/2 rotations along the x-axis (y-axis) for Q 1 or Q 2 , respectively.The 15 germs for this gate set are and the fiducial gates In contrast to the identity gate I, the null-gate has no physical idling time.We choose the sequence lengths L = (1, 2, 4, 8, 16), which results in a total of 1760 sequences.
For the CZ 12 gate-set which contains {I, X 1 , Y 1 , X 2 , Y 2 , CZ 12 }, the germs and fiducial gates are which results in a total of 1644 sequences.All the gates used in the GST experiment are composed of CROT and ZCROT π/2 pulses and single-qubit z-rotations (see Supplementary Fig. S4).The sequences are executed on the device to gather outcome counts.After execution of these sequences, the measured spin-up and spin-down counts are analyzed with an H+S model (see Methods 4.5) to obtain the PTMs G exp of the gates.This PTM has the form where d is the Hilbert space dimension and P i are the two-qubit Pauli operators.The estimated PTMs are then compared with the ideal PTMs to obtain the error generators through functions in the pyGSTi package.
To verify the assumption we made for the off-resonant Hamiltonian phase error, we also perform GST with simulated data sets.First, we take the sequences used in the experiment and calculate the corresponding series of unitary propagators with the ideal Hamiltonian.Then, we evolve the input ground state with this series of unitaries to obtain the final output state and calculate probabilities in each outcome to generate simulated counts.Finally, the simulated counts are analyzed in the same way as the experimental counts.

Error Generators
For a noisy implementation G exp of the ideal quantum gate G ideal , we can model the imperfect gate as which is an ideal quantum gate followed by some noise process E. By inverting the ideal gate, we get the noise process as If we take the logarithm of this noise process and assume that noise is small i.e.E I. Using the approximation log X (X − I) with small (X − I) we get the error generator [30] L = log G exp G −1 ideal = log E E − I, (24) which is the approximated difference between the noise process E and the identity.If the gate is noise-free, i.e., E = I, then L = 0.This error generator can be written into a The coefficient of these error generator terms is obtained by [30] h where P are the two-qubit Pauli matrices.We extract these coefficients using internal functions in the pyGSTi package.These coefficients can be used to calculate the Jamiolkowski probability J (L) and the Jamiolkowski amplitude θ J (L), which gives the amount of incoherent and coherent errors respectively.For an error generator L being decomposed into list of {h P , s P } coefficients, these two metrics are [30] J (L) = tr[ρ J (L)(I − |Ψ Ψ|)] = P s P , (30) The Jamiolkowski probability and Jamiolkowski amplitude can be used to approximate the averaged gate infidelity related with error generator L. For small errors, the approximated average gate infidelity is [30]

Fig. 1 (
b) shows the stability diagram around this configuration.

Figure 1 : 1 (Q 2 )
Figure 1: Two qubit system.a False-colour scanning electron microscope image of a device identical to the one measured.The two quantum dots are formed below plunger gate electrodes P1 (Q 1 ) and P2 (Q 2 ) and the yellow circle indicates the charge sensor quantum dot.Quantum gates are implemented via electric dipole spin resonance in the gradient field of a micro-magnet (not shown) by applying microwave signals to the MW gate electrodes.b Charge stability diagram around the qubit operation condition.The number of electrons in two quantum dots is denoted as (N 1 , N 2 ).Readout and initialization of qubit Q 1 (Q 2 ) is performed at square A (B). Qubit operations are executed at the charge symmetry point (circle C) to achieve high-fidelity two-qubit gates.c Energy level diagram of the two-qubit system.Exciting one of the four frequencies rotates the target qubit conditioned on the state of the other qubit, which allows the implementation of controlled rotation (CROT) and zero-controlled rotation (ZCROT) gates.The label X indicates a π/2 pulse around the x-axis of the qubit Bloch-sphere.d Electric dipole spin resonance peaks.The measured spectra shows the transition frequencies of Q 1 when Q 2 is in |↓ (blue) and |↑ (purple) and frequencies of Q 2 when Q 1 is in |↓ (red) and |↑ (orange).

Figure 2 :
Figure 2: Calibration sequences for measuring the off-resonant Hamiltonian phase error.a, b Calibration sequences used to measure off-resonant Hamiltonian phase errors for ZCROT 12 and CROT 12 , where X denotes a half-π rotation.For all the sequences see Supplementary Fig. S1.c Phase shifts in the calibration sequences.Spin up probability measured in each calibration sequences are fitted with A cos(θ + θ i ) + B to obtain the phase shifts.With A the amplitude, B the offset, θ the phase of the second half-π pulses and θ i the phase shift.d The off-resonant Hamiltonian phase error associated with each pulse.The phase error φ m,σ associated with each transition frequency f m,σ will accumulate in the off-resonant state when applying each pulse.Here we introduce the += (−=) operators which add (subtract) the value on the right to (from) the variable on the left, a common syntax in modern programming languages.e Schematic of compensation procedure for the off-resonant Hamiltonian phase error for the example of the calibration sequence for CROT 12 .The phases accumulated on each basis states before the pulse is applied are shown in the columns.We then use this table to calculate the phase offset φ of f set and subtract this offset from the applied pulses' microwave phase to compensate for the effect of off-resonant Hamiltonian phase error.f The measured phase shifts in the calibration sequences after the phase compensation, demonstrating a significant improvement to the uncompensated case shown in (b).

Figure 3 :
Figure 3: Off-resonant Hamiltonian phase error compensation procedure and performance.

( 3 )
. b Results of twoqubit randomized benchmarking with and without the compensation procedure.The RB without the compensation procedure shows an averaged primitive gate fidelity of 97.95 ± 0.11%.The compensation procedure increases this fidelity to 99.41 ± 0.02%.c. d Coefficients of the CNOT 12 gate error generator decomposition with and without compensation.The solid bars in the plot represent the experimental result, while the dashed bars represent the simulations.Large IX and ZX Hamiltonian component occurs when the compensation procedure is not applied.The two dominant error terms in (c) are significantly reduced when applying the compensation procedure.Experiment and simulation results show good consistency.The error bars represent 1σ standard deviation from the mean.

Fig. 3 (
Fig.3 (c) shows the error generator of the CNOT 12 gate obtained by both experiment and simulated GST.The simulated data is obtained using an ideal Hamiltonian (see Eq. (2)) without introducing any noise (seeMethods 4.3).Without the compensation, there are large errors in the IX and ZX Hamiltonian elements, both in simulation and experimental results.The consistency between the experiment and simulation shows the off-resonant Hamiltonian phase error considered in the Hamiltonian given in Eq. (2) is indeed the dominant error we measured in the experiment.The two terms are significantly suppressed when using the compensation procedure, as shown in Fig.3 (d).Both experiment and simulation results exhibit this suppression, showing that the off-resonant Hamiltonian phase errors are understood and corrected as expected.The full GST gate metrics of the experiment with the phase error compensation protocol are shown in Supplementary TableS1.
linear combinationL = L H + L S + L C + L A c P,Q C P,Q + P,Q>P a P,Q A P,Q .(27)The terms in the linear combination correspond to error generators representing different error processes.The error generators are divided into four categories, Hamiltonian generator H P , stochastic Pauli generator S P , Pauli-correlation generator C P,Q and active generator A P,Q .We use the H + S model of pyGSTi package [29, 38], which only contains Hamiltonian and stochastic Pauli errors which have clear physical meanings.The Hamiltonian error generators represent a systematic over-or under-rotation of the qubit state on the Bloch-sphere in one of the rotation axes.On the other hand, the stochastic Pauli generators represent the contraction to one of the axes of the qubit Bloch sphere.
12pulses are unchanged.This results in a π phase difference between CROT 12 and ZCROT 12 pulses which is effectively equivalent to a CZ 12 gate.To verify the virtual CZ 12 , we perform a GST experiment with the CZ 12 gate-set.Fig.4(b)shows the estimated PTM of the virtual CZ 12 .The measured PTM is close to the ideal CZ 12 and has a fidelity of 99.49 ± 0.08% which we anticipate can be increased by further phase calibrations.This shows that the virtual CZ 12 is well implemented.
a Figure 4: Virtual CZ a Implementation of virtual CZ 12 .A virtual CZ 12 is executed by adding a π phase to the |↑↑ state in the table of phase error.Other controlled z-gates CZ ij can be implemented analogously.b Gate-set tomography result of the virtual CZ 12 showing the PTM obtained with experimental (solid line bars) and simulated (dashed line bars) datasets.c Comparison of stochastic noise in different gates.
p n where A t and B t absorbs the SPAM error and p is the depolarizing strength.The two-qubit Clifford gate fidelity is obtained byF C = (1 + 3p)/4.EachClifford element is composed of 2.57 primitive gates on average.We therefore calculate the primitive gate fidelity as F p = 1 − (1 − F C )/2.57.