Context Aware Fidelity Estimation

We present Context Aware Fidelity Estimation (CAFE), a framework for benchmarking quantum operations that offers several practical advantages over existing methods such as Randomized Benchmarking (RB) and Cross-Entropy Benchmarking (XEB). In CAFE, a gate or a subcircuit from some target experiment is repeated n times before being measured. By using a subcircuit, we account for effects from spatial and temporal circuit context. Since coherent errors accumulate quadratically while incoherent errors grow linearly, we can separate them by fitting the measured fidelity as a function of n. One can additionally interleave the subcircuit with dynamical decoupling sequences to remove certain coherent error sources from the characterization when desired. We have used CAFE to experimentally validate our single- and two-qubit unitary characterizations by measuring fidelity against estimated unitaries. In numerical simulations, we find CAFE produces fidelity estimates at least as accurate as Interleaved RB while using significantly fewer resources. We also introduce a compact formulation for preparing an arbitrary two-qubit state with a single entangling operation, and use it to present a concrete example using CAFE to study CZ gates in parallel on a Sycamore processor.

An important problem that many existing techniques face is that as quantum computers scale, complex intercomponent interactions arise, such as control crosstalk or temporal correlations caused by residual pulse tails.For example, experiments may use amplifiers with temperature dependent gain profiles, which could be linked to pulse duty cycle and cause circuit-dependent errors.These effects can make the performance of a quantum operation highly context dependent [9,[20][21][22], and there is a growing need for characterization methods which account for them.
In this paper, we describe a characterization method called Context Aware Fidelity Estimation (CAFE) which measures the fidelity between an experimentally implemented quantum operation and a reference unitary.In CAFE, we repeat a gate or a subcircuit from a target experiment n times and measure the average gate fidelity against the reference unitary raised to the n th power.For example, the subcircuit can be a stabilizer extraction circuit in a quantum error correction experiment, as shown in App.D. This procedure allows us to capture context-related errors which isolated characterizations do not capture [23,24].Since coherent errors accumulate quadratically as a function of n while incoherent errors grow linearly, we can separate their contributions to the infidelity.This is in contrast with RB and XEB, where random compiling is used to twirl coherent errors into incoherent ones.
Throughout the main text of this paper, we focus on using CAFE to study the performance of CZ gates implemented in parallel, as two-qubit operations have been shown to be a dominant error source across recent large-scale experiments on a number of experimental platforms, especially in the presence of stray interactions [25][26][27].In the Appendices, we present results from CAFE experiments characterizing parallel singlequbit operations as well as a circuit layer of a surface code stabilizer extraction experiment.

II. CONTEXT AWARE FIDELITY ESTIMATION
A typical CAFE experiment is performed in three steps, schematically shown in Fig. 1.First, a state |ψ is prepared from an m-qubit 2-design {ψ i } [28][29][30][31].These ensembles match the Haar-random distribution up to second moments, allowing us to measure fidelity.In App.A we present a method to construct shallow circuits that prepare arbitrary two-qubit entangled states for this purpose.Next, the m-qubit circuit of interest, referred to as FIG. 1.A schematic of the circuits used to measure the fidelity of n repetitions of the cycle circuit, which is selected to be a subcircuit from some target experiment.First, a state pulled from a m qubit 2-design, {ψi}, is prepared (blue).Second, the cycle circuit being characterized is applied the desired number of times n (pink).Third, we apply the inverse of the combined preparation circuit and reference unitary U ref to the n th power.Finally, we measure the resulting state in the computational basis (green).The fidelity between n repetitions of the applied operation and the n th power of the reference unitary can be found by averaging the probability of getting |0 ⊗m over all 4 m initial states.The faded operations represent the spatial context of the cycle circuit being characterized.
the cycle circuit, is repeated n times, along with operations on neighboring uninvolved qubits.The cycle circuit can include multiple operations, allowing close matching to the circuit context found in the target experiment.As an example, an experiment which includes dynamical decoupling (DD) would be robust to certain coherent errors, and including these DD gates in the cycle circuit allows CAFE to neglect contributions from these coherent errors, focusing on the errors which impact performance.Inserting DD gates is one of several ways for CAFE to separate coherent and incoherent contributions to gate errors, which we discuss further in Sec.IV.Finally, using the "reference unitary", we apply a circuit which ideally maps the state back to |0 ⊗m , before measuring the qubits in the computational basis.This final step allows us to validate the performance of different unitary characterization methods, as discussed further in Sec.V.The average gate fidelity of the operation, which we refer to as fidelity in the following, can be found by averaging over the experiments for all 2-design states {ψ i } [14]: We note that in the m = 2 case, the preparation and measurement stages only use a single entangling gate, so their contribution to the total error rate is far smaller than the circuit being characterized for most values of n.In general, since the information we extract from CAFE comes from fitting fidelity over different values of n, reasonable SPAM errors do not significantly impact the characterization results, as they only cause a constant offset in the fidelity curve.While the number of 2-design states scales exponentially with qubit count, random circuits can approximate the states in polynomial depth [29], which could be used for performing CAFE on m > 2 qubits.

III. COMPARING CAFE WITH RANDOMIZED BENCHMARKING
In this section, we compare our CAFE approach to the widely used Interleaved Randomized Benchmarking (IRB) protocol for estimating the fidelity of noisy CZ gates.The error model for these gates, which is fully described in App.E, contains coherent errors together with amplitude and phase damping as incoherent noise.

IV. BUDGETING COHERENT AND INCOHERENT ERRORS
Using CAFE, one can accurately estimate the fidelity of any unitary operation and report this single number as a performance metric, which is useful for validation and for estimating the performance of different quantum algorithms [1].However, such a metric alone provides very little information as to how to improve the gate fidelity in practice.A central feature of CAFE is that one can extract actionable information about the origins of gate error by budgeting the coherent and incoherent contributions.To do so, one can fit the fidelity decay curve to a physical model, or modify the cycle circuit to echo out different parts of the gate errors, for example by leveraging Dynamical Decoupling (DD) pulses.
We note that such budgeting is helpful for directing research focus, as interventions for coherent and incoherent errors tend to be different.

A. Separating coherent and incoherent errors through fitting to a model
The CAFE experiment and resulting error budget is valid for any m-qubit unitary, but we will again focus on the two-qubit CZ case to simplify the discussion in the main text.We present a more general deriva-FIG.3. Experimental characterization of CZ gates performed in parallel using CAFE.Data represents the cycle circuit fidelity compared to the ideal CZ unitary for different cycle repetitions, along with fits using the model presented in Eq. 4 and resulting gate budgets in the inset table.We plot the data for three specific gates -at the 10th, 50th and 90th percentile on a Sycamore device in terms of average gate infidelity -to illustrate the accuracy of the modeling over a wide range of data.Error bars represent the standard deviation of binomial distributions scaled by a factor of 5 to be visible.tion in App.B. We assume that the cycle unitary is an excitation-preserving two-qubit gate close to a CZ where Ũ (0, 0, 0) = CZ, and the swap, single-qubit phase, and controlled-phased miscalibration angles are assumed to be small, such that ∆θ, ∆γ, ∆φ 1.To simplify further the expressions here, we also assume that the noisy quantum channel implementing the cycle circuit can be described by a two-qubit depolarizing channel which outputs a totally mixed state with probability p depol and otherwise applies the cycle unitary Ũ .With such a channel, the average gate fidelity for n repetitions of the cycle circuit is given by where we have explicitly included the SPAM errors spam , which are assumed to vary slowly over the timescale of an experiment.We note that in this model we made a steady-state assumption about the gates in our system, but modeling transient behavior could be achieved with a more advanced fit.Using the expression in Eq. 4, we can model the experimental data to obtain the gate fidelity F, in addition to the incoherent errors incoh and coherent errors coh of the quantum operation.To get these parameters in a way that is robust to SPAM errors, we use In numerical simulations, which are presented in App.E, this analysis procedure was shown to be valid and robust to different unitary errors, as well as amplitude and phase damping channels.We use this method to budget errors in Figs. 2 to 5. A similar budgeting approach for single-qubit X(π) gates, together with experimental results acquired on a Sycamore chip, are presented in App. C. As shown in Fig. 3, the simple model of Eq. 4 allows us to accurately fit the CAFE curves spanning a wide range of CZ gate fidelities executed in parallel on a Sycamore chip.Additional data showing the consistency of the resulting error budget with XEB is presented in App.H.
A useful and intuitive picture to analyze the CAFE data is to consider the incoherent and coherent errors as linear and quadratic contributions to the gate infidelity, respectively.This can be seen directly by expanding Eq. 4 up to terms O((∆θ) 4 , (∆γ) 4 , (∆φ) 4 , p depol 2 ) This approximate quadratic form can be found for different cycle unitaries under similar noise channels, and could be used directly in the budgeting procedure given low gate infidelities and shallow cycle repetitions n, or in cases lacking an accurate analytical model for the quantum channel, however we will stick to the model in Eqs. ( 5) to (7) in the rest of this paper.The quantity incoh estimates the average gate infidelity when no coherent control errors are present.This useful characterization metric is defined as R(E) in Ref. [32], where the authors show that it bounds the unitarity of the channel u(E): As such, we can also relate our incoherent error estimate to unitarity.For instance, in the case of depolarizing noise, the unitarity is and the incoherent error obtained from CAFE is, as derived in App.B,

B. Isolating Coherent Channels using Dynamical Decoupling
Exploiting the versatility of CAFE, we can modify the cycle circuit we are characterizing in order to isolate specific error channels.In particular, this is viable when the impact of the modifications is insignificant relative to the error channels being isolated.As a relevant example, we can leverage the fact that the cycle circuit is repeated n times by inserting dynamical decoupling gates in between repetitions.This approach of combining DD and CAFE, which we refer to as DECAF, is particularly useful to characterize the amount of certain coherent error present in the cycle circuit without necessitating full unitary tomography.We note that this is somewhat similar to the work in Ref. [5], with a single repetition of the cycle circuit, and the randomized unitaries replaced with specific DD pulses.
In the case of a CZ gate, adding an X gate to both qubits in the CAFE cycle circuit echos out the Using CAFE to validate different unitary characterizations of a CZ gate with 50th percentile infidelity.The different curves represent the fidelity between the experimental gate and an ideal CZ unitary (blue), a unitary extracted from an XEB experiment (green), and a unitary characterized with a newly introduced method called MEADD [35].In these three different CAFE experiments, only the final pre-measurement circuit layer changes (see the green box in Fig. 1).The improvement in fidelity can thus entirely be attributed to considering a gate unitary that is closer to the experimental CZ implementation.Error bars are scaled by a factor 5 to be visible.
single-qubit phase errors, in addition to mitigating lowfrequency noise, as demonstrated in App.F [33,34].As shown in Fig. 4, the DECAF data has higher fidelities and decreases more linearly in practice.Looking at the resulting fits over all characterized CZ gates, we see a significant decrease in the median coherent error from 5.5 × 10 −4 to 9.4 × 10 −5 , which highlights single-qubit phase miscalibrations, alongside a decrease of the median incoherent error from 7.2 × 10 −3 to 6.7 × 10 −3 , which indicates the level of low-frequency noise in the device.

V. VALIDATING UNITARY CHARACTERIZATIONS
One key difference between CAFE and other methods for extracting error rates is the final unentangling step before measurement.By doing all of the inversion in a single step, similar to RB, but using the reference unitary for the inversion, we can validate different unitary characterizations, while respecting the non-Clifford nature of most coherent error models.As such, we can use CAFE to benchmark unitary characterizations, simply by changing the final measurement step and seeing which predictions most accurately map the final state back to |0 ⊗m , in conjunction with the methods presented in Sec.IV A.
In Fig. 5, we compare the CZ unitary extracted by XEB to the one extracted using a unitary characterization method called Matrix-Element Amplification by Dynamical Decoupling (MEADD) [35], which is a characterization technique that we have developed based on Floquet characterization [36,37].MEADD allows us to isolate and precisely measure the different unitary parameters in any phased fSim gate (a general excitation number preserving two-qubit gate).We can see very clearly that the unitary predicted by MEADD is significantly better at predicting the coherent error than the unitary extracted by XEB.By increasing the amount of context around the gate, for example including some microwave operations or measurements on the surrounding qubits to include crosstalk or measurement-induced dephasing effects, we can see which characterizations break down in other contexts (see App. D), and build trust that the structures seen in our characterizations are the dominant effects on the processor impacting algorithm performance.

VI. CONCLUSION
CAFE separates itself from other gate characterization methods due to its simplicity and flexibility, most notably as a complementary tool to other gate characterizations that provide more granular output.We have found its ability to split coherent and incoherent errors more reliable than other methods, and the ability to experimentally test unitary characterizations has allowed us to design and evaluate novel characterization methods more effectively.
Its flexibility allows for many other as-of-yet unexplored variations, from creating fits which include the impact of leakage, to changing the cycle circuit roundby-round to echo out components in different ways, to analyzing the different input bases separately to extract details about the error structure.Our hope is that other researchers will be able to create their own modifications of the CAFE framework to study the errors facing their own systems.

VII. ACKNOWLEDGEMENTS
We are grateful to the Google Quantum AI team for building, operating, and maintaining software and hardware infrastructure used in this work.The authors would like to thank Will Livingston, Sergio Boixo, Ben Chiaro, Vinicius S. Ferreira, Abraham Asfaw, and Paul V. Klimov for discussions and feedback on the draft, and Catherine Erickson and Andreas Bengtsson for software support.
where d = 2 m is the dimension of the m-qubit Hilbert space.Using this projector, the fidelity takes the form In order to compute the fidelity expression in Eq.B7, one computes the eigenvalues of U † K α and sum them up directly to get tr(U † K α ).For example, we consider the case where the operation of interest can be described by a quantum channel that either applies the unitary Ũ , or totally depolarizes the qubits with probability p depol where I d is the d-dimensional identity operator.The fidelity of this channel after n consecutive applications is given by After subtracting spam from the RHS to account for experimental SPAM errors, the expression in Eq.B9 can be used to fit the CAFE data of any m-qubit unitary Ũ , subject to symmetric depolarizing noise, with respect to a reference unitary U .Note that in the case where the quantum operation under characterization, Ũ , corresponds exactly with the reference unitary U , the CAFE data should obey the following single-exponential decay Otherwise, when Ũ = U , coherent errors are present and the first term of B9 will introduce errors that scale quadratically with n to first order.
In the case we are interested in, we want to use CAFE to characterize a two-qubit gate, where d = 4, U = CZ and Ũ is a number-preserving fSim gate close to a CZ gate: with some residual SWAP-like error captured by the angle ∆θ, single-qubit phase error captured by ∆γ, and CPhaselike error captured by ∆φ, with ∆θ, ∆γ, ∆φ 1.Since CZ is diagonal, we can show that the eigenvalues of (U † ) n Ũ n are for n ∈ N. As such, tr[(U † ) n Ũ n ] = sum(λ) and substituting into Eq.B9 gives which is the expression in Eq. 4 of the main text, after subtracting the SPAM errors spam from the RHS to account for experimental imperfections.circuit for a distance-3 surface code experiment used in Ref. [26].In Fig. 8b, we show how this section of the circuit can be considered on its own and broken down into qubit groups.We can then run a parallel CAFE experiment on these qubit groups which uses the highlighted section as the cycle circuit.The resulting experimental data is shown in Fig. 8c.We can see that one of the m = 1 qubit groups experiences significant error, despite its individual single-qubit gates X and H showing good fidelities.This highlights the importance of context-aware characterization, as the error mode being experienced was only visible when the gates were run in conjunction.This experiment also highlights the flexibility of CAFE in being able to characterize multi-operation circuits simultaneously on qubit groups of varying size.

Appendix E: Numerical CAFE simulations
In this section, we present numerical simulations where we characterize a CZ gate using the CAFE approach presented in the main text.We simulate the same 2 4 = 16 circuits executed on the Sycamore device with noisy two-qubit unitaries, together with either depolarizing noise in App.E 1 or amplitude and phase damping noise in App.E 2. We also used the same low cycle repetitions n ∈ [0, 2, 4, 6, 8] which make the CAFE experiment have especially low execution time relative to other two-qubit benchmarking techniques.In Sec.III we showed that using CAFE to characterize a CZ gate can allow for a more accurate fidelity estimation than the widely used Randomized Benchmarking (RB) protocol while using significantly less experimental resources.Note that we also simulate the sampling of 2000 shots used experimentally, which places an upper bound on the standard deviation of all the CAFE data points presented in Figs. 3, 4, and 5 of which is smaller than all the markers used in the plots of the main text.Importantly, these simulations allow us to confirm the validity of the CAFE experiment and following fitting procedure to obtain accurate estimates of the following: the average gate fidelity of the quantum operation with regards to any reference unitary, the incoherent error contribution to the infidelity and the coherent error contribution.
The simulations were performed using the open-source software Cirq [38].In blue, using a quadratic fit for the fidelity at depths n ∈ [0, 2, 4], and in orange using the analytical expression of Eq. ( 4) for depths n ∈ [0, 2,4,6,8].Inset shows the median absolute errors (MAE) of the two error budgeting techniques from the true values inserted into the simulation.We note that MAE scales with the true error of the gates being considered.The true incoherent (coherent) errors are obtained from the channel's average gate infidelity when only depolarizing errors (unitary errors) are present in the simulation.

Depolarizing noise
In a first case, we consider that the noisy CZ gate we are trying to characterize is described by the quantum channel which outputs a totally depolarized state with probability p depol , and otherwise applies the excitation-preserving unitary where we allow for miscalibrations in all of the five angles that parametrize this gate, where Ṽ (0, 0, 0, 0, 0) = CZ.For the simulation, we then sample the depolarizing probability uniformly in the range from 0 to 0.05, and sample the miscalibrated angles from a normal distribution with zero mean and standard deviation of 0.05 rad such that Using this model, we can simulate the noisy CAFE experiment for different cycle repetitions n, and then fit these data points to obtain an error budgeting of the noisy CZ gate, which we can compare directly with the true parameter values that were drawn for the simulation.In Fig. 9, we present the average gate infidelity (1 − F), incoherent error contribution incoh , and coherent error contribution coh to this infidelity for 1000 independently sampled unitaries and depolarizing probabilities.We present two valid gate error budgeting approaches, one which fits the CAFE data with the analytical expression in Eq. 4 (orange points), and one which uses the simple quadratic form of Eq. 8 (blue points).These scatter demonstrate the validity of CAFE to characterize the fidelity of a quantum operation and budget its coherent and incoherent contributions with an imprecision smaller or equal to 0.001 on median for realistic gate fidelities.Note that the reported median absolute errors depend on the specific range of gate fidelities considered.For example, the MAE values decrease significantly when considering only gate infidelities < 1% in the ensemble.
The quadratic fit results shown in blue in Fig. 9 are an indication of how the CAFE framework can be deployed in order to estimate the fidelity of an operation in context.However, this simplest approach gives slightly biased results for incoh and coh , which is understood from the breakdown of the np depol  10.Similar simulations as presented in Fig. 9, now implementing realistic amplitude and phase damping incoherent noise instead of a depolarizing channel.The analytical fit, which still uses Eq. ( 4) that effectively lumps the incoherent noise into a p depol probability, works remarkably well even under this different noise channel.
approximation, where α here stands for any of the five miscalibrated CZ angles.Consequently, using a quadratic fit should be used carefully, for operations with high fidelity (> 99 %) and using shallow cycle repetitions n.In fact, for these simulations, we have found that using larger depths than n = 4 for the quadratic fit decreased the accuracy of the resulting error budgets.However, if the fidelity of the operation of interest is very high, for example with single-qubit gates, using a quadratic fit becomes an attractive strategy when lacking a proper analytical model of the error origins.
On the other hand, the analytical fit performs well for larger depths n since it does not rely on any approximation.Here, we have only used the depths n ∈ [0, 2,4,6,8] to be consistent with the experiment results of the main text.These results are an important validation for CAFE, but not necessarily surprising since we are using the same error model to simulate the experiment and to fit the resulting data.In the next sub-section, we show that this model also holds very well for different noise models such as amplitude and phase damping, provided they cause realistic gate infidelities of a few percent or less.

Amplitude and phase damping noise
To verify the robustness of the CAFE framework to different incoherent noise processes, we have also simulated a two-qubit CAFE experiment where the quantum channel is described by the same unitary Ṽ with randomly sampled angles as before, but now followed by a single-qubit amplitude and phase damping channel on both qubits.The total decay and phase-flip probabilities for both qubits were sampled from a normal distribution with a standard deviation of 0.03, before taking the absolute value {p decay,total , p phaseflip,total } ∝ N (0, 0.03 2 ) , (E6) and the individual decay and phase-flip probabilities of the two qubits were splitting these total probabilities with a ratio drawn from a normal distribution with mean of 0.5 and standard deviation of 0.1, such that the qubits have similar but distinct coherence properties.In Fig. 10, we show that the CAFE framework we developed again gives very accurate CZ characterizations and error budgets.In particular, using the analytical expression of Eq. ( 4) to fit this CAFE data works remarkably well given that the noise present in the gate cannot be fully described by the assumed depolarizing channel.Since the incoherent noise is small, but in the realistic range of creating about 0.1 % to 4 % gate infidelity, the fit is able to approximate well the incoherent error contribution into the single exponential of the model.This gives us confidence that we can leverage this model in realistic gate characterization scenarios.Moreover, the simulations show that the accuracy of using this model increases with increasing gate fidelities, which is a great indication for the continued usefulness of this method as gates and qubit coherences keep improving.In blue, we show simulations including both incoherent errors of strength incoh = 0.005 and coherent errors of coh = 0.002, and the dotted line shows the fit of this data using Eq. ( 4).As in the main text, only the even depths are used in the fit, the odd depths points are presented with x symbols simply to show their consistency with the model.Note: the green data is the same in both panels as we fix the incoherent errors to be zero.

Fitting CAFE data
In Fig. 11, we present examples of the CAFE data together with fits to the model of Eq. ( 4) realized on the simulation data to obtain the CZ error budgets presented in Figs. 9 and 10.We see that the model of Eq. ( 4) fits the CAFE data very well for realistic noise of strength incoh = 0.005 and coh = 0.002, even when the incoherent noise comes from an amplitude and phase damping channel.In Fig. 11, we also show simulation data using the same noise channel but an ideal unitary ( coh = 0) in red, and using the same unitary without any incoherent noise ( incoh = 0) in green.The linear and purely quadratic behaviors of these curves, respectively, can be easily understood from the quadratic form of Eq. ( 8) where the incoherent errors build up linearly with n, whereas the coherent errors build up quadratically.
Since we simulate the actual 16 circuits used to obtain the average gate fidelity at different depths, we capture some depth-dependent behaviors that are due to coherent effects in the single-and two-qubit unitaries (see for example the green 'x' at n = 1 in Fig. 11, which is actually higher than the point at n = 0).This is not an artifact of the finite sampling, but is due to the fact that the preparation and measurement circuits both require an imperfect CZ gate, which can coherently map the state closer or further away from the desired state |00 , depending on the specific imperfect unitaries.Similarly, part of the incoherent noise channel can anti-commute with the CZ unitary and produce back-and-forth behaviors between the odd and even depths n.We have found in simulation that using only the even depths avoids this issue, while performing similarly or better in terms of accuracy in error budgeting, compared to using all the even and odd depths.

Appendix F: Dynamical Decoupling in CAFE
In this Appendix, we show how applying an X gate to both qubits after the CZ in the cycle circuit decouples the fSim unitary from both of its single-qubit phases.First, we can show that interleaving a pair of fSim gates with parallel X gates gives a cycle unitary that also corresponds to an fSim gate.To see this, consider what happens when conjugating an fSim unitary by parallel Xs:  12. Error budgets of parallel CZ gates on a Sycamore processor.Data in blue is obtained from the CAFE experiment using the procedure detailed in the main text, where 3 samples of this data are shown in Fig. 3.For reference, we present similar gate error budget information as obtained from standard XEB in gray, where the incoherent error is extracted from the speckle purity and the coherent error is obtained from the difference with the total gate error.Data shows 103 different two-qubit gates.
FIG.5.Using CAFE to validate different unitary characterizations of a CZ gate with 50th percentile infidelity.The different curves represent the fidelity between the experimental gate and an ideal CZ unitary (blue), a unitary extracted from an XEB experiment (green), and a unitary characterized with a newly introduced method called MEADD[35].In these three different CAFE experiments, only the final pre-measurement circuit layer changes (see the green box in Fig.1).The improvement in fidelity can thus entirely be attributed to considering a gate unitary that is closer to the experimental CZ implementation.Error bars are scaled by a factor 5 to be visible.

FIG. 8 .
FIG.8.a.The stabilizer extraction circuit for a distance-3 surface code, with the section to be characterized boxed in orange.b.A layout for the CAFE experiment along with the qubit groups characterized.c.CAFE data for repetitions of the cycle circuit from a. on the qubit groups in b., with the m = 2 groups in green and the m = 1 groups in red.CAFE allows us to see that q11 (red, dotted) is experiencing a significant context-dependent error.

FIG. 11 .
FIG. 11.Example of fitting the CAFE curve obtained in simulations with the depolarizing noise channel used in App.E 1 (left) and the amplitude and phase damping noise channel used in App.E 2 (right).In blue, we show simulations including both incoherent errors of strength incoh = 0.005 and coherent errors of coh = 0.002, and the dotted line shows the fit of this data using Eq.(4).As in the main text, only the even depths are used in the fit, the odd depths points are presented with x symbols simply to show their consistency with the model.Note: the green data is the same in both panels as we fix the incoherent errors to be zero.
iγ u 11 e −iγ u 12 0 0 e −iγ u 21 e −iγ u 22 0 u 22 e −iγ u 21 0 0 e −iγ u 12 e −iγ u 11 0 FIG.12.Error budgets of parallel CZ gates on a Sycamore processor.Data in blue is obtained from the CAFE experiment using the procedure detailed in the main text, where 3 samples of this data are shown in Fig.3.For reference, we present similar gate error budget information as obtained from standard XEB in gray, where the incoherent error is extracted from the speckle purity and the coherent error is obtained from the difference with the total gate error.Data shows 103 different two-qubit gates.
4IG.4.Experimental results on characterizing CZ gates in parallel when dynamical decoupling gates are part of the cycle circuit (orange), alongside the standard CAFE data presented in Fig.3(blue).Interleaving the CZ gates with X gates on both qubits echoes out low-frequency Z noise, which contributes to incoh , and removes the sensitivity to single-qubit phase unitary errors, which contribute to coh .Performing the simple DECAF experiment thus provides valuable information about the error mechanisms in CZ gates.Error bars are scaled by a factor 5 to be visible.
Simulating the CAFE experiment for characterizing 1000 different CZ gates with realistically small coherent errors and depolarizing incoherent noise.Two error budgeting techniques obtained by fitting this CAFE data are presented.