• Photonics Research
  • Vol. 11, Issue 1, 81 (2023)
Yihong Zhang, Wenjun Yu, Pei Zeng, Guoding Liu, and Xiongfeng Ma*
Author Affiliations
  • Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
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    DOI: 10.1364/PRJ.473970 Cite this Article Set citation alerts
    Yihong Zhang, Wenjun Yu, Pei Zeng, Guoding Liu, Xiongfeng Ma, "Scalable fast benchmarking for individual quantum gates with local twirling," Photonics Res. 11, 81 (2023) Copy Citation Text show less

    Abstract

    With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled-(TX) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.

    1. INTRODUCTION

    Characterizing a quantum process has great importance in both the fundamental study and practical application of quantum information science. With the recent advent of noisy intermediate-scale quantum computing [1], benchmarking of quantum operations is critical for quantum control [2,3] as it provides an indicator for assessing experimental devices. Furthermore, it is essential for the development of high-precision quantum information processing instruments. Accurate benchmarking can reliably characterize the noise levels of quantum operations, and it plays a critical role in promoting fault-tolerant universal quantum computing [4,5]. In practice, we must evaluate the performance of a quantum circuit to verify whether a quantum algorithm or an error-correcting code is properly implemented in a quantum system.

    Numerous approaches have been proposed to characterize quantum processes. Conventional methods such as quantum process tomography [6] provide a full description of a channel. However, these methods are impractical for large-scale quantum systems as the required experimental resources increase exponentially with the number of qubits, even with state-of-the-art techniques such as compressed sensing [7,8]. Direct fidelity estimation [9] tackles the scaling problem and characterizes the quantum process in terms of average fidelity. Unfortunately, the result inevitably contains extra errors from the state preparation and measurement (SPAM) and hence often over-estimates the noise levels. In reality, SPAM errors usually grow rapidly with the system size so that it is hard to characterize the quantum process accurately for large-scale quantum systems with direct fidelity estimation.

    Randomized benchmarking (RB) and its variants have been proposed to avoid simultaneously the scaling problem and SPAM errors [1016]. Standard RB estimates the average error rate of a specific gate set under the assumption of gate-independent or weakly-dependent noise. The gate set is normally selected to be the Clifford group and has been widely implemented in experiments [1724]. Otherwise, to characterize a specific Clifford gate, a variant called interleaved RB was proposed. It utilizes random Clifford gates interleaved with the target gate [25]. The random gates here are considered to be the twirling gates for reference, whose fidelity should be measured separately to infer the fidelity of the target gate. The interleaved RB method is efficient and scalable in principle. However, it suffers from two severe problems in practice. The first is the compiling overhead for twirling operations. In reality, any operation needs to be compiled to one- and two-qubit gates native to the quantum system. Note that twirling gates are randomly picked from a gate set, like the Clifford group. In general, the average number of native gates used for compiling a single sample grows dramatically with the system size. The second is the gate-dependent noises introduced by twirling gates. Note that different twirling operations in a gate group can vary a lot in the depths of compiled circuits. For example, a local operation such as a Pauli gate can be implemented by a single layer circuit, while a complex entangling operation requires a deep circuit with massive native gates. The strong gate-dependent noises caused by the uneven compilations may bring inaccuracy to the fidelity estimation [26,27]. Consequently, the compiling overhead and gate-dependent noises introduced by twirling gates limit the scalability of the RB method in experiments.

    Recently, there have been several variants of RB attempting to address the two compiling problems. For example, character benchmarking employs the character theory so that the quality parameters can be extracted from the local twirling operations [28]. Unfortunately, for the gate groups with an exponentially increasing number of quality parameters, this method requires an exponential amount of SPAM settings. Besides, character benchmarking is still caught in the aforementioned compiling problems for the final inverse gate and can be hardly applied for a generic multi-qubit quantum operation. Another inspiring attempt called cycle benchmarking aims to estimate the fidelity of the target gate by interleaving it with the Pauli gate set. However, it is restricted to the Clifford gates [29]. In addition, cycle benchmarking requires numerous repetitions for the gates with large cyclic numbers, which is common for multi-qubit gates. Hence, this method cannot efficiently benchmark a wide class of gates. The cross-entropy benchmarking (XEB) characterizes the fidelity of a generic quantum gate reflected by linear cross-entropy using local Clifford gate twirling [30]. However, the Haar measure assumption in XEB may lead to poor fidelity estimation when the size of the target gate is large. Thus, how to efficiently and reliably estimate the fidelity of a large-scale quantum process from a universal gate set remains an open problem.

    This work proposes two scalable and efficient protocols to tackle the compiling problems as well as the SPAM error issues simultaneously, which we label character-cycle benchmarking (CCB) and character-average benchmarking (CAB), respectively. The protocols utilize local twirling to reliably characterize the fidelity of an individual multi-qubit quantum operation. We employ the Pauli and the local Clifford gates for twirling and extend the applicable gate set to non-Clifford gates via the local gauge transformation. The efficiency and reliability of the protocols are shown by rigorous mathematical derivations and by numerical simulations under realistic physical assumptions.

    2. CHARACTER CYCLE BENCHMARKING

    Let us denote the quantum operation of a unitary matrix U acting on an n-qubit quantum state, ρ, by the calligraphic letter U, i.e., U(ρ)=UρU1, and the noisy implementation by U˜. One can evaluate the quality of U˜ by the process fidelity of the noise channel Λ=U1U˜, F(Λ)=1d2i=04n1λi,where λi=d1Tr(PiΛ(Pi)) is the Pauli fidelity associated with the Pauli operator PiPn, and d=2n is the dimension of the quantum system. Here, Pn denotes the n-qubit Pauli group, containing the tensor product of the identity operation I and three Pauli matrices X,Y,Z.

    In practice, it is costly to figure out all the parameters λ0,λ1,,λ4n1 as their number increases exponentially with n. Instead, one can estimate the process fidelity via repeatable sampling of λ0,λ1,,λ4n1. Concretely, one samples a sufficient number of Pauli operators {Pj} and averages the corresponding {λj}, F(Λ)1MΣ{Pj}λj,where M is the number of samples and the summation takes over the sample set.

    Here, we propose a CCB protocol that employs the key techniques of the cycle [29] and character benchmarking [28]. Specifically, we extract different Pauli fidelities through applying specific initial states and measurements and utilize the character theory to fully separate the SPAM errors. The schematic circuit of the CCB protocol is shown in Fig. 1(a). Let us start with the Clifford case, where the target gate belongs to the n-qubit Clifford group Cn. The inner random gate layer consists of the target gate U and its inverse gate U1, interleaved with two random Pauli gate layers. The Pauli gates are the reference gates employed to perform local Pauli twirling over the generic quantum noise channel Λ and turn it into ΛP=14nΣ{PjPn}Pj1ΛPj,where Λ contains the errors of Pauli gates and target gate U; ΛP is a Pauli channel satisfying ΛP(ρ)=jpjPjρPj, and pj is the Pauli error rate related to Pj.

    Illustrations of circuit and procedures used in (a) CCB and (b) CAB protocols. The orange boxes represent the target gate U and its inverse gate U−1. The blue and green boxes represent the random Pauli gate and random local Clifford gate. The yellow boxes denote the inverse gate for the m inner gate layers in the light blue box. Here, Pk(i) is a single-qubit Pauli gate on qubit k, and P(i)=P1(i)⊗⋯⊗Pn(i) is an n-qubit Pauli gate.

    Figure 1.Illustrations of circuit and procedures used in (a) CCB and (b) CAB protocols. The orange boxes represent the target gate U and its inverse gate U1. The blue and green boxes represent the random Pauli gate and random local Clifford gate. The yellow boxes denote the inverse gate for the m inner gate layers in the light blue box. Here, Pk(i) is a single-qubit Pauli gate on qubit k, and P(i)=P1(i)Pn(i) is an n-qubit Pauli gate.

    Note that the introduction of the inverse target gate U1 is the major difference between CCB and cycle benchmarking. In cycle benchmarking, we need to repeat U for multiples of l times, where l is the cyclic number of U, i.e., Ul=I. In general, l can be quite large for a wide class of Clifford gates, which is prohibitive for the experiments. For example, the five-qubit quantum error-correcting encoding circuit requires l=124. The CCB protocol improves the efficiency and application scope via substituting a single U1 for multiple repetitions of U in cycle benchmarking. In many quantum platforms, such as superconducting quantum processors, the inverse gates of the native gates are also native. Typical examples include single-qubit gates, CZ, and iSWAP. Thus, the inverse gates of native gates are normally easy to implement. More generally, if U is composed of several native gates, the difficulty to implement U and U1 is often the same. Based on this consideration, the introduction of U1 does not increase the implementation difficulty of CCB in most cases.

    In CCB, the randomization of Pauli gates in the inner gate layers will generate a composite channel U1ΛP()UΛP, where ΛP and ΛP() are the Pauli-twirled channels corresponding to gates U and U1, respectively. Note that this composite channel is a Pauli channel for Clifford gate U. The fidelity we aim to estimate in the CCB protocol is defined as the CCB fidelity, Fccb=F(U1ΛP()UΛP),which contains the fidelities of U and U1. When the noise channel of U is the same as that of U1, which is valid for most of the experimental platforms, the CCB fidelity is simplified as Fccb(Λ)=F(U1ΛPUΛP).

    Equation (5) is a lower bound of the process fidelity F(Λ) in terms of the expectation value, as proved in Appendix B.2. In the following context, we will employ Eq. (5) as our CCB fidelity metric model and our arguments apply to the general model of Eq. (4) as well. The difference between Fccb and F is normally small because the physical realizations of the qubits in one experimental platform are similar and the qualities of these qubits will not differ too much. Note that if ΛP is a depolarizing channel, then Fccb=F. Thus, the CCB fidelity can be seen as a reliable metric for the noise channel Λ.

    The procedure of the CCB protocol is as follows. Sample a Pauli operator Pj and initialize state |s such that Pj|s=|s.Apply a gate sequence Sccb composed of a Pauli gate P(0), m inner gate layers denoted by Sm, and inverse gate Sm1.Perform measurement Pj and then calculate the Pj-weighted survival probability fj(m,Sccb)=χj(P(0))·Tr(PjSccb(ρs)), where χj(P(0))=1 if Pj commutes with P(0) and 1 otherwise.Repeat steps (2) and (3) for several times for different m and fit the Pj-weighted fidelity to f^j(m)=Ajλj2m.Repeat steps (1)–(4) for several times and finally estimate the CCB fidelity as Fccb=avejλj.

    Here, the estimated fidelity Fccb includes the errors from the local reference gate set Pn. To remove these extra errors, one can employ the interleaved RB technique, by performing additional CCB with a target gate of identity I to estimate the reference fidelity FccbI. Then, one can infer the fidelity of the target gate as Fccb/FccbI. In practice, the errors of local gates are often negligible, and hence, we focus on Fccb in the following discussions.

    Note that our inverse gate Sm1 is a Pauli gate and thus will not introduce extra gate compiling overhead. In contrast, character benchmarking for a single multi-qubit Clifford gate [28] requires a global inverse gate and a complicated compiling process. This may cause strong gate-dependent errors and lead to inaccuracy for fidelity estimation, especially for multi-qubit quantum operations. The CCB protocol maintains the local structure of reference and inverse gates and thus avoids the compiling problems.

    In the CCB protocol, one needs to average Pauli fidelities λj to estimate Fccb. The sampling complexity for the CCB protocol is given by the following theorem.

    Theorem 1 (informal version). For ann-qubit quantum noise channel, to estimate the CCB fidelity within the confidence interval[F^ccbϵMϵb,F^ccb+ϵM+ϵb]with probability greater than1δ, one needs to sampleMPauli fidelities where each Pauli fidelity is estimated viaKrandom sequences. The confidence probability of the estimation is given by Pr(|F^ccbF¯ccb|ϵM+ϵb)1δ,where ϵMO(logδM) and ϵbO(K1)+O((logδK)3/2).

    Here, the total number of samples, or sample complexity, depends on M and K. If the number of random sequences for each Pauli fidelity is the same, then the sample complexity is simply given by MK. Theorem 1 shows that the sample complexity only depends on fidelity precision ϵM,ϵb, and confidence level δ. The independence on system size n reflects the strong scalability of the CCB protocol. A more detailed description of the result is shown in Theorem 2.

    3. LOCAL GAUGE TRANSFORMATION

    Now, let us extend the applicable gates for the CCB protocol to non-Clifford gates. One can introduce local gauge transformation L to the twirling gate set, PnLPnL1, where L is an arbitrary local unitary operation LU(2)n. Note that the transformed twirling gate set LPnL1 is still local. Then, we can show that the applicable target gate set becomes LCnL1, where Cn is the n-qubit Clifford gate set.

    To benchmark a gate LUL1 from gate group LCnL1, we insert local gates L and L1 between the twirling gates and the target gates in the original CCB circuit, as shown in Fig. 2(a). Here, L=i=1nLi, where Li can be an arbitrary single-qubit gate. In practice, the local gates are absorbed into twirling gates and target gates and do not need to be implemented individually as manifested in Fig. 2(b). The character gate LP(0) and the twirling gate LP(1)L1 will be merged into a single gate in implementation as well. Details of the derivation are shown in Appendix B.3.

    Illustrations of the noisy CCB circuit with local gauge transformation. For simplicity, we show the case that the local gates are noiseless. The grey dashed boxes denote the noise channel Λ. The orange boxes represent the target gate U and its inverse gate U−1. The blue boxes represent the random Pauli gates. The green boxes represent the inserted local gates L and L−1, where L=⊗i=1nLi. The yellow box denotes the inverse gate for the m inner gate layers in the light blue boxes. In practice, we implement gates in circuit (b) while absorbing local gates L and L−1 into twirling and target gates. Here, the target gate following gauge transformation becomes LUL−1. Note that circuit (a) is equivalent to a CCB circuit with target gate U and noise channel ℒ−1Λℒ. Thus, it can be implemented to estimate Fccb(ℒ−1Λℒ), which is close to F(Λ).

    Figure 2.Illustrations of the noisy CCB circuit with local gauge transformation. For simplicity, we show the case that the local gates are noiseless. The grey dashed boxes denote the noise channel Λ. The orange boxes represent the target gate U and its inverse gate U1. The blue boxes represent the random Pauli gates. The green boxes represent the inserted local gates L and L1, where L=i=1nLi. The yellow box denotes the inverse gate for the m inner gate layers in the light blue boxes. In practice, we implement gates in circuit (b) while absorbing local gates L and L1 into twirling and target gates. Here, the target gate following gauge transformation becomes LUL1. Note that circuit (a) is equivalent to a CCB circuit with target gate U and noise channel 1Λ. Thus, it can be implemented to estimate Fccb(1Λ), which is close to F(Λ).

    As shown in Fig. 2, the CCB circuit with local gauge transformation L and noise channel Λ is equivalent to the original CCB circuit with noise channel L1ΛL. Thus, one can obtain Fccb(L1ΛL), which is close to the process fidelity F(L1ΛL). As process fidelity is gauge-invariant, that is, F(L1ΛL)=F(Λ), one can estimate the process fidelity of Λ as the performance indicator of gate LUL1.

    Now, let us check out what kinds of quantum gates belong to the set S={LUL1|LU(2)n,UCn}.

    First, notice that if a unitary US, then for any LU(2)n, LUL1S. As any unitary is generated by a Hamiltonian, that is U=eiH where H is Hermitian, one can conclude that if eiHS, then for any LU(2)n, LeiHL1=eiLHL1S.

    Take a step forward. If a controlled-eiHS, then through local gauge transformation IL, (IL)controlledeiH(IL)1=controlledeiLHL1S. The arguments also apply to the case of multi-controlled gates.

    The two observations inspire us to first represent Clifford gates in the form of eiH or multiple controlled-eiH, then replace H with LHL1 to find other gates in S. Take CZ as an example. CZ=eiπ|1111|= controlled-eiπ2Z. Through local gauge transformation, one can transform |11 to any product state |ψϕ and transform eiπ2Z to any π-rotation eiπ2σ·θ, where σ=(X,Y,Z) and θ is a unit vector. Thus, for any two-qubit product state |ψϕ, we have eiπ|ψϕψϕ|S. In addition, any controlled-π rotation, such as controlled-H and controlled-TX, belongs to S.

    Reversely, controlled-S= controlled-eiπ4Z is a controlled-π2 rotation. As any controlled-π2 rotation is not Clifford, one can conclude that controlled-S does not belong to S. Similarly, Tofolli = controlled-controlled-eiπ2X is a controlled-controlled-π rotation. As any controlled-controlled-π rotation is not Clifford, one can conclude that Tofolli does not belong to S either. It is interesting to decide whether a quantum gate belongs to S in a more general case, and we leave it for future work.

    4. CHARACTER-AVERAGE BENCHMARKING

    We can take the CCB protocol one step further. Observe that in CCB, one needs to implement the fitting procedures for each sampled Pauli operator Pj to estimate Pauli fidelity λj. Each estimation requires specific initial state, measurement, and independent randomization procedures. We can further simplify these procedures by introducing the local Clifford group C1n. Recall that in a qubit system, the Clifford twirling depolarizes a channel via averaging the error rates in X,Y,Z bases [13]. Then for an n-qubit system, the twirling over C1n would partially depolarize a channel and average out 4n Pauli fidelities λj into 2n terms. These 2n values can be obtained from the Zn basis measurement only with additional data post-processing.

    Based on the local Clifford twirling, we propose the CAB protocol as an improvement of the CCB protocol. The schematic circuit of CAB is shown in Fig. 1(b), with the detailed procedures described in Box 1. Like the CCB protocol, we can extend the target gate set beyond the Clifford group by employing local gauge transformation. Here, to suppress statistical fluctuations, we remove the character technique. Detailed description and analysis of the CCB and CAB protocols are presented in Appendix B.

    Similar to the CCB protocol, the randomization over m gate layers inside the blue box in Fig. 1(b) will generate a Pauli channel ΛP(m)=(U1ΛPUΛP)m. The local Clifford gates in the beginning and end of the circuit jointly perform local unitary 2-design twirling, which transforms the Pauli channel ΛP(m) into a partially depolarizing channel ΛC(m). Here, the quantum channel ΛC(m) contains less independent parameters than the original ΛP(m). It holds the unique value of fidelity for every disjoint Pauli subset in Rn={{I},{X,Y,Z}}n. The Pauli fidelities in ΛC(m) can be seen as the average values of those in ΛP(m), μk2m=Pjσkλj2m/|σk|, where {λj2m} are the Pauli fidelities of the channel ΛP(m). The local Clifford twirling here averages multi exponential decays into one exponential decay and captures all the information of the noise channel, as shown in Eq. (11). The comparison between Fcab and Fccb is shown in Lemma 3 in Appendix B.5, while the CCB protocol employs a sampling method as in Eq. (2), which only contains partial information of the noise channel. Thus, one can intuitively conclude that the CAB protocol is more efficient than the CCB protocol, as demonstrated in later simulations.

    5. SIMULATION

    In numerical simulations, we characterize a two-qubit controlled-(TX) gate and a five-qubit quantum error correcting encoding circuit, respectively. Furthermore, we simulate the noise channel for the target gate with a realistic error model that contains a Pauli channel, an amplitude damping channel, and a correlation channel. In the simulation, the Pauli fidelities of the Pauli channel are randomly sampled from a normal distribution N(μ,σ), which we call the N(μ,σ)-Pauli channel. Here, the error parameter μ reflects the quality of the Pauli channel and σ implies the discrepancy of the channel, i.e., the differences among Pauli fidelities. The detailed descriptions for the error models and simulations are presented in Appendix D.

    For the controlled-(TX) gate, we take (IT)·Pn(IT1) as the twirling gate set, where T is the π/8-phase gate, T=exp(iπZ/8), and IT is the local gauge transformation. We simulate the CAB and CCB protocols on the controlled-(TX) gate with 8 different noise channels. For each noise channel, we take 40 independent simulations for both CAB and CCB protocols. In CCB simulations, we sample M=10 Pauli operators to estimate Fccb.

    Figure 3(a) shows Fcab and Fccb versus the error rate r=1F for the controlled-(TX) gate with different noise channels. We observe that when the standard deviation of error parameters σ grows, the error bars of Fcab and Fccb become larger. Intuitively, the discrepancy of the Pauli fidelities is one of the key reasons for the fluctuations of Fcab and Fccb. The fluctuations for the estimations will reach the minimum level when the noise channel is completely depolarizing. Besides, the error bar of CAB is smaller than the error bar of CCB. This shows that under the same estimation accuracy, the sampling complexity, i.e., the amount of sampling sequences in total, of the CAB protocol is smaller than that of the CCB protocol, especially when the discrepancy of the noise channel is large. In Fig. 3(b), we take one of the eight noise channels as an example and show the three fitting curves of Eq. (10) for the CAB protocol. The resulting CAB fidelity is Fcab=95.99%, which is very close to the theoretical value of process fidelity F=95.98%.

    Simulation results for the controlled-(TX) gate with eight different noise channels, each containing a N(μ,σ)-Pauli channel, an amplitude damping channel, and a correlation channel. The parameters for the amplitude damping channels and correlation channels are set to be the same for the eight channels. For the Pauli channel, the error parameters are set to {(μ,σ)} = {(0.995, 0.001), (0.990, 0.002), (0.980, 0.003), (0.970, 0.004), (0.960, 0.005), (0.950, 0.006), (0.940, 0.007), (0.930, 0.008)}. (a) The fidelity estimations with different error rates in 40 independent simulations. The green dashed line represents the theoretical fidelities. The two insert scatter plots show the fluctuations of estimated fidelities over different simulations. (b) Take the fifth simulation with a process fidelity of 95.98% as an example. The fitting curves of Eq. (10) for Qk=IZ,ZI,ZZ. The decay parameters derived from the curves are λIZ=0.9580, λZI=0.9550, λZZ=0.9577.

    Figure 3.Simulation results for the controlled-(TX) gate with eight different noise channels, each containing a N(μ,σ)-Pauli channel, an amplitude damping channel, and a correlation channel. The parameters for the amplitude damping channels and correlation channels are set to be the same for the eight channels. For the Pauli channel, the error parameters are set to {(μ,σ)} = {(0.995, 0.001), (0.990, 0.002), (0.980, 0.003), (0.970, 0.004), (0.960, 0.005), (0.950, 0.006), (0.940, 0.007), (0.930, 0.008)}. (a) The fidelity estimations with different error rates in 40 independent simulations. The green dashed line represents the theoretical fidelities. The two insert scatter plots show the fluctuations of estimated fidelities over different simulations. (b) Take the fifth simulation with a process fidelity of 95.98% as an example. The fitting curves of Eq. (10) for Qk=IZ,ZI,ZZ. The decay parameters derived from the curves are λIZ=0.9580,  λZI=0.9550,  λZZ=0.9577.

    For the five-qubit error correcting encoding circuit, which is a Clifford gate, we take the Pauli group Pn as the twirling gate set. For the simplicity of simulation, we set the Pauli channel to be a depolarizing channel Λdep(ρ)=pρ+(1p)I/d where p=0.98. The setting of the amplitude damping and correlation channels remains the same. Furthermore, we simulate the CAB and XEB protocols to characterize the noisy five-qubit encoding circuit. For each protocol, we run 40 independent simulations. In each simulation, we take the sampling number of gate sequences as K=50,,500 for each sequence length m. The box plot of Fcab versus K is shown in Fig. 4(a). We can see that when K grows, the fluctuations of Fcab become smaller. When K is not too large, like K=50, the fluctuation is already small enough, which implies that the CAB protocol works well with few sampling sequences needed.

    Simulation results for the five-qubit quantum error correcting encoding circuit with a noise channel composed of a depolarizing channel Λdep(ρ)=pρ+(1−p)I/d, where p=0.98, an amplitute damping channel, and a correlation channel. The theoretical process fidelity is F=94.70%. (a) Box plot of the CAB fidelities versus sampling number K with 40 independent simulations. The red boxes represent the distributions of the CAB fidelity estimations with respect to K. The orange points represent the distributions of CAB fidelities. (b) Box plots of the CAB and XEB fidelities versus K with 40 independent simulations. The green dashed line represents the theoretical process fidelity. The plots of CAB in (a) and (b) are the same with different scaling.

    Figure 4.Simulation results for the five-qubit quantum error correcting encoding circuit with a noise channel composed of a depolarizing channel Λdep(ρ)=pρ+(1p)I/d, where p=0.98, an amplitute damping channel, and a correlation channel. The theoretical process fidelity is F=94.70%. (a) Box plot of the CAB fidelities versus sampling number K with 40 independent simulations. The red boxes represent the distributions of the CAB fidelity estimations with respect to K. The orange points represent the distributions of CAB fidelities. (b) Box plots of the CAB and XEB fidelities versus K with 40 independent simulations. The green dashed line represents the theoretical process fidelity. The plots of CAB in (a) and (b) are the same with different scaling.

    In Fig. 4(b), we show the box plots of Fcab and XEB fidelities Fxeb versus the sampling number K. It is clear to see that compared with Fxeb, Fcab is much closer to the theoretical process fidelity F=94.7%. Meanwhile, the convergence of Fcab is much better than that of Fxeb. This implies that the required K for CAB is much smaller than that of XEB under the same estimation accuracy.

    To give a concrete example, we take 20 CAB simulations and 20 XEB simulations under the same noise channel. From the simulation results, we find that for CAB, when Kcab=20, the standard deviation over the 20 simulations is σcab=3.25×104, while for XEB, when Kxeb=20,000, the standard deviation is σxeb=4.29×104. This shows that to estimate the fidelities with standard deviations around 4×104, the required Kxeb is over 1000 times larger than Kcab. Thus, we can conclude that the performance of CAB protocol is three orders of magnitude better than that of XEB protocol in terms of the sampling complexity.

    The simulation results reveal the strong scalability and reliability of our protocols, especially the CAB protocol. The fluctuation of estimated CAB fidelity is small even for multi-qubit gates. We believe the CAB protocols can provide fast feedback in experimental designs and promote the development of universal fault-tolerant quantum computing.

    6. CONCLUSION AND DISCUSSION

    The characterization of large-scale individual quantum processes is crucial to the development of near-term quantum devices. However, to the best of our knowledge, there are no scalable and practical methods as of yet that could benchmark multi-qubit universal gate-set currently. In this work, we propose and demonstrate efficient and scalable randomized benchmarking protocols—CCB and CAB that can individually characterize a wide class of quantum gates including and beyond the Clifford set. The key technique of our protocols is using the local reference gate-set for twirling, which avoid the inaccuracy of the estimation caused by gate-compiling overhead and gate-dependent noises. The method of local gauge transformation offers a tool for characterizing non-Clifford gates. The sampling and measurement complexity are independent of the qubit number of gate, which means our benchmarking protocols can be generalized to large-scale quantum systems.

    Our protocols maintain the simplicity and robustness of the conventional RB method, and estimate the quantity of most interest—process fidelity of the target gates. We believe our protocols will promote the development of universal fault-tolerant quantum computing. Furthermore, it would also be interesting to extend our randomization and estimation methods for characterizing other properties such as unitarity and coherence, which we leave for future research.

    Acknowledgment

    Acknowledgment. We acknowledge B. Chen for the insightful discussions.

    APPENDIX A: PRELIMINARIES

    Representation Theory

    The representation theory works as a general analysis of every representation for abstract groups. Informally, the representations of a group can reflect its block-diagonal structures. Let G be a finite group and gG be a group element. The representation of G is defined as follows.

    Definition 1 (Group representation). Mapϕis said to be a representation of groupGon a linear spaceVif it is a group homomorphism fromGtoGL(V): ϕ:GGL(V),gϕ(g),  gG,whereGL(V)is the general linear group ofV, such thatg1,g2G: ϕ(g1)ϕ(g2)=ϕ(g1g2).

    Given representation ϕ on V, a linear subspace WV is called invariant if wW and gG: ϕ(g)wW.

    The restriction of ϕ to the invariant subspace W is known as a subrepresentation of G on W. One can further define the irreducible representation (or irrep for short) as follows.

    Definition 2 (Irreducible representation). Representationϕof groupGon linear spaceVis irreducible if it merely has trivial subrepresentations, i.e., the invariant subspaces forVare only{0}andVitself.

    The Maschke’s theorem provides an interesting property that each representation ϕ of a finite group G can be decomposed to the irreducible representations, gG: ϕ(g)σRGσ(g)mσ,where RG={σ} denotes the set of all the irreps of representation ϕ and mσ is the multiplicity of the equivalent irreps of σ. In this paper, we will focus on the non-degenerate representation case, i.e., mσ=1,σ.

    Definition 3 (Character function). Letσbe a representation over groupG, and the character ofσis the functionχσ:GCgiven bygG: χσ(g)=Tr[σ(g)].

    With the character function, we introduce the generalized projection formula used in character randomized benchmarking [28].

    Lemma 1 (Generalized projection formula [31]). Given a finite group, G, and its representation, ϕ, denoteσto be an irreducible representation contained inϕwith its character functionχσ:GC. The projector onto the support space ofσcan be written asΠσ=dσ|G|gGχσ(g)ϕ(g),wheredσis the dimension ofσ, dσ=χ(e)withebeing the identity element inG.

    Next, we will introduce twirling over a group G.

    Definition 4 (Twirling). For representationϕof groupGon linear spaceV, a random twirling for a linear mapΛ:VVoverGis defined asΛG=1|G|gGϕ(g)Λϕ(g).

    Using Schur’s Lemma [31], one can show the following proposition.

    Proposition 1.For any linear mapΛ:VV, the twirling over groupGand its representationϕcan be written asΛG=σRGTr(ΛΠσ)Tr(Πσ)Πσ,whereΠσdenotes the projector onto the support space ofσ, andRGdenotes the set of all irreps ofϕ.

    Representation for Quantum Channel

    Here, we introduce the quantum channel and three frequently used channel representations that our main results rely on. Denote the Hilbert space for n qubits as H and the set of linear operators on H as L(H). Quantum channels are defined as completely positive and trace-preserving (CPTP) linear maps on L(H). Given any quantum channel E:L(H)L(H), we can represent it in Kraus representation, OL(H): E(O)=l=1mKlOKl,where {Kl} are the Kraus operators satisfying l=1mKlKl=I.

    With the Kraus representation, the concatenation of the quantum channels or quantum gates is given by E2E1(O)=l2=1m2l1=1m1Kl2Kl1OKl1Kl2,where {K1} and {K2} are the Kraus operators for E1 and E2, respectively.

    To describe a long quantum circuit, the Kraus representation is not convenient. Here, we introduce another widely used representation—Liouville representation. The Liouville representation is defined on a set of trace-orthonormal basis on L(H). Often, we use the normalized Pauli group, i.e., Pauli group with a normalization factor. The n-qubit Pauli group is given by Pn=i=1n{Ii,Xi,Yi,Zi},where Xi,Yi,Zi are the single-qubit Pauli matrices. Then the normalized Pauli group is given by {σi=12nPi|PiPn}.

    Each pair of elements (σi,σj) in this group satisfies the following constraints under the Hilbert–Schmidt inner product, σi,σjPn/2n: Tr(σiσj)=δij.

    Any n-qubit operator can be decomposed over the 4n normalized Pauli operators. We can rewrite the density operator O on L(H) in a vector form: |O=iPnTr(σiO)σi.

    Moreover, any quantum channel can be represented as a matrix in the Liouville representation. To be specific, we can represent an arbitrary channel Λ acting on an operator O as follows: |Λ(O)=Λ|O.

    We can see the element of this matrix is given by Λij=σi|Λ|σj=Tr(σiΛ(σj)).

    Consequently, in the Liouville representation, the concatenation of two channels can be depicted as the product of two matrices: |Λ2Λ1(O)=Λ2|Λ1(O)=Λ2Λ1|O.

    The measurement operator can also be vectorized with the Liouville bra-notation according to the definition of the Hilbert–Schmidt inner product. For example, the measurement probability of a state ρ on a positive operator-valued measure {Fi} is given by pi=Fi|ρ=Tr(Fiρ).

    We call such a Pauli–Liouville representation as the Pauli transfer matrix (PTM) representation. An n-qubit quantum channel can also be described in the χ-matrix representation: Λ(ρ)=di,jχijσiρσj.

    The process matrix χ is uniquely determined by the orthonormal operator basis {σj}, where the first element is proportional to the identity matrix σ0=I/d, and d=2n is the dimension of the quantum system. Often, we take the normalized Pauli operators as the basis of the χ-matrix representation. If a channel is diagonal in this representation, we call it Pauli channel.

    Quantum Channel Fidelity

    The process fidelity of a channel Λ can be defined with its χ-matrix representation: F(Λ)=χ00(Λ).

    Here, the quantity χ00 is independent of the choices of operator basis {σj}. In the following, we set the operator basis to be normalized Pauli group. Define the Pauli fidelity of a quantum channel Λ, λj=d1Tr(PjΛ(Pj)),which is the diagonal term of the PTM representation of Λ. We can relate Pauli fidelities to the diagonal terms of χ-matrix representation via Walsh–Hadamard transformation: λj=i(1)i,jχii.

    Here, i,j=0 if Pi commutes with Pj and i,j=1 otherwise. Then, one can derive the process fidelity from Eq. (A23): F(Λ)=χ00(Λ)=1d2jλj,which can be viewed as another definition of process fidelity.

    There is a relation between the commonly used average fidelity Fave and the process fidelity [32]: Fave=(dF+1)/(d+1).

    Average fidelity is defined as Fave=dψTr(|ψψ|Λ(|ψψ|)),where the integral is implemented over Haar measure. These two fidelity measures are both well-defined metrics to quantify the closeness of a channel to the identity.

    Representation Theory in Randomized Benchmarking

    Now, we can use the representation theory to analyze the randomized benchmarking (RB) procedures. Let us start from a quick review of the standard RB protocol. Considering an n-qubit gate set G, RB is performed via sampling random gate sequences: Srb=Gm+1GmG1,where for 1im+1, Gi denotes the quantum operation in the PTM representation of a unitary matrix giG. Here, for 1im, gi is randomly sampled from group G and gm+1=(gmg2g1)1. Then, one applies the random gate sequence Srb to the input state ρψ0 and performs measurement Q0 for a sufficient number of times to estimate the average survival probability, f(m)=EG1GmQ0|G˜m+1G˜mG˜2G˜1|ρψ0,where G˜i denotes the noisy implementation of Gi, ρψ0 is the noisy preparation of the initial state |ψ, and measurement Q0 also includes errors. Here, we employ the gate independent noise assumption as used in most RB protocols. That is, the noise channels attached to the gate set {Gi} are the same, i, G˜i=ΛLGiΛR,where ΛL and ΛR are left and right noise channels. The randomization over the gate sequence Srb can be seen as performing twirling operation for the noise channels between Gi and Gi+1. Denote Λ=ΛRΛL: f(m)=Q|(EgGGΛG)m|ρψ,where the right noise channel ΛR of the first gate is absorbed into the state preparation error, |ρψ=ΛR|ρψ0 and the left noise channel ΛL of the last gate is absorbed into the measurement error, Q|=Q0|ΛL. According to Proposition 1, one can express fm in a more elegant manner, f(m)=σRGQ|Πσm|ρψλσm,where Πσ denotes the projector onto the irreducible subspace associated with σ, and λσ contains the trace information of the channel Λ on the subspace: λσ=Tr(ΛΠσ)Tr(Πσ).

    Because the twirling operation will not change the trace value of Λ, the process fidelity of Λ can be given by F(Λ,I)=4nσdσλσ,where dσ is the dimension of the irrep σ with dσ=Tr(Πσ). Here, F(Λ,I) is also known as the entanglement fidelity, or χ00. We call {λσ} the quality parameters because they reflect the noisy level of a channel.

    In the conventional RB protocol, the n-qubit Clifford group Cn is often picked as the target gate set, which we call Clifford RB. Any Clifford operation CCn satisfies {CPiC1  for  alli}=Pn, which is a transformation permuting Pauli operators. Note that in the PTM representation, Cn has only one nontrivial irrep. Thus, we only need to solve one single quality parameter. This is rather convenient, but it is hard to extend the conventional RB scheme to other groups with multiple nontrivial irreps due to the multi-variable fitting problem, as shown in Eq. (A31), which has poor confidence intervals for {Λσ}.

    To solve the fitting problem, in the following discussions, we employ the technique of character randomized benchmarking, which utilizes the generalized projection formula of Lemma 1 in the character theory.

    APPENDIX B: BENCHMARKING PROTOCOLS

    Character Cycle Benchmarking

    Here, we present further technical details of the CCB protocol. As shown in Lemma 1, we can rewrite the projection equation of Eq. (A6) for a quantum operation group {G} in the PTM representation: Πσ=dσ|G|gGχσ(g)G.Then one can add an additional gate, as a character gate, to construct the projector for extracting the quality parameter λσ associated with irrep σ. In CCB, we estimate the process fidelity of a target gate UCn via the twirling group G=Pn, which has 4n irreps supported by the Pauli operator {Pj}. The schematic circuit is given in Fig. 1(a) in the main text. The detailed procedures of CCB protocol are given in Box 2.

    Note that in the CCB procedures, P(0) is the character gate, which we will merge into the gate P(1) in practical implementation. In addition, P(0) is not included in computing the inverse gate Uinv. The average Pj-weighted survival probability of Eq. (B4) can be further evaluated by fi(m)=djQj|((EPtPnPt1U1ΛUPt)(EPrPnPr1ΛPr))m(EPPnχj(P)P)|ρψj.

    According to Proposition 1, we have ΛPEPrPnPr1ΛPr=PrPnPr|Λ|PrΠPr,ΛPUEPtPnPt1U1ΛUPr=PtPnPt|U1ΛU|PtΠPt=U1ΛPU,where ΠPr=|PrPr| and ΠPt=|PtPt| are the projectors onto the support spaces of Pr and Pt in the PTM representation, respectively. Then Eq. (B7) can be further simplified to fi(m)=Qj|(ΛPUΛP)m(djEPPnχj(P)P)|ρs.

    According to the generalized projection formula of Lemma 1, we have Πj=dj4nPPnχj(P)P,where j, the dimension of the subspace associated with the Pj is dj=1. Substituting Eq. (B10) and Eq. (B8) into Eq. (B9), we have fj(m)=Qj|(PtPnPt|U1ΛU|PtΠPt×PrPnPr|Λ|PrΠPr)mΠj|ρs=Qj|Πj|ρs(Pj|U1ΛU|PjPj|Λ|Pj)m.

    By fitting the survival probability fj(m) to the function of Eq. (B5), one can obtain the fitting parameters, λj=Pj|U1ΛU|PjPj|Λ|Pj,Aj=Qj|Πj|ρs.

    Note that quality parameter λj is one of the Pauli fidelities of channel U1ΛPUΛP. Then we can use the CCB protocol to sample Pauli operators {Pj} and estimate process fidelity F(U1ΛPUΛP,I), which is close to F(Λ,I), as we will see in Section B.2.

    CCB Fidelity

    Here, we will explain the physical meaning of the estimated process fidelity in the CCB protocol. As shown in Section B.1, the process fidelity we estimate in the CCB protocol is called the CCB fidelity: Fccb(Λ)=F((U1ΛPU)ΛP,I).Here, ΛP is the noise channel Λ twirled by the Pauli gate set defined in Eq. (B8). To compute the deviation between Fccb(Λ) and F(Λ,I), we employ the rearrangement inequality.

    Lemma 2 (Rearrangement inequality [33]). Consider two sets ofnreal numbers{x1,x2,,xn}, {y1,y2,,yn}, andx1x2xn,y1y2yn, one hasxny1++x1ynxτ(1)y1++xτ(n)ynx1y1++xnyn,where{xτ(i)}is an arbitrary permutation of{xi}.

    Any Clifford unitary UCn satisfies, i, UPiU1=Pu(i),where PiPn and the index permutation u(i) is determined by the operation U. Thus, the diagonal terms of the Pauli channel U1ΛPU are a permutation of those in channel ΛP.

    Denote {ωi} as the Pauli fidelities of ΛP and {ωu(i)} as the Pauli fidelities of the channel U1ΛPU. Here, we assume i, ωi0 throughout the paper. In practice, the values of ωi are normally close to 1 for a high-fidelity gate implementation. Using the rearrangement inequality, one has ωu(1)ω1++ωu(4n)ω4nω12++ω4n2.Then one can further derive that ωu(1)ω1++ωu(4n)ω4nω1++ω4n.

    From the definition of the process fidelity and CCB fidelity, one can conclude that Fccb(Λ)F(Λ,I).

    CCB with Local Gauge Freedom

    In this section, we give a detailed analysis of the CCB protocol with local gauge freedom. As shown in Fig. 5(a), we insert local gates L and L1 between the twirling gates Pi and the target gates U, U1. We define the local gate as L=i=1nLi, where Li can be an arbitrary single-qubit gate. Note that the local gates can be absorbed into Pi,U,U1, as shown in Fig. 5(b), and thus they do not need to be implemented individually. Likewise, the initial character gate LP(0) and the twirling gate LP(1)L1 can be treated as single gates in experiments as well.

    Illustrations of the CCB circuit with local gauge transformation. Circuit (a) is equivalent to the original CCB circuit in the main text if all of gates are ideal. The orange boxes represent the target gate U and its inverse gate U−1. The blue boxes represent the random Pauli gates. The green boxes represent the inserted local gates L and L−1, where L=⊗i=1nLi. The yellow box denote the inverse gate for the m inner gate layers in the light blue box. In practice, we implement gates in circuit (b) while absorbing local gates L and L−1 into twirling and target gates. Here, the target gate after gauge transformation becomes LUL−1.

    Figure 5.Illustrations of the CCB circuit with local gauge transformation. Circuit (a) is equivalent to the original CCB circuit in the main text if all of gates are ideal. The orange boxes represent the target gate U and its inverse gate U1. The blue boxes represent the random Pauli gates. The green boxes represent the inserted local gates L and L1, where L=i=1nLi. The yellow box denote the inverse gate for the m inner gate layers in the light blue box. In practice, we implement gates in circuit (b) while absorbing local gates L and L1 into twirling and target gates. Here, the target gate after gauge transformation becomes LUL1.

    For simplicity, we assume that the local twirling gates are noiseless and the noises of LUL1 and LU1L1 are the same. Now, we can analyze the relationship between the sequence length of gate layers and the survival probability. Given the noise channel of the target gate Λ, the averaged Pj-weighted survival probability is given by fj(m)=djQj|(EPtPnPt1L1LU1L1ΛLUL1LPt)mL1L(EPrPnPr1L1ΛLPr)L1L(EPPnχj(P)P)|ρs=djQj|(EPtPnPt1U1ΛLUPt)m(EPrPnPr1ΛLPr)(EPPnχj(P)P)|ρs,where ΛL=L1ΛL. Equation (B19) is equivalent to Eq. (B7), except for substituting Λ with ΛL. That means running the CCB protocol with the circuit in Fig. 5 would provide an estimation of the process fidelity F(ΛL,I), which is equivalent to F(Λ,I): F(ΛL,I)=d2Tr(ΛL)=d2Tr(Λ)=F(Λ,I).

    This accounts for the validity of the CCB protocol with local gauge freedom.

    In the Pauli–Liouville representation, the off-diagonal terms of channel Λ vanish after the Pauli twirling. When applying LPnL1 for twirling, the off-diagonal terms of Λ in the L-transformed Pauli–Liouville representation, defined on LPnL1/d, would vanish. Here, the local gauge transformation merely changes the representation of Λ, while maintaining the exponential decay form of the survival probability. From another point of view, the CCB protocol with local gauge freedom is characterizing the diagonal terms of χ-matrix of the noise channel under basis LPnL1 instead of Pn. As the average fidelity is irrelevant to the representation basis, all the analysis in previous subsections still applies.

    Character-Average Benchmarking

    Here, we will provide more details of the CAB protocol. The CAB protocol can be seen as an improvement based on the CCB protocol, which adds an additional local Clifford gate CC1n to the beginning and C1 to the end of the inner gate sequence, respectively, as shown in Fig. 1(b) in the main text. The detailed procedures of CAB protocol are given in Box 1 in the main text.

    Let us first analyze the irreps of C1n in the PTM representation. According to Eq. (A5), the PTM representation of one-qubit Clifford group C1 is the direct sum of the trivial representation, I, associated with the identity element, and a nontrivial irrep, ϒ, supported by the subspace defined on the Pauli matrices {X,Y,Z}. For an n-qubit local Clifford group C1n, there are 2n irreps: Rn={I,ϒ}n.

    According to Proposition 1, the twirling over C1n can be written as ΛC=σkRnTr(ΛΠσk)Tr(Πσk)Πσk,where Πσk denotes the projector onto the support space of σk in the PTM representation, Tr(Πσk)=3π(σk) is the dimension of σk, and π(σk) is the counting of ϒ in σk. In the following, we will sometimes abuse the irrep notation and treat σk as the Pauli operator set that defines the support space of the irrep σk in the PTM representation.

    It is obvious to see that channel ΛC is a partially depolarizing channel, which has the same diagonal values in the subspace associated with the irrep σk for each k. We call the diagonal parameter as the local Clifford eigenvalue.

    Definition 5 (Local Clifford eigenvalue). The twirling channelΛCovern-qubit local Clifford groupC1has2nquality parameters{γk}, which are defined as the local Clifford eigenvalues: γk=Tr(ΛΠσk)Tr(Πσk)=1|σk|PjσkPj|Λ|Pj,where γk can be seen as an average value of the Pauli fidelities {Pj|Λ|Pj|Pjσk}.

    We will give a concrete example to show how C1n twirls an arbitrary channel. For a two-qubit local Clifford group C1C1, let us denote its irreps as σ1={II}={II},σ2={Iϒ}={IX,IY,IZ},σ3={ϒI}={XI,YI,ZI},σ4={ϒϒ}={XX,XY,XZ,YX,YY,YZ,ZX,ZY,ZZ},and the dimensions of these irreps are Tr(Πσ1)=|σ1|=30,Tr(Πσ2)=|σ2|=31,Tr(Πσ3)=|σ3|=31,Tr(Πσ4)=|σ4|=32.

    Then the twirling of C12 over a channel, Λ, according to Eq. (B22), is given by ΛC=Tr(ΛΠσ1)Πσ1+Tr(ΛΠσ2)3Πσ2+Tr(ΛΠσ3)3Πσ3+Tr(ΛΠσ4)9Πσ4,which has four local Clifford eigenvalues {Tr(ΛΠσ1),Tr(ΛΠσ2)/3,Tr(ΛΠσ3)/3,Tr(ΛΠσ4)/9}.

    Now return to the n-qubit case. The randomization over an inner gate layer in CAB is the same as CCB and generates a composite Pauli channel: Λin=U1ΛPUΛP,where ΛP is a Pauli twirling channel defined in Eq. (B8). The initial and last random local Clifford gates C,C1 in CAB together perform a local Clifford twirling over the inner Pauli channel: ΛC(m)=ECC1nC1ΛinmC=σkRnTr(ΛinmΠσk)Tr(Πσk)Πσk.

    Note that ΛC(m) is a partially depolarizing channel, as we mentioned above. Thus one can extract local Clifford eigenvalues via performing corresponding measurement observables Pσk. In the CAB protocol, by measuring in Zn basis, we can infer the measurement results of the 2n observables {I,Z}n, which span all the irrep spaces in Rn. We assume that the measurement Zn is performed with negligible errors that will not influence the fidelity estimations too much. If one wants to completely remove the SPAM errors, an additional character gate from the gate set {I,Z}n can be added to the beginning of the CAB gate sequence, which is similar to the CCB protocol.

    The survival probability of Eq. (9) in the main text can be derived as for Pauli operator Qk{I,Z}n: fk(m)=Qk|ΛC(m)|ρs,which contains fidelity information for irrep σk such that Qkσk. Substituting Eqs. (B8) and (B28) to Eq. (B29), we have fk(m)=Qk|ΠQkΠσkTr(ΛinmΠσk)Tr(Πσk)|ρs=Qk|ρsPjσk(Pj|U1ΛU|PjPj|Λ|Pj)mTr(Πσk)=Qk|ρsPjσkλj2mTr(Πσk),where λj is the Pauli fidelity of the channel U1ΛPUΛP defined in Eq. (B12). By fitting the survival probability fk(m) to the function fk(m)=Akμk2m, one can solve the quality parameters {μk} and estimate the CAB fidelity as Fcab=4nσkRndσkμk,where dσk=Tr(Πσk) is the dimension of irrep σk.

    Note that our fitting method averages multi-exponential decays {λj2m} into one exponential decay μk2m, which leads to μkavePjσkλj, as proven in Section B.5. We will further prove that the CAB fidelity is the upper bound of the CCB fidelity in Section B.5.

    Fitting Analysis for CAB

    Here, we will analyze the fitting results in the CAB protocol and explain the CAB fidelity Fcab in detail. As shown in Eq. (B30), the survival probability in CAB is given by fk(m)=Qk|ρs|σk|Pjσkλj2m,where |σk|=Tr(Πσk). Fit fk(m) to the function, fk(m)=Akμk2m,with Ak>0,μk>0. Take the natural logarithm for both sides in Eq. (B34), y=β0+β1m,where y=lnfk(m),β0=lnAk,β1=2lnμk.

    Next, we will employ the least-squares estimation for the linear regression of Eq. (B34). Set {m1,m2,,mq} as inputs, and assume m1<m2<<mq without loss of generality. The regression matrix M and the observed values Y are given by M=(1m11m21mq),Y=(y1y2yq).

    Using the least-squares estimation, one can solve the optimum parameters for the model β^=(β^0β^1)=(MTM)1MTY.

    The fitting parameter of interest in CAB is μk: μk=exp(β^12),where β^1=qi=1qmiyii=1qmii=1qyiqi=1qmi2(i=1qmi)2.

    Then, we have the following lemma.

    Lemma 3.Denote{μk}as the fitting parameters we solve in the CAB protocol, given in Eq. (B38). For eachμk, we haveμk1|σk|Pjσkλj,and the CAB fidelity is the upper bound of the CCB fidelity,Fcab(Λ)Fccb(Λ).

    Proof. For a simple linear regression using the least-squares estimation, there are some observations above the fitting curve, while the others are below the curve. Then one can conclude that there exist two adjacent observations whose slope in between is smaller than the fitting slope of the curve. Thus for the simple linear regression in CAB, the fitting slope holds β^1mini(yi+1yimi+1mi).

    According to the Chebyshev sum inequality [33], i, 1|σk|Pjσkλj2mi+1=1|σk|Pjσkλj2miλj2(mi+1mi)(1|σk|Pjσkλj2mi)(1|σk|Pjσkλj2(mi+1mi)).

    According to Eq. (B43), we can further derive that yi+1yi=ln(Pjσkλj2mi+1Pjσkλj2mi)ln(1|σk|Pjσkλj2(mi+1mi))2(mi+1mi)ln(1|σk|Pjσkλj),where the last inequality comes from the convexity of function x2(mi+1mi).

    Substituting Eq. (B44) into Eq. (B42), we have β^12ln(1|σk|Pjσkλj).

    Since β^1=2lnμk, we can conclude that μk1|σk|Pjσkλj.

    Besides, the dimensions of irreps of the local Clifford gate C1n satisfies σkRn|σk|=σkRnTr(Πσk)=4n.

    Thus, Fcab=4nσkRn|σk|μk4nσkRnPjσkλj=Fccb(Λ),which completes the proof.

    APPENDIX C: STATISTICAL ANALYSIS

    Here, we analyze the statistical fluctuation of the CCB protocol with finite sampling. Recall that the CCB fidelity is given by Eq. (B13), Fccb(Λ)=14njλj,where λj is the Pauli fidelity of channel U1ΛPUΛP related to the Pauli operator Pj. It is impractical to solve all the Pauli fidelities using the character RB method when qubit number n becomes large. One can sample a finite number of Pauli irreps to estimate the process fidelity of the channel. The reliability of these estimates is expressed by the confidence levels.

    In the CCB protocol, the Pauli fidelity is obtained by fitting the survival probability and the gate sequence length, which is very hard for the statistical analysis. For simplicity, we take two points in the fitting diagram to analyze the fluctuation of the slope. In what follows, we use the notation where x^ is an estimator of a quantity, x¯, where the bar denotes that either an expected value or a sample average has been taken over realizations of random variable x.

    We begin by describing the CCB protocol in a statistical way. System.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElement

    The main statistical errors are divided into two parts in the above protocol. The first comes from the sampling randomness of the Pauli fidelity estimation λ^j for each Pj in Eq. (C4). The second comes from the sampling randomness of the Pauli operators {Pj} in Eq. (C5). We shall calculate the confidence levels for these two sampling randomness, respectively.

    We first calculate the bias of the Pauli fidelity estimation: Δλj:=|ES1,S2[λ^j]λ¯j|,where λ¯j denotes the theoretical value of the Pauli fidelity, λ¯j=(f¯j(m2)f¯j(m1))1m2m1.

    Here, the expectations of the probability estimators are taken over the gate sequences, f¯j(m1)=Es1Sm1[f^j(m1,s1)],f¯j(m2)=Es2Sm2[f^j(m2,s2)].

    To calculate the expectation value of the ratio estimator in Eq. (C6), we take the expectations over the gate sequence sets S1,S2 on both sides of Eq. (C4), ES1,S2[λ^j]=ES2[f^j(m2)1m2m1]ES1[f^j(m1)1m2m1].

    The expectations for f^j(m1) and f^j(m2) can be separated since the random variables s1 and s2 are independent. By denoting b^:=f^j(m2),b¯:=f¯j(m2),a^:=f^j(m1),a¯:=f¯j(m1),t:=1m2m1,we have ES1,S2[λ^j]=ES1[a^t]ES2[b^t].

    Supposing that δa:=a^a¯a¯1,δb:=b^b¯b¯1,and using the second-order approximation of the Taylor expansion at δa=0,δb=0 for a^t,b^t, respectively, we have a^t=a¯t(1+δa)t=a¯t(1tδa+t(t+1)δa2+O(δa3)),b^t=b¯t(1+δb)t=b¯t(1+tδb+t(t1)δb2+O(δb3)).

    Take expectations over s1,s2 for Eq. (C13), ES1[a^t]=a¯t(1+t(t+1)Var[a^]a¯2+O(δa3)),ES2[b^t]=b¯t(1+t(t1)Var[b^]b¯2+O(δb3)).

    Then the expectation value of the ratio estimator is given by ES1,S2[λ^j]=(b¯a¯)t(1+t(t+1)Var[a^]a¯2)(1+t(t1)Var[b^]b¯2)+O(δa3,δb3)=λ¯j(1+t(t+1)Var[a^]a¯2+t(t1)Var[b^]b¯2)+O(δa3,δb3).

    Substituting Eq. (C15) into Eq. (C6), we can derive the bias of the Pauli fidelity estimation, Δλj=λ¯j(t(t+1)Var[a^]a¯2+t(t1)Var[b^]b¯2)+O(δa3,δb3).

    Recall that the assumptions of Eq. (D17) are established with specific failure probabilities. To calculate the confidence intervals for the aforementioned assumptions, we apply Bernstein’s inequality.

    Lemma 4 (Bernstein’s inequality [34]). Consider a set ofnindependent random variables{X1,,Xn}withi,Xib. LetX=1niXi, denoteV2=n1i=1nVar(Xi), thenϵ>0, Pr(|XE[X]|>ϵ)2exp(nϵ2/2V2+bϵ/3).

    Assuming that m,f^j(m)1, we derive the following proposition.

    Proposition 2.Letf^j(m)be the estimator ofPj-weighted probabilityf¯j(m)for fixedK, m, andj. Thenϵ>0, Pr(|f^j(m)f¯j(m)|>ϵ)2exp(Kϵ2/2Var[f^j(m,s)]+ϵ/3).

    Then for the estimators a^ and b^, we can derive the corresponding confidence intervals Pr(|a^a¯|>ϵ1)2exp(K1ϵ12/2Var[f^j(m1,s1)]+ϵ1/3):=ϒa(K1,ϵ1),Pr(|b^b¯|>ϵ2)2exp(K2ϵ22Var[f^j(m2,s2)]+ϵ2/3):=ϒb(K2,ϵ2).

    Substituting Eq. (C19) into Eq. (C16), we obtain the following lemma.

    Lemma 5.For any given Pauli operatorPjPnand some fixedm1,m2,K1,K2, the bias of the Pauli fidelity estimation is upper bounded byΔλj1(m2m1)2|1+m2m1K1f¯j2(m1)Var[f^j(m1,s1)]+1m2+m1K2f¯j2(m2)Var[f^j(m2,s2)]|+O(ϵ13,ϵ23):=ϵb(m1,m2;K1,K2;ϵ1,ϵ2),with a failure probability bounded byϒj(K1,K2;ϵ1,ϵ2)=ϒa(K1,ϵ1)+ϒb(K2,ϵ2).

    The notation ϵb(m1,m2;K1,K2;ϵ1,ϵ2) is abbreviated as ϵb in the following analysis. Next, we compute the confidence interval for the CCB fidelity F^ccb defined in Eq. (C5). We denote F¯ccb=1M{Pj}λ¯j,F¯ccb=14njλ¯j.

    Assume the Pauli fidelity 0λ¯j1 for all Pj. One can apply the Hoeffding’s inequality directly.

    Lemma 6 (Hoeffding’s inequality [35]). Consider a set ofnindependent random variables{X1,,Xn}andi,aiXibi. LetX=1niXi, thenϵ>0, Pr(|XE[X]|>ϵ)2exp(2n2ϵ2i=1n(biai)2).

    Then we can derive that ϵM>0, Pr(|F¯ccbF¯ccb|>ϵM)2exp(2MϵM2).

    According to Lemma 5, we have Pr(|F^ccbF¯ccb|>ϵb)Mϒj(K1,K2;ϵ1,ϵ2),where F^ccb is the estimator of F¯ccb, as defined in Eq. (C5). By combining Eqs. (C23) and (C24) and applying the union bound, we can compute the confidence interval for the process fidelity estimator Pr(|F^ccbF¯ccb|>ϵM+ϵb)2exp(MϵM2/2)+Mϒj(K1,K2;ϵ1,ϵ2).

    Let us assume that 0Var[f^j(m1,s1)]1,0Var[f^j(m2,s2)]1,and then Eq. (C25) can be simplified to Pr(|F^ccbF¯ccb|>ϵM+ϵb)2exp(MϵM2/2)+2Mexp(K1ϵ12/21+ϵ1/3)+2Mexp(K2ϵ22/21+ϵ2/3).

    We further assume that 12<f¯j(m1)<1,and then we can derive the following theorem.

    Theorem 2.Consider a CCB implementation with sampling numbersK1,K2at sequence lengthsm1,m2with estimation errorsϵ1,ϵ2, respectively, and the expected survival probabilityf¯j(m1)that satisfiesj,1/2<f¯j(m1)<1. The estimated CCB fidelity, F^ccb, is given by the average overMPauli fidelities{λj}. The confidence probability for the CCB fidelity falling into the estimated interval[F^ccbϵMϵb,F^ccb+ϵM+ϵb]is greater than1δ: Pr(|F^ccbF¯ccb|ϵM+ϵb)1δ,with ϵb4K11m2m1(1m2m1+1)+O(ϵ13,ϵ23),and 2exp(2MϵM2)+2Mexp(K1ϵ12/21+ϵ1/3)+2Mexp(K2ϵ22/21+ϵ2/3)=δ.

    A simplified informal version of the above theorem is presented in the main text as Theorem 1.

    APPENDIX D: SIMULATION

    Here, we will present the noise model for the simulation in the main text and give more details on the two-qubit controlled-(TX) gate and the five-qubit error correcting circuit. In addition, we compare a CAB process and an interleaved character randomized benchmarking (ICRB) process [28,36] for benchmarking the two-qubit CZ gate. We provide its simulation details and the results at the end of this part.

    Error Model

    The noise channel, Λt, we consider here for the target gate is composed of a Pauli channel Λ0, an amplitude damping channel Λ1, and a qubit–qubit correlation channel Λ2, Λt=Λ0Λ1Λ2. We consider the noise channel Λref for the twirling gate set Pn is a gate-independent Pauli channel, which is negligible compared with Λt.System.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElement

    Simulations for the Controlled-<inline-formula><math display="inline" id="m749" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></math></inline-formula> Gate

    The controlled-(TX) gate can be decomposed as CTX=(IT)CNOT(IT1).

    Then, we can take IT as the local gauge transformation, and the twirling gate set turns to Pctx=(IT)P2(IT1).

    Consider noise channel Λt=Λ0Λ1Λ2 in Section D.1 for the noisy controlled-(TX) gate. We randomly sample the Pauli fidelities of Λ0 from a normal distribution N(μ,σ), denoted as a N(μ,σ)-Pauli channel, where μ and σ are the mean value and standard deviation. The parameters for Λ1 are set to α1=α2=0.005. The parameter for Λ2 is set to β12=0.01. In the following discussions, we label the noise channel for the controlled-(TX) with Λt(μ,σ), since Λ1 and Λ2 remain the same in all the simulations.

    We simulate the CAB and CCB protocols for the noisy controlled-(TX) gate with eight different noise channels {Λt(μi,σi)}, and the noisy implementations of CTX and CTX1 are given by CT˜X=CTXΛt(μ,σ),CTX˜1=CTX1Λt(μ,σ).

    The error parameters are taken as {(μi,σi)}= {(0.995, 0.001), (0.990, 0.002), (0.980, 0.003), (0.970, 0.004), (0.960, 0.005), (0.950, 0.006), (0.940, 0.007), (0.930, 0.008)}. Take the N(0.998,0.001)-Pauli channel as the noise channel of the twirling gate set Λref and then denote the noisy implementation of the twirling gate set as P˜ctx=ΛrefPctx.

    Take the N(0.998,0.001)-Pauli channel as the SPAM error channel Λspam and then denote the noisy implementations of the initial state |ψ and measurement Q as ρψ=Λspam(|ψψ|),Q˜=QΛspam.

    The simulation procedures for CAB run as follows. System.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElement

    The simulation procedures for CCB run as follows. System.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElement

    Simulations for the Five-Qubit Error Correcting Circuit

    Here, we take the five-qubit stabilizer encoding circuit shown in Fig. 6 as the target gate U, which only contains Clifford gates.

    Five-qubit stabilizer encoding circuit.

    Figure 6.Five-qubit stabilizer encoding circuit.

    Simulation results of CAB and ICRB for benchmarking a CZ gate. (a) The error bar of the fidelity obtained from 50 experiments via CAB and ICRB protocols. The x-axis represents the amount of single-shot measurements, and the y-axis represents the fidelity. The term “shot" represents the number of single shots associated with one sample sequence. One can see that CAB and ICRB can both obtain an accurate estimation of the process fidelity of a CZ gate. (b) The standard deviation of the fidelity obtained from 50 experiments via CAB and ICRB protocols. The x-axis represents the amount of single-shot measurements, and the y-axis represents the standard deviation of the fidelity. The figure shows that CAB has an advantage over ICRB with regard to standard deviation.

    Figure 7.Simulation results of CAB and ICRB for benchmarking a CZ gate. (a) The error bar of the fidelity obtained from 50 experiments via CAB and ICRB protocols. The x-axis represents the amount of single-shot measurements, and the y-axis represents the fidelity. The term “shot" represents the number of single shots associated with one sample sequence. One can see that CAB and ICRB can both obtain an accurate estimation of the process fidelity of a CZ gate. (b) The standard deviation of the fidelity obtained from 50 experiments via CAB and ICRB protocols. The x-axis represents the amount of single-shot measurements, and the y-axis represents the standard deviation of the fidelity. The figure shows that CAB has an advantage over ICRB with regard to standard deviation.

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