- Photonics Research
- Vol. 11, Issue 1, 81 (2023)
Abstract
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 [10–16]. 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 [17–24]. 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.
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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 acting on an -qubit quantum state, , by the calligraphic letter , i.e., , and the noisy implementation by . One can evaluate the quality of by the process fidelity of the noise channel ,
In practice, it is costly to figure out all the parameters as their number increases exponentially with . Instead, one can estimate the process fidelity via repeatable sampling of . Concretely, one samples a sufficient number of Pauli operators and averages the corresponding ,
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 -qubit Clifford group . The inner random gate layer consists of the target gate and its inverse gate , 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 and turn it into
Figure 1.Illustrations of circuit and procedures used in (a) CCB and (b) CAB protocols. The orange boxes represent the target gate
Note that the introduction of the inverse target gate is the major difference between CCB and cycle benchmarking. In cycle benchmarking, we need to repeat for multiples of times, where is the cyclic number of , i.e., . In general, 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 . The CCB protocol improves the efficiency and application scope via substituting a single for multiple repetitions of 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 is composed of several native gates, the difficulty to implement and is often the same. Based on this consideration, the introduction of 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 , where and are the Pauli-twirled channels corresponding to gates and , respectively. Note that this composite channel is a Pauli channel for Clifford gate . The fidelity we aim to estimate in the CCB protocol is defined as the CCB fidelity,
Equation (5) is a lower bound of the process fidelity 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 and 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 is a depolarizing channel, then . Thus, the CCB fidelity can be seen as a reliable metric for the noise channel .
The procedure of the CCB protocol is as follows.
Here, the estimated fidelity includes the errors from the local reference gate set . To remove these extra errors, one can employ the interleaved RB technique, by performing additional CCB with a target gate of identity to estimate the reference fidelity . Then, one can infer the fidelity of the target gate as . In practice, the errors of local gates are often negligible, and hence, we focus on in the following discussions.
Note that our inverse gate 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 to estimate . The sampling complexity for the CCB protocol is given by the following theorem.
Here, the total number of samples, or sample complexity, depends on and . If the number of random sequences for each Pauli fidelity is the same, then the sample complexity is simply given by . Theorem 1 shows that the sample complexity only depends on fidelity precision , and confidence level . The independence on system size 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 to the twirling gate set, , where is an arbitrary local unitary operation . Note that the transformed twirling gate set is still local. Then, we can show that the applicable target gate set becomes , where is the -qubit Clifford gate set.
To benchmark a gate from gate group , we insert local gates and between the twirling gates and the target gates in the original CCB circuit, as shown in Fig. 2(a). Here, , where 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 and the twirling gate will be merged into a single gate in implementation as well. Details of the derivation are shown in Appendix B.3.
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
As shown in Fig. 2, the CCB circuit with local gauge transformation and noise channel is equivalent to the original CCB circuit with noise channel . Thus, one can obtain , which is close to the process fidelity . As process fidelity is gauge-invariant, that is, , one can estimate the process fidelity of as the performance indicator of gate .
Now, let us check out what kinds of quantum gates belong to the set .
First, notice that if a unitary , then for any , . As any unitary is generated by a Hamiltonian, that is where is Hermitian, one can conclude that if , then for any , .
Take a step forward. If a controlled-, then through local gauge transformation , . 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 or multiple controlled-, then replace with to find other gates in . Take as an example. controlled-. Through local gauge transformation, one can transform to any product state and transform to any -rotation , where and is a unit vector. Thus, for any two-qubit product state , we have . In addition, any controlled- rotation, such as controlled- and controlled-, belongs to .
Reversely, controlled- controlled- is a controlled- rotation. As any controlled- rotation is not Clifford, one can conclude that controlled- does not belong to . Similarly, Tofolli = controlled-controlled- is a controlled-controlled- rotation. As any controlled-controlled- rotation is not Clifford, one can conclude that Tofolli does not belong to either. It is interesting to decide whether a quantum gate belongs to 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 to estimate Pauli fidelity . Each estimation requires specific initial state, measurement, and independent randomization procedures. We can further simplify these procedures by introducing the local Clifford group . Recall that in a qubit system, the Clifford twirling depolarizes a channel via averaging the error rates in bases [13]. Then for an -qubit system, the twirling over would partially depolarize a channel and average out Pauli fidelities into terms. These values can be obtained from the 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 gate layers inside the blue box in Fig. 1(b) will generate a Pauli channel . The local Clifford gates in the beginning and end of the circuit jointly perform local unitary 2-design twirling, which transforms the Pauli channel into a partially depolarizing channel . Here, the quantum channel contains less independent parameters than the original . It holds the unique value of fidelity for every disjoint Pauli subset in . The Pauli fidelities in can be seen as the average values of those in , , where are the Pauli fidelities of the channel . 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 and 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-() 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 , which we call the -Pauli channel. Here, the error parameter reflects the quality of the Pauli channel and 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- gate, we take as the twirling gate set, where is the -phase gate, , and 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 Pauli operators to estimate .
Figure 3(a) shows and versus the error rate for the controlled- gate with different noise channels. We observe that when the standard deviation of error parameters grows, the error bars of and become larger. Intuitively, the discrepancy of the Pauli fidelities is one of the key reasons for the fluctuations of and . 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 , which is very close to the theoretical value of process fidelity .
Figure 3.Simulation results for the controlled-
For the five-qubit error correcting encoding circuit, which is a Clifford gate, we take the Pauli group as the twirling gate set. For the simplicity of simulation, we set the Pauli channel to be a depolarizing channel where . 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 for each sequence length . The box plot of versus is shown in Fig. 4(a). We can see that when grows, the fluctuations of become smaller. When is not too large, like , the fluctuation is already small enough, which implies that the CAB protocol works well with few sampling sequences needed.
Figure 4.Simulation results for the five-qubit quantum error correcting encoding circuit with a noise channel composed of a depolarizing channel
In Fig. 4(b), we show the box plots of and XEB fidelities versus the sampling number . It is clear to see that compared with , is much closer to the theoretical process fidelity . Meanwhile, the convergence of is much better than that of . This implies that the required 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 , the standard deviation over the 20 simulations is , while for XEB, when , the standard deviation is . This shows that to estimate the fidelities with standard deviations around , the required is over 1000 times larger than . 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
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 be a finite group and be a group element. The representation of is defined as follows.
Given representation on , a linear subspace is called
The restriction of to the invariant subspace is known as a
The Maschke’s theorem provides an interesting property that each representation of a finite group can be decomposed to the irreducible representations, :
With the character function, we introduce the
Next, we will introduce twirling over a group .
Using Schur’s Lemma [
Here, we introduce the quantum channel and three frequently used channel representations that our main results rely on. Denote the Hilbert space for qubits as and the set of linear operators on as . Quantum channels are defined as completely positive and trace-preserving (CPTP) linear maps on . Given any quantum channel , we can represent it in
With the Kraus representation, the concatenation of the quantum channels or quantum gates is given by
To describe a long quantum circuit, the Kraus representation is not convenient. Here, we introduce another widely used representation—
Each pair of elements in this group satisfies the following constraints under the Hilbert–Schmidt inner product, :
Any -qubit operator can be decomposed over the normalized Pauli operators. We can rewrite the density operator on in a vector form:
Moreover, any quantum channel can be represented as a matrix in the Liouville representation. To be specific, we can represent an arbitrary channel acting on an operator as follows:
We can see the element of this matrix is given by
Consequently, in the Liouville representation, the concatenation of two channels can be depicted as the product of two matrices:
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 on a positive operator-valued measure is given by
We call such a Pauli–Liouville representation as the Pauli transfer matrix (PTM) representation. An -qubit quantum channel can also be described in the -matrix representation:
The process matrix is uniquely determined by the orthonormal operator basis , where the first element is proportional to the identity matrix , and is the dimension of the quantum system. Often, we take the normalized Pauli operators as the basis of the -matrix representation. If a channel is diagonal in this representation, we call it
The process fidelity of a channel can be defined with its -matrix representation:
Here, the quantity is independent of the choices of operator basis . In the following, we set the operator basis to be normalized Pauli group. Define the
Here, if commutes with and otherwise. Then, one can derive the process fidelity from Eq. (
There is a relation between the commonly used average fidelity and the process fidelity [
Average fidelity is defined as
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 -qubit gate set , RB is performed via sampling random gate sequences:
Because the twirling operation will not change the trace value of , the process fidelity of can be given by
In the conventional RB protocol, the -qubit Clifford group is often picked as the target gate set, which we call Clifford RB. Any Clifford operation satisfies , which is a transformation permuting Pauli operators. Note that in the PTM representation, 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. (
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
Here, we present further technical details of the CCB protocol. As shown in Lemma 1, we can rewrite the projection equation of Eq. (
Note that in the CCB procedures, is the character gate, which we will merge into the gate in practical implementation. In addition, is not included in computing the inverse gate . The average -weighted survival probability of Eq. (
According to Proposition 1, we have
According to the generalized projection formula of Lemma 1, we have
By fitting the survival probability to the function of Eq. (
Note that quality parameter is one of the Pauli fidelities of channel . Then we can use the CCB protocol to sample Pauli operators and estimate process fidelity , which is close to , as we will see in Section B.2.
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:
Any Clifford unitary satisfies, ,
Denote as the Pauli fidelities of and as the Pauli fidelities of the channel . Here, we assume , throughout the paper. In practice, the values of are normally close to 1 for a high-fidelity gate implementation. Using the rearrangement inequality, one has
From the definition of the process fidelity and CCB fidelity, one can conclude that
In this section, we give a detailed analysis of the CCB protocol with local gauge freedom. As shown in Fig.
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
For simplicity, we assume that the local twirling gates are noiseless and the noises of and 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 , the averaged -weighted survival probability is given by
This accounts for the validity of the CCB protocol with local gauge freedom.
In the Pauli–Liouville representation, the off-diagonal terms of channel vanish after the Pauli twirling. When applying for twirling, the off-diagonal terms of in the -transformed Pauli–Liouville representation, defined on , would vanish. Here, the local gauge transformation merely changes the representation of , 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 -matrix of the noise channel under basis instead of . As the average fidelity is irrelevant to the representation basis, all the analysis in previous subsections still applies.
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 to the beginning and to the end of the inner gate sequence, respectively, as shown in Fig.
Let us first analyze the irreps of in the PTM representation. According to Eq. (
According to Proposition 1, the twirling over can be written as
It is obvious to see that channel is a partially depolarizing channel, which has the same diagonal values in the subspace associated with the irrep for each . We call the diagonal parameter as the
We will give a concrete example to show how twirls an arbitrary channel. For a two-qubit local Clifford group , let us denote its irreps as
Then the twirling of over a channel, , according to Eq. (
Now return to the -qubit case. The randomization over an inner gate layer in CAB is the same as CCB and generates a composite Pauli channel:
Note that is a partially depolarizing channel, as we mentioned above. Thus one can extract local Clifford eigenvalues via performing corresponding measurement observables . In the CAB protocol, by measuring in basis, we can infer the measurement results of the observables , which span all the irrep spaces in . We assume that the measurement 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 can be added to the beginning of the CAB gate sequence, which is similar to the CCB protocol.
The survival probability of Eq. (
Note that our fitting method averages multi-exponential decays into one exponential decay , which leads to , 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.
Here, we will analyze the fitting results in the CAB protocol and explain the CAB fidelity in detail. As shown in Eq. (
Next, we will employ the
Using the least-squares estimation, one can solve the optimum parameters for the model
The fitting parameter of interest in CAB is :
Then, we have the following lemma.
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
According to the Chebyshev sum inequality [
According to Eq. (
Substituting Eq. (
Since , we can conclude that
Besides, the dimensions of irreps of the local Clifford gate satisfies
Thus,
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. (
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 is an estimator of a quantity, , where the bar denotes that either an expected value or a sample average has been taken over realizations of random variable .
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 for each in Eq. (
We first calculate the bias of the Pauli fidelity estimation:
Here, the expectations of the probability estimators are taken over the gate sequences,
To calculate the expectation value of the ratio estimator in Eq. (
The expectations for and can be separated since the random variables and are independent. By denoting
Supposing that
Take expectations over for Eq. (
Then the expectation value of the ratio estimator is given by
Substituting Eq. (
Recall that the assumptions of Eq. (
Assuming that , we derive the following proposition.
Then for the estimators and , we can derive the corresponding confidence intervals
Substituting Eq. (
The notation is abbreviated as in the following analysis. Next, we compute the confidence interval for the CCB fidelity defined in Eq. (
Assume the Pauli fidelity for all . One can apply the Hoeffding’s inequality directly.
Then we can derive that ,
According to Lemma 5, we have
Let us assume that
We further assume that
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- gate and the five-qubit error correcting circuit. In addition, we compare a CAB process and an interleaved character randomized benchmarking (ICRB) process [
The noise channel, , we consider here for the target gate is composed of a Pauli channel , an amplitude damping channel , and a qubit–qubit correlation channel , . We consider the noise channel for the twirling gate set is a gate-independent Pauli channel, which is negligible compared with .System.Xml.XmlElementSystem.Xml.XmlElementSystem.Xml.XmlElement
The controlled- gate can be decomposed as
Then, we can take as the local gauge transformation, and the twirling gate set turns to
Consider noise channel in Section D.1 for the noisy controlled- gate. We randomly sample the Pauli fidelities of from a normal distribution , denoted as a -Pauli channel, where and are the mean value and standard deviation. The parameters for are set to . The parameter for is set to . In the following discussions, we label the noise channel for the controlled- with , since and remain the same in all the simulations.
We simulate the CAB and CCB protocols for the noisy controlled- gate with eight different noise channels , and the noisy implementations of and are given by
The error parameters are taken as {(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 -Pauli channel as the noise channel of the twirling gate set and then denote the noisy implementation of the twirling gate set as
Take the -Pauli channel as the SPAM error channel and then denote the noisy implementations of the initial state and measurement as
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
Here, we take the five-qubit stabilizer encoding circuit shown in Fig.
Figure 6.Five-qubit stabilizer encoding circuit.
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
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