The fault-tolerant quantum error correction code (FTQEC)^[1,2] is necessary for a quantum computer to reliably operate and obtain an exponential speed-up over the classical algorithm in some fields. The FTQEC requires that the operation of the quantum process is of high quality and that the error is below a certain threshold.^[3] One of the main challenges for FTQEC is to characterize the noise that affects a quantum system. Particularly, quantum process tomography (QPT)^[4–6] is a method to analyze the noise of a quantum channel.^[7] However, QPT is infeasible for large quantum systems. Fortunately, the evaluation of the average fidelity^[8,9] is an efficient method to partially characterize the noise of quantum channels or gates.^[10,11] The advantage of the average fidelity estimation method^[12] is that it does not require large amounts of resources and is insensitive to state preparation and measurement errors (SPAM).^[13]

Randomized benchmarking^[14–23] is an experimental scheme to measure the average fidelity. In this scheme, the average fidelity of a completely positive trace-preserving (CPTP) noise channel is measured by using a set of unitary operators that approximate the entire Clifford group, which is called a unitary 2-design.^[24] A unitary 2-design is a special case of the unitary t-design^[25] for t = 2. For t = 1, it corresponds to the case of a private quantum channel,^[26] whereas t = 4 corresponds to the conditions of the state-distinction problem.^[27,28]

In this paper, we present the first systematic analysis of the average fidelity estimation using the unitary 2t-design for a twirled t-tensor product noisy channel. Then, we extend this method to the scenario of (m + 1)-stage noise channels. Furthermore, we analyze the (m + 1)-stage sequence of quantum gates under their corresponding noise, such as gate-independent and time-independent noise, weakly gate-dependent and time-independent noise and gate-independent and weakly time-dependent noise. It can be recognized as the estimation of average gateset fidelity for quantum circuits. In particular, we give a lower bound on the average fidelity of the twirled channel under the unitary error caused by nonideal experimental implementations and SPAM.

The method of using unitary 2t-design for the twirling of noises has the following advantages. First, compared to the unitary 2-design in randomized benchmarking, the unitary 2t-design for the twirling of noises is more general to evaluate the average fidelity and provides more information about the noise. Here, t corresponds to the high dimensionality of a channel or the experiment times using the method of moment superoperator.^[29] Second, using the method of moment superoperator, it is convenient to provide the analysis of the time-dependent noise. Finally, a unitary design is similar to a spherical design in terms of Jamiolkowski isomorphism^[30,31] and the frame potential.^[32,33] Some constructions of a spherical 2t-design can be used in the unitary 2t-design to estimate the average fidelity. Compared with the Clifford group in the previous randomized benchmarking, the construction of the unitary 2t-design for the twirling of noises is more flexible and extensible.

The paper is organized as follows. In Section 2, we show the process of average fidelity estimation using the unitary 2t-design to twirl a t-tensor product noisy channel. Then, we introduce the method of moment superoperator to estimate the time-dependent noise estimation. In Section 3, we extend the fidelity estimation of an (m + 1)-stage noise channel using unitary 2t-design for twirling of channels. We also analyze the (m + 1)-stage sequence of quantum gates under their corresponding noise, such as gate-independent and time-independent noise, weakly gate-dependent and time-independent noise and gate-independent and weakly time-dependent noise. In Section 4, we provide a bound on the unitary error caused by nonideal experimental implementations and SPAM. Finally, we conclude the paper in Section 5.

Consider a completely positive trace-preserving (CPTP) noise channel as shown in Fig. 1.

Fig. 1.

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	Fig. 1. A CPTP quantum channel for any dn-dimensional input density operator ρ = |ψ⟩ ⟨ψ|.

We assume that the CPTP noise channel can be written as with being the linear operator for any dⁿ-dimensional input density operator ρ = |ψ⟩ ⟨ ψ|, where is the space of dⁿ-dimensional complex linear operators. We define the superoperator representation of the CPTP noise channel Λ based on Ref. [34], i.e., 2 where the superoperator has a d²ⁿ-dimensional matrix representation and denotes the usual sum over the diagonal elements of the matrix. The advantage of such representation is that the cascade of two arbitrary dⁿ-dimensional channels, Λ₁ and Λ₂, corresponds to their matrix multiplication, i.e., 3

Then, the average channel fidelity^[8] can be defined by 4 where F(|ψ⟩, Λ, E_|ψ⟩) is with respect to the fidelity with input state |ψ⟩, CPTP noise channel Λ and positive-operator valued measure (POVM) element E_|ψ⟩ = |ψ⟩ ⟨ψ|; μ_Haar(·) is a uniform distribution; and the integrals are with respect to the unitarily invariant Haar measure. The above average channel fidelity can be recognized as the probability that the output state Λ(|ψ⟩ ⟨ψ|) passes the test of being the input state |ψ⟩ averaged over all input states. Note that it is insensitive to the SPAM, if the POVM element E_|ψ⟩ = |ψ⟩ ⟨ ψ| is perfect.

Now, we define the unitary t-design as follows. Let t be a natural number and be the set of unitary operators in dⁿ-dimensional Hilbert space. A finite subset is called a unitary t-design if 5 holds for any polynomial p_(t,t)(U) homogeneous of degree t in the matrix elements of and their complex conjugates of U^*. We can use the superoperator representation of unitary operators to redefine the unitary t-design , i.e., 6 where .

Then, we consider a t-tensor product CPTP noise channel model using the unitary 2t-design to estimate the corresponding average fidelity in Fig. 2. The noise channel Λ^⊗t is a d^tn × d^tn matrix composed of t dⁿ-dimensional CPTP noise channels. In addition, if there exists a d^l-dimensional CPTP noise channel Λ_j for l < n, we can apply a d^n–l-dimensional ancillary quantum system to fit the model. Then, we set t-copies of the dⁿ-dimensional unitary operations U and U^† randomly chosen from a unitary 2t-design at both ends of the noise channels. Note that the above model can be used to estimate the average fidelity of a non-local noise channel, when t = 1.

Fig. 2.

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	Fig. 2. A t-tensor product CPTP noise channel model using unitary 2t-design is composed of three linear mappings Λ⊗t, U⊗t, and (U†)⊗t. For any dtn-dimensional input density operator ρ⊗t, the representation of the unitary operation and the noise channel are d2tn × d2tn matrices, where .

If a set of unitary operators is a unitary 2t-design to twirl a CPTP noise channel , it satisfies 7 where is a linear operator in the d^tn-dimensional Hilbert space satisfying . Let Δ_μHaar,2t(·) be the Haar-averaged mapping using unitary 2t-design for a twirled noise channel, i.e., 8 where is a fixed density operator in d^tn-dimensional Hilbert space. Then, the Haar-average superoperator representation can be presented as 9 where the superoperator representation of t-tensor product noise channel is represented as and .

Therefore, the average fidelity of the twirled channel in Fig. 2 is given by 10 with the perfect POVM element E_|ψ⟩^⊗t = |ψ⟩^⊗t⟨ψ|^⊗t for any input state = |ψ⟩^⊗t and E_|ψ0⟩^⊗t = |ψ₀⟩^⊗t⟨ψ₀|^⊗t for a fixed input state |ψ₀⟩^⊗t in a d^tn-dimensional Hilbert space. The above equality indicates that we can also obtain the average channel fidelity by uniformly averaging over all unitary operators with a fixed input state instead of averaging over all input states.

In addition, we introduce a method of moment superoperator,^[29] which is derived from the action of t-copies of the random circuit on nt-qubit density operators. It will be used to analyze the time-dependent noise in Section 3. Using Liouville representation,^[19,22] we define moment superoperator kets shown in Fig. 3 as 11 in a d^2nt-dimensional space, where is an orthonormal basis and j ∈ {1,…,d^2tn} with respect to the Hilbert Schmidt inner product. Let ⟨⟨ M^⊗t | = | M^⊗t ⟩⟩^†, we have 12 For a quantum channel, we have 13 Thus, the representation of the linear mapping Λ^⊗t(·) is given by 14

Fig. 3.

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Fig. 3. Moment superoperator of a d2nt-dimensional space can be visualized as an array of 2nt qubits, so that t copies support a ket in the state space on nt qubits, and the remaining t copies support the corresponding bra. The ovals Ui,i + 1 correspond to a unitary operator U that acts nontrivially on two neighboring qubits i and i + 1. The dashed rectangles indicate that the 2t qubits that correspond to a local moment space Hlt indicate the random circuit of two neighboring qubits that correspond to a d2-dimensional space.

The previous estimation can only analyze the time-dependent noise channel in one experiment. However, the noises in different experiments are assumed to have the identical change, which obviously affects the estimation result. The advantage of the moment superoperator is that it provides more information to characterize the time-dependent noise for t-copies of experiments.

If the set of unitary operators construct a unitary 2t-design, the average fidelity using the unitary 2t-design to twirl a channel is given by 15 for a fixed input density operator and a perfect POVM element E_{|ψ0 ⟩^⊗t} = |ψ₀⟩^⊗t ⟨ψ₀|^⊗t in d^tn-dimensional Hilbert space. Following this method, if we can implement an exact unitary 2t-design to twirl a noise channel, we can directly obtain the average channel fidelity using the normalized sum D times of the results.

From Schur’s lemma,^[35] the Haar-averaged superoperator Δ_μHaar,2t can be expressed as a depolarizing channel 16 with strength parameter 17

However, the size of a unitary 2t-design grows exponentially with the increase of parameters d and n. Hence, we generally use an (m + 1)-stage sequence of unitary operators (selected from a unitary 2t-design) to twirl the noise channels to estimate the relationship between the average channel fidelity and parameter m in Fig. 4.

Fig. 4.

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	Fig. 4. (m + 1)-stage twirled noisy channels using unitary 2t-design.

We assume that each for j ∈ {1,…,m + 1} is a t-tensor product gate-dependent and time-dependent CPTP noise channel. If the fixed input density operator is , we can obtain the corresponding output density operator, i.e., 18 where is the sequence s(m + 1) mapping of (m + 1)-stage CPTP noise channels each with and , j ∈ {1, …, m + 1} from the unitary 2t-design at both ends. Its superoperator representation can be given by 19

If U_sj is selected from a unitary 2t-design , then we obtain the average fidelity of (m + 1)-stage noise channel using the unitary 2t-design to twirl the noisy channels, i.e., 20

Following Eq. (16), the Haar averaged superoperator of each noise channel is as follows: 21 with 22 where we assume that . Then, the Haar-averaged superoperator of the (m + 1)-stage noise channel using the unitary 2t-design can be written as 23

3.1. The case of gate-dependent and time-dependent noise

We redraw (m + 1)-stage noise circuits using the unitary 2t-design for an actual experiment by inductively defining new gates, which are uniformly selected from the unitary 2t-design shown in Fig. 5.

Fig. 5.

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	Fig. 5. (m + 1)-stage noise circuits with each gate uniformly chosen from the unitary 2t-design for an actual experiment.

Note that if i ≠ k, U_si,i and U_sk,k are independent chosen from a unitary 2t-design for each i, k ∈ {2, …, m + 1}. Hence, for each j ∈ {1, …, m}, is independent from each other. However, depends on the first m gates for j ∈ {1, …, m}.

Therefore, we can use the unitary 2t-design to estimate the average gateset fidelity of (m + 1)-stage noise circuits by considering a perturbative expansion of each noise about the ideal average noise Λ^⊗t. We define the difference between and Λ^⊗t for j ∈ {1, …, m + 1} as 24 Note that each , j ∈ {1, …, m + 1} is a Hermiticity-preserving, trace-annihilating linear superoperator. Then, its surperoperator representation is 25 The ideal average strength parameter can be written as 26 where is the Haar-averaged superoperator of using the unitary 2t-design for twirling of noise. Then, we obtain the difference between p_sj,j and p, i.e., 27 where 28

Under the above conditions, this approach enables us to fit the experimental fidelity decay sequence to a model with fit parameters. The noise channel in Fig. 5 can be written as Using the unitary 2t-design for the twirling of noise, we obtain 30 The average gateset fidelity for any input density operator ρ^⊗t can be written as 31 where is the fidelity of channel with the input state |ψ₀⟩^⊗t, and the perfect POVM is E_|ψ0⟩^⊗t = |ψ₀⟩^⊗t ⟨ψ₀|^⊗t.

3.2. The case of gate-independent and time-independent noise

If is a gate-independent and time-independent noise channel, the average channel fidelity is the zeroth order fitting model for p_sj,j = p. Then, we obtain 32 with 33 34 Hence, the ideal case where the average gateset fidelity of each gate-independent and time-independent noise exponentially decays in p is shown in Fig. 6.

Fig. 6.

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Fig. 6. Average fidelity estimation of (m + 1)-stage quantum circuits using unitary 2t-designs. In these examples, we assume that we implement perfect quantum devices and have on SPAM, where d = 2, n = 1, and t = 6. We assume that the fidelity of the first noise channel is 0.99. The strength parameter is p(0) = 0.9925 in the zeroth-order case and p(1) = 0.9775 in the first-order case. In the first-order case, the perturbation of the strength parameter δ p(1) is from [10−4,10−3].

A unitary 2t-design is similar to a spherical 2t-design in terms of the Jamiolkowski isomorphism and the frame potential.^[32,33] The unitary operators in a unitary 2t-design are elements of the unitary group instead of points on a unit sphere. Therefore, we can use the partial results of a spherical 2t-design to construct a unitary 2t-design to estimate the average channel fidelity. If we can implement an exact unitary 2t-design to twirl the noise, we must only take D times of the experiment to estimate the average fidelity.

However, the size of the unitary 2t-design D grows exponentially with the increase in d and n. We generally take an (m + 1)-stage sequence of gates with their corresponding noise using a benchmarking protocol, which requires selecting a sequence of gates at random from the unitary 2t-design and repeating for many sequences to take the average of the results.

In an actual experiment, we take k times of the experiments to estimate the average fidelity, i.e., 47 where |ψ₀⟩^⊗t is the fixed input state and E_|ψ0⟩^⊗t = |ψ₀⟩^⊗t ⟨ ψ₀|^⊗t is the perfect POVM. For an exact unitary 2t-design , the ideal average fidelity is 48 Note that the gate G_{sm + 1} depends on the first m random gates. By Hoeffding’s inequality,^[36] we obtain 49 where the resulted fidelity of every experiment lies in [a,b]⊂ (0,1). We assume that is the desired confidence level. Then, we can obtain 50

From the above benchmarking analysis, two main deficiencies will affect the final result. The first one is the pseudo-random selection of quantum gates, which will cause the deviation of the experimental average fidelity from the ideal value mainly because of the incorrect estimation of the number of experiments k caused by the inaccuracy of b – a. The other deficiency is the unitary error. Imperfect quantum gate G_{sm + 1,m + 1} preparation and undesirable POVM will lead to errors in estimation results. Here, we neglect the effect of nontrace-preserving mappings.

The unitary error has an important effect on the estimation of average fidelity. It is mainly caused by undesirable experimental implementations and SPAM. Our approach is to bound the fidelity of unitary error using the method of moment superoperator. We use the moment superoperator because the unitary error in each gate is different from one another.

The quantum gate G_{sm + 1,m + 1} in (m + 1)-stage sequence of random gates with corresponding noise depends on other m gates that are randomly selected from the unitary 2-design, where . For undesirable experimental implementations, there is a dⁿ-dimensional unitary operator U_Gs with eigenvalues e^iθ_Gs,k for θ_Gs,k ∈ [−π,π] and k ∈{1, …, dⁿ}, which satisfy 51 where . For SPAM, there is a dⁿ-dimensional unitary operator U_SPAMs with eigenvalues e^{i θ_SPAMs,k for θ_SPAMs,k} ∈ [−π,π] and k ∈ {1, …, dⁿ}, which satisfy 52 where is the s-th sequence of measurement with a fixed input state |ψ₀⟩ and . Then, we obtain 53

Let U_s be the total unitary error, such that 54 with eigenvalues e^{i θ_s,k} = e^{i (θ_Gs,k – θ_SPAMs,k)} for θ_s,k ∈ [−π,π] and k ∈ {1, …, dⁿ}. Thus, the corresponding average fidelity impact factor of unitary error γ_s^[34,37] can be written as 55 It is obvious that γ_s ≤ 1. Our approach is to provide a lower bound on . We assume that θ_s = max_{k ∈ {1, …, dⁿ}}|θ_s,k|. Then, we assume that θ_s,1 = θ_s and θ_s,k = −θ_s/(dⁿ − 1) for all k ∈ {2, …, dⁿ}. This assumption can minimize γ_s. We can verify this result by implementing a perturbation δθ, i.e., 56 For large d and n, we obtain 57 Then, we have 58 Using the unitary 2-design to twirl the channels, we obtain 59 with the fixed input density operator ρ₀ = |ψ₀⟩ ⟨ψ₀| in a dⁿ-dimensional Hilbert space and strength parameters q_sj,j. Then, the average fidelity for t-copies of sequences can be written as 60

We now numerically analyze the average fidelity of (m + 1)-stage noise circuits with unitary errors using the unitary 2t-design by considering t copies single-qubit noise models. We provide an example that the average fidelity of each single-qubit noise model is 0.99488 and its deviation range is (10⁻⁵,10⁻⁴). We assume time-dependent unitary errors that correspond to rotations. The unitary error is Hamiltonian in dⁿ-dimensional Hilbert space with eigenvalues e^{i θ_k} for θ_k ∈ [−π,π] and k ∈ {1, …, dⁿ}. For trace-preserving noise, the unitary error satisfies

For the convenience of comparison, we make the following assumptions. (1)

Assume |θ_k|, k ∈ {1, …, dⁿ − 1} uniformly random chosen from (10⁻⁵,10⁻⁴) with random positive and negative signs;

Note that the above assumption ensures that the maximum eigenvalue of the unitary error is 10⁻⁴. Numerical values for the average fidelity are shown in Fig. 7. Our first order average fidelity estimation of weakly time-dependent and gate-independent noise satisfies the results of data analysis.

Fig. 7.

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Fig. 7. Average fidelity of the (m + 1)-stage noise circuit. In this example, we assume d = 2, n = 1, and t = 6 since the noise model is unital. The blue line is the curve fitting 1.054 × 0.9525m. The green dotted line is the estimation of average fidelity in the case of gate-independent and weakly time-dependent noise with the lower bound on the average fidelity impact factor of the unitary error.

In this work, we have presented a first systematic analysis of the average fidelity estimation using the unitary 2t-design for the twirling of noises. We provide a basic model using the unitary 2t-design to twirl a noise channel. It efficiently estimates the average channel fidelity if we can implement an exact unitary 2t-design. We also present a model of (m + 1)-stage sequence of noise channels using the unitary 2t-design to estimate the average channel fidelity. It can be recognized as the average gateset fidelity estimation for quantum circuit using a benchmarking protocol, which requires selecting a sequence of randomly selected gates from the unitary 2t-design. We analyze such cases of the gate-independent and time-independent noise, weakly gate-dependent and time-independent noise, and gate-independent and weakly time-dependent noise. For the time-dependent noise, the method of moment superoperator plays an important role in noise analysis in different times of experiments. Finally, we provide a bound on the unitary error caused by non-ideal experimental implementations and SPAM.

Although much insight into the application of unitary designs has been obtained, many questions remain unresolved as interesting open problems, such as the construction of unitary 2t-designs or the errors produced by pseudo-randomness of the benchmarking protocol.