Measurement-based interleaved randomised benchmarking using IBM processors

In the measurement-based model, quantum computing is realised by performing adaptive single-qubit measurements on an entangled resource state such as the cluster state [8–11]. In particular, 2D cluster states are entangled resource states for universal quantum computing [10] and fault-tolerant quantum computing can be achieved using 3D cluster states [60]. Here we concentrate on linear cluster states, which are entangled resource states for single-qubit measurement-based quantum computing [61]. A n-qubit linear cluster state is prepared on a linear array of n qubits by preparing each qubit in the state $\left|+\right\rangle =\left(\left|0\right\rangle +\left|1\right\rangle \right)/\sqrt{2}$ and then applying controlled phase gates CZ = diag(1, 1, 1, − 1) between neighbouring qubits.

We briefly summarise measurement-based processing with linear cluster states [56]. The measurement-based protocol for implementing unitary operations with a n-qubit linear cluster state is illustrated in figure 2. The first qubit is prepared in the input state, ${\rho }_{\mathrm{in}}=\left|{\psi }_{\mathrm{in}}\right\rangle \left\langle {\psi }_{\mathrm{in}}\right|$, to which the implemented unitary operation is to be applied. The remaining qubits are prepared in the state $\left|+\right\rangle$ (Step 1), and qubits are then entangled by applying controlled phase gates between neighbouring qubits (Step 2). Each qubit i ∈ {1,...,n − 1} is then measured at an angle ϕ_i in the Pauli XY plane, that is in the basis $\left\{\left|{}^{+}\ {\phi }_{i}\right\rangle ,\left|{}^{-}\ {\phi }_{i}\right\rangle \right\}$ with $\left|{}^{\pm }\ {\phi }_{i}\right\rangle =(\left|0\right\rangle \pm {e}^{-{\rm{i}}{\phi }_{i}}\left|1\right\rangle )/\sqrt{2}$. Each measurement is equivalent to applying the unitary operation

$$\begin{eqnarray}&&{U}_{{m}_{i}}({\phi }_{i})={X}^{{m}_{i}}{{HR}}_{z}({\phi }_{i}),\end{eqnarray} \tag{ 1 }$$

to $\left|{\psi }_{\mathrm{in}}\right\rangle$, where m_i ∈ {0, 1} is the measurement outcome (occurring with probability ${p}_{{m}_{i}}=\tfrac{1}{2}$ for all m_i ), X is the Pauli X operation, H is the Hadamard gate and ${R}_{z}({\phi }_{i})={e}^{-{\rm{i}}Z{\phi }_{i}/2}$ is a z-rotation by the angle ϕ_i . Hence, these measurements result in the n^th qubit being prepared in the output state, ${\rho }_{\mathrm{out}}={U}_{{\boldsymbol{m}}}({\boldsymbol{\phi }}){\rho }_{\mathrm{in}}{U}_{{\boldsymbol{m}}}^{\dagger }({\boldsymbol{\phi }})$ (Step 3), with

$$\begin{eqnarray}&&{U}_{{\boldsymbol{m}}}({\boldsymbol{\phi }})={U}_{{m}_{n-1}}({\phi }_{n-1})\cdots {U}_{{m}_{1}}({\phi }_{1}),\end{eqnarray} \tag{ 2 }$$

and where ϕ and m denote ordered lists of angles and measurement outcomes (occurring with probability ${p}_{{\boldsymbol{m}}}=\tfrac{1}{{2}^{n-1}}$ for all m ) respectively.

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Even though the implemented unitary operation is probabilistic and depends on the measurement outcomes, any deterministic quantum computation can be realised by employing adaptive measurement feedforward [11, 61]. In particular, unitary operations which are deterministic up to a single known Pauli gate can be implemented by performing the measurements on qubits 1 to n − 1 sequentially and allowing future measurement angles to depend on past measurement outcomes. The desired quantum computation is then realised deterministically by applying the required and known Pauli correction to qubit n.

We now give explicit measurement-based implementations of the Hadamard gate and the T = diag(1, e^iπ/4) gate. When all measurement outcomes are zero, the 2-qubit linear cluster state with the measurement angle ϕ₁ = 0 implements the Hadamard gate and the 3-qubit linear cluster state with the measurement angles ${\phi }_{1}=\tfrac{\pi }{4}$ and ϕ₂ = 0 implements the T gate. When non-zero measurement outcomes occur, the desired gate can still be implemented by applying the appropriate Pauli correction to the final qubit [61]. In particular, the Pauli X operation must be applied to implement the Hadamard gate when m₁ = 1. For the T gate, the Pauli X operation must be applied when m₁ = 0 and m₂ = 1, the Pauli Y operation must be applied when m₁ = 1 and m₂ = 1, and the Pauli Z operation must be applied when m₁ = 1 and m₂ = 0. Hence the Hadamard gate and the T gate can be realised deterministically with fixed measurement angles and a Pauli correction, and in these specific cases do not require adaptive measurement feedforward. However, implementations of more complicated deterministic operations, such as arbitrary single-qubit rotations, require both adaptive measurement feedforward and a Pauli correction [10].

A unitary t-design is a pseudorandom ensemble of unitary operators of which the statistical moments match those of the uniform Haar ensemble either approximately or exactly up to some finite order t. The expectation of any $\rho \in { \mathcal B }({\left({{\mathbb{C}}}^{d}\right)}^{\otimes t})$ with respect to the Haar measure on the unitary group U(d), where d = 2 for single qubits, is defined by

$$\begin{eqnarray}&&{{\mathbb{E}}}_{H}^{t}(\rho )=\int {U}^{\otimes t}\rho {\left({U}^{\otimes t}\right)}^{\dagger }\,{dU}.\end{eqnarray} \tag{ 3 }$$

Thus formally an ensemble of unitaries {p_i , U_i } is an exact unitary t-design if for all $\rho \in { \mathcal B }({\left({{\mathbb{C}}}^{d}\right)}^{\otimes t})$

$$\begin{eqnarray}&&{{\mathbb{E}}}_{H}^{t}(\rho )=\displaystyle \sum _{i}{p}_{i}{U}_{i}^{\otimes t}\rho {\left({U}_{i}^{\otimes t}\right)}^{\dagger },\end{eqnarray} \tag{ 4 }$$

and {p_i , U_i } is an -approximate t-design if there exists an such that for all $\rho \in { \mathcal B }({\left({{\mathbb{C}}}^{d}\right)}^{\otimes t})$

$$\begin{eqnarray}&&(1-\epsilon ){{\mathbb{E}}}_{H}^{t}(\rho )\leqslant \displaystyle \sum _{i}{p}_{i}{U}_{i}^{\otimes t}\rho {\left({U}_{i}^{\otimes t}\right)}^{\dagger }\leqslant (1+\epsilon ){{\mathbb{E}}}_{H}^{t}(\rho ),\end{eqnarray} \tag{ 5 }$$

where the matrix inequality A ≤ B holds if B − A is positive semidefinite [62, 63]. We are primarily interested in unitary 2-designs, which are sufficient for randomised benchmarking [28].

In addition to being entangled resource states for measurement-based quantum computing (see section 2.1), linear cluster states can be used to implement single-qubit t-designs by entirely foregoing adaptive measurement feedforward and Pauli corrections, that is by fixing the measurement angles ϕ and considering the ensemble of unitaries {p _m , U _m ( ϕ )} for all m . In particular, the 6-qubit linear cluster state with the measurement angles ϕ₁ = 0, ${\phi }_{2}=\tfrac{\pi }{4}$, ${\phi }_{3}=\arccos \sqrt{1/3}$, ${\phi }_{4}=\tfrac{\pi }{4}$ and ϕ₅ = 0 implements an exact 2-design [62] and the 5-qubit linear cluster state with the measurement angles ϕ₁ = 0, ${\phi }_{2}=\tfrac{\pi }{4}$, ${\phi }_{3}=\tfrac{\pi }{4}$ and ϕ₄ = 0 implements an approximate 2-design with = 0.5 [56]. In our previous work we implemented both the exact measurement-based 2-design and the approximate measurement-based 2-design on IBM processors [56]. Neither implementation passed our test for a 2-design under the test conditions set. However, the test results showed that the approximate 2-design implementation more closely resembled a 2-design than the exact 2-design implementation as a result of reduced noise for the smaller 5-qubit cluster state. This is why we use the approximate 2-design, and not the exact 2-design, in our randomised benchmarking experiments in this work (presented in section 5). We note that the implications of performing randomised benchmarking with an -approximate 2-design, as opposed to an exact 2-design, are not yet well understood theoretically. However, numerical investigations have shown that the estimated fidelity obtained when using an exact 2-design can differ from the estimated fidelity obtained when an exact 2-design is not used [35]. We find that the results obtained with the approximate 2-design are consistent with those obtained using quantum process tomography.

Unitary 2-designs can be used in interleaved randomised benchmarking, which provides an estimate of the Haar-averaged fidelity of a noisy implementation of a unitary operation or gate [38]. The Haar-averaged fidelity of a noisy implementation of an ideal gate G is defined by

$$\begin{eqnarray}&&{F}_{G}(\varepsilon ,\widetilde{\varepsilon })=\int {\left(\mathrm{Tr}\left(\sqrt{\sqrt{\varepsilon (\psi )}\widetilde{\varepsilon }(\psi )\sqrt{\varepsilon (\psi )}}\right)\right)}^{2}\,d\psi ,\end{eqnarray} \tag{ 6 }$$

where $\varepsilon (\psi )=G\left|\psi \right\rangle \left\langle \psi \right|{G}^{\dagger }$ is the channel representing the ideal implementation of G and $\widetilde{\varepsilon }(\psi )$ is the channel representing the noisy experimental implementation of G [55]. The average gate error is then given by $1-{F}_{G}(\varepsilon ,\widetilde{\varepsilon })$, which is useful for quantifying the overall reliability of the implementation. However, for some applications, such as determining thresholds for fault-tolerant quantum computing, the worst case error is required [64]. For these applications, the average gate error can be used to obtain bounds on the worst case error [24, 65–67].

We briefly review the interleaved randomised benchmarking protocol proposed by Magesan et al [38], in which the fidelity of individual Clifford gates [68] can be estimated. However, we relax the restriction to Clifford gates and review a variation of the protocol in which any single-qubit unitary 2-design ${ \mathcal U }$ can be used to estimate the fidelity of any single-qubit unitary operation or gate G. Section III A of [69] explains that the interleaved method of [38] holds for the variation of the protocol reviewed here. The only benefit of the restriction to Clifford gates is that the inverse of a sequence of Clifford gates can be efficiently computed on a classical computer as a result of the Gottesman–Knill theorem [70]. Restricting the protocol to single-qubit gates has the same benefit, since any single-qubit system can be efficiently simulated on a classical computer [61]. The key assumptions of the protocol are that noise from any gate is time-independent, noise from the 2-design ${ \mathcal U }$ is gate-independent and noise from the inverse of any sequence of gates is independent of the sequence.

Randomised benchmarking relies heavily on a unitary 2-design's ability to transform an arbitrary noise channel into a depolarising channel [29, 30, 38], a property which was studied extensively in our previous work [71]. In interleaved randomised benchmarking, two experiments are performed, namely an experiment to determine the reference depolarising noise parameter p_ref and an experiment to determine the interleaved depolarising noise parameter p_int. These parameters are then used to estimate the Haar-averaged gate fidelity using

$$\begin{eqnarray}&&{F}_{G}\approx 1-\displaystyle \frac{d-1}{d}\left(1-\displaystyle \frac{{p}_{\mathrm{int}}}{{p}_{\mathrm{ref}}}\right),\end{eqnarray} \tag{ 7 }$$

where d = 2 for single qubits.

The reference depolarising noise parameter p_ref is determined as follows:

• 1.  
  Prepare the qubit in an arbitrary, but fixed, initial state $\rho =\left|\psi \right\rangle \left\langle \psi \right|$.
• 2.  
  Choose m unitary operators uniformly at random from the 2-design ${ \mathcal U }$ and apply the resulting sequence of operators, U_m ⋯ U₂ U₁, to the state ρ.
• 3.  
  Compute and apply the inverse of the sequence of gates applied in step 2.
• 4.  
  Measure the qubit in the basis $\{\left|\psi \right\rangle \left\langle \psi \right|,I-\left|\psi \right\rangle \left\langle \psi \right|\}$. This measures the probability that the initial state is unchanged by the sequence applied in step 2 followed by its inverse (known as the survival probability).
• 5.  
  Repeat steps 1 to 4 for different sequences of a fixed length m and average the survival probability over the different sequences to obtain the sequence fidelity F_ref(m) for a given sequence length m. Guidance on choosing the number of repetitions is given in [65, 72].
• 6.  
  Repeat steps 1 to 5 for different sequence lengths m and extract the reference depolarising noise parameter p_ref by fitting the resulting data for F_ref(m) versus m to the exponential decay model

  $$\begin{eqnarray}&&{F}_{\mathrm{ref}}(m)={A}_{\mathrm{ref}}\,{p}_{\mathrm{ref}}^{m}+{B}_{\mathrm{ref}}.\end{eqnarray} \tag{ 8 }$$

  The parameters A_ref and B_ref absorb state preparation and measurement errors, as well as the error of the inverse applied in step 3. The relation between the number of repetitions chosen in step 5 and confidence intervals for the extracted parameters is discussed in [65].

The interleaved depolarising noise parameter p_int is determined as follows:

• 1.  
  Prepare the qubit in the same fixed initial state $\rho =\left|\psi \right\rangle \left\langle \psi \right|$.
• 2.  
  Choose m unitary operators uniformly at random from the 2-design ${ \mathcal U }$ and apply the interleaved sequence GU_m ⋯ GU₂ GU₁ to the state ρ.
• 3.  
  Compute and apply the inverse of the sequence of gates applied in step 2.
• 4.  
  Measure the qubit in the basis $\{\left|\psi \right\rangle \left\langle \psi \right|,I-\left|\psi \right\rangle \left\langle \psi \right|\}$. This once again measures the survival probability.
• 5.  
  Repeat steps 1 to 4 for a fixed m and average the survival probability over the different sequences to obtain the sequence fidelity F_int(m).
• 6.  
  Repeat steps 1 to 5 for different m and extract the interleaved depolarising noise parameter p_int by fitting the resulting data for F_int(m) versus m to the exponential decay model

  $$\begin{eqnarray}&&{F}_{\mathrm{int}}(m)={A}_{\mathrm{int}}\,{p}_{\mathrm{int}}^{m}+{B}_{\mathrm{int}}.\end{eqnarray} \tag{ 9 }$$

We implemented our measurement-based interleaved randomised benchmarking protocol, with the adjustments discussed in section 4, first on the ibm_hanoi quantum processor, a 27-qubit superconducting IBM quantum computer. Details of the qubits used can be found in supplementary material I. We used the 5-qubit approximate measurement-based 2-design proposed in our previous work [56] (see section 2.2) to estimate the fidelity of the 2-qubit measurement-based implementation of the Hadamard gate and the 3-qubit measurement-based implementation of the T gate given in section 2.1 on the ibm_hanoi quantum processor. To this end, we implemented reference sequences with lengths m ∈ {1, 2, 3} and interleaved sequences with m ∈ {1, 2, 3}. For each m (reference or interleaved), we prepared the required linear cluster state, performed the appropriate single-qubit measurements on all but the final qubit, and then performed full quantum state tomography on the final qubit for each of the possible measurement outcomes to infer the output state for each corresponding random sequence. As an example, the quantum circuit for the implementation of the interleaved sequence for the 3-qubit T gate with m = 1, which requires a 7-qubit linear cluster state, is shown in figure 3. The largest number of qubits was used for the implementation of the interleaved sequence for the 3-qubit T gate with m = 3, which required (m(k + ℓ) + 1) = (3(4 + 2) + 1) = 19 qubits, as shown in figure 1(a).

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Since controlled phase gates are not supported at the hardware level on IBM quantum processors, linear cluster states were prepared using Hadamards and controlled not gates, instead of Hadamards and controlled phase gates, as recommended by Mooney et al [73]. This prevented redundant Hadamards, which would have unnecessarily increased noise, from being introduced during transpilation. Since qubits can only be measured in the computational basis, $\{\left|0\right\rangle ,\left|1\right\rangle \}$, on IBM processors, single-qubit measurements at an angle ϕ in the Pauli XY plane were carried out by applying R_z (ϕ), followed by a Hadamard, and measuring the qubit in the computational basis.

To obtain the data required to determine the sequence fidelity (F_ref(m) or F_int(m)) for a given m, the relevant circuit was run 3(2ⁿ )/16 times with 8192 shots (see section 4) on the ibm_hanoi quantum processor (where n is the number of single-qubit measurements performed on the (n + 1)-qubit linear cluster state in the implementation). The data (counts) obtained in repeated runs were combined and grouped according to measurement outcomes. The output density matrix for each random sequence was then constructed from the tomography data obtained for the corresponding set of measurement outcomes. To ensure that the density matrices constructed from the data are physical (i.e. that they are Hermitian and have a trace of 1) we employed qiskit's built-in method [74], which uses maximum-likelihood estimation to find the closest physical density matrix to a density matrix constructed from tomography data. The appropriate inverse was then applied to each density matrix and the survival probability was extracted from each resulting density matrix. The sequence fidelity (F_ref(m) or F_int(m)) presented for a given m in section 5.2 is the average of these survival probabilities, and the error is given by the standard deviation.

In interleaved randomised benchmarking experiments, sequence fidelities are typically determined for sequence lengths up to m = 80 or longer [38, 46]. Here, the resources required to implement longer sequences prevented us from considering sequence lengths beyond m = 3. For example, the implementation of the interleaved sequence for the 3-qubit T gate with m = 3 required a 19-qubit linear cluster state and the relevant circuit had to be run 3(2¹⁸)/16 = 49152 times to obtain the data required to determine the sequence fidelity. Since this is already extremely resource intensive, longer sequences would not be feasible, as the number of runs required grows exponentially with the length of the sequence. Hence, even though the adjustments discussed in section 4 have enabled us to obtain data for a proof-of-concept demonstration, dynamic circuit execution remains essential for the efficient scaling of our measurement-based interleaved randomised benchmarking protocol in future.

Sequence fidelities obtained to estimate the fidelity of the 2-qubit measurement-based implementation of the Hadamard gate and the 3-qubit measurement-based implementation of the T gate (a universal single-qubit set), using the 5-qubit approximate measurement-based 2-design of [56], on the ibm_hanoi quantum processor are shown in figure 4. The large uncertainties in the sequence fidelities reflect the gate-dependence of noise from the approximate measurement-based 2-design [56]. A Monte Carlo method which takes these uncertainties into account was used to fit the reference and interleaved sequence fidelities to the exponential decay model given by equations (8) and (9) respectively. The fitting procedure used, the fitting constraints imposed and the estimation of Haar-averaged gate fidelities from the fitting parameters are discussed in supplementary material II. The estimated fidelity of the 2-qubit measurement-based implementation of the Hadamard gate is (0.977± 0.073) and the estimated fidelity of the 3-qubit measurement-based implementation of the T gate is (0.972 ± 0.070). The large uncertainties in the estimated fidelities can be attributed both to the large uncertainties in the sequence fidelities and to the small number of sequence fidelities. It is unclear to what extent the use of an -approximate 2-design, as opposed to an exact 2-design, contributed to the large uncertainties in the estimated fidelities. It is also not clear how is related to these uncertainties or how the relation is affected by the Monte Carlo method used to estimate the fidelities.

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We now compare the fidelities estimated using our measurement-based interleaved randomised benchmarking protocol to fidelities calculated from process tomography results. To this end, we performed quantum process tomography on each of the three respective sets of qubits of the ibm_hanoi quantum processor on which the 2-qubit Hadamard gate and the 3-qubit T gate were implemented in the interleaved sequences, and used the results to calculate the Haar-averaged gate fidelity of each of the three implementations of the 2-qubit Hadamard gate and the 3-qubit T gate. Further details about the process tomography method used, the implementation of the method and the calculation of Haar-averaged gate fidelities from process tomography results are given in supplementary material III. Fidelities obtained for the three different implementations of the 2-qubit Hadamard gate, on the three different sets of qubits used in the interleaved sequences, range from 0.948 to 0.972. For the 3-qubit T gate, the fidelities obtained from process tomography results range from 0.928 to 0.939.

For both gates, the uncertainty of the gate fidelity calculated using process tomography results is smaller than the uncertainty of the gate fidelity estimated using our measurement-based interleaved randomised benchmarking protocol. We also note that the estimated fidelities are slightly larger than those obtained using process tomography results. Nevertheless, the agreement between our estimated gate fidelities and the gate fidelities calculated from process tomography results is somewhat remarkable considering that we used a very weak approximate 2-design [56], the implementation of this 2-design on the IBM quantum processors did not even pass our test for an approximate 2-design for all states [56], noise from this 2-design is not entirely gate-independent [56] and the fitting parameters were inferred from very few sequence fidelities (only three). This clearly demonstrates the robustness of interleaved randomised benchmarking. It also provides motivation for considering weak approximate 2-designs, such as the one proposed in our previous work [56]. It is unclear to what extent the use of a weak -approximate 2-design, as opposed to an exact 2-design, contributed to the over-estimation of gate fidelities. However, one would expect that in general, an increase in results in an increase in the positive difference between the estimated gate fidelity and the gate fidelity determined from process tomography results.

We next investigate the extent to which our measurement-based interleaved randomised benchmarking protocol is able to detect noise variations in different measurement-based implementations of a gate. To this end, we artificially increase noise in the measurement-based implementations of the Hadamard gate and the T gate. One option is to perform the single-qubit measurements in these measurement-based implementations at measurement angles which deviate from the required measurement angles. However, this has the disadvantage that it is not easy to predict whether a measurement-based gate will be more or less noisy than the measurement-based 2-design used to estimate its fidelity. We therefore artificially increase noise in the measurement-based implementations of the Hadamard gate and the T gate by increasing the length of the linear cluster states used in the implementations.

Note that the 3-qubit linear cluster state with the measurement angles ϕ₁ = 0 and ϕ₂ = 0 implements the identity operation when all measurement outcomes are zero. By appending this measurement-based implementation of the identity operation to the 2-qubit Hadamard gate and the 3-qubit T gate given in section 2.1, we obtain measurement-based implementations of the Hadamard gate and the T gate with longer cluster states. In particular, when all measurement outcomes are zero, the 4-qubit linear cluster state with the measurement angles ϕ₁ = 0, ϕ₂ = 0 and ϕ₃ = 0 implements the Hadamard gate and the 5-qubit linear cluster state with the measurement angles ${\phi }_{1}=\tfrac{\pi }{4}$, ϕ₂ = 0, ϕ₃ = 0 and ϕ₄ = 0 implements the T gate. Furthermore, provided that all measurement outcomes are zero, the 6-qubit linear cluster state with the measurement angles ϕ₁ = 0, ϕ₂ = 0, ϕ₃ = 0, ϕ₄ = 0 and ϕ₅ = 0 also implements the Hadamard gate and the 7-qubit linear cluster state with the measurement angles ${\phi }_{1}=\tfrac{\pi }{4}$, ϕ₂ = 0, ϕ₃ = 0, ϕ₄ = 0, ϕ₅ = 0 and ϕ₆ = 0 also implements the T gate. When non-zero measurement outcomes do occur, the desired gate can be realised by simply applying the appropriate Pauli correction to the final qubit [61], that is, these implementations all use fixed measurement angles and a Pauli correction and do not require adaptive measurement feedforward.

We estimated the fidelity of the 4-qubit and 6-qubit measurement-based implementations of the Hadamard gate and the 5-qubit and 7-qubit measurement-based implementations of the T gate, using the 5-qubit approximate measurement-based 2-design of [56], on the ibmq_brooklyn quantum processor, a 65-qubit superconducting IBM quantum computer. Details of the qubits used are provided in supplementary material IV. We once again implemented reference sequences with lengths m ∈ {1, 2, 3} and interleaved sequences with m ∈ {1, 2, 3}, for each of the gates considered. The ibmq_brooklyn quantum processor was used for these experiments, instead of the ibm_hanoi quantum processor, as the ibm_hanoi quantum processor has too few qubits to implement some of the required sequences, such as the interleaved sequence for the 7-qubit T gate with m = 3, which requires a 31-qubit linear cluster state, as shown in figure 1(b). Sequences were implemented, classical post-processing was done to determine the survival probability for each random sequence and sequence fidelities were calculated in the same way as in the experiments on the ibm_hanoi quantum processor (see section 5.1).

The only difference in the experiments on the ibmq_brooklyn quantum processor is that for the interleaved sequences for the measurement-based gates defined here, we need not perform quantum state tomography for each of the possible measurement outcomes to infer the output state for each random interleaved sequence. This can be understood as follows. For the 2-qubit Hadamard gate and the 3-qubit T gate there is a one-to-one correspondence between the outcomes of single-qubit measurements in the measurement-based implementations and the random byproducts that result from omitting the Pauli corrections. This results in a one-to-one correspondence between random measurement outcomes and random interleaved sequences, since the random interleaved sequences consist of random unitary operators interleaved with random byproducts. In contrast, there are 8, 16, 32 and 64 possible outcomes for the single-qubit measurements performed in the respective 4-qubit, 5-qubit, 6-qubit and 7-qubit measurement-based implementations defined here, but there are only four possible byproducts, corresponding to the four possible Pauli corrections, for each implementation. As a result, there is no longer a one-to-one correspondence between random measurement outcomes and random interleaved sequences. Therefore, to infer the output state for each random interleaved sequence, we need not perform quantum state tomography for each of the possible measurement outcomes, since tomography data obtained for different measurement outcomes which correspond to the same random interleaved sequence can be grouped.

We note that, for the interleaved sequences for the measurement-based gates defined here, it is in fact not feasible to perform quantum state tomography for each of the possible measurement outcomes, since the number of possible measurement outcomes grows exponentially with the length of the cluster state. For example, the circuit implementing the interleaved sequence for the 7-qubit T gate with m = 3 would need to be run 3(2³⁰)/16 ≈ 201 million times to obtain the data required to perform tomography for each of the possible measurement outcomes. By performing tomography for each random interleaved sequence, and grouping tomography data obtained for different measurement outcomes corresponding to the same random sequence, we were able to drastically reduce the number of times each circuit had to be run. In particular, to obtain the data required to determine the interleaved sequence fidelity for a given m, for the 4-qubit, 5-qubit, 6-qubit and 7-qubit measurement-based gates defined here, the relevant circuit was run the same number of times as for the 3-qubit T gate (see section 5.1), since the number of possible byproducts, and therefore the number of random interleaved sequences, is the same as for the 3-qubit T gate.

Sequence fidelities obtained to estimate the fidelity of the 4-qubit Hadamard gate and the 5-qubit T gate, using the 5-qubit approximate measurement-based 2-design of [56], on the ibmq_brooklyn quantum processor are shown in figure 5. Furthermore, sequence fidelities obtained to estimate the fidelity of the 6-qubit Hadamard gate and the 7-qubit T gate are shown in figure 6. Table 1 shows the estimated gate fidelities, as well as the range of Haar-averaged gate fidelities calculated from process tomography results (see supplementary material III). Both the estimated gate fidelities and the gate fidelities obtained from process tomography results decrease as the length of the cluster state in the implementation increases, which reflects the expected increase in noise resulting from increasing the length of the cluster state. This shows that, even with very little data, our measurement-based interleaved randomised benchmarking protocol is able to detect large noise variations in different measurement-based implementations of a gate.

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