Single-Qubit Randomized Benchmarking Protocol

Introduction

Hello! In our last lesson, we derived the Lindblad master equation, which provides a rigorous theoretical framework for modeling continuous-time, Markovian noise processes. We saw how a complex system-environment interaction can be distilled into an equation governing the system's density matrix, characterized by a set of Lindblad "jump" operators.

Today, we shift our focus from theoretical modeling to experimental characterization. How can we measure the average effect of all these noise sources on our quantum gates without needing to know the precise details of the underlying Lindblad operators? This lesson introduces a powerful and widely-used technique to answer that question: single-qubit randomized benchmarking (RB).

Our learning outcome is to describe the protocol for single-qubit randomized benchmarking, including the role of the Clifford group in generating gate sequences and the model for fitting the fidelity decay curve.

We will cover:

The step-by-step experimental procedure for RB.
The crucial theoretical role of the Clifford group, which makes the protocol work.
The derivation of the exponential decay model used to extract the average gate fidelity from the experimental data.

This topic combines concepts from quantum information theory, group theory, and experimental practice, providing a holistic view of how we quantify noise in real-world quantum processors.

The Randomized Benchmarking Protocol

The core idea of randomized benchmarking is to apply a long sequence of random quantum gates that, in the absence of noise, should compose to the identity operation. Any deviation from the initial state at the end of the sequence is therefore a direct measure of the accumulated error from the gates.

Hands-on introduction to randomized benchmarking

To begin, let's get a precise description of the protocol. The following resource, a lecture note from SciPost Physics, provides an excellent and detailed walkthrough. Please read the section that outlines the standard RB procedure.

Please read the text from the beginning of the subsection "1.1 General description of standard RB and noise assumptions" up to the end of the list of seven steps (pages 4-5). This will walk you through the logic of constructing and measuring a random sequence.

As you've just read, the protocol can be summarized as follows:

Choose a sequence length m.
Generate a random sequence of m gates ${C_{1}, C_{2}, \dots, C_{m}}$ . These gates are chosen uniformly at random from the single-qubit Clifford group. We'll discuss why we use this specific group shortly.
Calculate the inverse gate, $C_{m + 1} = (C_{m} \circ \dots \circ C_{1})^{- 1}$ . Since the Clifford gates form a group, this inverse is guaranteed to also be a single Clifford gate.
Execute the full circuit: Prepare a qubit (e.g., in state $∣0 ⟩$ ), apply the sequence of $m + 1$ gates, and then measure the qubit in the computational basis.
Estimate the "survival probability": Repeat the execution many times to estimate the probability of measuring the qubit in its initial state $∣0 ⟩$ . This gives you the survival probability for this specific random sequence.
Average over many sequences: Repeat steps 2-5 for a large number, $K$ , of different random sequences, all of the same length m. The average of these survival probabilities gives the average sequence fidelity, $F_{seq} (m)$ .
Repeat for different lengths: Repeat the entire procedure for several different sequence lengths m to obtain a set of data points $(m, F_{seq} (m))$ .
Fit the data: Plot $F_{seq} (m)$ versus m and fit it to a decaying exponential model to extract the average error rate.

The elegance of this protocol is that it averages out the specific errors of individual gates, giving a single, reliable number that characterizes the performance of the entire gate set.

The "Magic" of the Clifford Group: Twirling

Why do we use the Clifford group? The choice is not arbitrary; it is the theoretical heart of the RB protocol. The Clifford group has a special property: it is a unitary 2-design. While the formal definition is abstract, its practical implication for RB is profound.

When we average over random sequences of Clifford gates, this process—known as twirling—effectively averages the physical noise channel $Λ$ acting on the qubit. Because the Clifford group is a 2-design, this twirling transforms any sufficiently weak, gate-independent noise into a simple depolarizing channel.

A depolarizing channel has a very simple action: with probability $p$ , it does nothing, and with probability $(1 - p)$ , it completely randomizes the state (i.e., maps it to the maximally mixed state $I /2$ ).

Hands-on introduction to randomized benchmarking

The following section of the same lecture notes provides the theoretical background for this simplification. It connects the concepts of average gate fidelity, twirling over the Clifford group, and the resulting depolarizing channel.

Please read Section 1.2, "Average gate fidelity, twirling and the depolarizing channel" (pages 7-9). Focus on understanding the three key 'Statements' and their implications:

Let's summarize the crucial logic from that section:

Statement 1: Average gate fidelity is invariant under twirling. This means that the average fidelity of our complex, true noise channel is the same as the fidelity of the much simpler, twirled (depolarizing) channel.
Statement 2: The Clifford group is a unitary 2-design. This is the mathematical property that guarantees that twirling over the finite Clifford group is equivalent to twirling over the entire, infinite group of all unitary operations.
Statement 3: The twirling of a quantum channel is a depolarizing channel. This is the key result. The average effect of any Markovian noise channel, when subjected to the RB procedure, is mathematically equivalent to a simple depolarizing channel.

Single-Qubit Randomized Benchmarking and Interleaved Randomized Benchmarking — This figure illustrates the core concepts of randomized benchmarking. (a) shows the circuit structure, with a random Clifford sequence followed by an inverse. (c) shows the resulting experimental data: the sequence fidelity decays as the sequence length `m` increases. The solid lines are fits to the exponential decay model we are about to derive. Image credit: Asaad et al., npj Quantum Information (2016).

This simplification is what allows us to extract a single error parameter from a complex physical reality. Instead of trying to characterize every possible error, we average them all into a single effective parameter, the depolarization probability.

The Fitting Model: From Twirling to Exponential Decay

Now we can connect the theoretical result of twirling to the experimental data. How does the noise being equivalent to a depolarizing channel lead to the characteristic exponential decay shown in the figure above?

The average effect of one noisy Clifford gate is equivalent to an ideal Clifford gate followed by a depolarizing channel, $Λ_{dep}$ . A sequence of m such gates corresponds to applying $Λ_{dep}$ m times. A key property of the depolarizing channel is that composing it with itself m times yields another depolarizing channel, but with a depolarizing probability that scales as $p^{m}$ .

Hands-on introduction to randomized benchmarking

This final reading section puts all the pieces together to derive the famous RB fitting formula.

Please read Section 1.3, "Average sequence fidelity and the definition of the average error rate" (pages 9-12). Follow the derivation that leads to Equation (28), and pay close attention to the final relationship between the decay parameter p and the average error rate r in Equation (30).

The derivation you just read shows that the average sequence fidelity, after averaging over many random sequences of length m, takes the form:

F_{seq} (m) = A_{0} p^{m} + B_{0}

This is the exponential decay model that we fit to our experimental data points $(m, F_{seq} (m))$ .

p: This is the key parameter extracted from the fit. It's the depolarizing parameter of the effective noise channel per Clifford gate.
A_0 and B_0: These constants absorb all the errors related to the initial state preparation and the final measurement. This is a critical feature of RB: the protocol is robust against State Preparation and Measurement (SPAM) errors. The decay rate p is independent of SPAM imperfections.

Once p is determined from the fit, the average gate fidelity $\overset{ˉ}{F}$ and the average error rate r for the Clifford gates can be calculated. For a single qubit (where the dimension $d = 2$ ), the formulas are:

Average Gate Fidelity: $\overset{ˉ}{F} = p + \frac{1 - p}{d} = \frac{p + 1}{2}$
Average Error Rate: $r = 1 - \overset{ˉ}{F} = \frac{1 - p}{2}$

This gives us what we wanted: a single number, r, that quantifies the average error of our quantum gate set.

Test your understanding!

An RB experiment on a single qubit yields a fitted decay parameter of $p = 0.99$ . What is the average error rate r per Clifford gate? If a typical Clifford gate is composed of, on average, 1.5 elementary gates (e.g., Hadamard, Phase gates), what is a rough estimate for the error rate of the elementary gates?

Show answer

Given $p = 0.99$ and $d = 2$ for a single qubit, the average error rate r per Clifford gate is:

r = \frac{1 - p}{2} = \frac{1 - 0.99}{2} = \frac{0.01}{2} = 0.005 or 0.5%

If we assume errors add incoherently, and a Clifford has on average 1.5 elementary gates, a rough estimate for the error per elementary gate, $r_{elem}$ , would be:

r \approx 1.5 \times r_{elem} ⟹ r_{elem} \approx \frac{r}{1.5} = \frac{0.005}{1.5} \approx 0.0033 or 0.33%

This is a simplification; more sophisticated methods like interleaved RB are needed for precise characterization of individual gates. But it gives the correct intuition.

A Deeper Look: Statistical Confidence

A natural question arises from the protocol: how many random sequences, K, do we need to average over for each length m? Early experiments used numbers like 30-100 based on empirical results. Your preference for original sources points us to a key paper that provided the theoretical justification for why this relatively small number is sufficient.

Randomized Benchmarking with Confidence

The following paper, "Randomized Benchmarking with Confidence," analyzes the statistical variance of the RB protocol. It demonstrates that for the low-error-rate regime of modern quantum devices, the variance of the measured survival probabilities across different random sequences is very small.

First, skim Section III.A "Confidence interval for randomized benchmarking" (page 5) to see the context of why the number of sequences matters. Then, you can look at the main result in Section VI.B, specifically Theorem 13 (page 20). You don't need to follow the full derivation, but notice that the variance \sigma^2_m scales with the error rate squared (r^2).

The key insight from this paper is that the variance $σ_{m}^{2}$ of the survival probabilities across different random sequences is very low, scaling as $O (m^{2} r^{2})$ . Since the error rate r is typically small (e.g., $1 0^{- 3}$ to $1 0^{- 4}$ ), the variance is tiny. This means that even a modest number of random sequences K is enough to get a very good estimate of the true mean $F_{seq} (m)$ , providing strong statistical confidence in the results.

Conclusion

In this lesson, we have dissected the protocol and theory of single-qubit randomized benchmarking. It is a cornerstone technique for assessing the performance of quantum hardware.

Key Takeaways:

The RB Protocol: A sequence of m random Clifford gates followed by an inverse allows for the measurement of accumulated error. The process is repeated for different m to map out a decay curve.
The Clifford Group as a 2-Design: This is the key theoretical underpinning. Averaging over random Cliffords (twirling) simplifies any gate-independent Markovian noise into an effective depolarizing channel.
The Fitting Model: The sequence fidelity follows an exponential decay, $F_{seq} (m) = A_{0} p^{m} + B_{0}$ . The decay constant p is insensitive to SPAM errors and can be used to calculate the average error rate per Clifford gate: $r = (1 - p) /2$ .
Efficiency: RB is statistically efficient, requiring a manageable number of random sequences to achieve high confidence, due to the low variance of the measurement outcome in the low-error regime.

Preview of the next lesson:

Theory is one thing, but implementation is another. In our next lesson, we will put this knowledge into practice. We will implement a single-qubit randomized benchmarking experiment on a noisy simulator to extract the average gate fidelity. This will involve writing code to generate the Clifford sequences, run the simulated experiment, and perform the fit to extract the error rate, solidifying the concepts we learned today.