# Why apply the inverse operations in Randomized Benchmarking, when we can easily simulate Clifford operations?

I am going to give an hour-long talk and I expect I'll be getting a few questions on standard methods of benchmarking Noisy Intermediate-Scale Quantum (NISQ) computers. Are there any good review papers I should look at for quantum benchmarking? If it matters, I'm more a physics guy than I am a computer science guy.

My understanding is that qubit count is technically a benchmark, as are error rates for various gates and measurements. How one obtains fidelity and error rates? From what I've pieced together, Randomized Benchmarking is sort of a standard. You run a bunch of randomly chosen Clifford gates, then apply the inverse operators and compare the result to the original state. Something like "For trials of 10 gates the probability of successfully returning the input state was $$0.5$$, then with 16 gates it was $$0.3$$..."

This actually begs another question: why apply the inverse operations in Randomized Benchmarking when we can easily simulate Clifford operations? If the original Clifford gates go from $$|0\rangle^{\otimes M}$$ to some $$M$$-qubit state $$|\psi\rangle$$, we then de-compute back to what should be $$|0\rangle^{\otimes M}$$ but on a real NISQ device will be some slightly different state. Why not just use $$|\psi\rangle$$ as our state of comparison because we can easily obtain it? I've heard Clifford circuits are efficiently simulable.

For the review, you can have a look at this tutorial by Kliesch and Roth.

For the technical points on RB, I think salix's answer captures the basic idea. However, I have some additional remarks, since I am not agreeing with everything they said.

The inversion is costly. In standard Clifford RB, $$n$$-qubit Clifford unitaries are used which have to be compiled into the native gate set. This means that the circuits produced by a standard RB protocol with sequence length $$m$$ have gate count $$O(m n^2/\log(n))$$. In practise, this is the reason why standard Clifford RB is basically not used for more than $$n=2$$ qubits: The circuits are too long, thus the effective noise is way to strong and your signal decays too fast to be estimated. What people mostly do, is that they apply random generators directly (instead of $$n$$-qubit Cliffords. This is ok, because you converge to a 2-design). In this way, you have a better control over the circuit depth. However: If you're still doing the inversion, then this gate, as a product of sufficiently many generators, will be a global $$n$$-qubit unitary and you agaub need a long circuit to implement it.

Leaving out the inversion. Yes, you can leave out the inversion, good observation. Instead, you can effectively invert "by post-processing". As salix pointed out, this is what XEB does, however, here it is not done for Cliffords, but for rather general unitaries, which is why the post-processing is very costly. With Cliffords, this would be much more efficient. In the RB context, this is described for instance in Sec. VIII of Helsen et al.: "A general framework for randomized benchmarking", and more prominently in Helsen et al. "Estimating gate-set properties from random sequences"

RB as a benchmark. In an idealized scenario (namely gate-independent noise), the decay rate produced by a standard Clifford randomized benchmarking protocol is directly linked to the average gate fidelity of the Clifford unitaries, averaged over the group. There are also interleaved RB schemes to estimate the average gate fidelity of a fixed target unitary. However, in a more realistic scenario, there are certain problems in interpreting the decay rate as average gate fidelity. First, there is technically no direct relation. Second, and more severly, there is a "gauge freedom" in RB, meaning that the "noise" in the random sequences in actually not well-defined, but somewhat "gauge-dependent". The rates, however, are not. Hence, it can be problematic to relate those two concepts. This is discussed in Proctor et al., Wallman, Merkel et al., and Helsen et al.. IMO, this problem is not really resolved. I think it would be more responsible to see the decay rate as an independent benchmarking quantity which reflects the quality of your gates -- and not connect it to AGF (cf. benchmarks like 3DMark or whatever for classical computers).

The aim of RB is to remove the SPAM errors (state preparation and measurement errors) and only characterize the gates errors. This is achieve with measuring the success probability for different circuits depths. Thank to some mathematical properties (Clifford twirling for RB) The noise in these sequence is depolarizing, meaning that this success probability will decay exponentially with the depth of the circuit. The error rate is obtained simply by fitting this exponential decay.

For your second questions: The inverse operation of a Clifford sequence is also a Clifford (the set of Clifford gate form a group), so, the inversion only cost a single gate, in the case you are benchmarking a Clifford gate. The why it is done is mostly because then the sequence is equal to identity, and measuring the error is straightforward. There exist benchmarking where there is no inversion at the end. The cross-entropy benchmarking, or XEB, do exactly this and compare the output probability result to the ideal one calculated directly.