[tagrun]

That sentence actually reads "Schrödinger–Feynman algorithm becomes exponentially more computationally expensive with increasing circuit depth" which is true (because the paths in a path integral in a discrete setting would grow combinatorically, but don't have to sort to path integrals to approximate the unitary in a "quick" and dirt way, which clearly doesn't scale well --in fact, if you avoid such "clever" tricks [which is only beneficial in some limited regime] and do it in the naive way, it will scale linearly). It's not the only game you can play on a classical computer, as IBM points out (for which the upfront cost is much higher).

Figure 4b is about error estimation, They use XEB which is exponentially faster than, say doing full quantum process tomography, which is also true. That's the whole reasoning behind XEB, which gives far less information about the error channels, but you still have a fair estimate on the overall fidelity.

None of these have anything to do with the complexity of the actual computation done on the quantum computer though.

[sanxiyn]

Indeed these don't have anything to do with quantum computers, but it does have something to do with quantum supremacy, because quantum supremacy is a claim about both quantum computers and classical computers.

Google chose an algorithm exponential in circuit depth as the best classical algorithm in order to establish quantum supremacy. IBM demonstrated (as you agree) it is in fact not the best classical algorithm. IBM is entirely correct to point this out.

[tagrun]

Again, no.

It is a trivial fact that on a classical computer, you can simulate a quantum computer in time that grows linearly in circuit depth, in principle (as in the case of the "naive" way I mentioned above). No one in doing quantum algrothims ever claimed this was the case, and claiming otherwise is just a silly mistake.

Preskill coined the word "quantum supremacy", not Martinis. Even if someone from Martinis' lab misspoke, you can't pin it on Preskill.

Take Shor's algorithm, which has been the poster boy of quantum computing for decades. It gives exponential speed up in the number of qubits, not circuit depth.

See other examples here: https://quantumalgorithmzoo.org/ The complexity is in the number of qubits, "cirucit depth" is a non-concept in quantum algorithms.

No one cares about the circuit depth in quantum algorithms, because in principle, you can always reduce the circuit depth to 1 on a quantum computer: quantum computers aren't necessarily made of basic logic gates, you can implement any unitary in by for example smoothly pulsing the fields in a correct way in a single go.

"Depth" is a concept which usually comes into play when you try to measure the quality of the implementation of some given gate U. You repeat U many many times say M, and see how some measured quantity decays as a function of M, which gives a measure of the fidelity. But this has nothing to do with the quantum algorithm's complexity, it's just for "benchmarking" a physical implementation of the gate.