This illustrates the disconnect of mathematicians trying to do physics. Not only his error model is unrealistic (you essentially need a frequency dependent power spectral density function to represent fluctuations in control which decays according to a power law at high frequencies, sometimes called 1/f noise but the power doesn't have to be 1; also you don't get that kind of strong spatial correlations in real systems, cross-talk among spins more of a realistic problem but it is not what he's doing either), his argument is based on something that is just wrong:
> that the effort required to obtain a low enough error level for any implementation of universal quantum circuits increases exponentially with the number of qubits, and thus, quantum computers are not possible.
That's what dynamical decoupling (DD) or pulse sequences are for: you can get arbitrarily high quality quantum gates (typically, but not always, at the cost of increasing the gate times; removing the most significant first order error typically increases the gate time by less than an order magnitude, think 2x-4x) without increasing the number of physical qubits at all. People don't just rely on surface codes, anyone serious about implementing a quantum computer use surface codes after DD to reduce the infidelity to the threshold required for them. Which is why you don't need hundreds of physical qubits to have a single robust logical qubit.
For those who are not familiar, DD is like one of the oldest tricks in the bag, it's nothing like a new cutting edge type quantum error correction code. In fact, the oldest form of DD, spin echo, precedes any real discussion about quantum computers by a decade.
DD is possible essentially because quantum operations don't commute so errors don't simply add up as they do with classical errors; this makes it possible to obtain a better gate by carefully combining noisy gates such that (significant) errors cancel.