### Paperz

Tuesday, March 3rd, 2020Soon, all anyone will want to talk about is quarantines, food shortages, N95 masks, the suspension of universities and of scientific conferences. (As many others have pointed out, this last might actually be a boon to scientific productivity—as it was for a young Isaac Newton when Cambridge was closed for the bubonic plague, so Newton went home and invented calculus and mechanics.)

Anyway, before that all happens, I figured I’d get in a last post about quantum information and complexity theory progress.

Hsin-Yuan Huang, Richard Kueng, and John Preskill have a nice preprint entitled Predicting Many Properties of a Quantum System from Very Few Measurements. In it they take shadow tomography, which I proposed a couple years ago, and try to bring it closer to practicality on near-term devices, by restricting to the special case of non-adaptive, one-shot measurements, on separate copies of the state ρ that you’re trying to learn about. They show that this is possible using a number of copies that depends logarithmically on the number of properties you’re trying to learn (the optimal dependence), not at all on the Hilbert space dimension, and linearly on a new “shadow norm” quantity that they introduce.

Rahul Ilango, Bruno Loff, and Igor Oliveira announced the pretty spectacular-sounding result that the Minimum Circuit Size Problem (MCSP) is NP-complete for multi-output functions—that is, for Boolean functions f with not only many input bits but many outputs. Given the 2^{n}-sized truth table of a Boolean function f:{0,1}^{n}→{0,1}, the original MCSP simply asks for the size of the smallest Boolean circuit that computes f. This problem was studied in the USSR as early as the 1950s; whether it’s NP-complete has stood for decades as one of the big open problems of complexity theory. We’ve known that if you could quickly solve MCSP then you could also invert any one-way function, but we’ve also known technical barriers to going beyond that to a flat-out NP-hardness result, at least via known routes. Before seeing this paper, I’d never thought about whether MCSP for many-output functions might somehow be easier to classify, but apparently it is!

Hamoon Mousavi, Seyed Nezhadi, and Henry Yuen have now taken the MIP*=RE breakthrough even a tiny step further, by showing that “zero-gap MIP*” (that is, quantum multi-prover interactive proofs with an arbitrarily small gap between the completeness and soundness probabilities) takes you **even beyond the halting problem** (i.e., beyond Recursively Enumerable or RE), and up to the second level of the arithmetical hierarchy (i.e., to the halting problem for Turing machines with oracles for the original halting problem). This answers a question that someone asked in the comments section of this blog.

Several people asked me for comment on the paper What limits the simulation of quantum computers?, by Yiqing Zhou, Miles Stoudenmire, and Xavier Waintal. In particular, does this paper refute or weaken Google’s quantum supremacy claim, as the paper does *not* claim to do (but, rather coyly, also does not claim *not* to do)? Short answer: No, it doesn’t, or not now anyway.

Longer, more technical answer: The quoted simulation times, just a few minutes for quantum circuits with 54 qubits and depth 20, *assume Controlled-Z gates rather than iSWAP-like gates.* Using tensor network methods, the classical simulation cost with the former is roughly the square root of the simulation cost with the latter (~2^{k} versus ~4^{k} for some parameter k related to the depth). As it happens, Google switched its hardware from Controlled-Z to iSWAP-like gates a couple years ago precisely because they realized this—I had a conversation about it with Sergio Boixo at the time. Once this issue is accounted for, the quoted simulation times in the new paper seem to be roughly in line with what was previously reported by, e.g., Johnnie Gray and Google itself.

Oh yeah, I enjoyed Quantum Homeopathy Works. Cool result, and the title is actually a pretty accurate description of the contents.

To end with a community announcement: as many of you might know, the American Physical Society’s March Meeting, which was planned for this week in Denver, was abruptly cancelled due to the coronavirus (leaving thousands of physicists out their flights and hotel rooms—many had even already arrived there). However, my colleague Michael Biercuk kindly alerted me to a “virtual March Meeting” that’s been set up online, with recorded talks and live webinars. Even after the pandemic passes, is this a model that we should increasingly move to? I wouldn’t have thought so ten or fifteen years ago, but today every schlep across the continent brings me a step closer to shouting “yes”…