For people who are mildly interested in the topic of quantum error correction, we have a short discussion and a Javascript app that shows some of the basic ideas in our MOOC (which is now free). https://www.futurelearn.com/courses/intro-to-quantum-computing For people with serious interest, there are two topics to cover: decoherence, and quantum error correction. (We'll leave aside fault-tolerant execution of algorithms, which requires yet more work.) My favorite source on decoherence is "Introduction to decoherence and noise in open quantum systems," by Todd Brun and Daniel Lidar, which is the opening chapter in the compendium _Quantum Error Correction_ (Cambridge, 2013). Unfortunately, AFAIK, it's not publicly available in PDF. Other sources include John Preskill's lecture notes (which are freely available) at http://www.theory.caltech.edu/~preskill/ph219/ph219_2018-19 and many books on quantum computing, such as the classic Nielsen & Chuang (Cambridge, 2000). Be warned that this does take moderately good linear algebra chops, and if you want to really understand decoherence, some partial differential equations. If you want to learn about quantum error correction, my favorite source is Devitt, Nemoto & Munro, "Quantum Error Correction for Beginners," arXiv:0905.2794v3 [quant-ph]. Already over a decade old, but still outstanding. But "for beginners" assumes you have a decent quantum background, just nothing on QEC. ----- Okay, a little on mohta-san's concerns... There are multiple sources of errors and multiple types. * bit flip: a symmetric channel, with 0-->1 and 1-->0 errors equally likely * phase flip: ditto, but for a qubit's phase * amplitude damping: In this mode, 1-->0 errors are more likely than 0-->1 errors, because the former happens when a photon is emitted and thpe latter when a photon is absorbed; moving to the lower-energy state is more common. This is the harder one to understand. https://en.wikipedia.org/wiki/Amplitude_damping_channel http://www.thphys.nuim.ie/staff/jvala/Lecture_12.pdf We can also talk about various gate errors, and errors in measurement. One that particularly worried me is, in fact, one of the things that Peter Shor simplified away in his first QEC paper: what happens when the errors on individual qubits aren't independent? *ALL* of these are handled properly by quantum error correction. In fact, they can all be treated as types of memory errors, which (thankfully) simplifies the math. This is over-simplified, but QEC consists of several phases: 1. Encoding a logical state 1.5 <<errors happen>> 2. Extracting syndrome information a. calculating the syndromes in superposition b. measuring the syndrome qubits, which *forces the state into one with discrete errors* 3. Using that information to correct the state. Steps 2 & 3 are repeated over the lifetime of the algorithm execution, interleaved with the operations that execute logical gates. (Well, in codes known as topological codes, surface codes, toric codes and color codes, that's not quite true. Those codes also have no classical equivalent, whereas other codes derive very directly from classical error correction.) Step 2a involves quantum operations, and so *adds* errors into the state, meaning an interesting question is when the error correction process as a whole *removes* more errors than the extraction adds. Monroe's group claims to have exceeded that threshold. Mohta's concern seems to be two-fold; first, that errors aren't independent, and second, that forcing of the state (actually, a *partial* collapse of the quantum state) doesn't take you a state where applying correction works. As it happens, the first problem worried me enough that I wrote about it in my Ph.D. thesis. See Sec. 2.3.5 (p. 55) of https://arxiv.org/abs/quant-ph/0607065 Fundamentally, coherent noise over a group of qubits (e.g., some coherent radio noise from outside, or shared dependence on a slightly incorrect classical oscillator, or something) results in all of the suppressed *to second order* -- to the *square* of the induced rotation angle (since it's less than 1.0, that means reducing the error). A similar argument applies to the amplitude damping that is the harder error type mentioned above: the syndrome extraction forces us back to a state where, *with high probability*, the correction we choose to apply makes the state better (== closer to the original) rather than worse. Some QEC schemes operate by applying multiple recursive levels of the same (or, possibly, different) encoding schemes. Yes, that results in a big growth in resources used, but a) we're only talking about 2 or 3 levels, not some huge number; and b) the errors are actually suppressed at a rate that exceeds the resource consumption, so it all works out in the end. I won't go into the math of that here. Fundamentally, quantum error correction *works*, and over the course of 2020-2025, we will see it demonstrated increasingly clearly in experimental systems. Barring something really extraordinary, this is all I will have to say on the topic on ietf@ietf. People who are interested are welcome to attend the QIRG meeting (Monday afternoon Bangkok time during IETF) and ask about this during the open mic. Since we have other business to attend to on the already-published agenda, I will defer discussion of this draft and this topic to the very last thing, but I expect there will be some time available. See y'all online in a couple of weeks! —Rod
