Revision history for comment 764

Edited by Robin Blume-Kohout Feb 27 2017 13:30 UTC

@Chris: as Ben says, the model for measurement errors is "You measure in a basis that's off by a small rotation".

@Ben: I don't think either of the techniques you mention will directly resolve the paper's concern/confusion. That concern is with the post-QEC state of the system. That state isn't invariant under re-expressing measurement noise as a small unitary followed by measurement in the right basis. And repeated measurement in the same (wrong) basis won't fix the putative problem either.

@James: You're right that this has been thought about before. However, the paper isn't complaining that asymptotic FT fails; it's complaining that the O(p^2) error scaling that *is* supposed to hold for distance-3 codes fails. It's true that repeated measurement would be necessary to achieve that scaling, but it's not sufficient to fix the putative problem.

The basic problem here is misinterpretation of the "success" condition for QEC. This paper assumes that the metric of success is the overlap between the post-QEC state $|\Psi\rangle$ and the predefined $|0_L\rangle$ logical state. But it's not. It's actually kinda hard to precisely define what "success" means in a completely rigorous way. The simplest way I know of is to consider the sequence: (1) logical prep in one of the 4 BB84 states; (2) $N$ rounds of QEC; (3) logical measurement (in whichever of X or Z commutes with the desired initial prep). Then, look at the probability of correctly deducing the initial state, and fit it to $(1-p)^N$.

If your syndrome measurements are slightly rotated as in this paper, you're actually performing good QEC in a slightly deformed code. Which means that $|0_L\rangle$ moves around, depending on the measurement error. FTQEC still works, as measured by the operational criterion I gave above, but the naive criterion "Overlap with the $|0_L\rangle$ that I intended to implement" doesn't reveal it.

Robin Blume-Kohout commented on What determines the ultimate precision of a quantum computer? Feb 27 2017 13:30 UTC

@Chris: as Ben says, the model for measurement errors is "You measure in a basis that's off by a small rotation".

@Ben: I don't think either of the techniques you mention will directly resolve the paper's concern/confusion. That concern is with the post-QEC state of the system. That state isn't invariant under re-expressing measurement noise as a small unitary followed by measurement in the right basis. And repeated measurement in the same (wrong) basis won't fix the putative problem either.

@JRW: You're right that this has been thought about before. However, the paper isn't complaining that asymptotic FT fails; it's complaining that the O(p^2) error scaling that *is* supposed to hold for distance-3 codes fails. It's true that repeated measurement would be necessary to achieve that scaling, but it's not sufficient to fix the putative problem.

The basic problem here is misinterpretation of the "success" condition for QEC. This paper assumes that the metric of success is the overlap between the post-QEC state $|\Psi\rangle$ and the predefined $|0_L\rangle$ logical state. But it's not. It's actually kinda hard to precisely define what "success" means in a completely rigorous way. The simplest way I know of is to consider the sequence: (1) logical prep in one of the 4 BB84 states; (2) $N$ rounds of QEC; (3) logical measurement (in whichever of X or Z commutes with the desired initial prep). Then, look at the probability of correctly deducing the initial state, and fit it to $(1-p)^N$.

If your syndrome measurements are slightly rotated as in this paper, you're actually performing good QEC in a slightly deformed code. Which means that $|0_L\rangle$ moves around, depending on the measurement error. FTQEC still works, as measured by the operational criterion I gave above, but the naive criterion "Overlap with the $|0_L\rangle$ that I intended to implement" doesn't reveal it.