# The essential content of the Threshold Theorem for Fault-Tolerant Quantum Computation

Nice posting. As an engineer, I feel like the theory is pretty well developed, but we need more work on the implementation, including optimization of syndrome (stabilizer) extraction circuits, and better ways to optimize for differing error rates. How do we match to a specific architecture and workload, to achieve a particular, desirable logical gate error rate?

Thanks for the question! The question is as difficult as it is important, so I'll try to do it justice.

I absolutely agree that there is a great deal of work left to be done on the implementation side of the matter. For this reason, there cannot yet be a single answer to the question of matching architecture and workload to observed logical error rate: we simply do not know enough about the technology that will underlie any working quantum computer. From the perspective of fault-tolerance thresholds, I would say that the biggest challenge is to produce strong but correct assumptions about the form of logical errors that affect real devices. Though I did not make this point in my blog post, such assumptions bear a strong relationship with the relevant threshold value. Another point I did not make is that the threshold value also depends on the choice of code, which restricts the set of possible fault-tolerant circuits Q' that can be produced given a target circuit Q.

From what I have seen of your research (e.g. this PDF), I guess that you agree that the problem of establishing performance targets for actual devices is architecture-dependent. I would add to this story that computer architectures are based on models of computing and that logical error rates are derived ultimately from the underlying model. In my opinion, the job of the scientist (as opposed to the job of an engineer like yourself) is to establish the properties of this underlying model. The real value of the Threshold Theorem is, to me, that it places important demands on the kind of model on top of which a fault-tolerant architecture can be built. In other words, I think the real importance of the Threshold Theorem is that it establishes the possibility of fault-tolerance if noise is well-behaved enough. Whereas this is an important implication, it is of limited (but non-zero) value for engineers.

Although I focussed in this blog post on the mathematical definition of thresholds, I think it is misguided to treat the Threshold Theorem as an indicator of a performance target of any kind. It is that thorny first category of noise promises that is the important message of the Threshold Theorem -- in my opinion, of course! I think it is work like yours that will establish the proper performance targets for real devices, not the abstract considerations of 'the' Threshold Theorem.

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