Threshold theorem

In

The key question that the threshold theorem resolves is whether quantum computers in practice could perform long computations without succumbing to noise. Since a quantum computer will not be able to perform gate operations perfectly, some small constant error is inevitable; hypothetically, this could mean that quantum computers with imperfect gates can only apply a constant number of gates before the computation is destroyed by noise.

Surprisingly, the quantum threshold theorem shows that if the error to perform each gate is a small enough constant, one can perform arbitrarily long quantum computations to arbitrarily good precision, with only some small added overhead in the number of gates. The formal statement of the threshold theorem depends on the types of error correction codes and error model being considered. Quantum Computation and Quantum Information, by Michael Nielsen and Isaac Chuang, gives the general framework for such a theorem:

Threshold theorem for quantum computation^[6]: 481: A quantum circuit on n qubits and containing p(n) gates may be simulated with probability of error at most ε using

$${\displaystyle O(\log ^{c}(p(n)/\varepsilon )p(n))}$$

${\displaystyle p<p_{\rm {th}}}$

Threshold theorems for classical computation have the same form as above, except for classical circuits instead of quantum. The proof strategy for quantum computation is similar to that of classical computation: for any particular error model (such as having each gate fail with independent probability p), use error correcting codes to build better gates out of existing gates. Though these "better gates" are larger, and so are more prone to errors within them, their error-correction properties mean that they have a lower chance of failing than the original gate (provided p is a small-enough constant). Then, one can use these better gates to recursively create even better gates, until one has gates with the desired failure probability, which can be used for the desired quantum circuit. According to quantum information theorist Scott Aaronson:

"The entire content of the Threshold Theorem is that you're correcting errors faster than they're created. That's the whole point, and the whole non-trivial thing that the theorem shows. That's the problem it solves."^[7]

Threshold value in practice

Current estimates put the threshold for the

• Quantum error correction schemes
• Physical and logical qubits
• Fault tolerance

Notes