As I understand it, a Quantum Computer has qubits, where the system is in a state such as:

$$c_0 |S_0\rangle +\ c_1 | S_1\rangle \ + ... +\ c_n|S_n\rangle $$

where $c_0, c_1, ..., c_n$ are complex values that represent the entangled state that the qubits are really in.

Furthermore, we can manipulate these states in certain ways, such as by applying a unitary matrix, such as rotate around an axis.

However, these operations are performed by physical devices, and hence will not be perfect. For example, a quantum gate may be designed to rotate by $\theta$, but in practice, might rotate by $\theta + \epsilon$, for some small but nonzero $\epsilon$.

Some proposed Quantum Computations take up millions of such individual operations; what would prevent the small errors from each of these millions of operations accumulating, and actually overwhelm the desired result?

Note that I am not talking about decorrelation; where some physical qubit gets a completely unexpected value. Quantum Error Correction is supposed to handle that; however, QEC works by comparing various physical qubits; if they all drift off slightly, I don't see how QEC can correct for that.

Now, with conventional digital gates, they are designed so that each bit doesn't have to be exact; as long as it is close, the gate will act as designed (and produce an output which also might not be precisely correct, but close enough for the next gate).

Is there something similar for Quantum Computers?

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    $\begingroup$ Try to google quantum error correction $\endgroup$ – Thomas Sep 9 '17 at 1:19
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    $\begingroup$ Did you actually read about QEC? It is indeed not obvious that qbit errors can be corrected, because you cannot copy a qbit. This is why before the discovery of QEC codes (like Shor's code) it was not obvious that error correction can be done. $\endgroup$ – Thomas Sep 9 '17 at 15:46
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    $\begingroup$ Again, I think you should take a look at how QECs work. The Shor code corrects arbitrary single qbit errors by encoding a logical qbit into 9 physical qbits. This just generalize to N logical qubits which get encoded as 9N physical qbits. $\endgroup$ – Thomas Sep 9 '17 at 20:54
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    $\begingroup$ Google fault tolerant quantum codes $\endgroup$ – Thomas Sep 9 '17 at 21:39
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    $\begingroup$ @poncho Quantum Error Corrections collapses analogue errors onto digital errors. But you'll really have to read up on that! This site does not replace a textbook. $\endgroup$ – Norbert Schuch Sep 9 '17 at 21:44

Quantum error correction codes involve repeatedly measuring an observable corresponding to an error syndrome. It's true that the expected value of the syndrome can vary continuously, and that this varying process corresponds to continuously varying amplitudes behind the scenes. But measuring the syndrome forces the system into either the "no error" state or the "one complete error" state (and the syndrome tells you how to fix it).

There are a lot more details, but that's the key idea. Measurement quantizes continuous errors. The next key idea is making sure you don't measure the encoded data as part of measuring the error syndrome, and that you get syndrome information for both bit-flip and phase-flip errors.

  • $\begingroup$ can you please comment on connections (if any) of this type of measurement with the Quantum Zeno effect? $\endgroup$ – ZeroTheHero Oct 22 '17 at 2:30

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