Quantum computation and entropy

[otennert]

As I explained, what you say above is correct: the entropy has increased, as the system is in a mixed state after the (non-selective!) measurement.

 

[vanhees71]

Do the error correction operations involve the ancillas? It didn't appear that way from your previous description; it appeared that those operations only operate on the qubits that are intended to store the desired information.

I'm not an expert in quantum computing, but as far as I understand it, the idea is to have some decoherence but to realize each q-bit of your quantum computer by many "physical q-bits" to have this possibility of "error correction". The holy grail are of course to engineer as protected as possible q-bits like superconductor topological structures.

 

[gentzen]

Do the error correction operations involve the ancillas? It didn't appear that way from your previous description; it appeared that those operations only operate on the qubits that are intended to store the desired information.

Yes, the error correction operations always involve ancillas. I personally think that "Chapter 5 Quantum error correction" in N. David Mermin "Quantum Computer Science - An Introduction" is a good place to start learning about this topic. The working of "standard" error correction schemes is nicely illustrated in Figure 5.3:

Fig 5.3 How to detect and correct the three possible single-bit-flip errors shown in Figure 5.2. One requires two ancillary Qbits (the upper two wires), each initially in the state $\ket{0}$, coupled to the codeword Qbits by cNOT gates. After the cNOT gates have acted each ancilla is measured. If both measurements give 0, then none of the erroneous NOT gates on the left have acted and none of the error-correcting NOT gates on the right need to be applied. If the upper measurement gate shows $x = 1$ and the lower one shows $y = 0$, then the uppermost of the three erroneous NOT gates has acted on the left. Its action is undone by applying the uppermost of the three NOT gates on the right. The other two possible 1-Qbit errors are similarly corrected.

Here the boxes around M with an x or y in a small box on top denote measurement operations, whose (binary) result is denoted x and y respectively. The boxes around X to the power of some combination of optionally negated x and y represent an X operation if the exponent evaluted to 1 (or true), and an identity operation if the exponent evaluated to 0 (or false).

The working of "non-standard" MWI like error correction schemes is nicely illustrated in Figure 5.4:

Fig 5.4 Automation of the error-correction process of Figure 5.3. The three controlled gates on the right - one of them a doubly controlled Toffoli gate with multiple targets - have precisely the same error-correcting effect on the three codeword Qbits as does the application of NOT gates contingent on measurement outcomes in Figure 5.3. The final state $\ket{\Psi}$ of the ancillas (which is also the state that determines the action of the three controlled gates on the right) is $\ket{00}$ if none of the erroneous NOT gates on the left has acted. It is $\ket{10}$ if only the upper erroneous NOT gate has acted, $\ket{11}$ if only the middle one has acted, and $\ket{01}$ if only the lower one has acted.

Note that both figures contain two ancilla qubits, which are both initialized to $\ket{0}$. This known state is what allows them to absorb entropy from the main qubits. With measurements, the state of the ancilla qubits is known once again after they absorbed the entropy, and hence the ancillas are "ready" to absorb entropy once again. Without measurement, the ancillas still aborbed the entropy, but they cannot be reused yet.

 

[gentzen]

So they are unitary? But:

So they are not unitary?

I'm confused.

This is why I keep asking for a reference.

The book by Mermin mentioned in my previous answer is an excellent first introduction. Scott Aaronson's old lecture notes are a good follow-up to learn in very few pages which closely related topics Mermin did not even mention:

• Lecture 27: Quantum Error Correction (8 pages, 15 pages in 2.0)
• Lecture 28: Stabilizer Formalism (9 pages, 11 pages in 2.0)
• Lecture 29: Experimental Realizations of QC (9 pages, removed in 2.0)

Scott's newer lecture notes 2.0 are more detailed (for example, its Figure 27.5 is the analog to Figure 5.3 from Mermin in my previous answer), but the lecture on experimental realization of QC has been removed. Nielsen / Chuang is great on the fundamentals and the physics. It is old (2000), but chapter "7 Quantum computers: physical realization" is still very instructive to read. By the way, the same is true for the experimental details from Ashcroft/Mermin, despite being outdated. The temptation is to omit experimental details, because they will soon be outdated. Trying to keep them up-to-date is hopeless, so the temptation to remove them once they became outdated is even stronger. Ashcroft/Mermin's decision to keep their book unchanged is very wise, but it took me a long time to understand this.

 

[antonantal]

Very good resources @gentzen. Thanks for sharing!