New answers tagged error-correction
0
votes
Commuting measurements performed in the different order?
You can indeed perform the measurement of a set of commuting observables in any order you like.
The issue with what you suggest is that you will be measuring 4 observables, $X_1, X_2, Z_1$ and $Z_2$, ...
- 665
2
votes
Accepted
Commuting measurements performed in the different order?
There is a more important intermediate question. Can you perform the measurements
$$
X_1,X_2,Z_1,Z_2
$$
and still record the same results?
You cannot divide the two-qubit observables into two one-...
- 60.8k
0
votes
Accepted
Why doesn't `PAULI_CHANNEL_1` require approximation in Stim sometimes?
Behind the scenes, stim tries to find an accurate conversion by either noticing it's a simple case or by using a few iterations of Newton's method. If it manages to find an accurate conversion, it ...
- 40.8k
3
votes
Accepted
Smallest qudit error correcting/detecting codes
I hope I understood the question, but here is the short answer of what I found: For single error-detecting codes, the smallest possible is $n = 3$ for any local qudit dimension (except for qubits, in ...
- 318
2
votes
Accepted
Proof for how logical operators generated systematically will satisfy Pauli commutation
The only written proof I know of this is in Gottesman's lecture notes/draft textbook at https://www.cs.umd.edu/class/spring2024/cmsc858G/. This is slightly more general than in your question, as it ...
- 188
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Obtaining the stabilizer outcome by small operators
Much as I'm a ZX-calculus fan, as in Craig's answer, I guess the most "standard" way to see this is using the stabilizer formalism, e.g. as in Nielsen-Chuang. I'll add a very quick reminder ...
- 188
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Obtaining the stabilizer outcome by small operators
As a sufficient condition for when you can get the observable you want, you can use commutation. Take both cases (ii) and (iii). The $ZZ$ term commutes with all the terms to its right by virtue of ...
- 60.8k
4
votes
Accepted
What is the definition of color codes?
Ideally, using a definition one should be able to answer the question; is this circuit implementing the color code?
Given the two definitions in your question, it would not be clear how to argue if ...
- 1,714
2
votes
Accepted
Smallest codes with transversal cliffords
This recent preprint High-distance codes with transversal Clifford and T-gates contains $[n, 1, d]$ code families that either admit a transversal implementation of the single-qubit Clifford group or ...
- 1,714
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What does it mean to "measure a stabilizer"?
As a supplement to the accepted answer, let's emphasize on the difference between "measure a stabilizer" and, the "measurement" that everyone is familiar with. And this is the ...
- 81
1
vote
Accepted
Can fault tolerant computation be performed in $1$d with strictly local gates?
The paper "Logical Qubit in a Linear Array of Semiconductor Quantum Dots" has a construction of a fault tolerant code with a threshold using nearest neighbor interactions on a 1d line of ...
- 40.8k
4
votes
Accepted
Showing that $V_{S} \subseteq \Pi_{S}$?
The image $\mathrm{Im}(\Pi)$ is defined as the subspace
$$
\mathrm{Im}(\Pi) := \{|v\rangle \, \mid \, \exists\, |w \rangle, \quad |v\rangle = \Pi |w\rangle\}
$$
you can check that this is indeed a ...
- 6,226
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To output syndrome measurement results
I'm not familiar with this package, but from a quick look I see that the code outputs parity check matrices. Given a parity check matrix $H$ and an error vector $E$ (a pauli string in symplectic ...
- 1,714
2
votes
Accepted
Are the preparations of a logical state in a surface code random and do we need to update the stabilizers after preparation?
TL;DR: Initializing data qubits in a physical $X$ (respectively, $Z$) eigenstate and switching on the stabilizers reliably$^1$ prepares a logical $X$ (respectively, $Z$) eigenstate.
Syndrome
Suppose ...
- 24.2k
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