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# Tag Info

## 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
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### 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
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### 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
1 vote

### 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
0 votes

### 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
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### 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
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### 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
2 votes

### 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 ...
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
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### 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
1 vote

### 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 ...
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