# Tag Info

1 vote

### MWPM algorithm for depolarizing-like noise channel (i.e. non-zero probability of $Y$ errors)

First of all, I think that the protection would not be "ruined". I mean this because even if the $X$ and $Z$ graphs are independently decoded, whenever a qubit is determined to have suffer ...

### MWPM algorithm for depolarizing-like noise channel (i.e. non-zero probability of $Y$ errors)

The simplest thing you can do is to tell the decoder to ignore the existence of Y errors and decoding the X and Z subgraphs totally independently. This is not optimal, it will require better ...
• 28.8k

### Concatenated quantum code

$\newcommand{\ket}[1]{|#1\rangle}$ The short answer is given on the same slide "optimal decoding must pass information between the levels" Preparation calculations To understand, let us work ...
1 vote

### Does a $[[5,1,2]]$ CSS code exist?

A $[[5,1,2]]$ code occurs as a member of a family of hypergraph product codes with parameters $[[2d^2-2d+1,1,d]]$ codes : $[[5,1,2]],[[13,1,3]],[[25,1,4]],\cdots$. These are all CSS codes. Here are (...
• 1,485
Accepted

### Does a $[[5,1,2]]$ CSS code exist?

$S=\langle ZZZZZ, XXXXI, IXXXX, XXIXX\rangle$ with logicals $L_X = XXIII, L_Z= IZIZI$ should do the trick.
Accepted

### Does every code have transversal Pauli group?

I don't think so - consider e.g. the 'diagonal' representation $$\phi:SU(2) \rightarrow GL((\mathbb{C}^2)^{\otimes 4})$$ $$U \mapsto U^{\otimes 4}.$$ The Clebsch-Gordan series tells us that, for spin ...
• 502

### Does every code have transversal Pauli group?

Every code that can be implemented by a stabilizer circuit (this includes stabilizer codes, gauge codes, floquet codes, etc) has this type of subset-transversal Pauli gate. In such a code, the X, Y, ...
• 28.8k
Accepted

### Does every code have a strongly transversal Pauli group?

The [[4,1,2]] surface code, or any code with an even number of data qubits, either doesn't have a transversal X or doesn't have a transversal Z. Because logical X has to anticommute with logical Z, ...
• 28.8k
1 vote

### Weight enumerators for Hermitian operator

This is just fleshing out some of the themes from Adam Zalcman's answer, which I have already accepted. The standard proof that $B_j\geq A_j$ for a projection $H$ uses the spectral theorem to ...
• 2,252
Accepted

### Basic question on the difference between Minimum Weight Perfect Matching and maximum likelihood

The job of a decoder is, given a syndrome 𝑆 caused by an error 𝐸 , to apply a correction 𝐶(𝑆) that will fix the error 𝐸 . The difference between decoders is solely contained in the function 𝐶(𝑆)...
• 502
Accepted

### Weight enumerators for Hermitian operator

TL;DR: No. We can actually blow $A_j$ up to infinity while simultaneously sinking $B_j$ negative. Sneaky plan Coefficients $A_j$ cannot be negative, but if $H$ squares to identity and anticommutes ...
• 18.2k
Accepted

### Weight enumerators for Hermitian operator (wrong $B_j$ definition)

TL;DR: No. Suppose that $H=\alpha H'$ for some scale factor $\alpha\in\mathbb{R}$. The key observation is that $A_j$ is independent of $\alpha$, but $B_j$ is linear in $\alpha$. We can use this to ...
• 18.2k

### Is there a way to perform a defect-free logical CNOT on the toric code?

I cannot find any information on this [...] Does lattice surgery provide an option? If you go to google scholar and search "lattice surgery" it brings up the relevant papers. The very first ...
• 28.8k

### How are logical operators performed and measured on the surface code, in a defect-free way?

I am still not sure how to perform [...] $X_L$ or $Z_L$ To perform logical X or logical Z in any code, apply the Pauli operators of the observable. This is always fault tolerant, in any stabilizer ...
• 28.8k

### How to encode an unknown state into stabilizer code?

Basically, think of any circuit between two qubits that you might want to apply. In this case, a SWAP where one of the qubits is prepared in $|0\rangle$. (This is implemented by two controlled-nots, ...
• 51.1k

### How to encode an unknown state into stabilizer code?

Encoding circuits exist for all stabilizer codes. You can find a procedure to construct them in literature [1,2]. I previously outlined this procedure, with a worked out example for the Steane code. [...

### How to correct error during the syndrome measurement or recovery process?

There are two-related concepts that we to understand: "quantum error-correction" and "fault-tolerant quantum computing". Consider the scenario where Alice wants to sends some ...
Accepted

### Is it possible to use quantum state to store and read information without destorying it?

Reading information from a quantum system is possible only via a measurement, which mathematically can be described as applying a Hermitian operator (also called "observable") on the Hilbert ...
• 1,604
Accepted

### Do non-stabilizer codes have integer weight enumerator?

No. Weight enumerators with the given normalization are not necessarily integers for non-stabilizer codes. In fact, the first discovered non-stabilizer code provides a counterexample. However, the ...
• 18.2k