New answers tagged error-correction
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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 ...
2
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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 ...
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3
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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 ...
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1
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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 (...
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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.
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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
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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, ...
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4
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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, ...
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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 ...
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2
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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
2
votes
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 ...
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2
votes
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
2
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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
2
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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 ...
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0
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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
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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.
[...
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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 ...
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2
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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 ...
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3
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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 ...
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2
votes
Accepted
For an error correction code $C$, does $\langle\psi| E|\phi\rangle/\langle\psi|\phi\rangle$ being constant imply the code has distance $2$?
No, it doesn't. Any code of distance $d>2$ is a counterexample since in any such code a single-qubit Pauli error sends every codeword to a subspace orthogonal to the code subspace, so $\langle\psi|...
- 18.2k
0
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Is the Eastin-Knill Theorem incorrect?
$ \mathcal{G} $ is a closed subgroup of $ \mathcal{T} $, which is a closed subset of a unitary group. Unitary groups are compact. Closed subsets of compact spaces are compact. Thus $ \mathcal{G} $ is ...
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1
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Bounds on success probability of an algorithm under depolarizing noise
It sounds like you're trying to make an exact estimate. I recommend saving yourself a lot of time and instead doing an approximate estimate. The approximation error will inevitably be less than the ...
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3
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Accepted
Equivalence between Quantum Error Correcting codes and uniqueness of the $[\![5,1,3]\!]$ code
TL;DR: 1. The code is unique under the stated equivalence. 2. No.
Weight of a logical operator is ill-defined
The crux of the issue is that the weight of a logical operator is ill-defined. Logical ...
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