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Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum ...

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Significance of Clifford operations from quantum error correction perspective

In the literature on QECC, Clifford gates occupy an elevated status. Consider the following examples which attest to this: When you study stabilizer codes, you separately study how to perform ...
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Quantum Belief Propagation decoding

I have been reading about a family of quantum error correction codes called Quantum Turbo Codes, which are the quantum analog of the well-known classical Turbo codes. This codes were introduced in ...
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topological error correction concepts

Reading about quantum topological error correction I’ve found some information, but I don’t have already clear the link with the topological concept (trivial and not trivial paths, how they are ...
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What are reliable references on analytical and/or numerical studies of threshold theorems under faulty quantum error correction?

By "faulty", I mean that you can have errors on the ancilla qubits, you can have faulty syndrome extraction, etc.
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Quantum error correction: necessary and sufficient condition

For quantum error correction, the necessary and sufficient condition is given in standard texts as: $\langle \phi| E^{\dagger}_{a} E_{b} |\psi \rangle = C_{ab} \langle \phi|\psi \rangle $ $|\psi\...
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Shor code: phase flip error

For Shor's error correcting code, what is the intuition behind saying that the following circuit corrects the phase flip error? I realize that the circuit is trying to compare phases of the three 3-...
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The Pauli group for $n$-qubits is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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Computing with Logical Qunits

What exactly is a logical (non-physical? error corrected?) qunit? Can quantum systems be built exclusively w/ logical qunits?
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What is a Bacon-Shor code and what is its significance?

I'm at the AQC conference at NASA and everybody seems to suddenly be talking about the Bacon-Shor code but there is no Wikipedia page and the pdf that I gave a link to does not really explain what it ...
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Are all $[[n, k, d]]$ quantum codes equivalent to additive self-orthogonal $GF(4)^n$ classical codes?

Theorem 2 of [1] states: Suppose $C$ is an additive self-orthogonal sub-code of $\textrm{GF}(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C^\perp/C$. ...
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What does “conjugation of coordinates” mean with respect to GF(4) (quantum) codes

In On the classification of all self-dual additive codes over $\textrm{GF}(4)$ of length up to 12 by Danielsen and Parker, they state: Two self-dual additive codes over $\textrm{GF}(4)$, $C$ and $...
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Zero-distance self-dual GF(4) quantum codes and constructing k > 0 codes from them

During a description of zero-dimensional self-dual $\text{GF}(4)$ quantum codes in "On self-dual quantum codes, graphs, and Boolean functions" by L.E. Danielsen, it states: A zero-dimensional ...
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Practical Implementations of QECCs in IBM Q Experience

I am learning how to program the IBM Q Experience quantum computers in order to learn more about how does it work and in order to perform some experiments in it. By doing so I was wondering what are ...
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How to find the fidelity between two state when one is an operator?

I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states Show that the fidelity between the ...
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Reference on MITxPRO Applications of Quantum Computing Professional Certificate Program

Recently I found out the Applications of Quantum Computing Professional Certificate Program that MITxPRO is offering for people interested in quantum computing. I saw that it is consisted of four ...
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Allowed CNOT gates for IBM Q 5 quantum computer

I trying to do some tests in the IBM Q5 computer of IBM quantm experience for some simple error correction protocols, but as I can see, some operations between the qubits are not allowed. For example,...
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Why do we use ancilla qubits for error syndrome measurements?

Consider the measurement of the syndrome for the standard 3-qubit code to correct bit flips: $$ \def\place#1#2#3{\smash{\rlap{\hskip{#1px}\raise{#2px}{#3}}}} \def\hline#1#2#3{\place{#1}{#2}{\rule{#3px}...
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Quantum Channel Models

The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is $\rho\...
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What is 'Surface Code'? (Quantum Error Correction)

I am studying Quantum Computing and Information. I have crossed with the 'Surface Code' phrase but I can't find a brief explanation of what it is and how it works. Hopefully you guys can help me with ...
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Violation of the Quantum Hamming bound

The quantum Hamming bound for a non-degenerate $[[N,k,d]]$ quantum error correction code is defined as: \begin{equation} 2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{...
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Degeneracy of Quantum Error Correction Codes

The feature of quantum error correcting codes called degeneracy is that they can sometimes be used to correct more errors than they can uniquely identify. It seems that codes exhibiting such ...
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Translation of color/toric code to a small network of solid-state spins

Within Quantum Error Correction and stabilizer codes, toric codes/surface codes are very tempting, mainly for their high error threshold. For more background please check up, in our Physics sister (...
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Lower bound for Degenerate Codes?

According to (Macchiavello, Palma, Zeilinger, 2001; pg82) a lower bound of the encoding Hilbert space of a non degenerate code is given by the quantum version of the Hamming bound: $$2^k \sum_{i=0}^t ...
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Intuition for Shor code failure probability

Consider the 9 qubit Shor code. This can detect and correct arbitrary single qubit errors, but if there are 2 or more single qubit errors before a correction round, the correction will fail. (In the ...
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What is the difference between “code space”, “code word” and “stabilizer code”?

I keep reading (e.g. Nielsen and Chuang, 2010; pg456 and 465) the following three phases; "code space", "code word" and "stabilizer code" - but am having a difficult time finding definitions of them ...
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Why is the Pauli group used for stabilizers?

When it comes to error correction, we take our stabilizers to be members of the Pauli group. Why is the Pauli group used for this and not, say, the group of all unitary matrices?
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What does quantum error correction code notation stand for?

I understand the notation for classical error correcting codes. E.g., "Hamming(7,4)" stands for a Hamming code that uses 7 bits to encode blocks of 4 bits. What does the notation for quantum error ...
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Is Gil Kalai's argument against topological quantum computers sound?

In a lecture, recorded on Youtube, Gil Kalai presents a 'deduction' for why topological quantum computers will not work. The interesting part is that he claims this is a stronger argument than the ...
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Why does the surface (quantum error correction) code have such a high threshold for errors?

Is there an intuitive explanation why the surface code fares so much better than older quantum error correction codes in terms of its high error threshold, with thresholds of up to a few percent ...
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Quantum entanglement and quantum error correction

I want to understand what quantum entanglement is and what role does it play in quantum error correction. NOTE: As per the suggestions of @JamesWootton and @NielDeBeaudrap, I have asked a separate ...
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How could Majorana particles be used to improve quantum computers?

This recent press release claiming that Improved measurements bring final proof of Majorana particles closer than ever, which summarizes the results of a recent paper in Nature simply entitled "...
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Why do error correction protocols only work when the error rates are already significantly low to begin with?

Quantum error correction is a fundamental aspect of quantum computation, without which large-scale quantum computations are practically unfeasible. One aspect of fault-tolerant quantum computing that ...
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What is the leading edge technology for creating a quantum computer with the fewest errors?

Which technological path seems most promising to produce a quantum processor with a greater quantum volume (preferring fewer errors per qubit over more qubits), than Majorana fermions? The preferred ...
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How does magic state distillation overhead scale compare to quantum advantages?

I'm interested in the model of quantum computation by magic state injection, that is where we have access to the Clifford gates, a cheap supply of ancilla qubits in the computational basis, and a few ...
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Is error correction necessary?

Why do you need error correction? My understanding is that error correction removes errors from noise, but noise should average itself out. To make clear what I'm asking, why can't you, instead of ...
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Which quantum error correction code has the highest threshold (as proven at the time of writing this)?

Which quantum error correction code currently holds the record in terms of the highest threshold for fault-tolerance? I know that the surface code is pretty good ($\approx10^{-2}$?), but finding exact ...
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What level of “confidence” of the result from a Quantum Computer is possible?

At a very basic level, reading or measuring a qubit forces it to be in one state or the other, so operation of a QC to gain a result collapses the state into one of many possibilities. But as the ...