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Questions tagged [stabilizer-code]

A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors.

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Why is the $N$-qubit stabilizer group abelian?

In Devitt et al. 2013's introduction to quantum error correction, the authors mention (bottom of page 12) how the stabilizer group for $N$ qubits is abelian. More specifically, here is the quote: ...
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Connection between stabilizer generators and parity check matrices in the Steane code

I'm working through Mike and Ike (Nielsen and Chuang) for self-study, and I'm reading about stabilizer codes in Chapter 10. I'm an electrical engineer with somewhat of a background in classical ...
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Compact way of describing the set of all stabilizer groups for fixed number of physical qubits and encoded logical qubits

Fix $n$, the number of qubits and $k$, the number of encoded logical qubits. We can find a set of $(n-k)$ operators that all mutually commute and moreover form a group $S$. Let's assume that the group ...
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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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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 ...
7
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1answer
61 views

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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2answers
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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,...
6
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1answer
82 views

Maximum number of Stabilizer Generators?

The Pauli group, $P_n$, is given by $$P_n=\{ \pm 1, \pm i\}\otimes \{ I,\sigma_x,\sigma_y,\sigma_z\}^{\otimes n}$$ Abelian subgroups of this which do not contain the element $(-1)*I$ correspond to a ...
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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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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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367 views

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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466 views

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?