14 votes

What is a Bacon-Shor code and what is its significance?

The key difference is that the Bacon-Shor code is a subsystem code, while the Shor code is a stabilizer code. They have the same stabilizer operators, but the error correction procedure is different. ...
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11 votes
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What is the difference between "code space", "code word" and "stabilizer code"?

Code spaces and code-words A quantum error correcting code is often identified with the code-space (Nielsen & Chuang certainly seem to do so). The code space $\mathcal C$ of e.g. an $n$-...
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9 votes
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Why is the Pauli group used for stabilizers?

There are some fairly simple reasons — beyond the merely historical — to use Pauli matrices instead of arbitrary unitary matrices. These reasons may not uniquely single out the Pauli group ...
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8 votes
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Connection between stabilizer generators and parity check matrices in the Steane code

There are a few conventions and intuition here, which perhaps it would help to have spelled out — $\def\ket#1{\lvert#1\rangle}\def\bra#1{\!\langle#1\rvert}$ Sign bits versus {0,1} bits The ...
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8 votes

How to calculate the distance of stabilizer code?

There are various ways that you might go about computing the distance. I'll give a fairly general strategy here, although I'm sure here are imprvements that can be made. Your starting point is a set ...
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8 votes
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Allowed CNOT gates for IBM Q 5 quantum computer

Yes, the physical implementation is the constraint. If you look at the image of the processor you'll notice the connections between qubits. This gives you an idea of how you can perform two qubit ...
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  • 1,669
7 votes
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Why do we use ancilla qubits for error syndrome measurements?

The key point of quantum error correction is precisely to correct the errors without collapsing the qubits, right? If we measure the encoded qubits we project the qubits to $\left|0\right>$ or $\...
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  • 3,587
7 votes
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How to find the stabilizer generators for a post-measurement state?

Yes, there exists a relatively straightforward algorithm for finding the stabilizer generators of the post-measurement state. TL;DR: Instead of "forgetting" about the stabilizer generators ...
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6 votes

Allowed CNOT gates for IBM Q 5 quantum computer

The five qubit IBM devices have a ‘bow tie’ architecture, which mean that it is only possible to interact certain pairs of qubits. These are shown in the answer of Andrew O. The interaction that can ...
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6 votes

Why do we use ancilla qubits for error syndrome measurements?

When you say "why not just measure the 3 encoded qubits directly", are you thinking that you could measure $Z_1$, $Z_2$ and $Z_3$, and that, from there, you can calculate the values $Z_1Z_2$ and $...
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6 votes
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Why is the $N$-qubit stabilizer group abelian?

It is not necessary to define the group as commuting —$\def\ket#1{\lvert#1\rangle}$ by virtue of every element in the group stabilising the state $\ket{\psi}$, this property follows. Because we ...
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6 votes
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Why do stabilizer codes in Nielsen and Chuang have Pauli X and Z matrices?

Any matrix can be decomposed into a sum of tensor products where each term is of the form $X^a Z^b$ (where $a$ and $b$ are bits; they can be 0 or 1). For example, the 16 matrices of the form $(X^{a_1} ...
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6 votes
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What is the Stabilizer of a Code?

A group action of a group $G$ on a set $X$ is a map $\phi:\, G\times X \rightarrow X$ such that $$ \phi(e,x) = x, \quad\text{and}\quad \phi(g, \phi(h,x) ) = \phi(gh,x), $$ for all $x\in X$ and $g,h\...
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5 votes

Compact way of describing the set of all stabilizer groups for fixed number of physical qubits and encoded logical qubits

There's good news and bad news. The good news is that your intuitions are essentially right, and that there is such a group action via the Clifford group. The bad news is, depending on what you want ...
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5 votes

What is the difference between "code space", "code word" and "stabilizer code"?

In a quantum error correcting code, you store a number of logical qubits, $k$, in a state of many physical qubits, $n$. A code word is a state of the physical qubits that is associated with a ...
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5 votes

What is the difference between "code space", "code word" and "stabilizer code"?

A code word (for a quantum code) is a quantum state that is typically associated with a state in the logical basis. So, you’ll have some state $|\psi_0\rangle$ that corresponds to the 0 state of the ...
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5 votes

Why is the Pauli group used for stabilizers?

Any operator from the Pauli group has two eigenspaces of equal size. So we known that by adding stabilizer generator from this group, we reduce the size of the stabilizer space by half. This means ...
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5 votes
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A question from Aaronson 2004 paper

The terms $II$ and $ZZ$ do not uniquely specify the state $|11\rangle$ because you could equally have the state $|00\rangle$. Indeed, you should not include the identity term in your stabilizer. Thus, ...
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5 votes

What is the motivation for using dual lattice in the surface code?

Primal and dual lattice We do not need to use the dual lattice. The observation that the primal lattice is sufficient to describe both the $X$- and $Z$-type stabilizer generators is correct. To that ...
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5 votes
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Stabilizer codes to Classical codes over GF(4)

Stabiliser codes map to linear subspaces of $\mathbb F_2^{2n}$ which are isotropic w.r.t. the standard symplectic form on this vector space. This is very useful for analysing these codes directly and ...
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5 votes
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Why do we require that the elements of a stabilizer group commute?

We do not require stabilizers to commute. We require them to jointly stabilize a non-trivial subspace. As a consequence, they commute. Suppose $P$ and $Q$ are anti-commuting $n$-qubit Pauli operators ...
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5 votes

Are ancillary qubits necessary to represent a syndrome measurement in the circuit model?

You can create circuits that perform the stabilizer measurements without using ancilla qubits. For example, you can take a data-qubit-only surface code and do a pattern of CNOT gates that temporarily ...
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5 votes

How to find the stabilizer generators for a post-measurement state?

Let $n$ be the observable you measured, and let $S$ be the set of stabilizer generators for the state before the measurement. Some of the old stabilizers generators anti-commute with the new ...
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5 votes
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Why Is This the Ground State of a Toric Code?

The crucial point it seems you are missing is to recognize that $A_s^2=1$ (for that matter, $B_p^2=1$ also) and therefore $(1+A_s)/2$ is a projector onto the $+1$ eigenspace of $A_s$. We would then ...
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  • 353
4 votes
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Maximum number of Stabilizer Generators?

Consider a subgroup $G $ of the Pauli group with at least one operator that acts non-trivially on some qubit. Given any qubit $j $, for which the group contains an operator $S_j $ which acts on $j $ ...
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4 votes

Degeneracy of Quantum Error Correction Codes

I don't have a complete answer, but perhaps others can improve on this starting point. There are probably 3 things to ask about the code: How degenerate is it? How hard is it to perform the ...
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4 votes
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What does "conjugation of coordinates" mean with respect to GF(4) (quantum) codes

I am confused what "conjugation of coordinates" means in this context. Conjugating coordinates of $\mathcal C$ is equivalent to setting some diagonal elements of Γ to 1. Read "Theorem 12, on page 8 ...
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4 votes
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What is a Bacon-Shor code and what is its significance?

Bacon Shor code is still very hot today! Assume you have some familiarity with Shor's code, here I just highlight the difference between these two. The Shor's code has distance 3 for both X and Z type ...
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4 votes
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Exercise 10.34 of Nielsen and Chuang : $-I$ not element of stabilizer group iff $g_j^2=I$ and $g_j \neq I$ where $g_j$ generators

Consulting my local copy of Nielsen & Chuang (10th anniversary edition, p. 457), the complete statement of the exercise is pretty much exactly as you have given it: Exercise 10.34. Let $\...
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4 votes
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All unitary errors are correctable

If you only have one $E_k$ (i.e., $k=1$ can only take one value), and this $E_k=U$ is unitary, then - as you point out in the comments - the first condition is always satisfied, and the error can be ...
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