18 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. ...
Simon Burton's user avatar
13 votes
Accepted

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 $n$-...
Niel de Beaudrap's user avatar
13 votes
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Transversal logical gate for Stabilizer (or at least Steane code)

Let $\mathcal{H}$ be the Hilbert space of a set of physical qubits and let $S$ be the stabilizer group of a stabilizer code $\mathcal{G} \subset \mathcal{H}$. A transversal operator $U$ on $\mathcal{H}...
Adam Zalcman's user avatar
  • 21.7k
11 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 ...
Niel de Beaudrap's user avatar
10 votes
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CSS codes are the only stabilizer codes with transversal CNOT?

TL;DR: There are a few inequivalent ways to define the transversal construction for a logical gate. The precise statement of the relationship between transversal CNOT and the CSS codes depends on the ...
Adam Zalcman's user avatar
  • 21.7k
9 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 ...
Niel de Beaudrap's user avatar
9 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 ...
DaftWullie's user avatar
9 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} ...
Craig Gidney's user avatar
  • 35.1k
9 votes
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What are reviews on the search for good quantum LDPC codes?

I'll try to answer your questions by giving the main motivations for implementing ldpc codes and the main practical challenges, along with fundamental papers. I only focus on quantum memory and ignore ...
Peter-Jan's user avatar
  • 1,379
8 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 ...
Niel de Beaudrap's user avatar
8 votes
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What is a Bacon-Shor code and what is its significance?

I'll add my own two cents. Personally, I find it best to understand the Bacon-Shor code in terms of how one would decode the error syndromes. Motivating example: Shor code Since OP seems to already ...
Imperishable Night's user avatar
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 ...
Andrew O's user avatar
  • 1,729
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 $\...
agaitaarino's user avatar
  • 3,807
7 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 ...
Markus Heinrich's user avatar
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 ...
Adam Zalcman's user avatar
  • 21.7k
7 votes

How to implement the Circuit of Steane's code for Quantum Error Correction?

I will do some of the work using a small python library I am writing to play with quantum stabilizer codes. It's called stac. Let's load up the Steane code. ...
Abdullah Khalid's user avatar
7 votes

CSS codes are the only stabilizer codes with transversal CNOT?

The XZZX surface code isn't a CSS code but has a transversal CNOT. The main caveat, which I haven't checked if it happens or not, is that the physical gates may include CZs and XCXs in addition to ...
Craig Gidney's user avatar
  • 35.1k
7 votes
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What does "lift" mean in the Lifted Product (LP) Code?

An $\ell$-lift is just a shorter term for an $\ell$-fold covering graph, a very general idea from graph theory and topology to obtain $\ell$ times larger graph $X^\alpha$ out of a small base graph $X$ ...
Pavel Panteleev's user avatar
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 $...
DaftWullie's user avatar
6 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 ...
DaftWullie's user avatar
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 ...
James Wootton's user avatar
6 votes
Accepted

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 $\...
Niel de Beaudrap's user avatar
6 votes

CSS Code in disguise

Yes. We will define a procedure for checking whether a given stabilizer code $\mathcal{G}$ is a Calderbank-Shor-Steane (CSS) code using the following Theorem. Stabilizer code $\mathcal{G}$ is a CSS ...
Adam Zalcman's user avatar
  • 21.7k
6 votes

CSS Code in disguise

If it were me, I'd write out the $k$ $N$-qubit stabilizer generators in a $k\times 2N$ binary matrix, where each row corresponds to a stabilizer generator, with the first $N$ bits being the positions ...
DaftWullie's user avatar
6 votes
Accepted

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 ...
Adam Zalcman's user avatar
  • 21.7k
6 votes

Why does fault-tolerant transversal phase gate $P$ only work with doubly-even codes?

It might help to think about what the logical states actually look like. Up to normalisation, the logical 0 is $$ |0\rangle_L=\left(\prod_x(I+g_x)\right)|00\ldots 0\rangle. $$ Now, if you want a ...
DaftWullie's user avatar
6 votes
Accepted

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\...
Markus Heinrich's user avatar
6 votes
Accepted

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 ...
nervxxx's user avatar
  • 520
6 votes

What is the smallest code for a single logical qubit

The [[5,1,3]] code that you mention is the smallest possible distance 3 quantum error correcting code. It is known as the "perfect" quantum code because it exactly saturates one of the size ...
DaftWullie's user avatar
6 votes
Accepted

What are the transversal gates of Shor's $[\![9,1,3]\!]$ code?

Superpositions are too entangled The logical computational basis of Shor's $9$-qubit code is $$ \begin{align} |0\rangle_L&=\frac{(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)(|000\rangle+|111\...
Adam Zalcman's user avatar
  • 21.7k

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