Questions tagged [error-correction]

Quantum error correction (QEC) is a collection of techniques to protect quantum information from decoherence and other quantum noise, to realise fault-tolerant quantum computation. Quantum error correction is expected to be essential for practical quantum computation in the face of noise on stored quantum information, faulty quantum gates, faulty state preparations, and faulty measurements. (Wikipedia)

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Transversal logical gate for Stabilizer (or at least Steane code)

I know that for Steane code, we can implement transversally some gates like cNOT, Hadamard and Pauli. What I am looking for is a resource in which it is explained why implementing those gate give rise ...
Marco Fellous-Asiani's user avatar
17 votes
1 answer
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What is the Helstrom measurement?

I have been reading the paper Belief propagation decoding of quantum channels by passing quantum messages by Joseph Renes for decoding Classical-Quantum channels and I have crossed with the concept of ...
Josu Etxezarreta Martinez's user avatar
2 votes
4 answers
1k views

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

I have referred this same question here 'Circuit for implementing Steane's code for Quantum Error Correction' . But the answer discusses the circuit to compute the syndrome and not clearly the ...
chetan waghmare's user avatar
20 votes
1 answer
4k views

Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
Josu Etxezarreta Martinez's user avatar
20 votes
1 answer
6k views

Twirling Quantum Channels: Pauli and Clifford Twirling

I am currently working through some papers related with approximations of more general quantum channels such as amplitude and phase damping channels to Pauli channels. The reason to do so is so that ...
Josu Etxezarreta Martinez's user avatar
5 votes
1 answer
754 views

How to compute the error threshold for the $9$-qubit Shor code?

I was trying to calculate the threshold of $9$-qubit Shor's code. The error channel is $$E=(1-p)I+p/3X+p/3Y+p/3Z.$$ But I got the threshold is equal to 1. How can I get the right threshold (I believe ...
QuantumDots's user avatar
3 votes
1 answer
75 views

Do the coefficients of the weight enumerator polynomial add up to $2^{n-k}$ for any $[\![n,k]\!]$ code?

Let $ C $ be an $ [\![n,k]\!] $ quantum error correcting code encoding $ k $ logical qubits into $ n $ physical qubits. Define the weight enumerator polynomial $ A(x) $ of the code as $$ A(x):=A_0+...
Ian Gershon Teixeira's user avatar
15 votes
1 answer
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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 ...
Discrete lizard's user avatar
9 votes
3 answers
1k views

CSS Code in disguise

Suppose I have a quantum error correcting code described by a set of stabilizers. Is there any easy way of checking whether or not it is equivalent to a code constructed using the CSS construction, ...
guest-20's user avatar
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How to implement if statement based on measurement results in qiskit?

I tried to implement three qubit bit flip code in qiskit and need to get the result of measurements and then apply recovery quantum operations conditioned on the measurement results. The following is ...
Jacey Li's user avatar
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Why is the action of controlled-Z unaltered by exchanging target control qubits?

In the book "Quantum Computer Science", when explaining the error correction code, it uses this picture and says "the action of controlled-z is unaltered by exchanging the target and ...
peachnuts's user avatar
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2 votes
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Are the coefficients of the weight enumerator polynomial of a stabilizer code always integers?

Consider an $ [\![n,k]\!] $ stabilizer code. Define the weight enumerator polynomial $ A(x) $ of the code as $$ A(x):=A_0+A_1x+\dots+A_nx^n $$ where $$ A_j:=\frac{1}{(2^k)^2} \sum_{p \in P_n,\,\mathrm{...
Ian Gershon Teixeira's user avatar
1 vote
2 answers
524 views

How do we create logical qubits in the surface code: understanding check

I am learning the basics of surface code theory through this paper I am trying to understand how we create multiple logical qubits. The goal of my post is to check my understanding. What I understood ...
Marco Fellous-Asiani's user avatar
28 votes
3 answers
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What are magic states?

I wonder what are magic states, and a magic state gadget. While I'm reading a paper, these terms frequently appear.
김동민's user avatar
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27 votes
4 answers
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Why can't there be an error correcting code with fewer than 5 qubits?

I read about 9-qubit, 7-qubit and 5-qubit error correcting codes lately. But why can there not be a quantum error correcting code with fewer than 5 qubits?
Aleph's user avatar
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13 votes
1 answer
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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 ...
Rob's user avatar
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6 votes
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Example non-stabilizer code?

A code is a non-stabilizer code if it is not equivalent by local unitaries to a stabilizer code. What is an example of a non-stabilizer code with distance $ d \geq 2 $? Are there any non-stabilizer ...
Ian Gershon Teixeira's user avatar
6 votes
1 answer
725 views

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: ...
glS's user avatar
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6 votes
1 answer
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Are applications with only polynomial speedup worth chasing after? (since error correction adds a heavy overhead)

A number of ML algorithms have demonstrated to have polynomial speed-up: But this (I'm assuming) is without error correcting qubits. How practical are algorithms that only exhibit polynomial speed-up ...
Steven Sagona's user avatar
5 votes
2 answers
565 views

Nielsen & Chuang Exercise Question on CSS code

I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that: suppose $C_1$ and $C_2$ are $[n,k_1]$ and $[n,k_2]$classical linear codes such that $C_2\...
Upstart's user avatar
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5 votes
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Quantum circuit for a three-qubit bit-flip code

I know a three-qubit bit-flip code has a common encoding circuit as follows, Further, as in page 35 in Gottesman's paper, the encoding circuit can also be constructed through stabilizer generators. ...
Jacey Li's user avatar
  • 585
4 votes
1 answer
476 views

Necessary and sufficient condition to define logical operation (stabilizer code)

My question is highly related to this topic It is about defining logical operation on a Stabilizer code. I call $S$ the stabilizer group of a code space $C$, and I assumed it is generated by a family $...
Marco Fellous-Asiani's user avatar
4 votes
1 answer
434 views

How the real IBM quantum computers apply arbitrary Rz(θ) gate rotation? [closed]

I want to ask the following questions: (1) The basis gate set of IBM quantum computers is { Id, Rz(θ),√X, X, CNOT, reset}. Somebody said that IBM didn’t really apply Rz(θ) gate on the machine. The ...
劉承瀚's user avatar
3 votes
1 answer
1k views

What is the standard noise channel that is applied in simulations?

I know there are various quantum noise channels, which include the depolarizing channel, the dephasing channel and the bit-flip channel; We can apply them in simulators easily. However, is there any ...
Henry_Fordham's user avatar
3 votes
1 answer
260 views

Stabilizer codes and 1,-1 coefficients

A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and local one which can be written as (a global scalar times) a linear combination of computational ...
Ian Gershon Teixeira's user avatar
3 votes
2 answers
424 views

How to calculate distance of $k=0$ stabilizer code?

This could be seen as a followup to the question "How to calculate the distance of stabilizer code?". Summarizing the accepted answer : distance is the minimum weight of the set $$E = \bigl\{...
unknown's user avatar
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3 votes
2 answers
310 views

Link between distance of a stabilizer code and number of errors it is able to correct

I am confused by a property. In the N&Chuang it is said that an $[n,k,2t+1]$ stabilizer code is able to correct up to $t$ errors. But for me if the code has distance $d$ it should be able to ...
Marco Fellous-Asiani's user avatar
3 votes
0 answers
104 views

Obtaining low weight stabilizer generators

Suppose I know a set of stabilizer generators of a qubit quantum code. Is there a systematic (and possibly efficient) way to transform this set of generators to a different set (generating the same ...
user1677907's user avatar
2 votes
1 answer
273 views

Analytical Approximation of Pseudothreshold for Steane Code

Is there an analytical way by which one can approximately find the pseudothreshold of the Steane code? My supervisor told me that there is a way, using the physical error rate and the number of gates ...
MatthewS1990's user avatar
2 votes
1 answer
387 views

Laflamme's perfect quantum error correcting code (5 qubit) - Not getting the correct syndrome bits

I am trying to implement 5 qubit encoding and decoding circuit which results into generation of syndrome bits that pin points the location of error so that we can correct it afterwards. Please have a ...
Sid's user avatar
  • 23
2 votes
2 answers
630 views

Mitigating the noise in a quantum circuit

I'm using Qiskit and I have a Quantum Circuit (say, circuit) that gives reasonable results when using the simulator, namely ...
Vitality's user avatar
  • 143
2 votes
3 answers
471 views

Is the $[\![7,1,3]\!]$ Steane code a surface code?

All surface codes are CSS codes. The smallest error correcting ($ d=3 $) CSS code is for 7 qubits. So we can be sure that the $ [\![5,1,3]\!] $ code is not equivalent to a surface code since no ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
124 views

enumerators polynomials and distance for a quantum error correcting code

I'm learning more about enumerator polynomials for quantum error correcting codes (QECCs). What is the relationship between enumerator polynomials and the distance $ d $ of an $ [[n,k,d]] $ quantum ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
380 views

how to go from matrix to tableau to circuit in qiskit or stim

I'm working with QECC using a non-python based platform. I'd like to move the results into python to do calculations that are better handled by packages like qiskit or stim. So the output of the non-...
unknown's user avatar
  • 2,122
2 votes
1 answer
132 views

How to read a graph representing the stabilizers and data qubits in surface code

I am trying to understand to properly read the graph below, provided in this post It is said that the black elements are the $Z$ stabilizers and the grey ones are the $X$ stabilizers. I am trying to ...
Marco Fellous-Asiani's user avatar
43 votes
2 answers
18k views

What is the "surface code" in the context of quantum error correction?

I am studying Quantum Computing and Information, and have come across the term "surface code", but I can't find a brief explanation of what it is and how it works. Hopefully you guys can ...
Ivanovitch's user avatar
26 votes
4 answers
2k views

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 ...
auden's user avatar
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24 votes
3 answers
2k views

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 the operation of a quantum computer to gain a result collapses the state into one of many ...
Rory Alsop's user avatar
24 votes
3 answers
965 views

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 ...
M. Stern's user avatar
  • 2,422
18 votes
5 answers
5k views

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 ...
user1271772 No more free time's user avatar
16 votes
1 answer
873 views

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{...
Josu Etxezarreta Martinez's user avatar
14 votes
2 answers
2k views

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 ...
Alexander Pozdneev's user avatar
13 votes
5 answers
719 views

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 ...
glS's user avatar
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12 votes
2 answers
693 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 $C^\perp/C$. ...
SLesslyTall's user avatar
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12 votes
2 answers
1k views

How to calculate the distance of stabilizer code?

How to calculate the distance of the stabilizer code [[n,k,d]]? It's better if you can make a 3-qubit example. And what's the relationship between d and Pauli group?
Zeo's user avatar
  • 151
11 votes
2 answers
777 views

Why can't there be an error detecting code with fewer than 4 qubits?

Essentially this boils down to: Is it possible to encode a single logical qubit in three physical qubits so that the resulting code has distance two? In other words, does a $[\![3,1,2]\!]$ code exist? ...
Ian Gershon Teixeira's user avatar
9 votes
2 answers
2k views

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,...
Josu Etxezarreta Martinez's user avatar
9 votes
2 answers
567 views

How does the size of a toric code torus affect its ability to protect qubits?

The Toric code Hamiltonian is: $\sum_{x,y}\left( \prod_{i\in p(x,y)} Z_{ixy} + \prod_{i\in v(x,y)} X_{ixy} \right),$ where the $v$ and $p$ are defined according to this picture (courtesy of James ...
user1271772 No more free time's user avatar
8 votes
2 answers
748 views

What is (formally) a transversal operator?

This question concerns about a formal definition of transversal operator. I understood that transversal operator are a group of operators which are efficient in terms of circuit depth and can be used ...
Daniele Cuomo's user avatar
8 votes
2 answers
562 views

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 ...
Josu Etxezarreta Martinez's user avatar