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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
3k views

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
2k 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
5k 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
9 votes
2 answers
1k views

CSS codes are the only stabilizer codes with transversal CNOT?

Given a stabilizer code $\mathcal{C}$ then $$ \mathcal{C} \text{ is CSS} \iff \text{CNOT} \text{ is transversal}. $$ The forward implication is well known, see for example Transversal logical gate ...
Eric Kubischta's user avatar
3 votes
1 answer
887 views

Example CSS codes and the properties "doubly even" and "self dual"

Steane's $ [[7,1,3]] $ code is a well known CSS code which is self-dual and doubly even. Indeed the entire family of $ [[2^{2m+1}-1,1,2^{m+1}-1]] $ quantum punctured Reed-Muller codes mentioned in ...
Ian Gershon Teixeira'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
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
  • 111
5 votes
1 answer
884 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
79 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
2k views

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
12 votes
3 answers
958 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
5 votes
1 answer
5k views

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
  • 585
4 votes
1 answer
2k views

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
  • 1,393
2 votes
1 answer
162 views

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
2 votes
2 answers
627 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
30 votes
3 answers
10k views

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
  • 937
28 votes
4 answers
4k views

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
  • 551
13 votes
1 answer
608 views

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
  • 2,335
10 votes
3 answers
1k views

Logical qubit initialization for the surface code

I am reading Fowler et al's paper on the surface code.. I do not understand how to initialize a logical qubit in an arbitrary state with the surface code. I do understand how to initialize the qubit ...
Peter-Jan's user avatar
  • 1,604
6 votes
1 answer
770 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
  • 26k
6 votes
2 answers
734 views

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
180 views

Smallest distance 9 self-dual CSS code?

The level-2 concatenated [[7,1,3]] Steane code, and the 4.8.8 color code are both self-dual [[49,1,9]] codes from the CSS family. Is there a distance 9 self-dual CSS code that has less than 49 qubits?
Balint Pato's user avatar
  • 1,063
6 votes
1 answer
477 views

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
1 answer
541 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
5 votes
2 answers
1k views

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
5 votes
2 answers
597 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
  • 1,432
4 votes
2 answers
443 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
  • 2,217
4 votes
2 answers
1k views

What is the pseudo threshold of a QECC using stabilizer formalism

Can someone explain what is the threshold and the pseudo threshold of a Quantum Error Correction Code , for instance the 9-qubit code, and how to calculate it using the stabilizer formalism simulation ...
El-Mo's user avatar
  • 103
4 votes
1 answer
514 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
4 votes
2 answers
424 views

What is the stabilizer group of a $|W\rangle$ state?

This question is related and complementary to this one: How to get the stabilizer group for a given state? What I want is to find the stabilizer group generators for the following state: $$|W\rangle ...
Dani's user avatar
  • 277
3 votes
0 answers
117 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
3 votes
2 answers
361 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
1 answer
270 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
1 answer
162 views

how to construct a ((7,2,3)) code and verify its properties

These two recent papers describe a process to construct (non-stabilizer) codes with "exotic" transversal gates : paper1 paper2 Most of the codes have distance $d=2$ which makes them less ...
unknown's user avatar
  • 2,217
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
2 answers
270 views

On the photonic implementation of Shor's code

The above picture comes from this paper. I can see that the standard Shor's code has been re-designed. I have two main doubts: I can't figure out in figure (b) how the setting inputs the state $\...
Daniele Cuomo's user avatar
2 votes
1 answer
305 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
2 answers
656 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
1 answer
437 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
3 answers
588 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
130 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
443 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
  • 2,217
1 vote
1 answer
379 views

Support for Qubit Leakage Errors in Stim

If I were interested in modeling qubit leakage errors using stim, how should I go about this? There are two scenarios of interest: Any leaked population can be treated as lost, until I reset the ...
shixian105's user avatar
2 votes
1 answer
146 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
19k 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
  • 3,459
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
1k 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
  • 2,457
19 votes
5 answers
6k 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