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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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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
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
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28 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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26 votes
4 answers
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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
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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
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21 votes
3 answers
4k views

Intuition about Knill-Laflamme QEC conditions

The Knill Laflamme QEC conditions are stated this way: We consider a code space $C$ and its associated projector $P_C$. We consider a noise acting on our system: $E(\rho)=\sum_a E_a \rho E_a^{\...
Marco Fellous-Asiani's user avatar
21 votes
1 answer
1k views

What is the intuition behind quantum t-designs?

I started reading about Randomized Benchmarking (this paper, arxiv version) and came across "unitary 2 design." After some googling, I found that the Clifford group being a unitary 2 design ...
Blackwidow's user avatar
21 votes
1 answer
538 views

How does magic state distillation overhead scale compare to quantum advantages?

I'm interested in the model of quantum computation by magic state injection, that is where we have access to the Clifford gates, a cheap supply of ancilla qubits in the computational basis, and a few ...
Emily Tyhurst'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
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
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
18 votes
3 answers
2k views

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 ...
Travis C Cuvelier'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
17 votes
1 answer
587 views

Reference that explains how to read 3d topological diagrams for surface code computations

I like making diagrams to describe computations. For the surface code, an excellent tool is 3d topological diagrams. Here is an example diagram (made by me in SketchUp): The basic idea is that white ...
Craig Gidney's user avatar
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16 votes
1 answer
958 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
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
14 votes
3 answers
3k views

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

I keep reading (e.g. Nielsen and Chuang, 2010; pg. 456 and 465) the following three phases; "code space", "code word" and "stabilizer code" - but am having a difficult time finding definitions of them ...
Quantum spaghettification's user avatar
14 votes
5 answers
737 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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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
1 answer
950 views

Does the Quantum Fourier Transform require universality?

Background: In most setups of fault-tolerant quantum computation, universality is achieved using Clifford gates such as $(S, H, \text{CNOT})$ and the $T$-gate. The Eastin-Knill theorem can be ...
Frederik Ravn Klausen's user avatar
13 votes
2 answers
979 views

Where do we put error correction code in quantum circuit?

First of all : I am a beginner in quantum computing. I would like to have a resource (or an answer if it is not complicated) explaining where we put the error correction codes in a quantum circuit. ...
Marco Fellous-Asiani's user avatar
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
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13 votes
2 answers
2k views

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 ...
Josu Etxezarreta Martinez's user avatar
13 votes
1 answer
845 views

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 ...
Tanmay Singal's user avatar
12 votes
2 answers
2k 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?
Quantum spaghettification's user avatar
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
12 votes
3 answers
956 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
12 votes
2 answers
738 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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11 votes
5 answers
938 views

What is quantum entanglement, and what role does it play in quantum error correction?

I want to understand what quantum entanglement is and what role does it play in quantum error correction. NOTE: As per the suggestions of @JamesWootton and @NielDeBeaudrap, I have asked a separate ...
Chinni's user avatar
  • 251
11 votes
3 answers
2k views

How is computation done in a 2D surface code array?

In a 2D surface code lattice, there are some data qubits and some measurement qubits. Suppose we want to do a 2-qubit computation, for example, let say, an X-gate on qubit-1 followed by a CNOT gate ...
Edifice's user avatar
  • 429
11 votes
2 answers
3k views

Characteristics of the IBM quantum computer

On the IBM Quantum Composer website, there are characteristics of qubit computers. For example, ibmq_16_melbourne. But there is no description anywhere of what: ...
alexhak's user avatar
  • 471
10 votes
1 answer
3k views

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
10 votes
2 answers
700 views

How could Majorana particles be used to improve quantum computers?

This recent press release claiming that Improved measurements bring final proof of Majorana particles closer than ever, which summarizes the results of a recent paper in Nature simply entitled "...
Discrete lizard's user avatar
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
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10 votes
2 answers
476 views

If Majorana qubits are analogous to surface codes, why do the diagrams use lines instead of squares?

If you go to some random Majorana paper or talk, you will find a diagram like this one. Note that the diagram is using lines. Making the lines longer should exponentially increase the error ...
Craig Gidney's user avatar
  • 39.6k
10 votes
0 answers
102 views

Entanglement-assisted hashing bound for asymmetric depolarizing channels

I reading the paper EXIT-Chart Aided Quantum Code Design Improves the Normalised Throughput of Realistic Quantum Devices, which proposes the use of QTCs in order to do quantum error correction for ...
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
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
9 votes
3 answers
547 views

Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$

My question is the following Let's assume I am using Steane concatenated code to do error correction. I consider that the stabilizers are extracted fault-tolerantly through the Steane method. The ...
Marco Fellous-Asiani'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
951 views

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}...
Quantum spaghettification's user avatar
9 votes
4 answers
3k views

2 ways to do the three qubits bit-flip code

I'm trying to understand the three qubits bit-flip code. I use the book of Phillip Kaye An introduction to quantum computing. In this book he introduce the three qubits bit-flip code with this ...
lufydad's user avatar
  • 481
9 votes
2 answers
726 views

Why does the surface (quantum error correction) code have such a high threshold for errors?

Is there an intuitive explanation why the surface code fares so much better than older quantum error correction codes in terms of its high error threshold, with thresholds of up to a few percent ...
user avatar
9 votes
2 answers
602 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
9 votes
2 answers
198 views

How is the decoherence rate connected to the error rate?

I'm reading about the threshold theorem, which states that "a quantum computer with a physical error rate below a certain threshold can, through the application of quantum error correction schemes, ...
VannyUW's user avatar
  • 91
9 votes
2 answers
343 views

Is there a good reason to use T-count minimization for circuits executed on current IBM open quantum systems (real hardware)?

As far as I understood from a series of papers, minimizing the T-count in Clifford+T circuits is essential for fault-tolerant ...
Psanfi's user avatar
  • 306
8 votes
2 answers
949 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
3 answers
1k views

What is the difference between Error Mitigation (EM) and Quantum Error Correction (QEC)?

From what I understood, EM is mostly post-processing and does not affect quantum circuit complexity whereas QEC has overheads and integrates as part of a quantum circuit. Is this correct? Also, which ...
MonteNero's user avatar
  • 2,778
8 votes
2 answers
969 views

Can you give an intuitive idea behind how the Minimum Weight Perfect Matching (MWPM) decoder work?

The Minimum Weight Perfect Matching (MWPM) decoder seems to be the most popular choice for decoding error syndromes in Surface Code quantum error correction. Can anyone give an intuitive idea of how ...
Aubrey Sharansky's user avatar

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