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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Is the Steane code the only $ [[7,1,3]] $ CSS code?

Is the Steane code the only $ [[7,1,3]] $ CSS code? This paper https://arxiv.org/abs/0709.1780 claims there are 10 non-equivalent $ [[7,1,3]] $ stabilizer codes. How many of these are CSS codes? Is it ...
Ian Gershon Teixeira's user avatar
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Why is it not a good idea to run Shor's algorithm on a big noisy quantum computer?

Suppose I try to run Shor's algorithm on a big noisy quantum computer. For a given input I repeat the circuit many times and collect statistics for the measurements of each qubit. Now based on the ...
Frederik Ravn Klausen's user avatar
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Error correction conditions for bit flip code

I am reading Nielsen and Chuang's chapter on error correction. I have a very basic question about this theorem: Theorem 10.8 (Error-correction conditions for stabilizer codes): Let $S$ be the ...
William Schober's user avatar
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What kind of quantum error is most likely to happen and are considered most in quantum error correction codes?

In quantum error correction literature, it is assumed that the possibility of happening a few-qubit error is bigger than a many-qubit error. Does this assumption have any physical evidence? I am ...
qdmj's user avatar
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How is coherent noise more detrimental than incoherent noise for QEC when QEC is supposed to decohere noise?

It's sometimes mentioned that unitary, coherent, noise can adversely impact the performance of QEC codes compared to purely stochastic Pauli noise. This motivates protocols such as randomized ...
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Ancilla qubit error spreading to multiple data qubits?

In this paper, the author considers a four qubit stabilizer circuit shown below. Note that the first four qubits from the top are data qubits and can be in any state (here we just put them in $\vert ...
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Decoding the Steane Code

The $[[5,1,3]]$ code is a perfect code basically meaning that the weight-0 and weight-1 error spaces completely fill out the $32$-dimensional Hilbert space. On the other hand, the $[[7,1,3]]$ Steane ...
Eric Kubischta's user avatar
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How can fault tolerance be understood from the perspective of detecting regions in a circuit?

The language of detectors and detecting regions in the context of quantum error correction is introduced in https://arxiv.org/pdf/2302.02192.pdf. Detecting regions seem to be capable of specifying ...
Ezequiel Rodriguez Chiacchio's user avatar
3 votes
2 answers
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How to calculate pseudo-threshold of the [[5,1,3]] code analytically to match simulation?

In this answer there is a simple method given to compute the "worst case pseudo-threshold" of the Shor code. I want to apply this to the $[[5,1,3]]$ perfect code. Namely, given a ...
Eric Kubischta's user avatar
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I am getting an error on STIM: An error case in a composite error exceeded that max supported number of symptoms (<=15)

I tried putting some ELSE_CORRELATED_ERROR on my custom made surface code circuit and I am getting this error while counting logical errors. What does it mean? And how can I fix it?
Omprakash Chandra's user avatar
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How does syndrome extraction in bit-flip codes using stabilizers work?

I'm trying to understand how syndrome extraction in bit-flip code works. I've been reading Quantum Error Correction: An Introductory Guide, and I was wondering how the syndrome extraction part in this ...
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Hilbert space of n qubits always has a decoherence free subspace if n is even?

I've heard that a subspace of an $ n $ qubit Hilbert space is called decoherence-free if it is not affected by errors of the form $ g^{\otimes n} $ where $ g \in SU(2) $. Let $ \textbf{2} $ denote the ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
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Circumventing the Eastin-Knill theorem by means of Shor's fault tolerant Toffoli gate

I have been reading about the Eastin-Knill theorem, which states that no quantum error correction code can can transversely implement a universal gate set. In this context, for example, the surface ...
Josu Etxezarreta Martinez's user avatar
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Is there a no-go theorem for or an upper bound on code threshold?

My understanding of the code threshold for an $ [[n,1,3]] $ code is something roughly like the setup given in How to compute the error threshold for the $9$-qubit Shor code? where the error channel is ...
Ian Gershon Teixeira's user avatar
4 votes
2 answers
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Codes saturating the bound d=(n+1)/2

An $ n $ qudit code always has distance bounded above by $ d \leq \frac{n+1}{2} $. The $ 5 $ qudit generalization of the Knill-Laflamme code with parameters $ [[5,1,3]]_q $ ( $ q $ denoting the size ...
Ian Gershon Teixeira's user avatar
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How does one create a long range CNOT gate on a square grid of qubits using constant depth circuits?

In this talk, the speaker shows a bunch of data qubits on the top and ancilla qubits at the bottom that need to be linked by CNOT gates. The grid here is $n\times n$. The procedure is to swap the ...
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Measuring observables in the $[\![5,3,1]\!]$ code with stim

Background This question was triggered when I wanted to learn about stim and how to use it to run simulations for error correction. The resources I used for learning about stim were basically Craig ...
Vincent's user avatar
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Can Wiesner's quantum money be realized (with logical qubits) today?

Consider Wiesner's quantum money scheme. With today's devices and today's error correction and mitigation schemes, how long can we hold $n$ logical qubits such that they are all (logically) in a ...
Mark Spinelli's user avatar
5 votes
1 answer
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What is the meaning of "error threshold" in quantum error correction?

I am having trouble understanding what people mean when they say "error threshold." I understand the answer here but it isn't code specific, it is only dependent on the number of physical ...
Eric Kubischta's user avatar
3 votes
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Why is the first logical unitary gate in this example fault tolerant?

From Arthur Pesah's blog on "Computing with Quantum Codes using Transversal Gates", found here: https://arthurpesah.me/blog/2023-12-25-transversal-gates/ He gives the following examples of ...
am567's user avatar
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What is the best way to parallelise processes across multiple cores when computing logical error rates with Stim and PyMatching?

I am currently trying to compute logical error rates for the surface code using Stim's detector error models and PyMatching for different distances and noise strengths. tl;dr : What is the best ...
Ezequiel Rodriguez Chiacchio's user avatar
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1 answer
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Why minimum weight in the minimum weight perfect matching?

I am studying quantum decoding, especially for minimum weight perfect matching. I wonder after finding perfect matching (syndrome graph referred by sparse Blossom), why do we need to select minimum ...
김동민's user avatar
3 votes
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Recent hardware advances towards fault-tolerant quantum computing and quantum error correction

Please excuse the broad nature of the question. I am interested in understanding recent developments by various hardware companies and their approaches towards quantum error correction. My knowledge ...
user1936752's user avatar
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How to track back which error occurred during DEPOLARIZE mechanism in a stim circuit

Suppose I create a circuit, which applies depolarizing error to a single qubit with probability 0.04. ...
Mainak Bhattacharyya's user avatar
5 votes
1 answer
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What are the properties of stabilizer codes that ensure $C(S) = N(S)$?

In Daniel Gottesman's thesis "Stabilizer Codes and Quantum Error Correction", he makes the following claim: The set of elements in $G$ that commute with all of $S$ is defined as the ...
am567's user avatar
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Why can a quantum code correct $t$ errors only when $d \geq 2t + 1$?

I am working from chapter 7 from notes for ph229 by J. Preskill. The notes define the distance of a quantum code as: The distance $d$ is the the minimum weight of a Pauli operator $E$ such that: $$\...
am567's user avatar
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A better name for "weakly self-dual CSS codes"

Does anyone know a better name than "weakly self-dual"/"self-orthogonal" for CSS codes where $H_X=H_Z$, for example the Steane code, and the color codes? Details In the discussion ...
Balint Pato's user avatar
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Lattice surgery and real-time decoding

This question pertains to the real-time classical processing required while running the surface code. Consider two logical patches $P$, $P'$. Let's assume that after $d$ rounds, a ...
control freak's user avatar
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Simulating any stabilizer code

While simulating any Stabilizer code on STIM do we need an explicit encoding circuit before performing circuit-level noise simulation and error-correction ?
Omprakash Chandra's user avatar
2 votes
2 answers
110 views

Implementing error correction (from physical qubit point of view)

I have worked with physical qubits, and I am fairly familiar with gates and sequences from initialisation to readout. I don't know much about error correction, and I love to learn how error correction ...
Rex's user avatar
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2 answers
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How to force PyMatching into the opposite equivalence class

PyMatching finds the minimum weight matching, which will belong to a particular equivalence class. But sometimes it is useful to know the minimum matching of each equivalence class. In the olden days, ...
James Wootton's user avatar
3 votes
2 answers
45 views

Equivalent definition of distance for nondegenerate code

Let $ \mathcal{C} $ be a nondegenerate quantum code. Is it true that $ \mathcal{C} $ has distance $ d $ if and only if $ d $ is the minimum nonzero weight of an error that preserves the codespace? For ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
39 views

Knill-Laflamme error correction conditions for codespace of dimension 1

The Knill-Laflamme theorem for the conditions of quantum error correction can be stated as follows: Let $ \mathcal{C} $ be a quantum error correcting code defined as a subspace of the $ n $ qubit ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
37 views

Are QC-codes really quasi-cyclic?

The definition of quasi-cyclic codes is that there is a constant $l$ such that a codeword right-shifted by $l$ positions is still a valid codeword. (defined e.g. in https://errorcorrectionzoo.org/c/...
SlowerPhoton's user avatar
1 vote
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LDPC in QKD difference from the classical context

The usage of LDPC codes for error correction in Quantum Key Distribution (QKD) is very different from the classical use. Say you have a word $\vec w$ that you want to transfer. In the classical sense, ...
SlowerPhoton's user avatar
1 vote
1 answer
60 views

What is the purpose of the "SHIFT_COORDS" command in Stim?

I have been looking at several circuits being generated by Stim in the literature, e.g. in https://arxiv.org/pdf/2302.02192.pdf, and I have stumbled upon the operation ...
Ezequiel Rodriguez Chiacchio's user avatar
3 votes
3 answers
931 views

Why do stabilizer cut the Hilbert space into two halves?

I am trying to follow the logic of Slide 8 in this deck. The result is that if you have $n-k$ stabilizers in the set of stabilizers, then the dimension of the +1 eigenspace of all the stabilizers is $...
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3D toric code (2D + measurement errors) has higher threshold error rate $p_{\text{th}}$ than 2D?

I am doing simulations of the toric code using the statistical mapping worked out by Preskill et al., Topological Quantum Memory, [arXiv:quant-ph/0110143], where we find the phase boundary of an Ising ...
JoJo P's user avatar
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5 votes
1 answer
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What are the limitations of Harvard's and Google experiment on quantum error-correction

I am currently reading the papers about the QEC (quantum error-correction) experiments of 2023 and while they are impressive (!), I would like to better understand what are their limitations. My goal ...
Marco Fellous-Asiani's user avatar
1 vote
0 answers
44 views

For a degenerate quantum error correcting code, what can we conclude about product of two distinct correctable errors with same syndrome?

It can be seen in this answer that for a stabilizer code, the two distinct correctable errors with the same syndrome acting together is always a stabilizer. However, what can we conclude in the case ...
FDGod's user avatar
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For an $[\![n,k,d]\!]$ degenerate stabilizer code, can product of two distinct correctable errors with same syndrome have weight less than $d$?

For an $[\![n,k,d]\!]$ degenerate stabilizer code, let's say there exist two distinct correctable errors, $E_a$ and $E_b$, which have the same syndrome. Could there exist a case such that $$ \text{...
FDGod's user avatar
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2 votes
1 answer
115 views

Why is $\text{dim}_{\mathbb{C}}Q(\chi) = 2^{n-(n-k)}$?

I am confused about a point in the proof of Theorem 27.3.6 which claims that $\text{dim}_{\mathbb{C}}Q(\chi) = 2^{n-(n-k)}$ in a "Concise Encyclopedia of Coding Theory" (page 663/664). The ...
am567's user avatar
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0 votes
2 answers
112 views

Integrating BPOSD into STIM

Pymatching doesn't seems to be working for Steane's code on STIM. I am wondering is there any other decoder that I could possibly use? I am trying to simulate circuit-level noise in syndrome ...
Omprakash Chandra's user avatar
1 vote
1 answer
51 views

How to find projection operators for spectral decomposition

I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information". The idea ...
am567's user avatar
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40 views

Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$

Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory". The exercise states: We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We choose the ordering $(...
am567's user avatar
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3D surface code and unionfind

I was reading some papers about the UnionFind Decoder but I don't get how matching can be used to decode on a 3D surface code. If I understood correctly every layer in our 3D surface code corresponds ...
Feder's user avatar
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1 vote
1 answer
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Obtaining both $X$ and $Z$ observables for a surface code

I am trying to reproduce the dataset in this paper [arXiv:1705.00857] using Stim. The paper decomposes the errors into a "pure-error", which is any error that generates the observed syndrome,...
lbello's user avatar
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2 votes
1 answer
108 views

Explicit examples of non-stabilizer (non-additive) quantum codes with recovery procedure

I am looking for resources that give an explicit example of a non-stabilizer code along with an explicit recovery procedure, i.e., syndrome measurement followed by correction operation for that code. ...
FDGod's user avatar
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3 votes
1 answer
116 views

How to devise a flag qubit scheme for concatenated codes?

This flag qubit idea was introduced by Rui Chao and Ben Reichardt, last year when I read the paper [arXiv:1705.02329] I thought I understood fully. But today as I revisit I found myself still a bit ...
AndyLiuin's user avatar
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1 vote
1 answer
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On DEMs in stim

I was looking into the stim library (very new to it) and was wondering if someone who has been using it or has experience with DEMs could explain a little of the following. What exactly are detector ...
Arjo's user avatar
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