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 ...
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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.
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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^{\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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♦
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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 ...
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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.
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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?
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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 ...
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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 ...
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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 ...
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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:
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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?
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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?
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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 $C^\perp/C$. ...
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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 ...
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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 ...
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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 ...
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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 "...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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,...
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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}...
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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 ...
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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 ...
user1039
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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 ...
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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, ...
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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 ...
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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 ...
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How to find the stabilizer generators for a post-measurement state?
My question is closely related to this one.
A bit of vocabulary and a reminder of basic properties:
I consider the total Hilbert space of the problem has dimension $2^n$.
I call a "well defined ...
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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 ...
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