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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 ...

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Measuring order ancilla qubits in surface code

Recently I have been reading about surface codes a little and one thing I came acros was a specific order in which gates should be applied before the ancilla is measured. See for instance figure 2 in ...
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Confusion on the definition of the phase-damping channel

I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action ...
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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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Stabilizer for quantum error correction code

I have some very basic questions about stabilizers. What I understood: To describe a state $|\psi \rangle$ that lives in an $n$-qubit Hilbert space, we can either give the wavefunction (so the ...
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What is the longest quantum circuit?

To date, what is the longest quantum computation ever performed? Length is measured in number of operations. EDIT --- I'm looking for a quantum computation with a clear ending and a clear output. ...
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684 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. ...
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Why does this error correcting code not work?

I was thinking of an error correcting code to correct 1-qubit errors. I came up with the following, which I guess has to have a mistake somewhere, but I am not able to find it. The code is the same ...
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63 views

Very basic question about surface codes

I'm currently reading the paper "Surface codes: Towards practical large scale quantum computing" and have a couple of very basic questions that if answered will help me contextualize and organize the ...
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48 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: ...
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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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3answers
92 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 ...
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1answer
40 views

Collective measurements: importance and realization

I am reading the paper Polar codes for classical-quantum channels by Wilde and Guha, and it is stated the fact that collective measurements are necessary in order to aciheve the Holevo symmetric ...
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EA-Turbo simulation package

I am working with the quantum turbo codes presented in this paper by Wilde, Hsieh and Babar, and it is claimed that a package to simulate such codes is available at ea-turbo. However, the hyperlink to ...
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How to deal with -LookupError: backend “ibmqx4” is not found?

I am using Jupyter notebook to write and running my qiskit codes (python 3.6) and every time I encounter the message: LookupError: backend "ibmqx4" is not found, right now the ibmqx4 computer is not ...
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Compact way of describing the set of all stabilizer groups for fixed number of physical qubits and encoded logical qubits

Fix $n$, the number of qubits and $k$, the number of encoded logical qubits. We can find a set of $(n-k)$ operators that all mutually commute and moreover form a group $S$. Let's assume that the group ...
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1answer
106 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 ...
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92 views

Physical meaning of twirling in Randomized Benchmarking

I was reading papers on Randomized Benchmarking, such as this and this. (more specifically, equation 30 in the second paper) It appears to be some kind of averaging but I would like to have a more ...
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What is quantum t-design (intuitive understanding)?

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 is a ...
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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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Quantum Belief Propagation decoding

I have been reading about a family of quantum error correction codes called Quantum Turbo Codes, which are the quantum analog of the well-known classical Turbo codes. This codes were introduced in ...
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68 views

topological error correction concepts

Reading about quantum topological error correction I’ve found some information, but I don’t have already clear the link with the topological concept (trivial and not trivial paths, how they are ...
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What are reliable references on analytical and/or numerical studies of threshold theorems under faulty quantum error correction?

By "faulty", I mean that you can have errors on the ancilla qubits, you can have faulty syndrome extraction, etc.
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Quantum error correction: necessary and sufficient condition

For quantum error correction, the necessary and sufficient condition is given in standard texts as: $\langle \phi| E^{\dagger}_{a} E_{b} |\psi \rangle = C_{ab} \langle \phi|\psi \rangle $ $|\psi\...
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Shor code: phase flip error

For Shor's error correcting code, what is the intuition behind saying that the following circuit corrects the phase flip error? I realize that the circuit is trying to compare phases of the three 3-...
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1answer
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The Pauli group for $n$-qubits is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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Computing with Logical Qunits

What exactly is a logical (non-physical? error corrected?) qunit? Can quantum systems be built exclusively w/ logical qunits?
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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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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 does “conjugation of coordinates” mean with respect to GF(4) (quantum) codes

In On the classification of all self-dual additive codes over $\textrm{GF}(4)$ of length up to 12 by Danielsen and Parker, they state: Two self-dual additive codes over $\textrm{GF}(4)$, $C$ and $...
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1answer
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Zero-distance self-dual GF(4) quantum codes and constructing k > 0 codes from them

During a description of zero-dimensional self-dual $\text{GF}(4)$ quantum codes in "On self-dual quantum codes, graphs, and Boolean functions" by L.E. Danielsen, it states: A zero-dimensional ...
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2answers
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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 ...
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1answer
34 views

How to find the fidelity between two state when one is an operator?

I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states Show that the fidelity between the ...
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Reference on MITxPRO Applications of Quantum Computing Professional Certificate Program

Recently I found out the Applications of Quantum Computing Professional Certificate Program that MITxPRO is offering for people interested in quantum computing. I saw that it is consisted of four ...
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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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Quantum Channel Models

The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is $\rho\...
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What is 'Surface Code'? (Quantum Error Correction)

I am studying Quantum Computing and Information. I have crossed with the 'Surface Code' phrase but I can't find a brief explanation of what it is and how it works. Hopefully you guys can help me with ...
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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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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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Translation of color/toric code to a small network of solid-state spins

Within Quantum Error Correction and stabilizer codes, toric codes/surface codes are very tempting, mainly for their high error threshold. For more background please check up, in our Physics sister (...
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Lower bound for Degenerate Codes?

According to (Macchiavello, Palma, Zeilinger, 2001; pg82) a lower bound of the encoding Hilbert space of a non degenerate code is given by the quantum version of the Hamming bound: $$2^k \sum_{i=0}^t ...
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Intuition for Shor code failure probability

Consider the 9 qubit Shor code. This can detect and correct arbitrary single qubit errors, but if there are 2 or more single qubit errors before a correction round, the correction will fail. (In 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; pg456 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 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 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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647 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 ...
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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 ...
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Quantum entanglement and 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 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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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 ...