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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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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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Get result measurement into a circuit with Qiskit

I'm trying to implement the three qubit bit flip code in Qiskit. Therefore, I need to get the result of a measure of two ancilla qubits, to deduce which gate I need to use do recover my logical qubit....
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Error syndromes and recovery procedure in bit flip code

This question relates to exercise 10.4 in Nielsen and Chuang. For syndrome diagnosis, the textbook provides an example where one has four projectors, by which, you can identify where a one qubit ...
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What is the definition of Bell state on a n-qubit system?

Question 1: The bell state for a 2-qubit system has been defined in Neilsen and Chuang's book as the set of maximally entangled states spanned by $\{|00\rangle + |11\rangle, |00\rangle - |11\rangle, |...
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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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How many equivalent logical operators are there on a [[n,k,d]] stabilizer code?

I know what are the logical operators which represent the Pauli operators in the subspace, used to preserve the code subspace. But I am wondering how many equivalent logical operators are there on a ...
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Where (in the algorithm) is the logical qubit encoded in Q.E.C?

I have a basic question about the encoding of logical information into physical qubits. I only know the 3 qubit code (I have very limited knowledge in QEC). The 3 qubit code is usually presented as: ...
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How does measurement calibration work?

One of the possible ways to improve the results of an experiment on the IBM machine using the Qiskit language is to use the measurement calibration methods. This is the link to the documentation. I ...
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Shor 9 qubit code — how are the observables measured and eigenvalues obtained during syndrome measurement?

Say we have the Shor 9 qubit code $$|\psi_L\rangle=\tfrac{1}{\sqrt{2^3}}((|000\rangle+|111\rangle)^{\otimes3}+(|000\rangle-|111\rangle)^{\otimes3}),$$ and we have a bit flip error. My lecture notes ...
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State of a system after the second qubit of a Bell state sent through a bit flip error channel

The second qubit of a two-qubit system in the Bell state $$|\beta_{01}\rangle= \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$$ is sent through an error channel which introduces a bit flip error with ...
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What quantum volume is needed to represent a single logical qubit?

The quantum volume metric $V_Q$ is a proposed metric for quantifying and comparing the performance of quantum computers1. The quantum volume is defined as $$V_Q = \max_{n<N} \left(\min\left[n, d(n)...
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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 the quantum Singleton bound compatible with the Toric Code?

Note: Cross-posted on Physics SE. The quantum Singleton bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qubits, and some subsystem $R$ of $m$ qubits that can '...
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Is there an inherent difference in need for error correction between quantum annealing and gate based methods?

When I read about computing using gate based methods, I mainly read about the difficulties with error rates, circuit depth (and connectivity) and not enough qubits. With computing using quantum ...
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How does the actual measurement collapsing an error to an orthogonal basis look like?

An error can be written as a linear combination of $\Bbb I$, $X$, $Z$, $XZ$ Pauli matrices. So when measuring an errand state we aim at collapsing the error into one of these four possibilities. How ...
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Estimating the depolarizing probability of depolarizing channels

When considering quantum error correction over depolarizing channels, the depolarizing probability $p$ such that an error of the kind $X,Y,Z$ will happen is used as a priori information in order to ...
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In error correction code why don't we imitate the Hamming code instead of the complicated Steane code?

Instead of the complicated Steane code, I don't understand why don't we use a much simpler and exact imitation of the classical Hamming code. So here is my idea: Let's have 7 qubits, and we will ...
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What does it mean to perform a measurement in correspondence with different projections?

In error correction, like the bit flip, you perform a measurement which corresponds to different projections so that the outcomes can teach you about the error. What does it mean? How do you actually ...
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How does Steane code use the classical Hamming code for error correction?

I know what the Hamming code is. But I don't understand how could this idea be applied in quantum computation since it's required to cover the case of superposition and entanglement. How could you ...
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Why does phase flip correction error? Why could any error be written as a linear combination of I, X, Z and ZX matrices?

I don't understand how it's being proven that error correction can be applied only to X, Z noises and this solves all errors? Does this have to be with this set being universal? (Z is exactly like ...
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Decoherence in quantum systems always produces $\vert0\rangle$

I was recently asked two questions concerning error in quantum computing: Is it possible for quantum computers to exhibit behavior similar to flip errors in classical computers where a state $\vert0\...
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In what situation are three rounds of syndrome measurement required for fault-tolerance in the surface code?

I have heard multiple times the claim that three rounds of syndrome measurement required for fault-tolerance in the surface code. I'm not sure what situation would require this, as I think less would ...
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A question on Eastin-Knill theorem

I am reading the paper Restrictions on Transversal Encoded Quantum Gate Sets, Bryan Eastin, Emanuel Knill. I am unable to understand the following lines in the proof. As the set of all unitary ...
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Why is it that the Pauli matrices and error correction operator act only on $|\psi\rangle\langle \psi|$ and not on state vector $|\psi\rangle$ itself?

I saw that the Pauli matrices really work, that's rotating a state by 180 degrees, only if you take the density matrix for example with X it only works if first we take X $|\psi\rangle$ and multiply ...
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Why can a point in anti-de Sitter space be modeled as a logical qutrit and how is its error correction done?

This isn't my area but the recent Quanta article How Space and Time Could Be a Quantum Error-Correcting Code struck me as interesting. They mention: In their paper[1] conjecturing that holographic ...
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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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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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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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Is the common depiction of a surface code to be taken literally as a real-space image of the actual hardware?

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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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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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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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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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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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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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-...