# 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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### How to compute the error threshold for the $9$-qubit Shor code?

I was trying to calculate the threshold of $9$-qubit Shor's code. The error channel is $$E=(1-p)I+p/3X+p/3Y+p/3Z.$$ But I got the threshold is equal to 1. How can I get the right threshold (I believe ...
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### Can Shor‘s code correct two- or three-qubit errors?

I have read some articles about Shor's code (e.g. this one). It is said that Shor's code can correct a single-qubit error. What about two qubit errors? Three qubit errors? It confused me a lot...
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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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### Are applications with only polynomial speedup worth chasing after? (since error correction adds a heavy overhead)

A number of ML algorithms have demonstrated to have polynomial speed-up: But this (I'm assuming) is without error correcting qubits. How practical are algorithms that only exhibit polynomial speed-up ...
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### Proper definition of logical operation in Q.E.C

My question is related to this topic I consider working with error correcting code on which I want to define logical operations. Let's assume I want to define a logical operation on logical qubit. In ...
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### Understanding transversal gates for the 7 qubit steane code

How can one derive the complete list of transversal operators for the 7-qubit Steane code? I can derive the Clifford operators that are transversal, but I do not understand an easy way to check for ...
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### Intuition behind Stabilizer code conditions

My question is very related to Intuition about Knill-Laflamme QEC conditions but in the particular case of Stabilizer code. In Nielsen&Chuang, the theorem 10.8 on page 466 gives error-correction ...
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### Is there a way to entangle to a dirty qubit?

Let's say I do something to a qubit, and I want to entangle it to a 2nd one, like this: ...
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### Understanding "Restrictions on Transversal Encoded Quantum Gate Sets"

I am trying to understand a part from the article "Restrictions on Transversal Encoded Quantum Gate Sets". in the article they talk about the importance of transversal encoded gates for ...
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### Considering quantum codes as codes over $F_2$

It is very common to look at stabilizer codes as codes over GF(2) or codes over GF(4). Mostly I have seen this for computations for distance of codes and syndromes. How do other notions like say ...
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### Why and how is quantum noise predictable?

I have been learning about quantum error correction using the zero-noise extrapolation method from this paper and have been pleased with the results. This method takes advantage of the fact that the ...
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### Quantum error correction: block length and error rate definitions

I encountered notions of block length and error rate for quantum error correcting codes, which literature seems to just assume. Can someone please give precise definitions for these? Usually we ...
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### Are there any implications of the automorphism group in QECC?

We often see that classically automorphism group of an error-correcting code plays a crucial role in many computational problems. Are there any important implications that depend on this in quantum ...
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### Nielsen&Chuang 5-qubit quantum error-correction encoding gate

$\newcommand{\bra}{\left<#1\right|}\newcommand{\ket}{\left|#1\right>}\newcommand{\bk}{\left<#1\middle|#2\right>}\newcommand{\bke}{\left<#1\middle|#2\middle|#3\right>}$ In ...
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### (Why) does every CSS code allow for transversal measurement?

I recently encountered the statement that any CSS code [that encodes single-qubit logical states] has a property that measurements can be performed qubit-wise, but the measurement outcome of every ...
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### Why is the action of controlled-Z unaltered by exchanging target control qubits?

In the book "Quantum Computer Science", when explaining the error correction code, it uses this picture and says "the action of controlled-z is unaltered by exchanging the target and ...
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### Is there a proof or example to show that a noiseless subsystem is not necessarily closed under addition?

In a text (section 3.6 page 92) about noiseless subsystems by D. Lidar, it is mentioned:'A subsystem is a tensor factor in a tensor product, and this does not have to be a subspace (e.g., in general ...
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### How is it not a contradiction that it is possible to build fault tolerant circuits with strictly contractive (e.g.: depolarizing noise) channels?

This paper discusses strictly contractive channels, i.e. channels that strictly decrease the trace distance between any two input quantum states. It is shown that if a quantum circuit is composed of ...
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### Dose Quantum threshold theorem apply to IBM Quantum Experience

According to Quantum threshold theorem, error rate can be arbitrarily low. But when I use IBM Quantum Experience to measure a simple $|+\rangle$, it gives result of $|1\rangle$ with probability of 52....
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### Is it possible to implement c_if statement based on a measurement in a single specified classical bit in qiskit?

I'm trying to write a 1-bit teleportation error correction code and there is one part in it where I need to add a gate that's dependent on a measurement from earlier in the circuit. The measurement ...
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### What is the certain error rate in a quantum channel?

Quantum error correction is a fundamental aspect of quantum computation. I have read some material about "Quantum Channel" and "Quantum error correction". I have known the formula ...
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### In the three-qubit bit flip code, why can the first bit flip without impacting the entanglement with the other qubits?

The principle of the three-qubit Bit Flip Code is straight forward at first sight. Using CNOT you basically encode $$a|0\rangle + b|1\rangle$$ to $$a|000\rangle + b|111\rangle$$ using ...
This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...