# 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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### Using ZX calculus to postselect the first round of a distance-2 rotated surface code

I'm translating gate model language to ZX calculus diagram using such notation, to describe the first round of measurement of such a distance-2 rotated surface code below. I wrote this diagram where ...
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### For any given parameters, does there always exist a quantum code which saturates the Quantum Hamming Bound?

We know that the Quantum Hamming Bound is as follows: $$\sum_{j=0}^t 3^j {n\choose j} \leq 2^{n-k}$$ where $n$ is the number of physical qubits, $k$ is the number of logical qubits, $t$ is the ...
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### How to address the 2 logical qubits on the toric code individually? In general, how to address $k$ logical qubits in a $[[n,k,d]]$ code independently?

Suppose I have a $[[n,k,d]]$-quantum error correction code. Let us take the toric code $(T^2=S^1 \times S^1)$ as an example. We have 2 logical qubits whose logical operators lie along the different ...
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### do local clifford gates preserve code distance?

It can be shown that clifford gates do not preserve distance. My question is what if you restrict to local clifford gates, is distance preserved by these? (by local I mean that they act on each qubit ...
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### Get parameters of a CU gate for implementing an erroneous CX gate with fidelity = 0.81

Let's say the definition of an erroneous CNOT gate is given as - CNOT is a 2-qubit quantum gate when the control qubit is at state 0, applies identity operation on the target qubit, when the control ...
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### Stim error when measuring detectors

I am attempting to compute the circuit for the syndrome extraction of the [[144, 12, 12]] code using stim so as to consider circuit-level noise. Unfortunately, when trying to obtain the detector error ...
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• 569
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### Z error detection circuit

In this video at 0:50, they have two circuits. One claims to detect a Z error in any of four qubits and one detects an X error in any of the four qubits. I don't understand the Z error detection ...
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### Does the number of stabilizers bound the number of correctable errors?

Suppose I have a quantum error correcting code with two stabilizers. Does this mean that I can potentially correct at most 3 distinct errors using this code? My reasoning is that for each error, we ...
• 522
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### Distance of the concatenated quantum error correcting code

If we have two quantum error-correcting qubit $[[n_1, 1, d_1]]$ and $[[n_2,1,d_2]]$ codes then the notes of Preskill says that the concatenation of the codes is a code of distance at least $d_1d_2.$ ...
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1 vote
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### how is logical error rate calculated?

I have a CSS code defined through binary matrices $H_X$ and $H_Z$. I also have the logicals $L_X$ and $L_Z$ as binary matrices. I decode $H_X$ and $H_Z$ independently as classical codes. The decoder ...
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1 vote
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### What is the motivation for quantum color codes?

I am a beginner in topological quantum codes and have been interested in quantum color codes recently. The 2-D quantum color codes have a very clear physical picture and can check stabilizers locally. ...
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### Encoding circuit for $[\![6, 4, 2]\!]$ code

For the famous $[\![4, 2, 2]\!]$ code, there is a circuit to encode two physical states into the logical state (from Roffe's work): As $[\![m, m-2, 2]\!]$ is a class of error-detection code, I am ...
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