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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Automorphism group of quantum codes

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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Eastin–Knill theorem

I am asked to explain why the Eastin–Knill theorem means that no quantum error correcting code can transversely implement a universal set of gates. But I cant find in any paper the reasons for this. ...
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Nielsen&Chuang 5-qubit quantum error-correction encoding gate

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ In ...
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How to measure syndromes in QEC

Shor's $9$ Qubit code. Imagine that we encode the state $|\psi \rangle =α|0\rangle+β|1\rangle$ using Shor's $9$ qubit code, then an $X$ error occurs on the 8th qubit of the encoded state $|E(\psi) \...
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48 views

Is there any difference between a quantum- and classically-controlled gate if I know my basis?

Consider an unrealistic 2-qubit plus 1 ancilla bit-flip error correction code (images generated by quirk), where I know by some means or other that an error may have happen on qubit0 (represented by ...
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66 views

Example of Quantum Error Correction [closed]

Shor's 9 Qubit code. Imagine that we encode the state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$ using Shor's 9 qubit code, then an X error occurs on the 8th qubit of the encoded state $...
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(Why) does every CSS code allow for transversal measurement?

I recently encountered the statement that for any CSS code [that encodes single-qubit logical states], "measurements can be performed qubit-wise, but the measurement of the outcome must be ...
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How to correct a phase error of the form $e^{ix}$

Is there a method to correct an error of the form of $e^{ix}$, acted on a two-qubit state without knowing the nature (the value of x) of the error? For example, $$\alpha|00\rangle +\beta|01\rangle+\...
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Where do errors occur during error syndrome and recovery?

I'm studying quantum error correction on Nielsen's Quantum Computation and Quantum Information, but I don't understand how it identifies the number of places where an error may occur. It uses ...
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47 views

Obtaining hermitian matrix using Knill and Laflamme condition?

Let $E$ be the set of all correctable errors and $E_a, E_b \in E$. Let $\lbrace \vert c_1\rangle, \vert c_2\rangle, \ldots \vert c_k\rangle\rbrace$ be the basis of codewords in the codespace. It is ...
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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, the terms 'magic state' and 'magic state gadget' frequently appear.
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Need mathematical Calculation of input(T1,T2,f) and output(Readout Error)

I would like to know about the mathematical relations among T1, T2, frequency, readout error, and single-qubit error? This screenshot is from the ibmq-16 Melbourne excel data file.
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What is the limit for approximate quantum error correction?

In their 2005 paper "Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes" that I have linked below, Crépeau, Gottesman and Smith give an approximate quantum error-...
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Finding the irreducible representation of Kraus operators of a dephasing channel

I would like to understand an example of finding a noiseless subsystem of a quantum channel from the irreducible representation of its Kraus operators. Assume we have $2$ dephasing channels acting on ...
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1answer
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Circuit of quantum error correction code

I read an interesting paper of QEC with only two extra qubits. In the paper, there is a circuit. Note that . Here are some contexts of the paper: "The stablizer is $XZZXI$. If the second gate ...
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What's the relation between the sign of error correction code and commute of operators?

For example, the 5-qubit QECC. If $X_i, Y_i, Z_i$ commutes with $M_i$, the eigenvalue will be +1. Otherwise, the eigenvalue will be -1. What's the relation between the commute and the sign of the ...
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How to calculate distance of $k=0$ stabilizer code?

This could be seen as a followup to the question "How to calculate the distance of stabilizer code?". Summarizing the accepted answer : distance is the minimum weight of the set $$E = \bigl\{...
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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 ...
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What is the standard noise channel that is applied in simulations?

I know there are various quantum noise channels, which include the depolarizing channel, the dephasing channel and the bit-flip channel; We can apply them in simulators easily. However, is there any ...
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Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)

This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
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How to create the logical $|0_L\rangle$ state for the Steane's 7-qubit code?

I don't know how to prepare using Qiskit the following state in order to implement a Steane's 7-qubit code circuit (I omit the normalization factor): \begin{align*} |0_L\rangle =&|0000000\rangle+|...
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Circuit for implementing Steane's code for Quantum Error Correction

I was reading about the Steane's code for QEC. However I could not find any implementation or circuit describing its creation. Can anyone share/explain how the Steane's code could be implemented. The ...
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A question about quantum noise model

Now, I try to understand some concept of Quantum Noise Model and Quantum Channel. The Quantum Noise can be represented as a quantum operator with their own probability. As far as I concerned,(I dont ...
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What is the pseudo threshold of a QECC using stabilizer formalism

Can someone explain what is the threshold and the pseudo threshold of a Quantum Error Correction Code , for instance the 9-qubit code, and how to calculate it using the stabilizer formalism simulation ...
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2answers
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Find the Kraus operators of a combined amplitude and phase damping channel

I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
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Question regarding part of the proof for the typical subspace theorem

Part three (going by N&C page 544) states that $$tr(S(n)\rho^{\otimes n})=tr(S(n)\rho^{\otimes n}P(n,\epsilon))+tr(S(n)\rho^{\otimes n}(I-P(n,\epsilon))).$$ Now I understand how the term on the ...
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Quantum Error Correcting Codes and Graphs

Couple of weeks ago I asked this question on theory CS but I didn't get an answer. So trying it here. I was reading combinatorial approach towards quantum correction. A lot of work in this is on ...
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Qubit relaxation time dependencies

The qubit relaxation time $T_1$ is an important parameter in order to describe the coherence time that a qubit presents, that is, the time in which its coherence quantum state is not compromised by ...
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1answer
131 views

How to do error mitigation on the ibmq_16_melbourne?

I'm trying to make a calibration matrix using CompleteMeasFitter as in here on IBMQ Melbourne, but I need to use all 15 qubits, which mean it has to run 32768 calibration circuit, far exceeding the 75 ...
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Algorithm to find stabilizer states

I'm working on a project in which I have to find stabilizer states based on a few criteria, the main one being that it has to have a certain amount of coherence, I'm using the following equation to ...
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How to calculate destabilizer group of toric and other codes

Some papers use the "destabilizer group" for more efficient simulations; see for example (page 3): https://www.scottaaronson.com/showcase2/report/ted-yoder.pdf "In addition to ...
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Other than distance, what other metrics can be used to compare quantum error correcting codes?

Using classical error correction (or channel coding) as a reference, I'd like to be able to compare QECC's from different constructions. Distance is a reasonable measure and you can argue that an $[[...
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What are generators for a “real” Clifford group?

I work with stabilizer codes using the real version of Pauli matrices: $X=((0,1),(1,0))$, $Z=((1,0),(0,-1))$, $Y=XZ$ (not $\imath XZ$). I know the encoders for these codes lie in the Clifford group ...
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Quantum channels and error correction research groups

I am wondering which are the top universities or research groups through the world that are doing research in the topic of quantum error correction or quantum channels (decoherence noise models). I ...
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1answer
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Quantum error correction on IBM devices?

I'm wondering if the gate errors on any of the IBM devices are low enough so that we could use some error correcting codes which do not require too many qubits? (such as 5- or 7-qubit codes). Even ...
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1answer
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Can Shor's 9 Qubit Code Correct for a measurement on the first qubit?

Start with Shor's codeword for $|0 \rangle$: $|\psi\rangle = \frac{1}{\sqrt{8}}(|000\rangle + |111\rangle)\otimes(|000\rangle + |111\rangle)\otimes(|000\rangle + |111\rangle)$. Now, assume that ...
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Where can I find tutorials about correcting gate errors in the quantum circuit?

I just have a question about where can I find tutorials/examples about correcting gate errors (amplitude errors and phase errors) in a quantum circuit? Are there any resources I can learn from? Thanks!...
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What is a difference between error rates and qubit/gate fidelity?

What is a difference between error rates and qubit/gate fidelity? A bit of maths in the explanation is fine but I am an A Level student doing a research project so definitions would be preferred.
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What is the stabilizer group of a $|W\rangle$ state?

This question is related and complementary to this one: How to get the stabilizer group for a given state? What I want is to find the stabilizer group generators for the following state: $$|W\rangle ...
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How to get the stabilizer group for a given state?

Let's say we have the GHZ state with 3 qubits: $$ |\mathrm{GHZ}\rangle = \dfrac{1}{\sqrt{2}}\Big(|000\rangle + |111\rangle \Big)$$ I want to find the stabilizer group of this state, that is, the $...
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Smallest Distance-5 Quantum Error Correction Code?

Is it known/proven what the smallest quantum error correction code is that can correct arbitrary two-qubit Pauli errors? I can think of the nested/concatenated 5-qubit code or a 25-qubit version of ...
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Is there any published research on quantum fault tolerant circuits that are designed for a specific task or algorithm?

I have searched for works that focus on designing a fault-tolerant quantum circuit for a specialized application and only found 'Error-corrected quantum annealing with hundreds of qubits' by Pudenz, ...
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Exponential Growth of Noise in Quantum Computers

I recently listened to a presentation for Introductory Quantum Computing. It was very general and meant to give the listeners an idea about the potential for Quantum Computing. The speaker ended the ...
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Physical Interpretation of Non-Trace Preserving Quantum Operations

In Chapter 8 of Nielsen and Chuang's Quantum Computation and Quantum Information, a mathematical framework is developed to describe the dynamics of open quantum systems. Suppose the initial state of ...