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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Calculating length of code words in quantum information(compression)

I was studying this article by Boestrom and Felbinger. We define the significant length of the codewords in the preparation of the communication protocol : $$L_c(w_i) = \lceil log_k(i) \rceil$$ We ...
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Why is a 15-qubit IBM quantum computer not working correctly?

I just wanted to implement an algorithm for adding two 2-bit binary numbers. And it works, but only on an IBM 32-qubit simulator. And on a real 15-qubit computer, ibmq_16_melbourne, it produces very ...
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Characteristics of the IBM quantum computer

On the IBM Quantum Composer website, there are characteristics of qubit computers. For example, ibmq_16_melbourne. But there is no description anywhere of what: ...
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'Best practices' for making Stim run as fast as possible?

I'm currently testing out an error-correction surface code circuit in Stim. I've tried a 19x19 surface code over 10k rounds of syndrome extraction sampled 256 times, and this takes about 165 seconds. ...
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Error syndrome on logical operators

The stabilizers and logical operators (for five qubit-codes) are given by $$ \hat{S}_1= \hat{X}_1\hat{Z}_2\hat{Z}_3\hat{X}_4 \\ \hat{S}_2= \hat{X}_2\hat{Z}_3\hat{Z}_4\hat{X}_5 \\ \hat{S}_3= \hat{X}_1\...
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Understanding the (2,3) threshold quantum secret sharing scheme in Cleve et al. 1999

In the paper "How to share quantum secret" by R.Cleve et al (arxiv). There is an example of secret sharing using qutrit in a (2,3) threshold scheme. $$\alpha\vert0\rangle+\beta\vert1\rangle+\...
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In Shor’s 9-qubit code, is the error syndrome for qubit 5 phase-flip the same as that for qubit 6 phase-flip?

The Shor’s 9-qubit code has the following stabilizers $\hat{S}_1= \hat{Z}_1\hat{Z}_2$ , $\hat{S}_2= \hat{Z}_2\hat{Z}_3$, $\hat{S}_3= \hat{Z}_4\hat{Z}_5$ $\hat{S}_4= \hat{Z}_5\hat{Z}_6$, $\hat{S}_5= \...
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Why is this topological gate mentioned in Raussendorf et al. 2007 a CNOT?

I read a Paper about quantum error corrections. I don't know why this is a CNOT gate. How to calculate this kind of CNOT gate as a topology form?
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Can quantum error correction work on any type of channel?

It says on wikipedia that quantum error correction can (at best) correct phase flips and bit flips. A popular form of representing a quantum channel is in its Kraus representation (scroll down to ...
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Quantum Error Correction implementation in Qiskit, OpenQASM or other languages based on Python

Where can we find example(s) of implementation code in a programming language of Quantum Error Correction, whether with the standard method of measurement or the automatic method without measurement (...
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How to write the general state of a corrupted n-qubit codeword in quantum error correction?

I am reading the chapter on quantum error correction in Quantum Computer Science by David Mermim. Which says, for a single qubit in a random state $|\Phi\rangle$ of the superposition of $|0\rangle$ ...
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Can one easily compute the weight enumerator of a CSS code constructed from two binary codes $M_1,M_2$?

Consider a CSS code, which is constructed from two binary codes $M_1$ and $M_2$, so its symplectic matrix is of the form: $$ \begin{pmatrix} M_1 & 0 \\ 0 & M_2 \end{pmatrix}.$$ Suppose one ...
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How to implement Majority Vote

I am trying to boost the success propability of standart phase estimation by repeating the procedure enought times and taking a majority vote that will be encoded in a quantum register. My problem is ...
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How to limit the error probability in large scale quantum computers

I am quite stumped by the fact that the roadmaps for quantum computers as given by IBM, Google, IonQ, etc. seem to imply a linear/exponential growth in the size of their quantum computers. Naively, I ...
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5 qubit codewords definition in terms of operators: Mermin

Book: Quantum computer science by David Mermin Chapter:5 Page-118 The 5-Qbit codewords are most clearly and usefully defined in terms of the $M_{i}$ (rather than writing out their lengthy explicit ...
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Applying a projector to a qubit in a qiskit circuit

I'd like to be able to apply $|0 \rangle \langle 0|$ to project a qubit to the state $|0 \rangle$ in the middle of qiskit circuit (see, for example, the attached circuit). I wonder if, in general, ...
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Stabilizer code: error detection why does it matter?

State vectors take more space when we have to specify certain states. In stabilizer formalism, we can specify these states in a much more compact way. But in error correction then we say that the ...
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How do we find the stabilizer generators for the three-qubit bit-flip code spanned by $|000\rangle$ and $|111\rangle$?

In Nielsen & Chuang's book "Quantum Computation and Quantum Information" section 10.5.6, page 467 there is the following statement Consider the familiar three-qubit bit-flip code ...
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How to do the counting when computing the fault tolerant threshold of quantum codes?

Here, I want to ask a basic question about how to compute the fault tolerance threshold of quantum codes. As I know, maybe, the most usual way is to do some simulations. Howvever, I am more intersted ...
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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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Necessary and sufficient condition to define logical operation (stabilizer code)

My question is highly related to this topic It is about defining logical operation on a Stabilizer code. I call $S$ the stabilizer group of a code space $C$, and I assumed it is generated by a family $...
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Transversal logical gate for Stabilizer (or at least Steane code)

I know that for Steane code, we can implement transversally some gates like cNOT, Hadamard and Pauli. What I am looking for is a resource in which it is explained why implementing those gate give rise ...
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Degenerated vs non degenerated code: for both there always exist Kraus bringing to different orthogonal subspaces?

Context of my question I call: $\mathcal{M}(\rho)=\sum_a M_a \rho M_a^{\dagger}$ an error map, $C$ the code space. A CPTP recovery operation exists if and only if, the Kraus operator of the error map ...
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Arbitrary error correctable iff Pauli error are: misunderstanding from Preskill notes

I am following Preskill notes. What I want to understand is why it is in general enough to be able to correct n-qubit Pauli errors to say that an arbitrary error can be corrected. I call: $\mathcal{M}(...
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Phase flip error correction on state $|0\rangle$

When I read documentations about quantum error correction, it generally speak about bit flip error, and phase flip on $|1\rangle$ state, so for example, let's say my initial state is $|\psi\rangle = a|...
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In quantum computation, if the fidelity is high enough to promise over 50% success rate, is quantum error-correction still needed?

Let's say we have many qubits and gates. The errors happen randomly, for example with a probability of 0.1% at each place(at this place, no quantum error correction is implemented). If the probability ...
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Standard form Shor's code

I'm trying to solve exercise 10.57 in Nielsen-Chuang, where you have to obtain the standard form check matrix of Shor's code. I followed the procedure laid out in the earlier chapter but then realised ...
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How to define POVM measurement operators for a composite quantum state

I have an evolved quantum composite state $\hat{\rho}^{\otimes{N}}$ that I retrieved from a quantum channel $\mathcal{E}$, Now I do know how to define a POVM for the evolved states $\hat{\rho}$ that ...
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What does it mean “the N uses of classical-quantum channel”?

I was reading a paper Quantum Polar codes by Mark M. Wilde, where he discusses the N uses of the channel in the classical-quantum channel setting. What does he mean by "multiple channel uses"...
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Which codes can implement transversal non-Clifford gates

A paper Three-dimensional surface codes: Transversal gates and fault-tolerant architectures discusses 3D surface codes and shows that CZ and CCZ gates are transversal in [these] codes. They give a ...
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CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book? The screenshot is shown below. I have no idea about that. Hope someone can give me some suggestions. By the way, if I ...
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Is there a good reason to use T-count minimization for circuits executed on current IBM open quantum systems (real hardware)?

As far as I understood from a series of papers, minimizing the T-count in Clifford+T circuits is essential for fault-tolerant ...
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Pauli Matrix tolerant and encoding

I'm stumped by these questions in Chuang's book. If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where $$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$ I rewrite its ...
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Question about quantum error correction and density matrixs

I am now studying QEC and feel confused. If I have a density matrix before the correction: The circuit is: $\rho = p^0(1-p)^3|\varphi\rangle \langle\varphi|+p^1(1-p)^2\sum_{i=1}^3X_i|\varphi\rangle \...
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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

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ 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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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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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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