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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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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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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How to implement the Circuit of Steane's code for Quantum Error Correction?
I have referred this same question here 'Circuit for implementing Steane's code for Quantum Error Correction' . But the answer discusses the circuit to compute the syndrome and not clearly the ...
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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Twirling Quantum Channels: Pauli and Clifford Twirling
I am currently working through some papers related with approximations of more general quantum channels such as amplitude and phase damping channels to Pauli channels. The reason to do so is so that ...
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Is Gil Kalai's argument against topological quantum computers sound?
In a lecture, recorded on Youtube, Gil Kalai presents a 'deduction' for why topological quantum computers will not work. The interesting part is that he claims this is a stronger argument than the ...
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How to implement if statement based on measurement results in qiskit?
I tried to implement three qubit bit flip code in qiskit and need to get the result of measurements and then apply recovery quantum operations conditioned on the measurement results. The following is ...
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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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How do we create logical qubits in the surface code: understanding check
I am learning the basics of surface code theory through this paper
I am trying to understand how we create multiple logical qubits. The goal of my post is to check my understanding.
What I understood ...
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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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What are magic states?
I wonder what are magic states, and a magic state gadget. While I'm reading a paper, these terms frequently appear.
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What is the leading edge technology for creating a quantum computer with the fewest errors?
Which technological path seems most promising to produce a quantum processor with a greater quantum volume (preferring fewer errors per qubit over more qubits), than Majorana fermions?
The preferred ...
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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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Nielsen & Chuang Exercise Question on CSS code
I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that: suppose $C_1$ and $C_2$ are $[n,k_1]$ and $[n,k_2]$classical linear codes such that $C_2\...
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Quantum circuit for a three-qubit bit-flip code
I know a three-qubit bit-flip code has a common encoding circuit as follows,
Further, as in page 35 in Gottesman's paper, the encoding circuit can also be constructed through stabilizer generators. ...
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How the real IBM quantum computers apply arbitrary Rz(θ) gate rotation? [closed]
I want to ask the following questions:
(1) The basis gate set of IBM quantum computers is { Id, Rz(θ),√X, X, CNOT, reset}. Somebody said that IBM didn’t really apply Rz(θ) gate on the machine. The ...
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Obtaining low weight stabilizer generators
Suppose I know a set of stabilizer generators of a qubit quantum code. Is there a systematic (and possibly efficient) way to transform this set of generators to a different set (generating the same ...
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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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Stabilizer codes and 1,-1 coefficients
A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and local one which can be written as (a global scalar times) a linear combination of computational ...
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Link between distance of a stabilizer code and number of errors it is able to correct
I am confused by a property.
In the N&Chuang it is said that an $[n,k,2t+1]$ stabilizer code is able to correct up to $t$ errors. But for me if the code has distance $d$ it should be able to ...
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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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Laflamme's perfect quantum error correcting code (5 qubit) - Not getting the correct syndrome bits
I am trying to implement 5 qubit encoding and decoding circuit which results into generation of syndrome bits that pin points the location of error so that we can correct it afterwards. Please have a ...
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Mitigating the noise in a quantum circuit
I'm using Qiskit and I have a Quantum Circuit (say, circuit) that gives reasonable results when using the simulator, namely
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Analytical Approximation of Pseudothreshold for Steane Code
Is there an analytical way by which one can approximately find the pseudothreshold of the Steane code? My supervisor told me that there is a way, using the physical error rate and the number of gates ...
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how to go from matrix to tableau to circuit in qiskit or stim
I'm working with QECC using a non-python based platform. I'd like to move the results into python to do calculations that are better handled by packages like qiskit or stim. So the output of the non-...
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How to read a graph representing the stabilizers and data qubits in surface code
I am trying to understand to properly read the graph below, provided in this post
It is said that the black elements are the $Z$ stabilizers and the grey ones are the $X$ stabilizers. I am trying to ...
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What is the "surface code" in the context of quantum error correction?
I am studying Quantum Computing and Information, and have come across the term "surface code", but I can't find a brief explanation of what it is and how it works. Hopefully you guys can ...
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Is error correction necessary?
Why do you need error correction? My understanding is that error correction removes errors from noise, but noise should average itself out. To make clear what I'm asking, why can't you, instead of ...
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Which quantum error correction code has the highest threshold (as proven at the time of writing this)?
Which quantum error correction code currently holds the record in terms of the highest threshold for fault-tolerance? I know that the surface code is pretty good ($\approx10^{-2}$?), but finding exact ...
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What level of "confidence" of the result from a quantum computer is possible?
At a very basic level, reading or measuring a qubit forces it to be in one state or the other, so the operation of a quantum computer to gain a result collapses the state into one of many ...
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What is a Bacon-Shor code and what is its significance?
I'm at the AQC conference at NASA and everybody seems to suddenly be talking about the Bacon-Shor code but there is no Wikipedia page and the pdf that I gave a link to does not really explain what it ...
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Violation of the Quantum Hamming bound
The quantum Hamming bound for a non-degenerate $[[N,k,d]]$ quantum error correction code is defined as:
\begin{equation}
2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{...
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Why do error correction protocols only work when the error rates are already significantly low to begin with?
Quantum error correction is a fundamental aspect of quantum computation, without which large-scale quantum computations are practically unfeasible.
One aspect of fault-tolerant quantum computing that ...
glS♦
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What does quantum error correction code notation stand for?
I understand the notation for classical error correcting codes. E.g., "Hamming(7,4)" stands for a Hamming code that uses 7 bits to encode blocks of 4 bits.
What does the notation for quantum error ...
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Are all $[[n, k, d]]$ quantum codes equivalent to additive self-orthogonal $GF(4)^n$ classical codes?
Theorem 2 of [1] states:
Suppose $C$ is an additive self-orthogonal sub-code of $\textrm{GF}(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C^\perp/C$. ...
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Why can't there be an error detecting code with fewer than 4 qubits?
Essentially this boils down to: Is it possible to encode a single logical qubit in three physical qubits so that the resulting code has distance two?
In other words, does a $[\![3,1,2]\!]$ code exist?
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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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Allowed CNOT gates for IBM Q 5 quantum computer
I trying to do some tests in the IBM Q5 computer of IBM quantm experience for some simple error correction protocols, but as I can see, some operations between the qubits are not allowed.
For example,...
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How does the size of a toric code torus affect its ability to protect qubits?
The Toric code Hamiltonian is:
$\sum_{x,y}\left( \prod_{i\in p(x,y)} Z_{ixy} + \prod_{i\in v(x,y)} X_{ixy} \right),$
where the $v$ and $p$ are defined according to this picture (courtesy of James ...
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What quantum channels are considered in quantum communication, and how does this choice affect the construction of error correction codes?
The so-called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is
$$\rho\...
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Intuition for Shor code failure probability
Consider the 9 qubit Shor code. This can detect and correct arbitrary single qubit errors, but if there are 2 or more single qubit errors before a correction round, the correction will fail. (In the ...
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Practical Implementations of QECCs in IBM Q Experience
I am learning how to program the IBM Q Experience quantum computers in order to learn more about how does it work and in order to perform some experiments in it. By doing so I was wondering what are ...
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CSS Code in disguise
Suppose I have a quantum error correcting code described by a set of stabilizers. Is there any easy way of checking whether or not it is equivalent to a code constructed using the CSS construction, ...
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Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$
My question is the following
Let's assume I am using Steane concatenated code to do error correction. I consider that the stabilizers are extracted fault-tolerantly through the Steane method. The ...
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What is (formally) a transversal operator?
This question concerns about a formal definition of transversal operator.
I understood that transversal operator are a group of operators which are efficient in terms of circuit depth and can be used ...
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CSS codes are the only stabilizer codes with transversal CNOT?
Given a stabilizer code $\mathcal{C}$ then
$$
\mathcal{C} \text{ is CSS} \iff \text{CNOT} \text{ is transversal}.
$$
The forward implication is well known, see for example Transversal logical gate ...
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Why do we need to keep of history of $d$ cycle for surface code in order to resist against data *and* measurement errors
I am trying to understand why we need to keep in memory $d$ clock cycles in order to correct efficiently for the surface code where the syndrome qubits are also faulty ($d$ is the code distance).
[...
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Lower bound for Degenerate Codes?
According to (Macchiavello, Palma, Zeilinger, 2001; pg82) a lower bound of the encoding Hilbert space of a non degenerate code is given by the quantum version of the Hamming bound:
$$2^k \sum_{i=0}^t ...
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Simulating flag qubits and conditional branches using Stim
In a quantum error correcting code using flag qubits, it's common to have flag measurements that tell you it's necessary to do some extra measurements for safety. So, for example, I want to say:
...
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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:
...
glS♦
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