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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Algorithm for finding Pauli stabilizers of a code

Given the zero logical $ |0_L\rangle $ and one logical $ |1_L\rangle $ for an $ [[n,1,d]] $ code is there a well known/ efficient algorithm for determining which Pauli operators stabilize the code?
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Stabilizers Of The H Code

How does one interpret the ... in the third and fourth stabilizers of the H code? Is this $[1, 2, 5, 6, 9, 10, \dots]$ or $[1, 2, 5, 6, 7, 8,\dots]$? Can someone please write this out for $k = 1, 2, 3,...
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How to simulate rectangular surface code with stim?

The Stim package is a tool for simulation and analysis of quantum stabilizer circuits. Stim's built-in surface code circuit generation, stim.Circuit.generated, can run different code tasks (rotated ...
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How to get the matched nodes using Matching.decode() in PyMatching?

In the module matching.py of PyMatching, it is possible to construct Matching objects to decode matching graphs using the minimum-weight perfect matching decoder. The ...
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Necessity of decoding in fault-tolerant quantum computation

On page 476 of Nielsen/Chuang's book it is stated: The basic idea of fault-tolerant quantum computation is to compute directly on encoded quantum states in such a manner that decoding is never ...
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Encoded Fusion Based Ring Resource State

In the article on Fusion Based Quantum Computing, they presented that a fusion of 4qubits encoded states, is creating an encoded ring resource state. My question is - when we make fusion (Type 2) to 2 ...
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Why are all the error cosets $Q.S$ given the erasure chain $\mathcal E$ and a syndrome $\sigma$ equiprobable? (Delfosse-Zémor)

In arXiv 1703.01517 (published here), a maximum likelihood decoding for qubit loss is explained. A quantum erasure channel erases each qubit independently with some probability $p$ and replaces it ...
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Is the assumption of constant per gate error probability $q$ in quantum computation proved , practical or valid?

The threshold theorem is valid provided that the error rate( per gate or per unit time), be under a constant threshold. But I have not read or known any proof about the constancy of theshold. Is “the ...
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Errors while encoding (3 qubit code)

I am relatively new to quantum error correction, so apologies if this question appears is naive. In the three qubit code, there seems to always be the assumption that errors occur after encoding the ...
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Fault-tolerant code and the error rate

Threshold theorem assumes that the error rate is bounded above by a constant. However, quantum decoherence is characterized by $$\langle E_{i}(t)|E_{j}(t) \rangle \propto e^{-t/\tau _{d}}$$ see here, ...
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what are the parameters of honeycomb code

The honeycomb (subsystem?) code seems to have some interesting properties. I'm trying to build up my understanding of these properties but I can't find a concrete description of its parameters. I see ...
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is cirq suitable for simulation of quantum error correction?

I experimented with stim and qiskit for QECC simulations. They both have limitations : for example stim has no support for non-clifford operations (qiskit does); qiskit has no support for multiple ...
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Bit flip error correction syndrome measurements

I'm coming across some confusion in chapter 10.1.1 of Nielsen and Chaung. In terms of the 'recovery' procedure, how can the result of the syndrome measurement be 0, 2 or 3? I am assuming that, for ...
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How to intuitively understand why $T$ gate can't be implemented transversally?

Suppose we have all to all connectivity and can implement fault-tolerant logical gates in the Clifford group. Why can't we just apply a physical $T$ gate to every physical qubit in the code block? (...
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QecSim vs Stim for simulating QEC

I am starting a quantum error correction (QEC) project and I would like to invest some time at the outset familiarizing myself with a nice software library that can be used to simulate different codes....
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Cloning for error correction by preparing state multiple times

In quantum error correction, we start with the fact that no-cloning theorem doesn't allow cloning of unknown states, hence we need to come up with other strategies. But often we know the circuit that ...
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Error correction without syndrome measurements

Active error correction is usually described as a three-step process -- measuring syndromes, decoding the error, and applying the correcting operator. There seem to be alternatives to this approach, ...
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How to perform basic correction based on syndrome measurements in CSS codes

I'm trying to implement the rotated surface code with 9 qubits. I'm willing to perform one round of syndrome measurements, based on which corrective X and Z gates will be applied on the 9 qubits. I'd ...
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Distance of one dimensional quantum error correcting code

Consider one dimensional quantum codes $[[ n,k=0]] $. One way to describe them is using stabilizer framework with $n$ independent Pauli matrices. Usually, one considers them in the graph state model ...
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Given a list of stabilizers (or parity check matrix), find an encoding circuit

I am trying to study Quantum LDPC code and have generated the A matrix by taking the hypergraph product of a classical LDPC code. Now my aim is to study those matrices by implementing the same on ...
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Does anybody know what a low-degree Markov field is?

In the paper Fast Estimation of Sparse Quantum Noise I saw the following description: quantum devices approaching the fault-tolerant regime will have very few significant errors (and therefore are ...
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Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
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Why doesn't reading the ancilla qubit in Quantum Error Correction kill entanglement?

In Quantum Error Correction we have ancilla qubits (in gray) that will be entangled with the pair of bits we want to check. Because we don't want to directly measure the "main" qubits (in ...
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Are there tools to calculate distance of a stabilizer code?

Does anyone know of any tools that calculate the distance of an arbitrary stabilizer code? I have some primitive ways of doing this which can handle codes up to ~32 stabilizers (number of qubits a ...
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Why should we wait an additional d-1 surface code cycles in one-cell logical qubit move process of surface code?

I was reading this paper. In section XIII-A, after we finish two surface code cycles for a one-cell logical qubit move, it says "To establish all stabilizer values in time, we then wait an ...
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Different codes with the same parameters

The parameters of a code are $ [[n,k,d]] $ where $ n $ is the number of physical qubits, $ k $ is the number of logical qubits, and $ d $ is the distance. There are lots of trivial codes with $ d=1 $ ...
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What are the tools available that are used for analysing Quantum LDPC code?

Want to learn how to simulate QLDPC code. what are the best tools available to do the same. can any one provide some sample python codes if possible for getting started with it.
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stim: how to control sign of syndrome measurements

I have a circuit that encodes the $[[5,1,3]]$ code, adds noise on one of the qubits and then measures the syndrome on 4 ancilla qubits : ...
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Is there any transversal logical $CNOT$ (or $CZ$), not involving all the physical qubits?

Well-known codes, such as Shor's code or the CSS [[7,1,3]], allows for transversal implementation of logical $CNOT$. On the other hand, there are codes, such as the prefect code [[5,1,3]], for which ...
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Fake Stabilizer States

Consider an $ n $ qubit stabilizer state $ |\psi> $ with stabilizer group $ S $. Then the projector onto this one dimensional stabilizer subspace is given by $$ \sum_{g \in S} g $$ $ |\psi> $ ...
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Example non-stabilizer code?

A code is a non-stabilizer code if it is not equivalent by local unitaries to a stabilizer code. What is an example of a non-stabilizer code with distance $ d \geq 2 $? Are there any non-stabilizer ...
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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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Does a $[\![3,1,2]\!]$ code exist?

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? Comments: No $[\![3,1,2]\!]$ ...
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Does every code have transversal Pauli group?

What is an example of a code for which the Pauli group cannot be implemented transversally? Mostly interested in an $ [[n,1,d]] $ code where $ d \geq 3 $. Edit: By transversal I mean that there exists ...
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What is the difference between Error Mitigation (EM) and Quantum Error Correction (QEC)?

From what I understood, EM is mostly post-processing and does not affect quantum circuit complexity whereas QEC has overheads and integrates as part of a quantum circuit. Is this correct? Also, which ...
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What informs which choice of stabilizer generators to measure?

A stabilizer code is a code with codespace defined as the joint $+1$ eigenspace of some set of stabilizers $\mathcal S = \{s_1, s_2,\ldots,s_j\}$ with eigenvalues $\pm 1$. Error correction involved ...
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How to train neural network decoder with stim?

Following Decoding Small Surface Codes with Feedforward Neural Networks, I am trying to train a similar neural network decoder with stim. Decoding is reduced to a ...
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How can we translate classical SNR into the quantum world?

I am using a LDPC codes to correct error encountered in the quantum world. In the LDPC package in python the one of the parameter is SNR. I would like to explore the possibility if I can translate ...
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High-weight errors from ion trap crosstalk

A basic assumption of various error correction threshold theorems is that the probability of an error exponentially decreases with the weight of the error. One of the selling features of ion traps is ...
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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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Qubit Resource Estimates for 2D Ising Model

I recently came across this paper where the 1d Transverse Field Ising Model (TFIM) with $n$ spins was simulated on a quantum computer. The estimated resources were $n^2$ for the number of gates and ...
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Support for Qubit Leakage Errors in Stim

If I were interested in modeling qubit leakage errors using stim, how should I go about this? There are two scenarios of interest: Any leaked population can be treated as lost, until I reset the ...
4 votes
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Is there a "simple" way to tell if a stabilizer code is degenerate?

Suppose I have a stabilizer code defined by $m$ independent Pauli strings. Is there a "simple" way to check if the code is degenerate or not? As test cases the $[[5,1,3]]$ code is not ...
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Confusion in Aaronson's CHP Paper

In section 3 of the paper (https://www.scottaaronson.com/papers/chp5.pdf) it states that bits $x_{ij}, z_{ij}$ determine the $j$th Pauli matrix $P_j$: $00$ means $I$, $01$ means $X$, $11$ means $Y$, $...
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QuTiP lecture: Single-Atom-Lasing

I have tried to run one of the examples in QuTip lectures: single atom lasing. Although I have used the prepared code represented in reference, I have received error and I have not been able to ...
4 votes
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What are the transversal gates of Shor's [[9,1,3]] code?

What are the transversal gates of Shor's $ [[9,1,3]] $ code? Since this is a stabilizer code we have transversal Pauli gates. Indeed, transversal $X^{\otimes 9}$ implements logical $\overline Z$ and ...
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Options for more complex error models on stim?

We're trying to simulate whether some initialization strings are preferable to others. For example, would initializing all qubits to $|0\rangle$ lead to a lower error rate? Is it possible to further ...
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what can be said about stabilizer codes derived by combining stabilizers and logicals

Let $C$ be an $[[n,k]]$ stabilizer code with stabilizers $(S_1,\cdots,S_m)$ and logicals $(\bar Z_1,\cdots,\bar Z_k)$ and $(\bar X_1,\cdots,\bar X_k)$. Let's also assume that the destabilizers $(D_1,\...
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Best implementation for logical CNOT on Shor's code?

As the Shor's code is a CSS code, it admits a transversal implementation of logical CNOT. An immediate implementation may perform 9 (reversed) CNOT, by respecting the order of the qubits. However. ...
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On the photonic implementation of Shor's code

The above picture comes from this paper. I can see that the standard Shor's code has been re-designed. I have two main doubts: I can't figure out in figure (b) how the setting inputs the state $\...

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