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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Integrating BPOSD into STIM
Pymatching doesn't seems to be working for Steane's code on STIM. I am wondering is there any other decoder that I could possibly use? I am trying to simulate circuit-level noise in syndrome ...
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How to find projection operators for spectral decomposition
I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information".
The idea ...
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Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$
Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory".
The exercise states:
We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We
choose the ordering $(...
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3D surface code and unionfind
I was reading some papers about the UnionFind Decoder but I don't get how matching can be used to decode on a 3D surface code. If I understood correctly every layer in our 3D surface code corresponds ...
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Obtaining both $X$ and $Z$ observables for a surface code
I am trying to reproduce the dataset in this paper [arXiv:1705.00857] using Stim.
The paper decomposes the errors into a "pure-error", which is any error that generates the observed syndrome,...
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Explicit examples of non-stabilizer (non-additive) quantum codes with recovery procedure
I am looking for resources that give an explicit example of a non-stabilizer code along with an explicit recovery procedure, i.e., syndrome measurement followed by correction operation for that code.
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How to devise a flag qubit scheme for concatenated codes?
This flag qubit idea was introduced by Rui Chao and Ben Reichardt, last year when I read the paper [arXiv:1705.02329] I thought I understood fully. But today as I revisit I found myself still a bit ...
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On DEMs in stim
I was looking into the stim library (very new to it) and was wondering if someone who has been using it or has experience with DEMs could explain a little of the following.
What exactly are detector ...
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Decoding algorithms for qLDPC codes
Is there any open source decoding algorithm implementation for qLDPC codes under circuit-based noise model?
From the paper arXiv:2308.07915, I know that it is possible to adapt BP-OSD to circuit-based ...
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Erasure errors in quantum error correction
Consider an $[n, k, d]$ classical code. This code can correct up to $d-1$ erasures. For example, if we have the code that maps $0\rightarrow 00$ and $1\rightarrow 11$, this code has distance $2$. ...
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Digitization of errors in QEC
In Nielsen and Chuang, it is stated that any error is given by a quantum channel with Kraus operators $E_i$. A pure state $\vert\psi\rangle\langle\psi\vert$ becomes $\sum_i E_i\vert\psi\rangle\langle\...
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$X$ and $Z$ stabilizer generators are on identical sites. What does $\otimes_{i=1}^n H_i$ do?
I am learning about the CSS construction in Nielsen and Chuang, and I'm trying to understand the effect of $\otimes_{i=1}^n H_i$ on a code with $X$ and $Z$ stabilizer generators on the same sites. (...
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Are there any additional drawbacks of large-weight stabilizer besides measurement and decoding costs?
Sometimes, in the discussion of why low-density parity check codes are useful, I have heard that stabilizer codes with large stabilizer generators have several drawbacks. For example, I have heard it ...
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Does qiskit make any rounding errors or anything similar, or is it a perfect (but slow) simulation of a quantum computer?
I ran the following circuit to calculate the real part of the Bargmann Invariant of a system of quantum states. The invariant is defined as
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In stim's `CORRELATED_ERROR` docs example, where does the 60% come from? [closed]
I do not see what exactly adds up to 60%.
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How to estimate the logical error rate function of different code distance parameters?
I know that if we have physical qubits in a grid of square shape with the same code distance of $X$ and $Z$, $d_X=d_Z=d$, then given physical qubit error rate $p$, we have logical estimate $$p_L\sim A\...
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For arbitrary single-qubit unitary gate, does there exist a code such that the given gate is transversal?
Background
Transversal gates are considered to be fault-tolerant logical operation as they won't lead to the propagation of local errors. Here are some examples:
CSS code has transversal Pauli and ...
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Quantum Error Correction on a hexagonal lattice
I have an exercise in QEC, where I have given a hexagonal lattice with periodic boundary conditions (wrapped around a torus), with a qubit at each vertex. I have also given the Stabilizer generators $...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding a proof in Nielsen & Chuang, specifically the identity in (10.20), which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger F_l P \sqrt{\rho}.$$
By ...
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What does "lift" mean in the Lifted Product (LP) Code?
I am watching this talk about Lifted Product Code (LP). For me, it's quite straightforward to see how to construct a HyperGraph Product Code (HGP) using two separate classical codes. However, I failed ...
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Effect of error correction gates in QCNN
In Iris Cong et. al. (2019) they propose a Quantum Convolutional Neural Network that utilizes mid-circuit measurements to control an error-correcting ansatz $V_j$. This is the equivalent of a pooling ...
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Can we get which error mechanisms acutually happens in Stim?
I'm working on making some decoders for surface codes.
I know there is a great stabilizer simulator, Stim, so I try using it.
Now, I have a question: can we get what type of error mechanisms actually ...
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Preparation of logical state with qLDPC code
I'm recently learning qLDPC code. Although I've seen literature talking about the code construction, parity check, possible logical operation and decoding. How exactly do I prepare e.g. a logical phi ...
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How to convert stim encoder circuit to a parity check matrix?
I would like to take an encoding noiseless circuit and output the corresponding parity check matrix in the the binary symplectic format $(H_x | H_z)$
I'm aware of the related question: How to get ...
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How to properly generate circuit measurement results from detector error model
I'm reading the paper of Google's neural network decoder for surface code, and I'm confused about how to generate the measurement results for pre-training dataset.
If I understand it correctly, the ...
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Current situation of quantum computing with respect to physical vs logical qubits
As an example I'm going to start with Google and IBM. Google has the Sycamore processor right now with 53 physical qubits. However I haven't found any info on how many logical qubits it can actually ...
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Recent references establishing connections between stabilizer error correction and entanglement distillation procedures
I know some concrete entanglement distillation protocols (from this review).
I also know there are connections between error-correction codes and entanglement distillation (again this review but also ...
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How to fix two flip-bit errors in a 3 qubit input
Aware of how a 1-bit flip can be fixed for 3-qubit input by having 2 Ancilla bits, encoding, and using the Toffoli gate to fix the error and decode. How can this idea be extended for a 2-bit flip (...
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If a quantum error correcting code can correct every single-qubit $X$ and $Z$ error, can it also correct every single-qubit $Y$ error?
Let $\mathcal{C}$ be a given arbitrary $n$ qubit quantum error correcting code which can correct any single qubit $X$ error and any single qubit $Z$ error, i.e., $\{X_i\}_{i=1}^n$ & $\{Z_i\}_{i=1}^...
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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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Qutrit Steane Code
There is a well known 5 qubit code $ [[5,1,3]] $ with stabilizer generators
$$
XZZXI \\
IXZZX \\
XIXZZ \\
ZXIXZ
$$
There is a corresponding $ [[5,1,3]]_3 $ code for qutrits given by
\begin{align*}
&...
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For any given parameters, does there always exist a quantum code which saturates the Quantum Hamming Bound?
We know that the Quantum Hamming Bound is as follows:
$$ \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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How is Pauli twirling so powerful?
So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to:
$\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$
where $\sigma^0 = \mathbb{I}, \...
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Quantum error correction circuit from stabilizer codes
I am trying to familiarize myself with QEC codes and their use in quantum communication. Right now, I am trying to implement some more widely known codes in Qiskit. My main problem is I cannot wrap my ...
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What are overlooked noise sources in circuit-level noise?
In the research of quantum error correction, noise is often classified into code capacity noise, phenomenological noise, and circuit-level noise. Among these, circuit-level noise is considered the ...
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Isn't measurement-free quantum error correction uneconomic, i.e. requires too much overhead?
For certain quantum computing systems, measurement is an operation that takes up >90% of the total quantum error correction time. Measurement-free quantum error correction has been proposed, where ...
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What is the clear definition of coherent versus incoherent errors?
The noise types in quantum computing are pivotal topics frequently discussed in academic papers.
Could you provide a clear definition of coherent versus incoherent errors? Additionally, I'm keen to ...
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Single-shot error correction for the surface code with measurement errors
I am trying to implement a single-shot error correction for the surface code with data + measurement errors (both with prob. p), using the (build-in) BP+OSD decoder. I am mostly following these papers:...
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How to interpret the circuit that measures stabilizers in the 5-qubit error correcting code
I would like to understand the exact role of the following circuit (By Vtomole - Own work, CC BY-SA 4.0) in the 5-qubit QECC.
The image attached (from Wikipedia) is captioned "Quantum Circuit ...
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lattice surgery in state picture
I was following Surface code quantum computing by lattice surgery.
A few questions about this paper have been asked in this forum, but I believe my question is new.
The main text took a 'state picture'...
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If $\bar{S}$ has dimension $n-k$, then $|E / \bar{S}|= 2^{n-k}$
This question revolves around understanding a claim made in "Quantum Error Correction via GF(4) Codes".
Background
$S \leqslant S^{\perp} \leqslant E$
Where $E$ is the quantum error group, ...
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Smallest distance 9 self-dual CSS code?
The level-2 concatenated [[7,1,3]] Steane code, and the 4.8.8 color code are both self-dual [[49,1,9]] codes from the CSS family. Is there a distance 9 self-dual CSS code that has less than 49 qubits?
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Understanding the operation of commutation of stabilizer operators
I want to show that the stabilizer operators ($M_{0}, M_{1}, M_{2}, M_{3}$) for the 5-qubit quantum error correcting code:
If $M_{1} = [XXZIZ]$ and $M_{2} = [XZIZX]$
They commute iff $[M_{1},M_{2}]=0$....
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Sample detection events with TableauSimulator
I've been manually driving stim.TableauSimulator to simulate more complex noise models like in a previous post.
The way I've set up my code is to replace instances ...
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Complexity of translation between Measurement Based and Circuit Based with Error Correction
As I know, translation between measurement-based and circuit-based is polynomial, not trivial, and might not be efficient in real time. It behaves this way because given some quantum circuit, we have ...
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A matrix that can be simultaneously diagonalisable can induce a decomposition of its space
Potentially "space" is not the correct word to use in the title, please correct if wrong
I am reading "Quantum Error Correction Via codes over Gf(4)" and I came across something I ...
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