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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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 ...
Srini Thirumalai's user avatar
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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\...
Zehong Fan's user avatar
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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 this 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 P_k^\dagger F_l P \...
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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 ...
Federico Tiblias's user avatar
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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 ...
orclecle's user avatar
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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 ...
Inm's user avatar
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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 ...
Marco Fellous-Asiani's user avatar
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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 ...
Jiakai Wang's user avatar
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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}, \...
JoJo P's user avatar
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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 ...
JoJo P's user avatar
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5 votes
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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 ...
Yunjie Wang's user avatar
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1 answer
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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:...
David Dentelski's user avatar
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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, ...
am567's user avatar
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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?
Balint Pato's user avatar
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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 $[M_{1},M_{2}]=0$....
am567's user avatar
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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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2 votes
2 answers
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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 ...
am567's user avatar
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The standard form of a CSS code is CSS?

I suspect that if the standard form of a code is $$H = \begin{pmatrix} H_X & 0 \\ 0 & H_Z \end{pmatrix}~, \quad(1)$$ then I can claim that the code is CSS. They way I'm thinking about ...
Jan Olle's user avatar
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What do the cosets of the group $E/Z(E)$ look like? (E is the quantum error group and Z(E) is the centre of E)

If I have the quantum error group $E$ which contains elements of the form: {$ \pm w_{1} \otimes \dots \otimes w_{n}, \pm i w_{1} \otimes \dots \otimes w_{n} $} The centre of $E$: $Z(E) = <iI> = $...
am567's user avatar
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Distance of concatenated codes: a counter example?

In the following answer to a recent question, it is claimed that when we concatenate two error correcting codes of distances $d_1$ and $d_2$, the greatest distance that we can prove without assuming ...
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How to find the order of the error group $E$

The following is taken from "Quantum Error Correction Via Codes Over GF(4)" Calderbank, Rains, Shor, Sloane. We are told that the group $E$ of tensor products $\pm w_{1} \otimes \dots \...
am567's user avatar
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Z error detection circuit

In this video at 0:50, they have two circuits. One claims to detect a Z error in any of four qubits and one detects an X error in any of the four qubits. I don't understand the Z error detection ...
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Does the number of stabilizers bound the number of correctable errors?

Suppose I have a quantum error correcting code with two stabilizers. Does this mean that I can potentially correct at most 3 distinct errors using this code? My reasoning is that for each error, we ...
JRT's user avatar
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Distance of the concatenated quantum error correcting code

If we have two quantum error-correcting qubit $[[n_1, 1, d_1]]$ and $[[n_2,1,d_2]]$ codes then the notes of Preskill says that the concatenation of the codes is a code of distance at least $d_1d_2.$ ...
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how is logical error rate calculated?

I have a CSS code defined through binary matrices $H_X$ and $H_Z$. I also have the logicals $L_X$ and $L_Z$ as binary matrices. I decode $H_X$ and $H_Z$ independently as classical codes. The decoder ...
unknown's user avatar
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What is the motivation for quantum color codes?

I am a beginner in topological quantum codes and have been interested in quantum color codes recently. The 2-D quantum color codes have a very clear physical picture and can check stabilizers locally. ...
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Encoding circuit for $[\![6, 4, 2]\!]$ code

For the famous $[\![4, 2, 2]\!]$ code, there is a circuit to encode two physical states into the logical state (from Roffe's work): As $[\![m, m-2, 2]\!]$ is a class of error-detection code, I am ...
Yunzhe's user avatar
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Quantum error correction terminology

I'm asking for help to clear up something, but instead of asking a question, I'm just writing my attempt of a definition below. I assume it's not this simple, and the below is not the correct ...
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