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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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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How to perform below operation in Qiskit?
I want to implement the below equation in Qiskit.
$(A \otimes B).\rho.(B^\dagger \otimes A^\dagger)$ where $\rho$ is a density matrix and $A$ and $B$ are CNOT gates.
$$ A=\begin{bmatrix}
1 & 0 &...
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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 do error correction after encoded Bell measurement?
Need some help with the concepts of encoded/logical bell measurement.
Please visualize the picture in your mind. Suppose I have a node with 7+7 qubits side by side, left 7 is $|0_{L}\rangle$ and right ...
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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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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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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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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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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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How to get parity check matrix from a circuit in stim
I am working on QECC and, differently from classical ECC where everything is generally described by the parity-check matrices, QECC generally involves the low-level description of the circuit instead, ...
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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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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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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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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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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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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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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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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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Simulating erasures with stim
We're trying to simulate erasure errors on the surface code using Stim. The threshold for erasure errors on the data qubits (after initialization) is 50%.
We followed the following post: How do I ...
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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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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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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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Commutation relationship and measurement results
There are things I do not understand about the following circuit, and I would appreciate it if you could explain.
...
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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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Smaller quasi-cyclic LDPC code
Recently, IBM proposed a new high-threshold and high-encoding-rate LDPC code family called quasi-cyclic code. In the paper, they give the examples with the smallest [[72, 12, 6]] code.
My question is ...
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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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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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Using LDPC for Information Reconciliation in QKD
I am confused about the role of LDPC in QKD information reconciliation.
Say Alice and Bob have raw key bitstrings $r_A$ and $r_B$, respectively. They might contain some errors and generally $r_A \neq ...
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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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Smallest qudit error correcting/detecting codes
Consider encoding a qubit into $n$ qubits. It is well known that the smallest error detecting code is in $n=4$ qubits and the smallest error correcting code is in $n=5$ qubits.
Is there a similar ...
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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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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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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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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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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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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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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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How horizontal loops on the toric code are all undetectable?
I understand that if we have, for example, the blue path doesn't make any check operator detect any error, but what about the red path? We have the check operator detecting since there is just one ...
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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 ...
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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> = $...
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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 \...
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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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Logical error rate for QECC using qiskit
I've been coding QECC with qiskit recently. As I'm not very skilled in programming I encountered a lot of questions.
Now I'm trying to generate the logical error rate against physical error rate graph ...
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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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How to use surface codes on a quantum circuit?
So far, I have used surface codes on Stim simulations only. The simulation is for a surface code and I want to understand how surface codes work when I comes to quantum circuit. How is this done for a ...
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Why does the qubit give random results in the circuit with rearranged CNOTs for Steane's seven qubit code in Stim?
The following is a part of the syndrome measurement circuit for Steane's seven qubit code in Stim(For ease of viewing, the TICK is omitted.). Since we are considering the detection of X errors, we use ...
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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 ...
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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 (http://theory.caltech.edu/~preskill/ph229/notes/chap7.pdf) says that the ...
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