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)
642
questions
1
vote
0
answers
38
views
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 ...
- 1,826
1
vote
1
answer
34
views
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. ...
- 73
2
votes
2
answers
102
views
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 ...
- 73
2
votes
0
answers
61
views
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 ...
- 21
3
votes
0
answers
24
views
When do we need to decode live the syndromes? Various examples to understand
I would like to precisely understand why we need to decode syndrome "live" when we want to perform $T$ gates. For this reason I consider different scenario (some are fictive but helpful to ...
- 1,242
0
votes
1
answer
64
views
Is there any programming project that implements one quantum error correction code or demonstrates it?
A gigantic list of all the error correction codes(Classical Domain, Quantum Domain, Classical-quantum Domain), see https://errorcorrectionzoo.org/all
There are lots of quantum error correction code. ...
- 141
0
votes
1
answer
64
views
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 ...
1
vote
0
answers
25
views
Leakage Detection Circuit
Leakage happens when the state of the system is no longer in the computational subspace that defines a qubit. In Preskill's notes on Fault Tolerant Quantum Computation (Pg 40), a circuit for leakage ...
- 43
2
votes
0
answers
35
views
What is the smallest system to apply nonlocal quantum LDPC codes?
I just learned that there are very promising nonlocal low-density parity check codes (LDPC) codes for quantum error correction. What is the smallest number of qubits necessary to carry out these ...
- 2,110
2
votes
0
answers
60
views
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 ...
- 383
0
votes
1
answer
42
views
Is it possible to adjust the weights of a pymatching graph after it is created?
I'd like to adjust the characteristics of a pymatching graph (specifically the edge weights of individual edges) to explore the effects on the decoding process.
While the ...
- 94
2
votes
1
answer
88
views
15-to-1 distillation protocol implemented with lattice surgery: don't we loose transversality by commuting Clifford toward measurements?
I am following Litinsky's paper explaining the 15-to-1 distillation protocol.
The usual "trick" done to distill, say the $|T \rangle \equiv T |+ \rangle$ magic state, is to use a code ...
- 1,242
4
votes
1
answer
153
views
Proof that for an $[\![n,k,d]\!]$ code we have $N(S)/S\simeq{\cal G}_k$ with $S$ stabilizer
I am reading about Quantum Error Correction and more specifically about the stabilizer formalism. Nielsen's textbook introduces the selection of logical Pauli as a kind of "ad hoc" process ...
1
vote
1
answer
74
views
How is a quantum error correcting code $C(E)=\langle\psi|E^\dagger E|\psi\rangle$ diagonalized?
UIn page 6 of the following paper: https://arxiv.org/pdf/0904.2557.pdf, in the proof of theorem 3:"Suppose equation (30) holds. We can diagonalize $C_{ab}$. This involves choosing new basis $\{...
- 45
3
votes
0
answers
68
views
If you had a 100 qubit fully fault-tolerant quantum computer, what would you do?
As quantum computers improve, eventually we may have error-corrected devices that have very low error rate. However, many applications (Shor's algorithm, quantum chemistry) appear to require thousands ...
- 163
1
vote
0
answers
37
views
The planar surface code vs the toric code
Where can I find a numerical or analytical comparison of the effect of open and closed boundaries on surface code error correction power, such as the logical error rate at the threshold, logical error ...
- 629
2
votes
0
answers
26
views
What are the advantages of building CSS codes out of dual-containing classical codes?
Is there any advantage to building CSS codes out of dual-containing classical codes, as opposed to general CSS codes? Other than satisfying the commutation conditions directly, are there any useful ...
- 21
0
votes
0
answers
42
views
Why does Stim/PyMatching evaluate the threshold error rate differently for even and odd distances?
For the toric code, the threshold error rate value for no measurement errors should be around $p_{th}\approx 0.109$. If simulated with even distances, one finds that this is true. However, when I ...
- 1
0
votes
1
answer
38
views
How can we know the expected value of data errors in Stim?
In Stim, after performing syndrome measurements circuit using stabilizer codes like surface codes, how can we know the expected value of errors occurring on each data qubit? Even if there are no ...
- 31
1
vote
1
answer
26
views
Is the topological cluster state a single-shot stabilizer code?
I saw in the literature claims that no single-shot 3D stabilizer code is known and that only 3D subsystem codes are known to have this feature.
However, isn't the topological cluster state (also known ...
- 629
1
vote
1
answer
72
views
15-to-1 distillation protocol: why is the probability of error of the distilled state $35p^3$ and not $\binom{15}{3} p^3=455p^3$?
The circuit performing the $15$-to-$1$ distillation of the magic state $T|+\rangle$ is shown below (figure taken from Litinski's paper).
The first gates encode the $|+\rangle$ of the 15 qubit code. A ...
- 1,242
2
votes
1
answer
98
views
Injecting arbitrary rotations into the surface code
I'm trying to find an exact resource count for specific QFTs.
A QFT generally needs small power-of-two rotations. These can be constructed out of Clifford+T, but I'm wondering if one can inject them ...
- 1,974
2
votes
1
answer
47
views
Efficiently computing weight enumerators of quantum codes
Are there any packages that efficiently compute weight enumerators (and dual weight enumerators) of quantum error correcting codes?
I'm interested in a general method that works for both stabilizer ...
- 2,820
2
votes
1
answer
58
views
Understanding entanglement distillation via stabilizer codes?
I am having some trouble understanding the distillation of EPR pairs using a stabilizer code. This idea goes back to the paper by Bennet, DiVincenzo, Smolin, and Wooters.
The idea (I think) is that $k$...
- 1,863
1
vote
1
answer
41
views
Why does adding a 50/50 Z error after every initialization, reset and measurement account for randomness of outcomes in Stim?
Stim generates multiple (say N many) noisy samples by first collecting one reference sample, then running a Pauli frame simulator N times to track which errors may flip which measurements and ...
- 33
1
vote
0
answers
60
views
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, ...
- 11
1
vote
1
answer
60
views
How to run Color code on STIM?
I am trying to simulate color code on STIM. I have provided my code below. But I am getting the following error
...
- 11
3
votes
0
answers
42
views
Smallest qudit error correcting/detecting codes
Consider an $((n,2,d))$ qubit code (i.e., encoding a qubit into $n$ qubits). The smallest error detecting code ($d=2$) is in $n=4$ qubits and the smallest error correcting code ($d=3$) is in $n=5$ ...
- 559
0
votes
0
answers
11
views
Which algorithms use amplitude of the measurement of a cricuit?
So my question is motivated by noise. If we have a circuit and we get multiple results.
In which algorithms does it matter to use which amplitude the results have?
For example, when we measure our ...
- 151
1
vote
1
answer
49
views
Effect of too many syndrome measurement cycles on surface code threshold
Assume I have a surface code with distance $d$ and an i.i.d error model with both single qubit depolarization and measurement errors, both with probability $p$.
In this case, one usually repeats the ...
- 629
2
votes
1
answer
45
views
Shor FTEC scheme vs Steane FTEC scheme and other schemes
In the common stabilizer-based error correction procedure, one uses one syndrome qubit to measure the eigenvalue of a weight "w" syndrome by "w" times entangling the syndrome ...
- 629
1
vote
1
answer
43
views
Majorana Quantum Error Correcting Code
I am looking at a paper that consists of a stabilizer code that has only one stabilizer $$-c_1c_2c_3c_4$$. Here $$c_i$$ denote the hermitian majorana modes.
By the Jordan Wigner Transform the ...
- 43
0
votes
0
answers
21
views
Why are the other qubits affected by a phase flip in Shor's 9-qubit code?
We have the following "concatenated" initial state in the 9-qubit Shor code:
$\alpha{(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)}/{2\sqrt{2}}+\beta{(|000\...
- 147
1
vote
1
answer
79
views
Simulating a small distance surface code with individual qubit addressing
I want to simulate a small-distance surface code with either STIM or Qiskit. I want individual control over all the qubits (including the ancillas) to induce controlled errors in my circuit. I want to ...
- 219
0
votes
1
answer
51
views
How to see the effects of the depolarizing channel in qiskit?
I am playing around with adding noise to quantum circuit simulation using qiskit_aer.noise.NoiseModel(). My code creates a quantum circuit and simulates it with ...
0
votes
1
answer
40
views
The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)
On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
- 21
0
votes
1
answer
57
views
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 &...
- 51
6
votes
1
answer
146
views
What are reviews on the search for good quantum LDPC codes?
Can anyone provide a link for an up-to-date review of the current situation in researching good quantum LDPC codes and prospects?
I am interested in questions like:
Does the quantum computing ...
- 629
2
votes
1
answer
51
views
Is there something like error-suppression codes?
My question is a bit on the soft side, and basically asks if there is something between error mitigation and full error correction?
On the one hand, as far as I understand, trying to increase the ...
- 1,455
1
vote
1
answer
61
views
Logical |+> State Preparation
From this paper (page 2), they said that
Surface code qubits can be easily initialized in the logical
+1-eigenstates Logical |0> and Logical |+> of Logical Z and Logical X by initializing all ...
- 65
3
votes
1
answer
128
views
List of codes which can perform non-Clifford gates fault-tolerantly
I am looking to make a comprehensive list of all known quantum error correction codes which can perform one non-Clifford gate fault-tolerantly.
Currently, I have found two codes.
3D Color code
Tri-...
- 559
0
votes
1
answer
82
views
How to apply error-correction to the swap test circuit?
Swap test is a simple quantum circuit to measure the inner product of two quantum states [Wiki: swap test], it only contains three quantum gates. However, due to the error in the real quantum computer ...
- 183
1
vote
0
answers
69
views
Why can Shor code fix arbitrary errors?
This is taken from Page 434 of Nielsen and Chuang:
To simplify the analysis, suppose noise of an arbitrary type is occurring on the first
qubit only; we’ll come back to what happens when noise is ...
- 147
1
vote
1
answer
52
views
For a $d=2t+1$ code, is a Pauli error of weight $> t$ equal to a logical error multiplied by a weight $\leq t$ Pauli error?
Consider a distance $d=2t+1$ stabilizer code.
Consider a Pauli error $E$ of weight $w(E)>t$.
Is it true in general that $E$ can always be written as $E' \times O_L$, where $O_L$ is a logical ...
- 1,242
3
votes
0
answers
62
views
Why is it necessary for the Fault-tolerant error correction condition to take into account two qubit errors?
The Fault-tolerant error correction (FTEC) condition states that for a code distance of 3, if there are no input errors, even if up to one gate fault (including a two qubit error occurring on a two ...
- 41
1
vote
1
answer
60
views
Error correction simulation after state injection
How can one do an error correction simulation after a state injection? I understand that we can keep the subspace with the syndrome measurements, but how can a logical error be verified, if the X and ...
- 629
1
vote
1
answer
39
views
Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
- 147
0
votes
1
answer
29
views
In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
- 147
2
votes
1
answer
38
views
How to disentangle the environment after the quantum erasure channel?
How does error correction disentangle the environment from the encoded state after the quantum erasure channel?
For example; the paper Codes for the Quantum Erasure Channel introduces a 4 qubit code ...
- 21
3
votes
2
answers
128
views
Is the honeycomb code a subsystem code?
The paper https://arxiv.org/abs/2110.09545.pdf claims that the honeycomb code falls outside the definition of a subsystem code.
But here: https://errorcorrectionzoo.org/list/subsystem, here: https://...
- 629