Gottesman's code #240

Krastanov · 2024-03-17T20:30:23Z

A couple of cleanups on top of #238 by @Fe-r-oz

Iterative Decoder Attempted Packages Used: 1. QuantumClifford 2. LDPCDecoders 3. Distances

#how to use # Create two different instances of the quantum Hamming code #code1 = QHamming(5) # r = 5 #code2 = QHamming(7) # r = 7 # Get the parity check matrices for each instance #H1 = parity_checks(code1) #H2 = parity_checks(code2)

Package to be used: Linear Algebra # Example usage #n_i = [2, 3] # Valid example #k_i = [1, 2] #d_i = [1, 1] #r_i = [1, 1] #code = HypergraphProduct(n_i, k_i, d_i, r_i) # Access and use functionalities: #println("Code block size (n):") #println(code_n(code)) #println("X parity-check matrix:") #println(parity_checks_x(code)) #println("Z parity-check matrix:") #println(parity_checks_z(code))

Create hypergraphproductcode.jl

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All modified and coverable lines are covered by tests ✅

Project coverage is 82.75%. Comparing base (51b0ecf) to head (96df6a8).

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@@            Coverage Diff             @@
##           master     #240      +/-   ##
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- Coverage   82.78%   82.75%   -0.03%     
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  Files          55       56       +1     
  Lines        3676     3706      +30     
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+ Hits         3043     3067      +24     
- Misses        633      639       +6

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Fe-r-oz · 2024-03-17T21:27:54Z

Kindly please see the output of j==4 which is

parity_checks(Gottesman(4))
+ XXXXXXXXXXXXXXXX
+ ZZZZZZZZZZZZZZZZ
+ X_X_X_X_YZYZYZYZ
+ X_X_YZYZ_X_XZYZY
+ X_YZ_XZYX_YZ_XZY
+ XZ_YXZ_YXZ_YXZ_Y

This does not match with the Stabilizer of [[16, 10, 3 ]] that is presented in his Gottesman thesis, Table 8.1 Page 91. More details can be found in Pages 90 to 95. There are fundamental changes in the main generators M1 to MJ. That's why he dedicated 3 pages to just checks

Here is what the [[16,10, 3]] Stabilizer is like:

+ XXXXXXXXXXXXXXXX
+ ZZZZZZZZZZZZZZZZ
+ _X_X_X_XZYZYZYZY
+ _X_XZYZYX_X_YZYZ
+ _XZYX_YZ_XZYX_YZ
+ _YXZ_YXZ_YXZ_YXZ

Krastanov · 2024-03-17T22:00:15Z

Aren't they the same up to a row operation?

Krastanov · 2024-03-17T22:03:57Z

There is really only one fundamental feature in these codes: Every single-qubit error needs to be detectable and it has to provide a unique syndrome (that is where the name "quantum Hamming code" comes from). The newly added test file explicitly checks.

Of course I might be missing something, but to convince me you need to at least explain what you mean by "there are fundamental changes in the main generators M1 to MJ".

Fe-r-oz · 2024-03-17T22:06:15Z

Aren't they the same up to a row operation?

Only first two rows are the same.

The third row to the last row are different which are a large number of rows.

Krastanov · 2024-03-17T22:10:50Z

Check the screenshot I put above. The two stabilizer tableaux you posted correspond to the same state (the same code) because they can be obtained from each other by performing row operations. Take the first row and multiply it in 3rd-to-6th rows and you get the other stabilizer tableau. Just like the matrix representing a system of linear equations does not change its solution after performing row operations.

Certain, when talking about codes it is worthwhile to also consider other features, like row-weights, but those also do not change between the two representations.

Fe-r-oz · 2024-03-17T22:11:45Z

Of course I might be missing something, but to convince me you need to at least explain what you mean by "there are fundamental changes in the main generators M1 to MJ".

But there is a fundamental design error according to my understanding of his thesis, M1..Mj , first two digits can't be X.

The stabilizer is very different from what he presented. You can think of it if you write it as a full blown matrix. This is clearly violating the symmetric structure!

Please see his comments below:

From my understanding of his thesis, M1 to Mj starts with I...not X. That's the characteristics of this family. There are many other changes in the stabilizer as well Please see pages 90 to 95

It would be better to produce the stabilizer that is in his thesis.

Krastanov · 2024-03-17T22:18:00Z

Feroz, I already answered in multiple ways:

the tableaux are equivalent up to a pretty simple row operation
the fundamental property of distinguishing all single-qubit errors is present

Here is a third version of the answer:

The difference between the two approaches is simply using not x instead of x when enumerating stabilizer.

From my understanding of his thesis, M1 to Mj starts with I...not X. That's the characteristics of this family. There are many other changes in the stabilizer as well Please see pages 90 to 95

Why should they? Presumably the answer is that this makes some enumeration a bit easier. Sure, that is a valid reason to pick a given set of generators. But I have shown you above how to prove that this set of generators is the same.

It would be better to produce the stabilizer that is in his thesis.

This type of statements need to be defended. Why is it better? Especially given that the thesis and the paper are basically the same algorithm with slightly different conventions (and it seems the thesis does not describe it in particularly explicit detail).

Fe-r-oz · 2024-03-17T22:22:21Z

This type of statements need to be defended. Why is it better? Especially given that the thesis and the paper are basically the same algorithm with slightly different conventions (and it seems the thesis does not describe it in particularly explicit detail).

It's the opposite, the thesis describe many checks and is explicit for straight 3 pages from 92 to 95 and the paper does not. Why did he then explained those checks if there was no need to? He could have described the same 3 paragraph algorithm in his thesis, but he explained in with many checks for even j and odd j for both [[8,3,3]] and [[16,10,3]] cases? There is more detail then just matter of change of conventions

Fe-r-oz · 2024-03-17T22:35:42Z

We can write the full blown H with 5 rows and 32 columns matrix from this stabilizer code just like the stabilizer Matrix.

Similarly, we can write the matrix for this stabilizer, and study whether its correct or not.

Example of [[8,3,3]], similarly, [[16,10,3]] also is very similar to this from his thesis as well.

We can compare the structures. We can also see if there are extended hamming matrices present or not.

Krastanov · 2024-03-17T23:03:35Z

There is more detail then just matter of change of conventions

Feroz, answer with yes or no, do you understand that two parity check matrices are equivalent if they can be converted from one to the other just by row operations?

Second question, yes or no, are the matrices from the thesis and from the paper equivalent?

Krastanov · 2024-03-17T23:49:12Z

I am preparing a small public website that will contain some visualizations related to codes like this. I would like to add the work you have contributed here. May I list you as one of the contributors? Currently I just have your name as Feroz, but most folks will be listed as FirstName LastName, position, institution, e.g. "Joe Smith, undergraduate, UMass Amherst". Let me know how to fill out that information about you if you are comfortable being listed.

Krastanov · 2024-03-18T01:04:32Z

Tests pass, including the new "fundamental feature" test. I am merging this and closing the other related pull request.

Thank you for contributing this! If you did not start this process, this would never have been added! And I learnt quite a bit from the bibliography you shared.

Krastanov · 2024-03-18T01:23:19Z

here are the performance plots:

Here they are up to j=6, but in the test suite there is a test that pushes it to j=12 by explicitly enumerating the errors

Fe-r-oz · 2024-03-18T07:11:22Z

Feroz, answer with yes or no, do you understand that two parity check matrices are equivalent if they can be converted from one to the other just by row operations?

Second question, yes or no, are the matrices from the thesis and from the paper equivalent?

I needed time to be sure about second so I verified the Row Operations of last three rows with XXXXXXXXXXXXXXXX. @Krastanov

yes 2) yes

Fe-r-oz · 2024-03-18T07:28:20Z

I am preparing a small public website that will contain some visualizations related to codes like this. I would like to add the work you have contributed here. May I list you as one of the contributors?

Alright 👍🏻

Feroz Ahmad, graduate, NUCES.

Thank you for contributing this! If you did not start this process, this would never have been added! And I learnt quite a bit from the bibliography you shared.

Thank you as well. 👍🏻

Fe-r-oz added 30 commits February 28, 2024 22:58

Update references.bib

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Iterative Decoder Attempted Packages Used: 1. QuantumClifford 2. LDPCDecoders 3. Distances

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Create QHammingcode.jl

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#how to use # Create two different instances of the quantum Hamming code #code1 = QHamming(5) # r = 5 #code2 = QHamming(7) # r = 7 # Get the parity check matrices for each instance #H1 = parity_checks(code1) #H2 = parity_checks(code2)

Merge pull request #1 from Fe-r-oz/Fe-r-oz-patch-1

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Create hypergraphproductcode.jl

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Rename QHammingcode.jl to qhammingcode.jl

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Fe-r-oz and others added 9 commits March 16, 2024 21:30

Implementation of non-CSS Gottesman codes aka quantum Hamming codes

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undoing whitespaces

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undoing whitespaces

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undoing whitespaces

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minor modification

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updating changelog

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cleanup

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cleanup

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add refereces in markdown docs

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fix gottesman reference

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fix gottesman reference

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Krastanov merged commit 615fbe5 into master Mar 18, 2024
12 of 15 checks passed

Krastanov deleted the gottesmancode branch March 18, 2024 00:09

Krastanov mentioned this pull request Mar 18, 2024

Zoo: [[2^j, 2^-j-2,3]] Gottesman /Quantum Hamming Code #238

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Gottesman's code #240

Gottesman's code #240

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Gottesman's code #240

Gottesman's code #240

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