Code database

[dsvandet]

People working on these topics

Andrew Cross, Drew Vandeth

Problem: Store and lookup quantum error correcting codes and their known properties in a database

• What are the codes, properties, and data representations?

• Need precise definitions throughout

Problem: Given a quantum code $C$ and a defintion of code equivalence determine if $C$ is equivalent a quantum code in the QEC code database.

• Equivalences could be: Permutation (of qubits), Conjugation by (local) Cliffords, Conjugation by (local) Cliffords and qubits permutation, ...

General families of codes

We choose subsystem stabilizer codes defined on qubits as the most general type of code of interest at the beginning of this project. These are codes that correspond to a subgroup of the n-qubit Pauli group. See Nielsen and Chuang, Quantum Information and Computation for an introduction to stabilizer codes.

• Stabilizer codes are a special case with G = [i,S]
  • Gottesman, Stabilizer codes and quantum error correction, quant-ph/9705052
  • Poulin, Stabilizer formalism for operator quantum error correction, quant-ph/0508131
• CSS codes are a special case where the stabilizer group S = [Sx, Sz] is generated by a X-type subgroup Sx and Z-type subgroup Sz
  • Calderbank, Shor, Good quantum error correcting codes exist, quant-ph/9512032
  • Steane, Error Correcting Codes in Quantum Theory, PRL 77, 793 (1996)
• Other important specializations or properties
  • Weakly self-dual CSS codes are CSS codes with a transversal Hadamard gate
    • e.g., Haah et al, Magic state distillation with low space overhead and optimal asymptotic input count, arXiv:1703.07847
  • GF(4)-linear codes are stabilizer codes with a transversal R gate
  • Doubly-even and triorthogonal CSS codes and their generalizations
    • Bravyi, Haah, Magic state distillation with low overhead, arXiv:1209.2426
• Other generalized families of codes are left for potential future incorporation into the software
  • qudit subsystem stabilizer codes
    • e.g., Ashikhmin and Knill, Nonbinary quantum stabilizer codes, quant-ph/0005008
  • non-stabilizer codes
    • CWS codes: Cross et al, Codeword stabilized quantum codes, arXiv:0708.1021
    • XS codes: Ni et al, A non-commuting stabilizer formalism, arXiv:1404.5327
    • Permutation-invariant codes: Ouyang, arXiv:1302.3247
    • Continuous variable codes

Code properties

1. The gauge group as set of Pauli operators that generate a subgroup of the n-qubit Pauli group
   • Low weight and/or "natural" generating sets motivated by geometry or symmetry
2. The stabilizer group is an abelian subgroup of the n-qubit Pauli group. It can be computed from the gauge group.
   • Low weight and/or "natural" generating sets motivated by geometry or symmetry
3. Code parameters
   • n = number of physical qubits
   • k = number of encoded qubits
   • d = minimum distance
   • r = number of gauge qubits
4. Representatives of logical Pauli operators
5. Weight enumerator(s)
6. Degree distribution(s)
7. Automorphism group(s)
8. Transversal gates, gates enacted by qubit permutations, etc.
9. Is the code CSS? GF(4)-linear? Etc.

Examples

The 5-qubit code

• S = [XZZXI, IXZZX, XIXZZ, ZXIXZ]
• G = [i, S]
• [[n, k ,d]] = [[5, 1, 3]], r = 0
• Logical X = XXXXX
• Logical Z = ZZZZZ
• W(x) = 1 + 15 x^4
• (other weight enumerators?)
• (degree distribution?)
• |Aut(S)| = 360 (generators?)
• Not CSS
• GF(4)-linear
• Logical H = (15)(12)(13) . H^5
• Logical R = R^5
• Logical S = (15)(12)(13) . S^5

Cycles are composed left to right.

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Notes

What is the definition of a CSS subsystem code?

1. The gauge group is generated by an X-type subgroup Gx and a Z-type subgroup Gz
2. The corresponding stabilizer is a CSS code
   This includes (1) but is it equivalent?