1. 11111 arXiv:0708.1021 Codeword stabilized quantum codes (CWS codes for short) Graeme Smith IBM TJ Watson Research Center Joint with: Andrew Cross John Smolin Bei Zeng See also: arXiv:0803.3232

2. Venn Diagram All codes CWS stabilizer classical

3. Outline Quantum Codes CWS Codes (Graph+Classical Code = Quantum Code) Error Correction Conditions Some Examples Encoding Circuits

4. Definition: A Quantum Error Correcting Code is a subspace of. “n qubits” Messages are vectors in dimensional space. dimensions. Corrects t errors if can recover original state after noise acts on t of the qubits. Goal: Large K, large t, small n. Quantum Codes

5. The Pauli Group Pauli group on n qubits: Weight = # of nonidentity factors These form a basis for the set of quantum errors.

6. Distance of a Code Might as well just worry about the Paulis Minimum distance d: Lowest weight Pauli that (nontrivially) maps the code to itself. ((n,K,d)) Can detect d-1 errors Can correct errors.

7. Stabilizer/Additive Codes Let S be an abelian subgroup of n-bit Paulis |S| = Let Gives an ((n,,d)) code We call it [n,k,d]

8. Stabilizer Codes are… Easy to find (cf GF(4)) Easy to encode Like classical linear codes Easy to think about Suboptimal --- ((5,6,2)) circa. ‘97

9. Nonadditive Codes Rains, Hardin, Shor, Sloane ‘97--- ((5,6,2)) Rains  Smolin, S., Wehner ’07  Embarrassing to only have distance 2.

10. ((9,12,3))‏ Our work was inspired by the ((9,12,3)) code of Yu, Chen, Lai and Oh quant-ph/0704.2122. This was the first nonadditive quantum code with distance > 2 that outperforms any known additive code (and even any possible additive code).

11. Codeword Stabilized Codes Ingredient 1: A graph state Ingredient 2: A Classical Code

12. What is a Graph State? A graph state is the +1 eigenvector of a maximal abelian subgroup of the Paulis. Generators: each has a single X and Z's on the nodes to which it’s connected. 1 2 3 4 5 XIIII IXZZI IZZXZ IZXZZ IIZZX

13. Codeword Stabilized Codes Ingredient 1: Graph state Ingredient 2: Classical code Basis for our Code: with

14. X-Z rule (lemma)‏ On a graph state X errors are equivalent to (possibly multiple) Z errors. We call these the induced errors. S i has only one X on bit i so the X's cancel This defines Cl(E)

15. X-Z rule X Z Z Z Z Z Z Z Y Y=XZ

16. Error detection conditions Since all induced errors are Z's, things are essentially classical ( degeneracy )‏

17. Example: [5,1,3] Stabilizer XZZXI IXZZX XIXZZ ZXIXZ XXXXX logical X ZZZZZ logical Z Generators of the stabilizer Can be made into a CWS code by adding in XXXXX to the stabilizer, and using 00000 and ZZZZZ as the codeword operators

18. On a cycle To correct single errors, need to detect double errors If the codewords are 00000 and 11111 a nondetectable error would have to be weight 5 The X-Z rule tells us all single errors on the ring are weight 1, 2, or 3 We need one weight 3 and one weight 2 But the only weight 2 errors are nonadjacent Z Z Z Z Z these cancel

19. Example: ((5,6,2)) Rains, Hardin, Shor, Sloane code “The symmetries discussed above generate a group of order 640. There is an additional symmetry which can be described as follows: First, permute the columns as k->k^3, that is exchange qubits 2 and 3. Next, for each qubit negate one of the Pauli matrices and exchange the other two, where the Pauli matrices negated are Z, Y, X, X, Y, respectively. This increases the size of the symmetry group to 3840. This group acts as the permutation group S5 on the qubits. This is the full group of symmetries of the code. That is, the full subgroup of the semidirect product of S5 and PSU2 5 that preserves the code [10].” !?

20. ((5,6,2)) code 00000 11010 01101 10110 01011 10101 Since weight 3 induced errors are adjacent, the weight can't change by 3 so none of these can be transformed into 00000. Since weight 2 errors are non-adjacent, they can't be transformed amongst each other.

21. ((9,12,3))‏ 9-cycle

22. ((10,18,3))‏ 10-cycle Linear programming bound is ((10,24,3))‏ If cycles are so great, why not a bigger one?

23. ((10,20,3))‏ Linear programming bound is ((10,24,3))‏ If one cycle is good, two must be better

24. Big search problem For all graphs of size n, search for best classical code of a given distance. Super-exponential Turns out to be quite doable for n up to 10 or maybe 11 ((10,24,3)) code meeting linar programming bound: Yu, Chen and Oh, 0709.1780

25. ((10,24,3))‏ ((10,24,3)) code is unique and meets linear programming bound

26. Example: Quantum Goethals-Preparata Codes Grassl and Rotteler found a systematic family of CWS codes in arXiv:0801.2150 Idea 1: Impose large amount of linear structure on the classical code. Do this by translating whole stabilizer codes rather than stabilizer states. Idea 2: Construct the code from the (linear part of) the Goethals and Preparata codes. The Goethals and Preparata codes are famous nonlinear classical binary codes. Result: ((2^m, 2^(2^m-5m+1), 8)) for m ≥ 6--- three extra qubits. (for 64 bit, 35 vs. 32 encoded bits)

27. Encoding Circuits H H H H H C

28. H H H H H C { } Classical Encodes Graph State

29. Encoding Circuits: [5,1,3] H H H H H

30. Future Work Find more codes! Particularly of higher distance Generalize to higher dimensions Looi, Yu, Gheorghiu and Griffiths 0712.1979 Understand strange classical error models What are the transversal gates?