Southwest Quantum Information and Technology

Eleventh Annual Meeting, February 19-22, 2009
Seattle, Washington

Full Program | Thursday | Friday | Saturday | Sunday

SESSION 3: Error Correction
Session Chair:
1:15-2:00	Barbara Terhal, IBM Research (invited)
	No-Go Results for a 2D Quantum Memory Based on Stabilizer Codes
	Abstract. We study the possibility of a self-correcting quantum memory based on stabilizer codes with geometrically-local stabilizer generators. We prove that the distance of such stabilizer codes in D dimensions is bounded by O(L^{D-1}) where L is the linear size of the D-dimensional lattice. In addition, we prove that in D=1 and D=2, the energy barrier separating different logical states is upper-bounded by a constant independent of L. This shows that in such systems there is no natural energy dissipation mechanism which prevents errors from accumulating. Our results are in contrast with the existence of a classical 2D self-correcting memory, the 2D Ising ferromagnet.
2:00-2:30	Bei Zeng, Massachusetts Institute of Technology
	Generalized Concatenated Quantum Codes
	Abstract. Quantum error-correcting codes play a central role in quantum computation and quantum information. While considerable understanding has now been obtained for a broad class of quantum codes, almost all of this has focused on stabilizer codes, the quantum analogues of classical additive codes. Nevertheless, there are a few known examples of nonadditive codes which outperform any possible stabilizer code. In previous work (a talk given at SQuInT 2008), my colleagues and I introduced the codeword stabilized ('CWS') quantum codes framework for understanding additive and nonadditive codes. Within this framework we found good new nonadditive codes using exhaustive or random search. However, these new codes have no obvious structure to generalize to other cases -- no nonbinary nonadditive code which outperforms any additive code has ever been found since the search space is getting too large. A systematical understanding of constructing good nonadditive CWS codes is still lacking. In this work we provide a systematical method of constructing nonadditive CWS codes by introducing the concept of generalized concatenated quantum codes. Compared to the usual concatenated quantum code construction, the role of the basis vectors of the inner quantum code is taken on by subspaces of the inner code. Using this generalized concatenation method, we systematically construct families of single-error-correcting nonadditive CWS codes, in both binary and nonbinary cases, which outperform any stabilizer codes. Particularly, we construct a ((90,2^{81.825},3)) qubit code as well as a ((840,3^{831.955},3)) qutrit code, which is the first known nonbinary nonadditive code that outperforms any stabilizer codes. For large block lengths, we show that these families of nonadditive codes asymptotically achieve the quantum Hamming bound. What is more, our new method can also be used to construct stabilizer codes. We show that many good stabilizer codes, e.g. quantum Hamming codes, can be constructed this way. Moreover, we found new stabilizer codes with better parameters than previously known, e.g. a [[36,26,4]] qubit code. Based on joint work with Markus Grassl, Peter Shor, Graeme Smith, and John Smolin.
2:30-3:00	Bryan Eastin, National Institute of Standards and Technology
	Restrictions on Transversal Encoded Quantum Gate Sets
	Abstract. Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. In this talk, I discuss new results showing that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.