Approaches to Quantum Gravity

146 F. Markopoulouis concerned with algorithmic searches for a NS given H and . However,ifweapply this method to the example theory of 9.2.1, it is straightforward to see that ithas a large conserved sector. 9Noiseless subsystems in our example theory.Are there any non-trivial noiseless subsystems in H? There are, and they are revealedwhen we rewrite H S in eq. (9.6) asH S = H n′S ⊗ HB S , (9.13)where H n′S := ⊗ n ′ ∈S Hn′ contains all unbraided single node subgraphs in S (the prime onn serves to denote unbraided) and H b S := ⊗ b∈S H b are state spaces associated to braidingsof the edges connecting the nodes. For the present purposes, we do not need to be explicitabout the different kinds of braids that appear in H b S .The difference between the decomposition (9.6) and the new one (9.13) is best illustratedwith an example (details can be found in [4]). Given the stateeq. (9.6) decomposes it as(9.14)while (9.13) decomposes it to(9.15).(9.16)With the new decomposition, one can check that operators in A evol can only affect theH n′S and that Hb S is noiseless under A evol. This can be checked explicitly by showing thatthe actions of braiding of the edges of the graph and the evolution moves commute.We have shown that braiding of graph edges are unaffected by the usual evolutionmoves. Any physical information contained in the braids will propagate coherently underA evol . These are effective coherent degrees of freedom. 10Note that this example may appear simple but the fact that the widely used system oflocally evolving graphs exhibits broken ergodicity (H splits into sectors, characterized by9 The noiseless subsystem method (also called decoherence-free subspaces and subsystems) is the fundamentalpassive technique for error correction in quantum computing. In this setting, the operators are called theerror or noise operators associated with . It is precisely the effects of such operators that must be mitigatedfor in the context of quantum error correction. The basic idea in this setting is to (when possible) encode initialstates in sectors that will remain immune to the deleterious effects of the errors associated with a givenchannel.The term “noiseless” may be confusing in the present context: it is not necessary that there is a noise in theusual sense of a given split into system and environment. As is clear from the definition above, simple evolutionof a dynamical system is all that is needed, the noiseless subsystem is what evolves coherently under thatevolution.10 The physical interpretation of the braids is beyond the scope of this paper. See [4], for an interpretation of thebraids as quantum numbers of the standard model.