Error correction for a proposed quantum annealing architecture

Recently, Lechner, Hauke and Zoller [1] have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise. Quantum annealing [2] is a method for solving combinatorial optimization problems by using quantum adiabatic evolution to find the ground state of a classical spin glass. Hoping to extend the reach of quantum annealing in practical devices, Lechner et al. [1] have proposed a scheme, using only geometrically local interactions, for simulating a classical spin system with all-to-all pairwise connectivity. Their scheme may be viewed as a classical low-density parity-check code (LDPC code) [3]; here we point out that the error-correcting power of this LDPC code makes the scheme highly robust against weakly correlated bit-flip noise. Lechner et al. propose representing N logical bits ~b = {bi, i = 1, 2, . . . , N} using K = ( N 2 ) physical bits ~g = {gij , 1 ≤ i < j ≤ N}, where gij encodes bi ⊕ bj and ⊕ denotes addition modulo 2. The K physical variables obey K−N+1 independent linear constraints. Hence only N−1 physical variables are logically independent; we may, for example, choose the independent variables to be {g12, g23, g34, . . . , gN−1,N}. The linear constants may be chosen to be weight-3 parity checks. If weight-4 constraints are also allowed then the parity checks can be chosen to be geometrically local in a two-dimensional array. Higher-dimensional versions of the scheme may also be constructed [1]; we will discuss only the two-dimensional coding scheme here, but the same ideas also apply in higher dimensions. While gij denotes the value of bi⊕ bj in the ideal ground state of the classical spin glass, we use g ij to denote the (possibly noisy) readout of the corresponding physical variable after a run of the quantum annealing algorithm. If the readout is not too noisy, we can exploit the redundancy of the LDPC code to recover the ideal value of {bi ⊕ bj} from the noisy readout ~g ′ with high success probability. Given an error model, we can determine the conditional probability p(~g′|~b) of observing ~g given ~b. Assuming that each ~b has the same a priori probability, we decode ~g by finding the most likely ~b: ~bdecoded = MLE(~g ) = ArgMax~b p(~g |~b), (1)