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Abstract: We study properties of stabilizer codes that permit a local description on aregular D-dimensional lattice. Specifically, we assume that the stabilizergroup of a code the gauge group for subsystem codes can be generated by localPauli operators such that the support of any generator is bounded by ahypercube of constant size. Our first result concerns the optimal scaling ofthe distance $d$ with the linear size of the lattice $L$. We prove an upperbound $d=OL^{D-1}$ which is tight for D=1,2. This bound applies to bothsubspace and subsystem stabilizer codes. Secondly, we analyze the suitabilityof stabilizer codes for building a self-correcting quantum memory. Anystabilizer code with geometrically local generators can be naturallytransformed to a local Hamiltonian penalizing states that violate thestabilizer condition. A degenerate ground-state of this Hamiltonian correspondsto the logical subspace of the code. We prove that for D=1,2 the height of theenergy barrier separating different logical states is upper bounded by aconstant independent of the lattice size L. The same result holds if there areunused logical qubits that are treated as -gauge qubits-. It demonstrates thata self-correcting quantum memory cannot be built using stabilizer codes indimensions D=1,2. This result is in sharp contrast with the existence of aclassical self-correcting memory in the form of a two-dimensional ferromagnet.Our results leave open the possibility for a self-correcting quantum memorybased on 2D subsystem codes or on 3D subspace or subsystem codes.



Author: Sergey Bravyi, Barbara Terhal

Source: https://arxiv.org/



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