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arXiv · 0810.1983

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes

Sergey Bravyi·Barbara Terhal
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Abstract

We study properties of stabilizer codes that permit a local description on a regular D-dimensional lattice. Specifically, we assume that the stabilizer group of a code (the gauge group for subsystem codes) can be generated by local Pauli operators such that the support of any generator is bounded by a hypercube of constant size. Our first result concerns the optimal scaling of the distance $d$ with the linear size of the lattice $L$. We prove an upper bound $d=O(L^{D-1})$ which is tight for D=1,2. This bound applies to both subspace and subsystem stabilizer codes. Secondly, we analyze the suitability of stabilizer codes for building a self-correcting quantum memory. Any stabilizer code with geometrically local generators can be naturally transformed to a local Hamiltonian penalizing states that violate the stabilizer condition. A degenerate ground-state of this Hamiltonian corresponds to the logical subspace of the code. We prove that for D=1,2 the height of the energy barrier separating different logical states is upper bounded by a constant independent of the lattice size L. The same result holds if there are unused logical qubits that are treated as "gauge qubits". It demonstrates that a self-correcting quantum memory cannot be built using stabilizer codes in dimensions D=1,2. This result is in sharp contrast with the existence of a classical self-correcting memory in the form of a two-dimensional ferromagnet. Our results leave open the possibility for a self-correcting quantum memory based on 2D subsystem codes or on 3D subspace or subsystem codes.

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Sergey Bravyi, Barbara Terhal. 2008-11-04. A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. https://doi.org/10.1088/1367-2630%2F11%2F4%2F043029

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