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

By Sergey Bravyi, Barbara M. Terhal. 2008

284 citations recorded by the source corpus.

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 size O(1). 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(LD-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, different logical states can be mapped into each other by a sequence of single-qubit Pauli errors such that the energy of all intermediate 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 (2D) 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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/d6eacf0fd0f17e533f83638f0dbda1b3bbed360c

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Quantum error correction for quantum memories (2013, unknown): “One can prove that any 2D stabilizer code has an energy barrier E barrier ¼ O ð 1 Þ : this result is obtained by showing that there always exist stringlike logical operators for a 2D stabilizer code (Bravyi and Terhal, 2009).”
Statistical mechanics of quantum error correcting codes (2020, background): “From the theorem it follows directly that `A + `A = 2k, the “cleaning lemma” [43, 44].”
Local stabilizer codes in three dimensions without string logical operators (2011, background): “8 Now the two anchors are located on (0, 0, 0) (after shift of the origin) and (a, b, c), and the B is of size (a+w)× (b+ w)× (c+ w).”
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence (2015, background): “In the case were the boundary is one dimensional, we may simply appeal to a known result in quantum coding theory, that a stabilizer code in one dimension with geometrically local generators has constant distance [46, 47].”
Topological order in an exactly solvable 3D spin model (2010, background): “This distance must obey the general upper bound dG = O(L (2)), where L is the smallest of the lattice dimensions, see Theorem 1 in [12].”
Quantum memories at finite temperature (2014, methodology): “It was shown by Bravyi and Terhal (Bravyi and Terhal, 2009) that local two-dimensional stabilizer states, which include the toric code model, cannot support a macroscopic energy barrier.”