[QIP-Sem] QIP seminar, Mon 9/8, 4:15 pm, 36-428, Sergey Bravyi

[Peter Shor]

MIT Quantum Information Processing seminar
Monday 9/8 at 4:15 pm in 36-428
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Sergey Bravyi (IBM)

Additive quantum codes with geometrically local generators

Abstract:

 We study properties of additive quantum error-correcting codes (QECC) that permit
 a local description on a regular D-dimensional lattice. Specifically,
 we assume that
 the stabilizer group of a code has a set of local generators such that
support of any
 generator can be bounded by a rectangular box of size $r=O(1)$.
Our first result concerns the optimal scaling of the distance $d$ with
 the linear size of
 the lattice $L$. We establish an upper bound $dle r L^{D-1}$ which is
 tight for $D=1,2$.
 We also prove a bound $d=O(1)$ for one-dimensional subsystem codes
 assuming that
the gauge group of a code has a set of local generators.
 Secondly we analyze suitability of various QECCs for building a
 self-correcting quantum
 memory. Any additive QECC with geometrically local generators can be
 naturally converted
 to a local Hamiltonian that penalizes states violating the stabilizer
 conditions. A degenerate
 ground state of this Hamiltonian corresponds to the logical subspace
 of a code. We prove
 that for $D=1,2$ the height of an energy barrier separating different
 logical states is upper
 bounded by a constant independent of the lattice size $L$. This result
 demonstrates that a
 self-correcting quantum memory cannot be build using additive QECC in
 dimensions $D=1,2$.

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