Locality bounds on hamiltonians for stabilizer codes

In this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. Weintroduce various notions of k-locality of a stabilizer code, inherited from the associated stabilizergroup. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establishbounds on the locality of any other Hamiltonian whose groundspace contains such a code, whetheror not its Pauli tensor summands commute. Our results provide insight into the cost of creating anenergy gap for passive error correction and for adiabatic quantum computing. The results simplify inthe cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-orderedstabilizer codes arising from surface cellulations.

[1]  Amílcar Sernadas,et al.  Quantum Computation and Information , 2006 .

[2]  H. Briegel,et al.  Graph states as ground states of many-body spin-1/2 Hamiltonians , 2006, quant-ph/0612186.

[3]  Steane,et al.  Simple quantum error-correcting codes. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[4]  David P. DiVincenzo,et al.  The complexity of stoquastic local Hamiltonian problems , 2006, Quantum Inf. Comput..

[5]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[6]  E. Farhi,et al.  Fe b 20 08 Perturbative Gadgets at Arbitrary Orders , 2008 .

[7]  H. Briegel,et al.  Quantum simulation of interacting high-dimensional systems: The influence of noise , 2007, 0706.0154.

[8]  D. DiVincenzo,et al.  Quantum simulation of many-body Hamiltonians using perturbation theory with bounded-strength interactions. , 2008, Physical review letters.

[9]  Howard E. Brandt,et al.  Quantum computation and information : AMS Special Session Quantum Computation and Information, January 19-21, 2000, Washington, D.C. , 2002 .

[10]  J. Eisert,et al.  Entanglement in Graph States and its Applications , 2006, quant-ph/0602096.

[11]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Quantum states far from the energy eigenstates of any local Hamiltonian. , 2003, Physical review letters.

[13]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[14]  H. Bombin,et al.  Homological error correction: Classical and quantum codes , 2006, quant-ph/0605094.

[15]  R. Ho.  Algebraic Topology , 2022 .

[16]  Michael H. Freedman,et al.  Projective Plane and Planar Quantum Codes , 2001, Found. Comput. Math..

[17]  Julia Kempe,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

[18]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.