The detectability lemma and quantum gap amplification

The quantum analog of a constraint satisfaction problem is a sum of local Hamiltonians - each (term of the) Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. Formalizing the intuitive connection between the ground (minimal) energy of the Hamiltonian and the minimum number of violated constraints is problematic, since the number of constraints being violated is not well defined when the terms in the Hamiltonian do not commute. The detectability lemma proved in this paper provides precisely such a quantitative connection. We apply the lemma to derive a quantum analogue of the classical gap amplification lemma of random walks on expander graphs. The quantum gap amplification lemma holds for local Hamiltonians with expander interaction graphs. Our proofs are based on a novel structure imposed on the Hilbert space, which we call the XY decomposition, which enables a reduction from the quantum non-commuting case to the commuting c...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Dorit Aharonov, Itai Arad, Zeph Landau, Umesh V. Vazirani
• Keywords: Algorithms, Gap Amplification Lemma, Quantum Gap Amplification, Quantum Hamiltonian Complexity, STOC 2009