Quantum algorithms for testing properties of distributions
Sergey Bravyi, Aram W. Harrow, et al.
STACS 2010
We consider a family of translation-invariant quantum spin chains with nearestneighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary two-qubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank-1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ) is upper bounded by 1/(n - 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
Sergey Bravyi, Aram W. Harrow, et al.
STACS 2010
Sergey Bravyi, David Gosset
Physical Review Letters
Sergey Bravyi, Barbara M. Terhal, et al.
New Journal of Physics
Sergey Bravyi, Anirban Chowdhury, et al.
PRX Quantum