A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians

A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians

The density matrix renormalization group method has been extensively used to study the ground state of 1D many-body systems since its introduction two decades ago. In spite of its wide use, this heuristic method is known to fail in certain cases and no certifiably correct implementation is known, leaving researchers faced with an ever-growing toolbox of heuristics, none of which is guaranteed t...

A polynomial-time algorithm for the ground state of one-dimensional gapped Hamiltonians

A (deterministic) polynomial-time algorithm is proposed for approximating the ground state of (general) one-dimensional gapped Hamiltonians. Let , n, η be the energy gap, the system size, and the desired precision, respectively. Neglecting -dependent subpolynomial (in n) and constant factors, the running time of the algorithm is n ) for η = n−O(1) and is n for η = n−o(1).

A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians Supplementary material

Definition 1. Given a vector |v〉 ∈ H, by a Schmidt decomposition across the (i, i + 1) cut we shall mean a decomposition |v〉 = ∑j=1 λj|aj〉|bj〉 with {|aj〉} (respectively {|bj〉} ) a family of orthonormal vectors of H[1,i] (respectively H[i+1,n]) and with λj ≥ λj+1 > 0 for all 1 ≤ j ≤ D. The vectors |aj〉 will be called the left Schmidt vectors across that cut, and the vectors |bj〉 the right Schmid...

A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians

Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including rec...

Entanglement and Ground States of Gapped Hamiltonians