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 Schmidt vectors; D is the Schmidt rank across the cut.