2 this modified system is unfrustrated, its ground state en- ergy is zero (all the terms are positive semi-definite). The unfrustrated ground state belongs to the intersection of the ground state subspaces of each original H$_{k,k}$$_{+1}$ and is annihilated by all the projector terms. We now choose to focus on a class of Hamiltonians for which each ground states. The paper is organized as follows. First, in Section II we show that the question of non-frustration for qu- dit chain Hamiltonians with general nearest-neighbor in- teractions can be simplified to only Hamiltonians that are sums of projector terms[14]. We then analytically show under what conditions zero energy ground states for this system exist. Second, in Section III we show how to search for and approximate the ground states numer- ically and analyze the efficiency of finding the required MPS. We identify an interesting class of unfrustrated qu- dit chain Hamiltonians, on which our MPS methods do not work well. Led by our numerical work, we conjecture that these ground states are highly entangled. Finally, we summarize our results and conclude with an outlook to further work in Section IV. Figure 1: A qudit chain with nearest neighbor interactions H$_{k,k}$$_{+1}$ given by (2). The matrices Γ [ k ] are a local MPS de- scription (1) of the state. P$_{k,k}$$_{+1}$ = r ∑ p =1 | v p k,k $_{+1}$〉〈 v p k,k $_{+1}$| (5) is a random rank r projector acting on a d $^{2}$-dimensional Hilbert space of two qudits, chosen by picking an or- thonormal set of r random vectors (a different set for every qudit pair – we are not assuming translational in- variance). We now find conditions governing the existence of zero energy ground states (from now on, called solutions in short). We do so by counting the number of solutions possible for a subset of the chain, and then adding an- other site and imposing the constraints given by the Hamiltonian. Suppose we have a set of s$_{n}$ linearly independent solu- tions for the first n sites of the chain in the form We investigate chains of d -dimensional quantum par- ticles (qudits) with nearest-neighbor interactions. The Hamiltonian of the system, similar to MPS, with i$_{k}$ = 1 , . . . , d and α$_{k}$ = 1 , . . . , s$_{k}$ ; here and below all the repeated indices are summed over. The Γ ’s satisfy the linear independence conditions 1 ψ i$_{1}$,...,i n α n = Γ i$_{1}$, [1] α$_{1}$ Γ i$_{2}$, [2] α$_{1}$α$_{2}$ . . . Γ i n$_{−}$ $_{1}$, [ n − 1] α n$_{−}$ $_{2}$,α n$_{−}$ $_{1}$Γ i n , [ n ] α n$_{−}$ $_{1}$,α n (6) Γ i k , [ k ] α k$_{−}$ $_{1}$,α $_{k}$x$_{α}$$_{k}$ = 0 , ∀ i$_{k}$$^{, α$_{k}$}$− 1 ⇐⇒ x$_{α}$$_{k}$ = 0 , ∀ α$_{k}$. ψ i$_{1}$,...,i $^{n}$+1 α $^{n}$+1 = ψ i$_{1}$,...,i $^{n$_{−}$}$1 α $^{n$_{−}$}$1 Γ i n , [ n ] α n$_{−}$ $_{1}$,α $_{n}$Γ i n $_{+1}$, [ n +1] α n ,α $^{n}$+1 . (7) 〈 v p n,n $_{+1}$| i$_{n}$i$_{n}$$_{+1}$ 〉 Γ i n , [ n ] α n$_{−}$ $_{1}$,α $_{n}$Γ i n $_{+1}$, [ n +1] α n ,α $^{n}$+1 = 0 (8) We now add one more site to the chain, impose the con- straint P$_{n,n}$$_{+1}$ and look for the zero-energy ground states for n + 1 sites in the form The unknown matrix Γ i n $_{+1}$, [ n +1] α n ,α $^{n}$+1 must satisfy for all values of $^{α$_{n}$}$− $_{1}$, α$_{n}$$_{+1}$ and p , with | v p n,n $_{+1}$〉 vectors defined in (5). This results in a system of linear equations with C$_{pα}$ n$_{−}$ $_{1}$,i n $_{+1}$α n = 〈 v p n,n $_{+1}$| i$_{n}$i$_{n}$$_{+1}$ 〉 Γ i n , [ n ] α n$_{−}$ $_{1}$,α n a matrix C$_{pα}$ n$_{−}$ $_{1}$,i n $_{+1}$α n Γ i n $_{+1}$, [ n +1] α n ,α $^{n}$+1 = 0 , (9) H = N − 1 ∑ k =1 H$_{k,k}$$_{+1}$ (2) H$_{k,k}$$_{+1}$ = E ( k ) 0 P (0) k,k +1 + ∑ p E ( k ) p P ( p ) k,k $_{+1}$, (3) II. WHEN IS A QUDIT CHAIN UNFRUSTRATED? is 2-local (each H$_{k,k}$$_{+1}$ acts non-trivially only on two neighboring qudits)(Fig. 1). Our goal is to find the nec- essary and sufficient conditions for the quantum system to be unfrustrated – its ground state is also a common ground state of all of the local terms H$_{k,k}$$_{+1}$ . The local terms can be written as where E ( k ) 0 is the ground state energy of H$_{k,k}$$_{+1}$ and each P ( p ) k,k +1 is a projector onto the subspace spanned by the eigenstates of H$_{k,k}$$_{+1}$ with energy E ( k ) p . The question of existence of a common ground state of all the local terms is equivalent to asking the same question for a Hamilto- nian whose interaction terms are with P$_{k,k}$$_{+1}$ = ∑ r p $_{=1}$P ( p ) k,k +1 projecting onto the excited states of each original interaction term H$_{k,k}$$_{+1}$ . When $_{H}$′ k,k +1 = $^{I$_{1}$$_{,...,k}$}$− 1 ⊗ P$_{k,k}$$_{+1}$ ⊗ I$_{k}$$_{+2}$$_{,...,N}$ , (4) y$_{α}$$_{k}$$_{−}$$_{1}$ Γ i$_{k}$, [ k ] α$_{k}$$_{−}$$_{1}$,α$_{k}$ = 0 , ∀ i$_{k}$, α$_{k}$ ⇐⇒ y$_{α}$$_{k}$$_{−}$$_{1}$ = 0 , ∀ α$_{k}$$_{−}$$_{1}$ 1 Note that this is not the standard MPS form, which also requires linear independence in the other direction, i.e. In this case s$_{k}$ would be the Schmidt rank for the partition of the qudits into (1 , . . . , k ) and ( k + 1 , . . . , n ) .