4 value and that increasing χ will not yield a lower value of energy. Therefore, the numerical results suggest that there are no ground states with zero energy. In the previous section we analytically showed that when d ≤ r ≤ d $^{2}$/ 4 there are many zero energy ground states. However, when we try to numerically find these states we see that the final energy converges to zero slowly. This is shown in all the Figures by the curves marked by circles. Out of these there are the critical cases, where r = d 2 4 . These correspond to the curves marked by closed circles in Figures 3 and 5. The numerical investigation of the case d ≤ r ≤ d $^{2}$/ 4 is interesting because it suggests that for large number of spins finding the ground state with small χ , tractable on a normal computer, is very hard. We interpret this as high amount of entanglement among the zero energy ground states and leave the analytical proof of this statement for a follow up paper. IV. CONCLUSIONS We have investigated the no-frustration conditions for a system of qudits on a line with d states per site and random rank r local projector Hamiltonians acting be- tween the nearest neighbor sites. We proved that there are no ground states with zero energy for r > d 2 4 and suf- ficiently large N . The system is not frustrated for r ≤ d 2 4 . This second parameter region further splits into two. For d ≤ r ≤ d 2 4 , many entangled zero energy ground states exist. On the other hand, for r < d we can also construct separable zero-energy ground states (see also Figure 2). We have verified the above numerically, in particular we have seen that when d ≤ r ≤ d $^{2}$/ 4 approximating the ground state energy (finding the ground states) is hard as the states seem to be highly entangled. Future work entails the investigation of the energy gap [16] and the amount of entanglement in the system as a function of the parameters of the chain. Furthermore, we would like to address how far from an eigenstate of the Hamiltonian is the wave function after the truncations are made (i.e. as a function of χ ). Finally, we would like to quantify the nature of the convergence to the ground state starting from an arbitrary state in this framework. 5; all the plots are on semi-log scale and the quantities being plotted are dimensionless. We see that for r < d the final energy converges to the zero energy ground state relatively fast with χ ≪ d N/ $^{2}$. This can be seen in all the figures by the lowest curves (marked by triangles). As can be seen the final energy ob- tained from imaginary time evolution tends toward zero with a steep slope, indicating that the ground state can be approximated efficiently with a small χ in MPS ansatz. The r > d $^{2}$/ 4 case, marked by squares, is shown by the top curves in all the figures. One sees that the final energy plateaus relatively fast in all three cases. This shows that the numerics have converged to a nonzero Figure 4: (Color online) This is a plot for d = 5, and projector ranks of 4, 6, 8. Exact description would require χ = 5 $^{10}$. Figure 3: (Color online) Ground state energy from imaginary time evolution vs. χ for different ranks of the Hamiltonian. This is a plot for d = 4, and projector ranks of 2, 4, 6. Exact description would require χ = d N 2 = 2 $^{20}$. V. ACKNOWLEDGMENTS RM would like to thank Salman Beigi, Michael Artin, Sam Gutmann and especially Alan Edelman for fruitful discussions. RM and PWS would like to thank the Na- tional Science Foundation for the support through grant number CCF-0829421. EF and JG were supported in part by funds provided by the U.S. Department of En- ergy under cooperative research agreement DE-FG02- 94ER40818, the W. M. Keck Foundation Center for Ex- Figure 5: (Color online) The case of d = 6, and projector ranks: 5, 7, 9, 11. Exact description in general would require χ = 6 $^{10}$.