Preparing Projected Entangled Pair States on a Quantum Computer

1. Preparing Projected Entangled Pair States on a Quantum Computer Martin Schwarz, Kristan Temme, Frank Verstraete University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria Toby Cubitt, David Peréz-García Mathematics and Quantum Information group, Departamento de Analisis Matematico, Facultad de CC Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Ciudad Universitaria, 28040 Madrid, Spain Phy. Rev. Lett. 108, 110502 (2012), arXiv:1104.1410

2. Overview • Some PEPS background • Growing PEPS • Run-time bound • Generalizations • Conclusion

3. PEPS properties • Projected entangled-pari states (PEPS) are a natural parameterization of multi-partite quantum states defined on lattices • Conjectured structure of ground states of gapped local Hamiltonians • Proven so by Hastings[1] for 1D case (Matrix Product States) • General PEPS preparation oracles are powerful resources![2] • as powerful as contracting tensor networks or computing their norm • PP-complete, for general PEPS given as classical input Key question raised by Verstraete, Wolf, Peréz-García, Cirac (2006)[3]: Is it possible to prepare PEPS in BQP under mild restrictions? If PEPS generation is hard, what subclass of PEPS are physical? [1] M. Hastings, Journal of Stat. Mech. 2007, P08024 (2007) [2] N. Schuch, M. Wolf, FV, and J.I. Cirac, PRL 98, 140506 (2007) [3] FV, M. Wolf, DPG, and J. Cirac, PRL 96, 220601 (2006)

4. PEPS definition • PEPS are quantum states defined on an arbitrary graph G=(V,E). • Edges of G are associated to maximally entangled pairs states providing virtual indices. • A tensor A(v) of rank k+1 is associated to each vertex v of degree k mapping kvirtual indices to one physical index. • PEPS is created by applying the maps A(v) to the virtual indices.

5. Injective PEPS • A PEPS is called injective, iff the map A(v) is left-invertible (after blocking sites), i.e. • Injectivity is a generic property of many interesting states (e.g. 2D AKLT, all lattices, etc. almost always injective) • Each injective PEPS is the unique ground state of a parent Hamiltonian[4] • There is a standard construction[4] for 2-local parent Hamiltonian H= projector onto the complementof the support of each 2-body reduced density matrix of PEPS [4] DPG, FV, J. I. Cirac, and M. M. Wolf, Quant. Inf. Comp 8, 0650 (2008)

6. Growing PEPS Key idea for preparing injective PEPS: • prepare a set of entangled pair states, one for each edge • construct parent Hamiltonian with entangled pairs as groundstate • treat “virtual“ PEPS indices as physical indices as well • grow the PEPS by growing the parent Hamiltonian, vertex by vertex • update parent Hamiltonian edge and vertex terms • add Hamiltonian term to restrict to the physical subspace with energy penalty larger than spectral gap D • project onto unique ground state vector of updated Hamiltonian • This step is probabilistic and requires some more work… • final ground state is the PEPS we want to prepare

7. Growing PEPS

8. Growing PEPS

9. Growing PEPS

10. Growing PEPS

11. Growing PEPS …

12. Projecting on next ground state • Projection onto the next ground state is performed using Phase Estimation[5] • We perform a binary measurement on the energy register (zero or non-zero) • If the outcome is zero, we have perpared the desired ground state • Else, we undo the measurement using the well-known Marriot-Watrous trick[6] and re-try, either starting from the original state or an orthogonal one • By Jordan´s Lemma, analysis reduces to a single 2x2 block, i.e. 2D subspace • Transition may also be effected adiabatically[7] (doesn‘t generalize to G-inj. case) unique GS ofHt+1 unique GS ofHt [5] M. Nielsen, I. Chuang. Quant. Comp. and Quant. Inf., Cambridge Univ. Press. (2000) [6] C. Marriot, J. Watrous, Comput. Complex. 14, pp. 122-152. (2003) [7] D. Aharonov, A. Ta-Shma, SIAM J. Comput. Vol. 37, No. 1, pp. 47-82, (2007)

13. Bounding the run-time • We need a lower bound for the transition probability • Let be the condition number of matrix A • Then, using the PEPS structure, we show that for each injective PEPS • Let k=max(k(A(v))) be the max. condition number over all PEPS projectors A(v) • Let D=min(D(Ht)) be the smallest spectral gap over all Ht generated • Then the algorithm produces PEPS p with probability at least 1-e in time

14. Generalization • The algorithm generalizes to G-injective PEPS[8], where symmetry group G is acting on virtual indices and A(v) is left-invertible on the G-invariantsubspace. A PEPS is called G-isometric, if all A(v)´s are isometries. • Problem: A(v)´s not injective  parent Hamiltonian has degenerate ground space!  How can we undo failed projections which are not rank-1? [8] N. Schuch, J.I. Cirac, DPG, Annals of Physics, Volume 325, Issue 10 (2010)

15. Growing G-injective PEPS • Generalization of the basic algorithm • A related G-isometric PEPS is prepared deterministically by known methods[9,10] to enter the G-invariant subspace first • The G-isometric PEPS is then transformed into the G-injective PEPS as before, maintaining the G-invariant subspace • To undo measurements, we crucially use the PEPS structure and the fact that A(v) is invertible on the the G-invariant subspace to show that in fact we can proceed as before! • Same run-time bound as basic algorithm. • Examples of G-isometric PEPS being prepared initially: • G=Z1: (trivial): product of entangled pairs (same as before) • G=Z2: well-known toric code state • The class of G-injective PEPS includes many physically interesting states with topologically order, such as quantum double models, etc. [9] M. Aguado, G. Vidal, Phys. Rev. Lett. 100, 070404 (2008) [10] FV, M. M. Wolf, and J. I. Cirac. Nature Physics, 5:633–636 (2009)

16. Conclusion • We have shown how to efficiently (in BQP) preparewell-conditionedinjective PEPS on a quantum computerwith high probability. • We have exploited the PEPS structure to construct a sequence of parent Hamiltonians with an induced sequence of unique ground states with lower-bounded overlap, and we have shown how to move through this sequence efficiently to produce the final PEPS. • The result generalizes to G-injective PEPS yielding the same run-time bound of (with k restricted to the G-invariant subspace) • Future directions: • implement quadratic speedup ink by using Quantum Rejection Sampling[11] • can we solve interesting computational problems in this PEPS framework faster?[12] [11] M. Ozols, M. Roetteler, J. Roland, arXiv:1103.2774 (2011) [12] I. Arad, Z. Landau, SIAM Journal on Computing 39, 3089 (2010)

17. References • M. Hastings, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007) • N. Schuch, M. Wolf, F. Verstraete, and J.I. Cirac, PRL 98, 140506 (2007) • F. Verstraete, M. Wolf, D. Perez-Garcia, and J. Cirac, PRL 96, 220601 (2006) • D. Perez-Garcia, F. Verstraete, J. I. Cirac, and M. M. Wolf, Quant. Inf. Comp 8, 0650 (2008) • D. Aharonov, A. Ta-Shma, SIAM J. Comput. Vol. 37, No. 1, pp. 47-82, (2007) • C. Marriot, J. Watrous, Comput. Complex. 14, pp. 122-152. (2003) • M. A. Nielsen, I. L. Chuang. Quantum Comp. and Quantum Inf., Cambridge Univ. Press. (2000) • N. Schuch, J.I. Cirac, D. Peréz-García, Annals of Physics, Volume 325, Issue 10 (2010) • M. Aguado, G. Vidal, Phys. Rev. Lett. 100, 070404 (2008) • F. Verstraete, M. M. Wolf, and J. I. Cirac. Nature Physics, 5:633–636 (2009) • M. Ozols, M. Roetteler, J. Roland, Quantum rejection sampling, arXiv:1103.2774 (2011) • I. Arad, Z. Landau, SIAM Journal on Computing 39, 3089 (2010)