1. 1 Dorit Aharonov School of Computer Science and Engineering The Hebrew University, Jerusalem, Israel Israel Quantum Hamiltonian Complexity Complexity What is it? What are The implications? Ground states Entanglement Why is it interesting?

2. 2 Modern Church Turing Thesis “ All physically reasonable computational models can be simulated in polynomial time by a Turing machine” Quantum computation: Only Model which threatens this thesis: Seem to have exponential power Corner stone of theoretical computer Science: ≈≈ Computational properties of Quantum are different ProbabilisticQuantum Polynomial time, Equivalence up to Polynomial reductions Post-

3. 3 Quantum computation  Physics Quantum Universality (BQP): The question of the Quantum Universality (BQP): The question of the computational power of the system: Is it fully computational power of the system: Is it fully quantum? quantum? Reductions: Equivalence between systems Reductions: Equivalence between systems from a Computational point of view from a Computational point of view Multiscale Entanglement (examples: QECCs) Multiscale Entanglement (examples: QECCs) Q. Hamiltonian complexity: apply to Cond. matter physics Quantum error correction: Meta stabilityQuantum error correction: Meta stability out of equilibrium out of equilibrium

4. 4 Condensed Matter Physics Local Hamiltonian, (e.g.AKLT) Ground states: What are their properties? Expectation values of various observables? How do two-point correlations behave? And what about the spectral gap?

5. 5        J=1 (red) J=-1 green (1 violation.) NP completeness – Reductions!!! (Polynomial time) Probabilistically checkable proofs (PCP) Inapproximability K-SAT formulas 3-coloring of a graph… Mathematical proofs.. Constraint Satisfaction problem (CSP): n variables, constraints on k-tuples

6. 6 Quantum Hamiltonian Complexity Deep connection between these two major Problems: Local Hamiltonians can be viewed as quantum CSPs Similar questions plus more complications: enter entanglement Power of various Hamiltonian classes… Entanglement properties of ground states… Provides a whole new lance through which to look at Quantum many body physics: The computational point of view.

7. 7 Constraints  Energy Penalties Solution  Ground state CSP  Is ground energy 0 or at least 1? CSP  estimate the ground energy of H. [Cook-Levin[‘69ish]: CSP is NP-complete  Estimating the ground energy is at least as hard as NP Estimating the ground energy is at least as hard as NP CSP & Hamiltonian

8. 8 The Local Hamiltonian Problem Input: Output: ground energy of H a +1/poly(n). [QMA: Like NP, Except both Verifier Circuit and Witness are quantum. are quantum. Input Witness] : U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2 Theorem [Kitaev’98]: The k-local Hamiltonian problem is QMA complete

9. 9 The Cook-Levin Theorem: Computation is local [Cook-Levin’79] History of a computation can be checked locally  can associate a CSP with the local dynamics Timesteps The verifier is mapped to a SAT formula  SAT is NP-complete

10. 10 The Circuit-to-Hamiltonian construction [Kitaev98, Following Feynman82] Hamiltonian whose ground state is the History. Timesteps : L 0 k-1k+1k Feynman’s particle on a line Reduction from any Qcircuit to a local Hamiltonian  Quantum Universality

11. 11 H(0)H(T) U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2 Adiabatic Computation ≈ Quantum Computation [A’vanDamKempeLandauLloydRegev’04] [A’vanDamKempeLandauLloydRegev’04] H(t) ≈ random walk on time steps! Markov chain techniques. L 0 k-1k+1k Spectral gap: Instead of, use a local Hamiltonian H(T) whose ground state is the History. Reduction  Quantum Universality Want adiabatic computation with γ(t)>1/L c from which to deduce answer. H(0) H(T) [FarhiGodstoneGutmanSipser’00] Adiabatic Computation: [FarhiGodstoneGutmanSipser’00] Ground state of H(0) ground state of H(T)

12. 12 2D, 2-local Ham’s (6-states) [A’vanDamKempeLandauRegevLloyd’04] 2-local Ham’s, in general geometry (qubits) (using Gadgets) [KempeKitaevRegev’06] [KempeKitaevRegev’06] 2D, 2-local Ham’s (Using Gadgets) [OliveiraTerhal’05] 1D, 2-local Ham’s ! (using 12 states) [A’IraniKempeGottesman’07] Adiabatic Computation is Quantum Universal (& estimating the ground energy is QMA complete) for much stricter families of Hamiltonians: Perturbation Gadgets: *** *** Reductions ≈ 1Dim result is surprising…

13. 13 Quantum Hamiltonian Complexity: Easy: In P, gs is MPS gs is MPS Hard: BQP complete, QMA complete, NP hard, etc. Constant gap: Correlations decay exponentially for all D [Hastings’05]. Small correlations  Little Entanglement! (data Hiding, Q expander states) In 1D: Limited entanglement too (area law). [Hastings’07]. MPS description of ground state of 1D gapped systems Efficient simulation of 1D gapped adiabatic [Hastings’09] Open: Can the ground state be found classically efficienlty? Open: Correlations & entanglement?

14. 14 3. How hard is local Hamiltonians for restricted Hamiltonians? For a 1/poly(n) gapped 1D system: QCMA hard [A’Ben-OrBrandaoSattath’08]. What if we know the ground state is an MPS? The classical analog (solving 1D CSPs) is easy… Quantumly: NP-hard [SchuchCiracVerstraete’08] 1. Hardness for interesting physical systems: Approximating ground energy of Hubbard model: QMA complete Solving Schrodinger’s eq. for interacting electrons: QMA-hard [SchuchVerstraete’07] 2. Ruling out various physical attempts: “Universal density functional” cannot be efficiently computable unless NP=QMA.[SchuchVerstraete’07] Back to the Hardness side… Some examples

15. 15 The PCP theorem X ≈ Verifier Verifier Verifier X NP PCP Witness\Proof Slightly longer Witness\Proof Gap amplification version [Dinur’07]        CSP Y  CSP Z Y satisfiable: Z is satisfiable Y is not : Z violated > 10%. (Hardness of approximation!!!)       

16. 16 Quantum PCP theorem? Quantum Ground Energy amplification?        What would be the implications? Hardness of Quantum approximations.. Hardness of Quantum approximations.. Ways to manipulate ground energies, Ways to manipulate ground energies, Maybe spectral gaps (adiabatic Fault-Tolerance?) Maybe spectral gaps (adiabatic Fault-Tolerance?) Mainly: Attempts to follow Dinur’s proof seem to encounter conceptual difficulties: No cloning theorem. No go for QPCP – sophisticated no cloning theorem… On the other hand, a proof might constitute a sophisticated version of QECCs.        Hamiltonian H  Hamiltonian H’ H Frustration free: So is H’ H is not : Ground energy of H’ large and detectable.

17. 17 Quantum Gap Amplification [A’AradLandauVazirani’09] ( A proof of an important ingredient in Dinur’s proof, but without handling the no-cloning issue)               Local terms Larger constraints, defined by walks on the graph Analyzing the ground energy of the new Hamiltonian H’: Requires a sophisticated reduction to a commuting case (The XY decomposition, pyramids, the detectability lemma)

18. 18 Open problems: Quantum PCP? Relations to adiabatic fault tolerance? or: Can we rule out quantum PCP (a sophisticated No-Cloning theorem?) Extending other important classical results, e.g.: 1. Remove degeneracy? (Q Valiant-Vazirani) [see A’BenOrBrandaoSattath’09 ]? 2. Frustration freeness? (QMA1 vs. QMA?)[see Aaronson’08] Rule out other physics programs similar to the universal density functional? Identify the complexity of other types of systems? (interacting electrons), Check the Post-Modern CT thesis for other known systems (field theory)? Much more on the computationally “easy” side: Area laws and entanglement vs. correlations in Dim>1? Finding the ground state for gapped 1D Hamiltonians? Computational power of commuting Hamiltonians?

19. 19 Thanks!