Dorit Aharonov School of Computer Science and Engineering The Hebrew University, Jerusalem, Israel

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

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

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

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

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. CSP & Hamiltonian 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

8. …. U5 Witness]: Input U4 U3 U2 U1 The Local Hamiltonian Problem Input: Output: ground energy of H <a or >a+1/poly(n). Theorem [Kitaev’98]: The k-local Hamiltonian problem is QMA complete [QMA: Like NP, Except both Verifier Circuit and Witness are quantum.

9. Time steps 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 The verifier is mapped to a SAT formula  SAT is NP-complete

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

11. Adiabatic Computation:[FarhiGodstoneGutmanSipser’00] Ground state of H(0)ground state of H(T) H(T) H(0) L 0 k-1 k k+1 …. U5 U4 U3 U2 U1 H(T) H(0) Adiabatic Computation ≈ Quantum Computation Want adiabatic computation with γ(t)>1/Lc from which to deduce answer. [A’vanDamKempeLandauLloydRegev’04] Instead of , use a local Hamiltonian H(T) whose ground state is the History. Reduction  Quantum Universality Spectral gap: H(t) ≈ random walk on time steps! Markov chain techniques.

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

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

14. Back to the Hardness side… Some examples 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] 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]

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

16.               Quantum Ground Energy amplification? Quantum PCP theorem? Hamiltonian H  Hamiltonian H’ H Frustration free: So is H’ H is not : Ground energy of H’ large and detectable. What would be the implications? Hardness of Quantum approximations.. Ways to manipulate ground energies, 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.

17.               Quantum Gap Amplification[A’AradLandauVazirani’09](A proof of an important ingredient in Dinur’s proof, but without handling the no-cloning issue) Larger constraints, defined by walks on the graph Local terms 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. Open problems: Computational power of commuting Hamiltonians? 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-ModernCT 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?

19. Thanks!