1. Product-State Approximations to Quantum Groundstates Fernando G.S.L. Brandão Imperial -> UCL Based on joint work with A. Harrow Paris, April 2013

2. Quantum Many-Body Systems Quantum Hamiltonian Interested in computing properties such as minimum energy, correlations functions at zero and finite temperature, dynamical properties, …

3. Quantum Hamiltonian Complexity …analyzes quantum many-body physics through the computational lens 1.Relevant for condensed matter physics, quantum chemistry, statistical mechanics, quantum information 2. Natural generalization of the study of constraint satisfaction problems in theoretical computer science

4. Constraint Satisfaction Problems vs Local Hamiltonians k-arity CSP: Variables {x 1, …, x n }, alphabet Σ Constraints: Assignment: Unsat :=

5. Constraint Satisfaction Problems vs Local Hamiltonians k-arity CSP: Variables {x 1, …, x n }, alphabet Σ Constraints: Assignment: Unsat := k-local Hamiltonian H: n qudits in Constraints: qUnsat := E 0 : min eigenvalue H1H1 qudit

6. C. vs Q. Optimal Assignments Finding optimal assignment of CSPs can be hard

7. C. vs Q. Optimal Assignments Finding optimal assignment of CSPs can be hard Finding optimal assignment of quantum CSPs can be even harder (BCS Hamiltonian groundstate, Laughlin states for FQHE,…)

8. C. vs Q. Optimal Assignments Finding optimal assignment of CSPs can be hard Finding optimal assignment of quantum CSPs can be even harder (BCS Hamiltonian groundstate, Laughlin states for FQHE,…) Main difference: Optimal Assignment can be a highly entangled state (unit vector in )

9. Optimal Assignments: Entangled States Non-entangled state: e.g. Entangled states: e.g. To describe a general entangled state of n spins requires exp(O(n)) bits

10. How Entangled? Given bipartite entangled state the reduced state on A is mixed: The more mixed ρ A, the more entangled ψ AB : Quantitatively: E(ψ AB ) := S(ρ A ) = -tr(ρ A log ρ A ) Is there a relation between the amount of entanglement in the ground-state and the computational complexity of the model?

11. NP ≠ Non-Polynomial NP is the class of problems for which one can check the correctness of a potential solution efficiently (in polynomial time) E.g. Graph Coloring: Given a graph and 3 colors, color the graph such that no two neighboring vertices have the same color 3-coloring

12. NP ≠ Non-Polynomial NP is the class of problems for which one can check the correctness of a potential solution efficiently (in polynomial time) E.g. Graph Coloring: Given a graph and 3 colors, color the graph such that no two neighboring vertices have the same color 3-coloring The million dollars question: Is P = NP?

13. NP-hardness A problem is NP-hard if any other problem in NP can be reduced to it in polynomial time. E.g. 3-SAT: CSP with binary variables x 1, …, x n and constraints {C i }, Cook-Levin Theorem: 3-SAT is NP-hard

14. NP-hardness A problem is NP-hard if any other problem in NP can be reduced to it in polynomial time. E.g. 3-SAT: CSP with binary variables x 1, …, x n and constraints {C i }, Cook-Levin Theorem: 3-SAT is NP-hard E.g. There is an efficient mapping between graphs and 3-SAT formulas such that given a graph G and the associated 3-SAT formula S G is 3-colarable S is satisfiable

15. NP-hardness A problem is NP-hard if any other problem in NP can be reduced to it in polynomial time. E.g. 3-SAT: CSP with binary variables x 1, …, x n and constraints {C i }, Cook-Levin Theorem: 3-SAT is NP-hard E.g. There is an efficient mapping between graphs and 3-SAT formulas such that given a graph G and the associated 3-SAT formula S G is 3-colarable S is satisfiable NP-complete: NP-hard + inside NP

16. Complexity of qCSP Since computing the ground-energy of local Hamiltonians is a generalization of solving CSPs, the problem is at least NP-hard. Is it in NP? Or is it harder? The fact that the optimal assignment is a highly entangled state might make things harder…

17. The Local Hamiltonian Problem Problem Given a local Hamiltonian H, decide if E 0 (H)=0 or E 0 (H)>Δ E 0 (H) : minimum eigenvalue of H Thm (Kitaev ‘99) The local Hamiltonian problem is QMA- complete for Δ = 1/poly(n)

18. The Local Hamiltonian Problem Problem Given a local Hamiltonian H, decide if E 0 (H)=0 or E 0 (H)>Δ E 0 (H) : minimum eigenvalue of H Thm (Kitaev ‘99) The local Hamiltonian problem is QMA- complete for Δ = 1/poly(n) (analogue Cook-Levin thm) QMA is the quantum analogue of NP, where the proof and the computation are quantum. Input Witness U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2

19. The meaning of it It’s widely believed QMA ≠ NP Thus, there is generally no efficient classical description of groundstates of local Hamiltonians Even very simple models are QMA-complete E.g. (Aharonov, Irani, Gottesman, Kempe ‘07) 1D models “1D systems as hard as the general case”

20. The meaning of it It’s widely believed QMA ≠ NP Thus, there is generally no efficient classical description of groundstates of local Hamiltonians Even very simple models are QMA-complete E.g. (Aharonov, Irani, Gottesman, Kempe ‘07) 1D models “1D systems as hard as the general case” What’s the role of the acurracy Δ on the hardness? … But first what happens classically?

21. PCP Theorem PCP Theorem (Arora et al ’98, Dinur ‘07) : There is a ε > 0 s.t. it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm -NP-hard even for Δ=Ω(m) -Equivalent to the existence of Probabilistically Checkable Proofs for NP. -Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n 1-ε -factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)

22. PCP Theorem PCP Theorem (Arora et al ’98, Dinur ‘07) : There is a ε > 0 s.t. it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm -NP-hard even for Δ=Ω(m) -Equivalent to the existence of Probabilistically Checkable Proofs for NP. -Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n 1-ε -factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)

23. PCP Theorem PCP Theorem (Arora et al ’98, Dinur ‘07) : There is a ε > 0 s.t. it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm -NP-hard even for Δ=Ω(m) -Equivalent to the existence of Probabilistically Checkable Proofs for NP. -Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n 1-ε -factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)

24. PCP Theorem PCP Theorem (Arora et al ’98, Dinur ‘07) : There is a ε > 0 s.t. it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm -NP-hard even for Δ=Ω(m) -Equivalent to the existence of Probabilistically Checkable Proofs for NP. -Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n 1-ε -factor))

25. Quantum PCP? The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E 0 (H)=0 or (ii) E 0 (H) > εm. -(Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits. -Equivalent to estimating mean groundenergy to constant accuracy (e o (H) := E 0 (H)/m) - And related to estimating energy at constant temperature - At least NP-hard (by PCP Thm) and in QMA

26. Quantum PCP? The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E 0 (H)=0 or (ii) E 0 (H) > εm. -(Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits. -Equivalent to estimating mean groundenergy to constant accuracy (e o (H) := E 0 (H)/m) - And related to estimating energy at constant temperature - At least NP-hard (by PCP Thm) and in QMA

27. Quantum PCP? The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E 0 (H)=0 or (ii) E 0 (H) > εm. -(Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits. -Equivalent to estimating mean groundenergy to constant accuracy (e o (H) := E 0 (H)/m) - And related to estimating energy at constant temperature - At least NP-hard (by PCP Thm) and in QMA

28. Quantum PCP? The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E 0 (H)=0 or (ii) E 0 (H) > εm. -(Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits. -Equivalent to estimating mean groundenergy to constant accuracy (e o (H) := E 0 (H)/m) - Related to estimating energy at constant temperature - At least NP-hard (by PCP Thm) and in QMA

29. Quantum PCP? The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E 0 (H)=0 or (ii) E 0 (H) > εm. -(Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits. -Equivalent to estimating mean groundenergy to constant accuracy (e o (H) := E 0 (H)/m) - Related to estimating energy at constant temperature - At least NP-hard (by PCP Thm) and in QMA

30. Quantum PCP? NP QMA qPCP ? ?

31. Previous Work and Obstructions (Aharonov, Arad, Landau, Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm (gap amplification) But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignment

32. Previous Work and Obstructions (Aharonov, Arad, Landau, Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm (gap amplification) But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignment (Bravyi, Vyalyi ’03; Arad ’10; Hastings ’12; Freedman, Hastings ’13; Aharonov, Eldar ’13, …) No-go for large class of commuting Hamiltonians and almost commuting Hamiltonians But: Commuting case might always be in NP

33. Going Forward Can we understand why got stuck in quantizing the classical proof? Can we prove partial no-go beyond commuting case? Yes, by considering the simplest possible reduction from quantum Hamiltonians to CSPs.

34. Mean-Field… …consists in approximating groundstate by a product state is a CSP Successful heuristic in Quantum Chemistry (Hartree-Fock) Condensed matter (e.g. BCS theory) Folklore: Mean-Field good when Many-particle interactions Low entanglement in state It’s a mapping from quantum Hamiltonians to CSPs

35. Approximation in NP (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.

36. Approximation in NP (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. X1X1 X3X3 X2X2 m < O(log(n))

37. Approximation in NP (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. X1X1 X3X3 X2X2 m < O(log(n)) E i : expectation over X i deg(G) : degree of G Φ(X i ) : expansion of X i S(X i ) : entropy of groundstate in X i

38. Approximation in NP (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. Then there are products states ψ i in X i s.t. E i : expectation over X i deg(G) : degree of G Φ(X i ) : expansion of X i S(X i ) : entropy of groundstate in X i X1X1 X3X3 X2X2 m < O(log(n))

39. Approximation in NP (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. Then there are products states ψ i in X i s.t. E i : expectation over X i deg(G) : degree of G Φ(X i ) : expansion of X i S(X i ) : entropy of groundstate in X i X1X1 X3X3 X2X2 Approximation in terms of 3 parameters : 1. Average expansion 2. Degree interaction graph 3. Average entanglement groundstate

40. Approximation in terms of average expansion Average Expansion: Well known fact: ‘s divide and conquer Potential hard instances must be based on expanding graphs X1X1 X3X3 X2X2 m < O(log(n))

41. Approximation in terms of degree No classical analogue: (PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σ β /deg(G) γ Parallel repetition: C -> C’ i. deg(G’) = deg(G) k ii. Σ’ = Σ k ii. Unsat(G’) > Unsat(G) (Raz ‘00) even showed Unsat(G’) approaches 1 exponentially fast

42. Approximation in terms of degree No classical analogue: (PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σ β /deg(G) γ Q. Parallel repetition: H -> H’ i. deg(H’) = deg(H) k ????? ii. d’ = d k iii. e 0 (H’) > e 0 (H)

43. Approximation in terms of degree No classical analogue: (PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σ β /deg(G) γ Contrast: It’s in NP determine whether a Hamiltonian H is s.t e 0 (H)=0 or e 0 (H) > αd 3/4 /deg(G) 1/8 Quantum generalizations of PCP and parallel repetition cannot both be true (assuming QMA not in NP)

44. Approximation in terms of degree Bound: Φ G < ½ - Ω(1/deg) implies Highly expanding graphs (Φ G -> 1/2) are not hard instances Obs: (Aharonov, Eldar ‘13) k-local, commuting models

45. Approximation in terms of degree 1-D 2-D 3-D ∞-D …shows mean field becomes exact in high dim Rigorous justification to folklore in condensed matter physics

46. Approximation in terms of average entanglement Mean field works well if entanglement of groundstate satisfies a subvolume law: Connection of amount of entanglement in groundstate and computational complexity of the model X1X1 X3X3 X2X2 m < O(log(n))

47. Approximation in terms of average entanglement Systems with low entanglement are expected to be easy So far only precise in 1D: Area law for entanglement -> MPS description Here: Good: arbitrary lattice, only subvolume law Bad: Only mean energy approximated well

48. New Classical Algorithms for Quantum Hamiltonians Following same approach we also obtain polynomial time algorithms for approximating the groundstate energy of 1.Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07) 2.Dense Hamiltonians, improving on (Gharibian, Kempe ‘10) 3.Hamiltonians on graphs with low threshold rank, building on (Barak, Raghavendra, Steurer ‘10) In all cases we prove that a product state does a good job and use efficient algorithms for CSPs.

49. Proof Idea: Monogamy of Entanglement Cannot be highly entangled with too many neighbors Entropy quantifies how entangled it can be Proof uses information-theoretic techniques to make this intuition precise Inspired by classical information-theoretic ideas for bounding convergence of SoS hierarchy for CSPs (Tan, Raghavendra ‘10, Barak, Raghavendra, Steurer ‘10)

50. Tool: Information Theory 1.Mutual Information 1.Pinsker’s inequality 1.Conditional Mutual Information 1.Chain Rule for some t<k

51. Conditioning Decouples Idea that almost works (c.f. Raghavendra-Tan ‘11) 1. Choose i, j 1, …, j k at random from {1, …, n}. Then there exists t<k such that i j1j1 j2j2 jkjk

52. Conditioning Decouples 2. Conditioning on subsystems j 1, …, j t causes, on average, error <k/n and leaves a distribution q for which j1j1 jtjt j2j2

53. Conditioning Decouples 2. Conditioning on subsystems j 1, …, j t causes, on average, error <k/n and leaves a distribution q for which which implies j1j1 jtjt j2j2

54. Conditioning Decouples 2. Conditioning on subsystems j 1, …, j t causes, on average, error <k/n and leaves a distribution q for which which implies By Pinsker’s: j1j1 jtjt j2j2

55. Conditioning Decouples 2. Conditioning on subsystems j 1, …, j t causes, on average, error <k/n and leaves a distribution q for which which implies By Pinsker’s: j1j1 jtjt j2j2 Choosing k = εn

56. Quantum Information? Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.

57. Quantum Information? Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. Bad news Only definition I(A:B|C)=H(AC)+H(BC)-H(ABC)-H(C) Can’t condition on quantum information I(A:B|C) ρ ≈ 0 doesn’t imply ρ AB is approximately product by meas. C (Ibinson, Linden, Winter ’08) Good news I(A:B|C) still defined Chain rule, etc. still hold I(A:B|C) ρ =0 implies ρ AB is product by measuring C (Hayden, Jozsa, Petz, Winter‘03) information theory

58. Really Good News: Informationally Complete Measurements There exists an informationally-complete measurement M( ρ ) = Σ k tr(M k ρ ) |k><k| s.t. for ρ, σ in D(C d ) and for all k and ρ 1…k, σ 1…k in D((C d )  k )

59. Proof Overview 1.Measure εn qudits with M and condition on outcomes. Incur error ε. 2.Most pairs of other qudits would have mutual information ≤ log(d) / ε deg(G) if measured. 3.Thus their state is within distance d 3 (log(d) / ε deg(G)) 1/2 of product. 4.Witness is a global product state. Total error is ε + d 6 (log(d) / ε deg(G)) 1/2. Choose ε to balance these terms. 5.General case follows by coarse graining sites (can replace log(d) by E i H(X i ))

60. Proof Overview Let … previous argument q : probability distribution obtained conditioning on z j1, …, z jt

61. Proof Overview σ : probability distribution obtained by measuring M on j 1, …, j t and conditioning on outcome info complete measurement

62. Conclusions Can approximate mean energy in terms of degree and amount of entanglement: Monogamy of entanglement in groundstates Mean field exact in the limit of large dimensions No-go against qPCP + “quantum parallel repetition” Tools from information theory are useful

63. Open Questions Go beyond mean field Is there a meaningful notion of parallel repetition for qCSP? Does every groundstate have subvolume entanglement after constant-depth-circuit renormalization? Find more classes of Hamiltonians with efficient algorithms (dis)prove qPCP conjecture! Mean field exact in the limit of large dimensions No-go against qPCP + “quantum parallel repetition” Tools from information theory are useful!