Fernando G.S.L. Brand ão UCL Based on joint work with A. Harrow and M. Horodecki

1. Fernando G.S.L. Brandão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info Information-Theoretic Techniques in Many-Body Physics Day 1

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. Constraint Satisfaction Problems vs Local Hamiltonians k-arity CSP: Variables {x1, …, xn}, alphabet Σ Constraints: Assignment: Unsat :=

4. Constraint Satisfaction Problems vs Local Hamiltonians qudit H1 k-arity CSP: Variables {x1, …, xn}, alphabet Σ Constraints: Assignment: Unsat := k-local Hamiltonian H: nqudits in Constraints: qUnsat := E0 : min eigenvalue

5. Classical vsQuantum Optimal Assignments Finding optimal assignment of CSPs is usually hard (NP-hard) Finding optimal assignment of quantum CSPs (groundstates) seems even harder (QMA-hard; See Daniel Nagaj’stalk) Main difference: Optimal Assignment can be a highly entangled state (unit vector in )

6. The Plan Today: Product-State Approximations to Groundstates - de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas., …) Tomorrow: Groundstates in 1D - area laws and matrix product states - information theory approach (decoupling, state merging, single-shot protocols, …)

7. Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H): today Small total error: Small extensive error:

8. Mean-Field… …consists in approximating the groundstate by a product state is a CSP

9. Mean-Field… …consists in approximating the groundstate by a product state is a CSP

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

11. Hamiltonian on the Complete Graph Consider a Hamiltonian on the complete graph G of size n Hij The Hamiltonian is permutation symmetric: with

12. Quantum de FinettiTheorems (remember Graeme Mitchison’s talk) (Stormer’69, Hudson, Moody ’76) Infinite Quantum de Finetti Theorem (Raggio, Werner ’89) Connection of Infinite Quantum de Finetti with Mean-Field (Caves, Fuchs, Sachs ’01) Proof infinite de Finetti using info-complete measurements (Koenig, Renner ’05) Finite Quantum de Finneti Theorem (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a permutation-symmetric state. Then

13. Product-States Approximation and de FinettiTheorem (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a permutation-symmetric state. Then

14. Product-States Approximation and de FinettiTheorem (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a permutation-symmetric state. Then By de Finetti:

15. Product-States Approximation and de FinettiTheorem (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ1,…,n be a permutation-symmetric state. Then By de Finetti: So Product-states achieve error 2d2/n for mean-energy

16. The Role of Permutation Symmetry To apply quantum de Finetti we need a permutation-invariant Hamiltonian. Can we relax this assumption? Can we show product states do a good job for models not on the complete graph?

17. Product-State Approximation without Symmetry (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E)and |E| local terms. Let {Xi} be a partition of the sites with each Xi having m sites. Ei: expectation over Xi Deg: degree of G S(Xi) : entropy of groundstate in Xi size m X2 X1

18. Product-State Approximation without Symmetry (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E)and |E| local terms. Let {Xi} be a partition of the sites with each Xi having m sites. Then there are states ψi in Xis.t. Ei: expectation over Xi Deg: degree of G S(Xi) : entropy of groundstate in Xi size m X2 X1

19. Approximation in terms of degree Implications to the quantum PCP problem (whether to compute is QMA-hard ): Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP”

20. Approximation in terms of degree Implications to the quantum PCP problem (whether to compute is QMA-hard ): Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP” Bound: ΦG < ½ - Ω(1/deg) implies product states work well on highly expanding graphs (ΦG -> ½) Obs: Restricted to 2-local models (Aharonov, Lior ‘13) k-local, commuting models

21. Approximation in terms of degree …shows mean field becomes exact in high dim ∞-D 1-D 2-D See (Cirac, Kraus, Lewenstein) for rotationally invariant systems 3-D

22. Approximation in terms of average entanglement Product-states do a good job if entanglement of groundstate satisfies a subvolume law: m < O(log(n)) X3 X2 X1

23. Approximation in terms of average entanglement If , Pinsker’s inequality shows product states give error

24. Approximation in terms of average entanglement If , Pinsker’s inequality shows product states give error In constrast, if merely , the theorem shows product states give error

25. When does it fail? E.g. Expander graph G(V, E) with expansion ΦG

26. Intuition: Monogamy of Entanglement Quantum correlations are non-shareable (see Aram Harrow’s and Thomas Vidick’s talks) Cannot be highly entangled with too many neighbors S(Xi) quantifies how much entangled Xi can be with the rest

27. Intuition: Monogamy of Entanglement Quantum correlations are non-shareable (see Aram Harrow’s and Thomas Vidick’s talks) Cannot be highly entangled with too many neighbors S(Xi) quantifies how much entangled Xi can be with the rest Proof uses information-theoretic techniques to make this intuition precise Inspired by classical information-theoretic ideas for bounding convergence of Sum-Of-Squares hierarchy for CSPs (Tan, Raghavendra’10; Barak, Raghavendra, Steurer ‘10)

28. Mutual Information Mutual Information Pinsker’s inequality Conditional MI Chain Rule Upper bound 4+5 for some t ≤ k

29. Quantum Mutual Information Mutual Information Pinsker’s inequality Conditional MI Chain Rule Upper bound 4+5 for some t ≤ k

30. But… …conditioning on quantum is problematic For X, Y, Z random variables No similar interpretation is known for I(X:Y|Z) with quantum Z

31. Conditioning Decouples Idea that almost works. Suppose we have a distribution p(z1,…,zn) 1. Choose i, j1, …, jkat random from {1, …, n}. Then there exists t<k such that Define So j1 i jk j2

32. Conditioning Decouples 2. Conditioning on subsystems j1, …, jtcauses, on average, error <k/n and leaves a distribution q for which , and so By Pinsker: Choosing k = εn jt j1 j2

33. InformationallyComplete Measurements There exists a POVM M(ρ) = Σktr(Mkρ) |k><k| s.t.for all k and ρ1…k, σ1…kin D((Cd)k) (Lacien, Winter ‘12, Montanaro ‘12)

34. Proof Overview • Measure εnqudits with Mand condition on outcomes.Incur error ε. • Most pairs of other qudits would have mutual information ≤ log(d) / εdeg(G) if measured. • Thus their state is within distance d3(log(d) / εdeg(G))1/2of product. • Witness is a global product state. Total error isε + d6(log(d) / εdeg(G))1/2.Choose ε to balance these terms. • General case follows by coarse graining sites • (can replace log(d) by EiH(Xi))

35. Proof Overview Let previous argument … q: probability distribution obtained conditioning on zj1, …, zjt

36. Proof Overview info complete measurement (σ: state obtained by measuring M on j1, …, jt and conditioning on the outcome). Choosing k = εn

37. Other Applications 1: New Classical Algorithms for Q. Hamiltonians Following same approach one obtains polynomial time algorithms for approximating the groundstate energy of Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07) Dense Hamiltonians, improving on (Gharibian, Kempe ‘10) 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.

38. Other Applications 2: New de Finetti Theorems - Classical de Finetti without symmetry: For p(x1,…,xn) with - Q. version using info-complete measurement - Q. version using locality constrained norms (see Aram’s talk) - Version replacing uniform randomness by Santa-Vazirani source (Ramanathan et al ‘13)

39. Thank you!

40. Fernando G.S.L. Brandão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info Information-Theoretic Techniques in Many-Body Physics Day 2

41. The Plan Yesterday: Product-State Approximations to Groundstates - de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas.) Today: Groundstates in 1D - matrix product states - area law and exponential decay of correlations - information theory approach (decoupling, state merging, single-shot protocols) (see NilanjanaDatta’s talk)

42. Quantum Many-Body Systems Quantum Hamiltonian Interested in computing properties such as minimum energy, correlations functions, etc…

43. Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H): today Small total error: Small extensive error:

44. Matrix Product States (Fannes, Nachtergaele, Werner ‘92) D : bond dimension • Only nD2 parameters. • Local expectation values computed in poly(D, n) time • Variational class of states for powerful DMRG • Generalization of product states (MPS with D=1)

45. Area Law in 1D • Let be a n-qubit quantum state • Entanglement Entropy: • Area Law: For all partitions of the chain (X, Y) X Y (Bekenstein ‘73, ….…, Eisert, Cramer, Plenio’10)

46. MPS Area Law X Y For MPS,

47. MPS Area Law X Y For MPS, If is s.t. then it has a MPS description of bound dim. D (Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06)

48. MPS Area Law X Y For MPS, If is s.t. then it has a MPS description of bound dim. D (Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06) (Approx. version) If is s.t. then it can be approximated by a MPS of bound dim. D up to error ε Def:

49. Exponential Decay of Correlations • Let be a n-qubit quantum state • Correlation Function: • Exponential Decay of Correlations: There is ξ > 0 s.t. for all cuts X, Y, Z with |Y| = l l X Z Y

50. MPS EDC ≈ Let Define (w.l.o.g. ) and let λj be the second largest eingenvalue of and λ:= max |λj| If λ is independent of n we say is a gMPS has (1/log(1/|λ|))-EDC