1. Is entanglement “robust”? Lior Eldar MIT / CTP Joint work with Aram Harrow

2. Highlights We prove a conjecture of Freedman and Hastings [‘12] that there exist quantum systems of “robust entanglement”- called NLTS. Tightly connected to the quantum PCP conjecture: one of the major open problems in quantum complexity theory. Connected to the conjecture on quantum locally- testable codes (qLTC). Implies that the folklore that quantum entanglement is “fragile” is an artifact of considering spatially- localized systems.

3. Quantum PCP Complexity perspective

4. Local Hamiltonians n qubits Sum of few-body terms, WLOG projectors. Fractional energy of a Hamiltonian:

5. In complexity theory Local Hamiltonian problem (LHP): Decide: is min Ψ E(Ψ)=0 or min Ψ E(Ψ)>1/poly. QMA - the quantum analog of MA: quantum prover, quantum verifier, bounded error. Theorem [Kitaev ‘98]: LHP is QMA-complete.

6. Classical constraint satisfaction problems (CSP) CSP = classical version of LH. All local terms are diagonal in standard basis. Like 3-SAT formulas. PCP theorem [AS,ALMSS,Dinur]: NP-hard even to distinguish minimal energy 0 from minimal energy > 0.1

7. Quantum PCP: hardness of LHP Approximate-LHP: Given LH, decide whether there exists a state t such that E(t) = 0 or whether E(t)>1/3 for all states t. Approximate LHP is NP-hard, and in QMA. Quantum PCP conjecture [AALV ‘09] : Approximate- LHP is QMA-hard.

8. NLTS Entanglement perspective

9. Quantify entanglement U U U UU U U Complexity of a state = minimal depth circuit |0>

10. “Robustly-quantum” systems 2: NLTS [Freedman, Hastings ‘13]: No low-energy trivial states. Definition: “trivial states” - depth = O(1). NLTS Conjecture: there is a constant c>0, and an infinite family of LH’s in which all “trivial states” have fractional energy at least c.

11. qPCP  NLTS If NLTS is false  NP to every Hamiltonian Suppose Ѱ = U |000..0>. qPCP NLTS U+U+ U K-Local measurement on Ѱ, k2 O(d) -local measurement on |00..0> H

12. Main Result Theorem [E., Harrow ‘15]: There exists an explicit infinite family of 7-local commuting Pauli Hamiltonians, and a constant c such that: If a circuit U generates a quantum state whose energy is at most c  U has depth at least Ω(log(# qubits)). Proof techniques: products of hyper-graphs, locally- testable codes, Hamiltonian (graph)-powering, degree- reduction, uncertainty principle

13. Perspective on NLTS

14. Example of highly-entangled states: quantum code-states Fact: |Ψ 1 >, |Ψ 2 > are quantum code-states, M local measurements then = Corollary : quantum code-states require large circuit depth. Proof : Let U be a depth-d circuit : |Ψ = U|00..0 Define LH : H = Σ i U|00| i U + H distinguishes Ψ from any orthogonal code-state but is 2 d -local  contradiction.  no codestate can be locally generated  Ω(log n) circuit lower-bound.

15. Most systems have fragile entanglement Many LHs have highly entangled ground states. But have tensor product states that are “almost” ground states. Approximating the ground energy is not “crucially” quantum. NLTS means: find (a family of ) LH’s for which the high- entanglement property is “robust”. Ground-state Spec(H)

16. Why are known LH’s not robust ? Topology ! Known systems are embedded on low-dimensional grids Approximate by cutting out boxes B: Fractional energy of is In a sense - question is unfair! Mystery: recent results show that expanding topology per-se is insufficient for NLTS! [BH’13, AE’13, H’12].

17. The construction

18. Our goal: 1. 1.Define a property of quantum states that arise as ground states of LHs. 2. 2.Show circuit lower bounds for this property. 3. 3.Show it is “robust” against constant- fraction energy of the parent LH.

19. Lower Bounds for Quantum Circuits

20. What is this property? Low vertex expansion Canonical image to remember: Moses parting the red sea ! Essentially: two large-measure sets, separated by large distance, require divine intervention (quantum circuits of logarithmic depth)

21. Vertex expansion: analog of Cheeger’s constant for distributions G = hypercube {0,1} n with edges between vertices of dist ≤ m V = {0,1} n E = { (x,y) : dist(x,y) ≤ m} For each S, ∂(S): – the set of strings of S adjacent to S c, union the set of strings in S c adjacent to S. The m-th vertex expansion:

22. Examples of vertex expansion High expansion: A single string (no S, with p(S)<1/2) Uniform distribution on the hypercube (m at least √n) Low expansion: The cat state Quantum code states Uniform superpositions over classical codes.

23. Bounded-depth quantum circuits induce high-expansion dist. Claim: Let U be a quantum circuit of depth d, and Consider its induced distribution on the first n qubits. Its expansion is at least 1/2 for m > 2 1.5d √n.

24. Classical case: Harper’s theorem Claim: Classical circuit C of depth d (bounded fan-in / fan- out), receives n uniform random bits, generates distribution D on n bits. Then D has no large separation. Proof: In D - identify two subsets S,T with measure >1/3. Consider the pre-images under C: C -1 (S), C -1 (T). They are at most O(√n) apart. (Harper’s theorem). So S,T are at distance 2 d √n.

25. Toy proof: CAT state CAT state  has low expansion. Let U be a circuit of depth d that generates CAT : |CAT> = U |00…0> Then H = U Σ i |0> from |cat> : |00…0> - |11…1> (because = 0) H is 2 d local.

26. cat metric explanation Local Hamiltonians examine “pairs” |x><y| Cannot affect pairs (x,y) if d(x,y) > locality !  d(U) > log(n).

27. Quantum circuits also limited by Harper Start with Ψ = U |00…0>, depth(U) = d. Ψ’s expansion minimized at set A. Write Ψ = |A>+|A c >. Consider the state Ψ’ = |A>-|A c >. AAcAc

28. Distinguishing Hamiltonian Put m = √n2 O(d) Consider H = U Σ i |0><0| i U +. Take the √n-degree Chebyshev polynomial of H, C(H). C(H) is m–local. By assumption: is small.  Ψ, Ψ’ have eigenvalues 1, and <1-1/poly in H.  Ψ, Ψ’ have eigenvalues 1, <0.7 in C(H). Claim: this is sufficient to conclude that the distribution of Ψhas high expansion. Why?

29. Recall CAT example: Local Hamiltonians: examine “pairs” of strings in density matrix. Cannot affect pairs whose distance exceeds locality !

30. Large energy gap  high expansion T A AcAc (A,A):same sign in both (A c,A c ):same sign in both ∂(A) – only place they differ  ∂(A) must have large (relative) mass  high expansion

31. Locally-Testable Codes are classical NLTS

32. Goal of this section So far: low expansion  high circuit depth Next goal: Hamiltonian where any low-energy state has low expansion.

33. Local testability Turns out that “local testability” is a “turnkey” property. Definition: Locally-testable codes A subspace C of GF(2) n. A local tester: set of check terms {C i } such that for every word w :

34. Example: Hadamard code Encode x in {0,1} n as the function c(w) = Express as a truth table of length 2 n. Rate is log(n). It is locally testable [BLR] ! Sample a,b at random Accept if and only if c(a) + c(b) = c(a+b). Reads only 3 bits from c LTC with ρ ≥ 1/3.

35. LTC = Classical NLTS We would like a locally defined system that preserves low expansion in the presence of noise. “toy” example: distributions on LTCs Uniform distribution on a code – low expansion Noisy uniform distribution on a code – could have high expansion. Noisy uniform distribution on an LTC – low expansion !

36. How do we make this property quantum ? A hypergraph product [Tillich-Zémor ‘09]: Takes in two classical codes C Produces a quantum code Q = C x TZ C. We show: If C is LTC  Q has a “residual” property of local testability. Informally, means clustering.

37. The hyper-graph product code [Tillich-Zémor ‘09] Parity-check code C, bi-partite graph G(A,B). Consider the transpose of C, C T – G(B,A). Generate two new linear codes: Bits are A x A, B x B Checks are X checks: B ⊗ I +I ⊗ A, Z checks: I ⊗ B+A ⊗ I] AxABxB New check: query C, C T

38. Last tool: uncertainty principle Suppose you have a quantum code with “residual” local testability. Noisy words cluster around the original code. But what if they all cluster around the same codeword ? One cluster  no low-expansion !! What is the answer ? The uncertainty principle ? There is a high-degree of uncertainty  at least two clusters.

39. Putting it all together Take the Hadamard code C Reduce its degree as a bi-partite graph: C  C’. Apply the hypergraph-product to C’: derive a quantum CSS code from a classical code. Argue: the local constraints of the code are an NLTS local Hamiltonian.

40. Constructed LH is NLTS! Fix a low-energy quantum state. It superposes nontrivially on at least two clusters. These clusters correspond to different code-words of an LTC with large distance. Distribution is low expansion So any U approximating state has d(U) > log(n).

41. Clustering around affine spaces ! Live “footage” from F 2 n Live “footage” from F 2 n _

42. Summary of NLTS-LH Hamiltonian corresponds to quantum CSS code. n qubits, O(n) local terms, each is 7-local. Evades previous no-go’s: Not expanding enough for [AE’13] Not 2-local so doesn’t contradict [BH’13]. Low girth so [H’12] doesn’t apply. Rate is only log(n). Distance is √n.

43. Take-home message

44. Results Entanglement is not “inherently fragile”. Entanglement is not “inherently fragile”. Local Hamiltonians can “expel” trivial states from the low side of the spectrum. Local Hamiltonians can “expel” trivial states from the low side of the spectrum. They need to have an expanding topology. They need to have an expanding topology. Expansion per-se is not enough ! Expansion per-se is not enough ! You need an extra structure. What is it ? You need an extra structure. What is it ? Local testability ! Local testability !

45. Future directions Find NLTS Hamiltonians that actually do something useful. Find NLTS Hamiltonians that actually do something useful. To break the log(n) lower-bound, one needs to abandon “light-cone” arguments and encode computational problems… To break the log(n) lower-bound, one needs to abandon “light-cone” arguments and encode computational problems… If you can make it QMA-hard – it’s the qPCP conjecture. If you can make it QMA-hard – it’s the qPCP conjecture. Try to find qLTCs – even with moderate locality. Try to find qLTCs – even with moderate locality.