A counterexample for the graph area law conjecture Dorit Aharonov Aram Harrow Zeph Landau Daniel Nagaj Mario Szegedy Umesh Vazirani arXiv:
Background: local Hamiltonians || H i,j || ≤ 1 Assume: degree ≤ const, gap := λ 1 – λ 0 ≥ const. Define: eigenvalues λ 0 ≤ λ 1 ≤ … eigenstates |ψ 0, |ψ 1, … Known: |AB - AB| ||A|| ||B|| exp(-dist(A,B) / ξ) “correlation decay” [Hastings ’04, Hastings-Kumo ‘05,...] Intuition: ((1+λ 0 )I – H) O(1) ≈ ψ 0 [Arad-Kitaev-Landau-Vazirani, ] X := tr[Xψ 0 ]
Area “law”? Conjecture: For any set of systems A ⊆ V Or even, with variable dimensions d 1, …, d n. gap Known: in 1-D correlation decay area law Hastings ‘04 Brandão-Horodecki ‘12 Hastings ’07, Arad-Kitaev-Landau-Vazirani ‘13 further implications: efficient description (MPS), algorithms
This talk dimensionNN33 Entanglement ∝ log(N) Qubit version: n qubits entanglement ∝ n c
Apparent detour: EPR testing |Ãi=|©i?|Ãi=|©i? |Ãi|Ãi Idea: |©i is unique state invariant under UU*. Result: Error ¸ with O(log 1/¸) qubits sent. Previous work used O(loglog(N) + log(1/λ)) qubits [BDSW ‘96, BCGST ’02] log(t) qubits UiUi Ui*Ui*
EPR testing protocol 1.Initial state: 2.Alice adds ancilla in state 3.Alice applies controlled U i i.e. ∑ i |ii| ⊗ U i 4.Alice sends A’ to Bob and Bob applies controlled U i * 5.Bob projects B’ onto t -1/2 ∑ i |i. |ψ AB steps state
Analyzing protocol Subnormalized output state (given acceptance) is Pr[accept] = ψ|M†M|ψ Key claim: || M - |ΦΦ| || ≤ λ Interpretation as super-operators: X = ∑ a,b X a,b |ab| |X = ∑ a,b X a,b |a ⊗ |b T(X) = AXB T|X = (A ⊗ B T )|X T(X) = UXU y T|X = (U ⊗ U * )|X identity matrix |Φ ||M(X)|| 2 ≤ λ||X|| 2 if tr[X]=0 || M - |ΦΦ| || ≤ λ
Quantum expanders A collection of unitaries U 1, …, U t is a quantum (N,t,λ) expander if whenever tr[X]=0 (cf. classical t-regular expander graphs can be viewed as permutations π 1,…,π t such that t -1 ||∑ i π i x|| 2 ≤ λ||x|| 2 whenever ∑ i x i = 0.) Random unitaries satisfy λ≈ 1 / t 1/2 [Hastings ’07] Efficient constructions (i.e. polylog(N) gates) achieve λ≤ 1 / t c for c>0. [various] Recall communication is log(t) = O(log 1/λ)
Hamiltonian construction Start with quantum expander: {I, U, V} dimensionNN33 HLHL HMHM HRHR H L = -proj span { |ψ ⊗ |1 + U|ψ ⊗ |2 + V|ψ ⊗ |3} H R = -proj span { |1 ⊗ |ψ + |2 ⊗ U T |ψ + |3 ⊗ V T |ψ} H M = (|12-|21)(12| - 21|) + (|13-|31)(13| - 31|)
ground state H L = -proj span { |ψ ⊗ |1 + U|ψ ⊗ |2 + V|ψ ⊗ |3} H R = -proj span { |1 ⊗ |ψ + |2 ⊗ U T |ψ + |3 ⊗ V T |ψ} H M = (|12-|21)(12| - 21|) + (|13-|31)(13| - 31|) ≅ X 1,1 X 1,2 X 1,3 X 2,1 X 2,2 X 2,3 X 3,1 X 3,2 X 3,3
constraints X 1,1 X 1,2 X 1,3 X 2,1 X 2,2 X 2,3 X 3,1 X 3,2 X 3,3 H L = -proj span { |ψ ⊗ |1 + U|ψ ⊗ |2 + V|ψ ⊗ |3} X1X1 X2X2 X3X3 UX 1 UX 2 UX 3 VX 1 VX 2 VX 3
constraints X1X1 X2X2 X3X3 UX 1 UX 2 UX 3 VX 1 VX 2 VX 3 H R = -proj span { |1 ⊗ |ψ + |2 ⊗ U T |ψ + |3 ⊗ V T |ψ} XXUXV UXUXUUXV VXVXUVXV
constraints H M = (|12-|21)(12| - 21|) + (|13-|31)(13| - 31|) XXUXV UXUXUUXV VXVXUVXV XU = UX XV = VX X ∝ I N ≅ |Φ N forces X 1,2 = X 2,1 and X 1,3 = X 3,1 expander has constant λ H has constant gap
qubit version n qubits entanglement ∝ n c still qutrits strategy: 1. use efficient expanders 2. use Feynman-Kitaev history state Hamiltonian 3. amplify by adding more qubits
in more detail 1.Let N = 2 n. Efficient expanders require poly(n) two-qubit gates to implement U i, given i as input. Ground state of the Feynman-Kitaev(-Kempe-...) Hamiltonian 2. History state: Given a circuit with gates V 1,..., V T A history state is of the form
amplification Circuit size is poly(n) gap ∝ 1/poly(n) (Highly entangled ground states are known to exist in this case, even in 1-D [Gottesman-Hastings, Irani].) idea: amplify H L and H R by repeating gadgets [Cao-Nagaj] gadgets: [Kempe-Kitaev-Regev ‘03] b a c -Z a Z c -Z b Z c -Z a Z b εZ 1 X a εZ 3 X c εZ 2 X b abc qubits ≈ in {|000,|111} subspace ε 3 Z 1 Z 2 Z
summary 1. EPR-testing with error λ using O(log 1/λ) qubits (Note: testing whether a state equals |ψ requires communication ≈Δ(ψ) := “entanglement spread” of ψ.) 2. Hamiltonian with O(1) gap and n Ω(1) entanglement. caveat: requires either large local dimension or large degree. Questions: O(1) degree, O(1) local dimension? Qutrits in the middle qubits? Longer chain in the middle? Lattices vs. general graphs
possible graph area law conjecture: entanglement ≤ ∑ v log(dim(v)) exp(-dist(v, cut) / ξ)