The complexity of the Separable Hamiltonian Problem

1. AndréChailloux, Université Paris 7 and UC BerkeleyOr Sattath, the Hebrew University The complexity of the Separable Hamiltonian Problem QIP 2012

2. Quantum Merlin-Arthur (QMA):Quantum analogue of NP • Merlin(prover) is all powerful, but malicious.Arthur(verifier) is skeptical, and limited to BQP. • A problem LQMA if: • xL ∃| that Arthur accepts w.h.p. • xL  ∀| Arthur rejects w.h.p.

3. QMA(2) • The same as QMA, but with 2 provers, that do not share entanglement. • Similar to interrogation of suspects:

4. QMA(2): what is “The power of unentanglement”? QMA(2) has been studied extensively: • There are short proofs for NP-Complete problems in QMA(2)[BT’07,ABD+’09,Beigi’10,LNN’11]. • Pure N-representability QMA(2) [LCV’07], not known to be in QMA. • QMA(k) = QMA(2) [HM’10]. • QMA ⊆PSPACE, while the best upper-bound is QMA(2) ⊆ NEXP [KM’01]. [ABD+’09] open problem: “Can we find a natural QMA(2)-complete problem?”

5. Main results We introduce anatural candidate for a QMA(2)-completeness: Separable version of Local-Hamiltonian. Theorem 1: Separable LocalHamiltonian is QMA-Complete! Theorem 2: Separable Sparse Hamiltonianis QMA(2)-Complete. • The Local Hamiltonian problem: • Given , ( acts on k qubits) • is there a state with energy <a • or all states have energy > b ? • A Hamiltonian is sparse if each row has at most polynomial non-zero entries.

6. The Separable Hamiltonian problem • Problem: Given , and a partition of the qubits, decide whether there exists a separable state with energy at most a or all separable states have energies above b? • The witness: the separable state with energy below a. • The verification: Estimation of the energy.

7. Separable Local Hamiltonian • Theorem 1:Separable LocalHamiltonianQMA. • First try: the prover provides the witness, and the verifier checks that it is not entangled. We don’t know how.

8. Separable Local Hamiltonian • Theorem 1:Separable LocalHamiltonianQMA. • Second try: The prover sends the classical description of all k local reduced density matrices of the A part and of the B part separately . • The prover proves that there exists a state which is consistent with the local density matrices. • The state can be entangled, but if exists, then also exists, where , and similarly . • The verifier uses the classical description to calculate the energy:

9. Consistency of Local Density Matrices Consistensy of Local Density Matrices Problem (CLDM): • Input: density matrices over k qubits and sets . • Output: yes, if there exists an n-qubit state which is consistent: . No, otherwise. Theorem[Liu`06]: CLDM QMA.

10. Verification procedure forSeparable Local Hamiltonian • The prover sends:a) Classical part, containing the reduced density matrices of the A part, and the B part.b) A quantum proof for the fact that such a state exists. • The verifier:a) classically verifies that the energy is below the threshold a, assuming that the state is.b) verifies that there exists such a state using the CLDM protocol.

11. Part II Separable Sparse Hamiltonian is QMA(2)-Complete.

12. Reminder: Kitaev’sproof that Local Hamiltonian is QMA-Complete • Given a quantum circuit Q, and a witness , the history state is: , . • Kitaev’s Hamiltonian gives an energetic penalty to: • states which are not history states. • history states are penalized for Pr(Q rejects ) • Only if there exists a witnesswhich Q accepts w.h.p., Kitaev’sHamiltonian has a low energy state.

13. Adapting Kitaev’s Hamiltonian for QMA(2) – naïve approach What happens if we use Kitaev’s Hamiltonian for a QMA(2) circuit, and allow only separable witnesses? Problems: • Even if , then is typically not separable. • Even if is separable , is typically entangled. For this approach to work, one part must be fixed during the entire computation.

14. Adapting Kitaev’s Hamiltonian for QMA(2) • We need to assume that one part is fixed during the computation. • Aram Harrow and Ashley Montanaro have shown exactly this! • Thm: Every QMA(k) verification circuit can be transformed to a QMA(2) verification circuit with the following form:

15. Harrow-Montanaro’s construction SWAP SWAP time

16. Adapting Kitaev’s Hamiltonian for QMA(2) SWAP SWAP The history state is a tensor product state:

17. Adapting Kitaev’s Hamiltonian for QMA(2) • There is a delicate issue in the argument: SWAP Non-local operator!Causes H to be non-local! SWAP

18. Adapting Kitaev’s Hamiltonian for QMA(2): Local vs. Sparse • Control-Swap over multiple qubits is sparse. • Local & Sparse Hamiltonian common properties: C-SWAP=.

19. Adapting Kitaev’s Hamiltonian for QMA(2) • The instance that we constructed is local, except one term which is sparse. • Theorem 2: Separable Sparse Hamiltonianis QMA(2)-Complete.

20. Summary • Known results: Local Hamiltonian & Sparse Hamiltonian are QMA-Complete. • A “reasonable” guess would be that both their Separable version are either QMA(2)-Complete, or QMA-Complete, but it turns out to bewrong*. • Separable Local Hamiltonian is QMA-Complete. • Separable Sparse Hamiltonian is QMA(2)-Complete. * Unless QMA = QMA(2).

21. Open problems • Can this help resolve whether Pure N- Representability is QMA(2)-Complete or not? • QMA vs. QMA(2) ?

22. Thank you! • We would especially like to thank Fernando Brandão for suggesting the soundness proof technique.

23. Pure N-Representability • Similar to CLDM, but with the additional requirement that the state is pure (i.e. not a mixed state). • In QMA(2): verifier receives 2 copies, and estimates the purity using the swap test: passes the swap test.

24. Adapting Kitaev’s Hamiltonian for QMA(2) • Theorem 2: Separable SparseHamiltonian is QMA(2)-Complete. • Why not: Separable Local Hamiltonian is QMA(2)-Complete? • If we use the local implementation of C-SWAP, the history state becomes entangled. • Only Seems like a technicality. SWAP SWAP