André Chailloux, Université Paris 7 and UC Berkeley Or Sattath, the Hebrew University QIP 2012

 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.

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

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?”

We introduce a natural candidate for a QMA(2)- completeness: Separable version of Local- Hamiltonian. Theorem 1: Separable Local Hamiltonian is QMA-Complete! Theorem 2: Separable Sparse Hamiltonian is QMA(2)-Complete.

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

Separable Sparse Hamiltonian is QMA(2)-Complete.

 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:

SWAP time

SWAP The history state is a tensor product state:

 There is a delicate issue in the argument: SWAP Non-local operator! Causes H to be non-local!

 Control-Swap over multiple qubits is sparse.  Local & Sparse Hamiltonian common properties: LocalSparse Compact description Simulatable Hamiltonian in QMA Separable Hamiltonian in QMA(2)

 The instance that we constructed is local, except one term which is sparse.  Theorem 2: Separable Sparse Hamiltonian is QMA(2)-Complete.

 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 be wrong*.  Separable Local Hamiltonian is QMA- Complete.  Separable Sparse Hamiltonian is QMA(2)- Complete. * Unless QMA = QMA(2).

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

 We would especially like to thank Fernando Brandão for suggesting the soundness proof technique.

 Theorem 2: Separable Sparse Hamiltonian 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

 Merlin is all powerful, but can be malicious.  Arthur is computationally bounded.  Arthur has an input x for some decision problem L, and Merlin wants to convince Arthur that x in L.  They agree on a protocol, such that if the output of the protocol is accept, Arthur says “yes”, and otherwise, Arthur says “no”.  If x in L, Merlin can convince Arthur about that. (completeness)  If x not in L, Merlin cannot fool Arthur about that.

 The protocol is complete if: for every x in L, Arthur accepts with high probability.  The protocol is sound if: for every x not in L, Arthur rejects with high probability.

 Limiting the prover can increase the size of the complexity class:  IP=PSPACE vs. MIP = NEXP.

 We introduce two variants of the Local- Hamiltonian problem, which are natural for the class QMA(2).  We show that Separable Sparse Hamiltonian is QMA(2)-complete.  We show that the Separable Local Hamiltonian is in QMA!

 Restricting the prover can increase the size of a complexity class: IP: =PSPACE MIP: =NEXP

 Merlin A and Merlin B are all powerful, do not share entanglement.  Arthur is skeptical, and limited to BQP.  A problem L ∊ QMA(2) if:  x  L  ∃ |  1  ⊗ |  2  that Arthur accepts w.h.p.  x  L  ∀ |  1  ⊗ |  2  Arthur rejects w.h.p.