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Transcript

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 LQMA if: xL ∃| that Arthur accepts w.h.p. xL ∀| Arthur rejects w.h.p. |𝜓〉 The same as QMA, but with 2 provers, that do not share entanglement. Similar to interrogation of suspects: |𝝍𝑨 〉 ⊗ |𝝍𝑩 〉 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 LocalHamiltonian. Theorem 1: Separable Local Hamiltonian is QMA-Complete! Theorem 2: Separable Sparse Hamiltonian Local Hamiltonian problem: isThe QMA(2)-Complete. Given 𝐻 = 𝑖 𝐻𝑖 , (𝐻𝑖 acts on k qubits) A is sparse each row has at is Hamiltonian there a state𝐻 with energyif <a most or all polynomial states have non-zero energy > bentries. ? Problem: Given 𝐻 = 𝑖 𝐻𝑖 , 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. Theorem 1: Separable Local HamiltonianQMA. First try: the prover provides the witness, and the verifier checks that it is not entangled. We don’t know how. Theorem 1: Separable Local HamiltonianQMA. 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 𝜌 which is consistent with the local density matrices. The state 𝜌 can be entangled, but if 𝜌 exists, then also 𝜌′ = 𝜌 𝐴 ⊗ 𝜌𝐵 exists, where 𝜌 𝐴 = 𝑡𝑟𝐵 (𝜌), and similarly 𝜌𝐵 . The verifier uses the classical description to calculate the energy: 𝑇𝑟 𝐻𝜌′ = 𝑖 𝑇𝑟(𝐻𝑖 𝜌′) Consistensy of Local Density Matrices Problem (CLDM): Input: density matrices 𝜎1 , … , 𝜎𝑚 over k qubits and sets 𝐴1 , … , 𝐴𝑚 ⊂ {1, … , 𝑛}. Output: yes, if there exists an n-qubit state 𝜌 which is consistent: ∀𝑖 ≤ 𝑚, 𝜌 𝐴𝑖 = 𝜎𝑖 . No, otherwise. Theorem[Liu`06]: CLDM ∈ QMA. 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 𝜌 exists. The verifier: a) classically verifies that the energy is below the threshold a, assuming that the state is 𝜌′ = 𝜌 𝐴 ⊗ 𝜌𝐵 . b) verifies that there exists such a state 𝜌 using the CLDM protocol. Separable Sparse Hamiltonian is QMA(2)-Complete. Given a quantum circuit Q, and a witness |𝜓〉, the history state is: 𝜂𝜓 ∼ 𝑇𝑡=0 𝑡 ⊗ 𝜓𝑡 , 𝜓𝑡 ≡ 𝑈𝑡 … 𝑈1 𝜓 . Kitaev’s Hamiltonian gives an energetic penalty to: states which are not history states. history states are penalized for Pr(Q rejects |𝜓〉) Only if there exists a witness which Q accepts w.h.p., Kitaev’s Hamiltonian has a low energy state. What happens if we use Kitaev’s Hamiltonian for a QMA(2) circuit, and allow only separable witnesses? Problems: Even if 𝜓 = 𝜓𝐴 ⊗ |𝜓𝐵 〉, then |𝜓𝑡 〉 is typically not separable. Even if ∀𝑡 |𝜓𝑡 〉 is separable , |𝜂𝜓 〉 is typically entangled. For this approach to work, one part must be fixed during the entire computation. 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: |+〉 ⊗ |𝜓〉 SWAP ⊗ |𝜓〉 SWAP time |+〉 ⊗ |𝜓〉 SWAP ⊗ SWAP |𝜓〉 The history state is a tensor product state: 𝑇 𝜂~ 𝑡 ⊗ 𝑈𝑡 … 𝑈1 𝜓 𝑡=0 ⊗ |𝜓〉 There is a delicate issue in the argument: |+〉 ⊗ |𝜓〉 SWAP ⊗ |𝜓〉 SWAP Non-local operator! Causes H to be non-local! Control-Swap over multiple qubits is sparse. Local & Sparse Hamiltonian common 1 properties: 1 1 C-SWAP= Compact description Simulatable Hamiltonian in QMA Local Sparse 1 . 1 1 1 Separable Hamiltonian in QMA(2) 1 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 QMAComplete. Separable Sparse Hamiltonian is QMA(2)Complete. * Unless QMA = QMA(2). Can this help resolve whether Pure NRepresentability is QMA(2)-Complete or not? QMA vs. QMA(2) ? We would especially like to thank Fernando Brandão for suggesting the soundness proof technique. 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: Pr(𝜎 ⊗ 𝜏 passes the swap test) = 𝑇𝑟 𝜎𝜏 ≤ 𝑇𝑟(𝜎 2 ). Theorem 2: Separable Sparse Hamiltonian is QMA(2)-Complete. Why not: Separable Local Hamiltonian is QMA(2)-Complete? SWAP SWAP If we use the local implementation of C-SWAP, the history state becomes entangled. Only Seems like a technicality.