the complexity of the separable hamiltonian problem

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The complexity of the Separable Hamiltonian Problem André Chailloux, Université Paris 7 and UC Berkeley Or Sattath , the Hebrew University QIP 2012

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Andr é Chailloux , Université Paris 7 and UC Berkeley Or Sattath , the Hebrew University. The complexity of the Separable Hamiltonian Problem. QIP 2012. Quantum Merlin-Arthur (QMA): Quantum analogue of NP. - PowerPoint PPT Presentation

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Page 1: The complexity of the  Separable Hamiltonian  Problem

The complexity of the Separable Hamiltonian Problem

André Chailloux, Université Paris 7 and UC BerkeleyOr Sattath, the Hebrew University

QIP 2012

Page 2: The complexity of the  Separable Hamiltonian  Problem

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.

Quantum Merlin-Arthur (QMA):Quantum analogue of NP

¿𝜓 ⟩

Page 3: The complexity of the  Separable Hamiltonian  Problem

QMA(2)

¿𝝍 𝑨⟩

¿𝝍𝑩⟩

The same as QMA, but with 2 provers, that do not share entanglement.

Similar to interrogation of suspects:

Page 4: The complexity of the  Separable Hamiltonian  Problem

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?”

Page 5: The complexity of the  Separable Hamiltonian  Problem

The Local Hamiltonian problem:Given , ( acts on k qubits)is there a state with energy <a or all states have energy > b ?

Main resultsWe 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.A Hamiltonian is sparse if each row has at most polynomial non-zero entries.

Page 6: The complexity of the  Separable Hamiltonian  Problem

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.

Page 7: The complexity of the  Separable Hamiltonian  Problem

Separable Local Hamiltonian 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.

Page 8: The complexity of the  Separable Hamiltonian  Problem

Separable Local Hamiltonian 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 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:

Page 9: The complexity of the  Separable Hamiltonian  Problem

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.

Page 10: The complexity of the  Separable Hamiltonian  Problem

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.

Page 11: The complexity of the  Separable Hamiltonian  Problem

Part IISeparable Sparse Hamiltonian is QMA(2)-Complete.

Page 12: The complexity of the  Separable Hamiltonian  Problem

Reminder: Kitaev’s proof 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 witness which Q accepts w.h.p., Kitaev’s Hamiltonian has a low energy state.

Page 13: The complexity of the  Separable Hamiltonian  Problem

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.

Page 14: The complexity of the  Separable Hamiltonian  Problem

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:

Page 15: The complexity of the  Separable Hamiltonian  Problem

Harrow-Montanaro’s construction

SWAP

SWAP

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time

Page 16: The complexity of the  Separable Hamiltonian  Problem

Adapting Kitaev’s Hamiltonian for QMA(2)

SWAP

SWAP

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|𝜂 ⟩ (∑𝑡=0𝑇

|𝑡 ⟩⊗𝑈 𝑡…𝑈 1|𝜓 ⟩)⊗∨𝜓 ⟩

The history state is a tensor product state:

Page 17: The complexity of the  Separable Hamiltonian  Problem

Adapting Kitaev’s Hamiltonian for QMA(2) There is a delicate issue in the

argument:SWA

PSWA

P

¿+⟩

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Non-local operator!

Causes H to be non-local!

Page 18: The complexity of the  Separable Hamiltonian  Problem

C-SWAP=.

Adapting Kitaev’s Hamiltonian for QMA(2): Local vs. SparseControl-Swap over multiple

qubits is sparse.Local & Sparse Hamiltonian

common properties:Local

Sparse

Compact descriptionSimulatableHamiltonian in QMASeparable Hamiltonian in QMA(2)

Page 19: The complexity of the  Separable Hamiltonian  Problem

Adapting Kitaev’s Hamiltonian for QMA(2) The instance that we constructed is

local, except one term which is sparse.

Theorem 2: Separable Sparse Hamiltonian is QMA(2)-Complete.

Page 20: The complexity of the  Separable Hamiltonian  Problem

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 be wrong*.

Separable Local Hamiltonian is QMA-Complete.

Separable Sparse Hamiltonian is QMA(2)-Complete.

* Unless QMA = QMA(2).

Page 21: The complexity of the  Separable Hamiltonian  Problem

Open problems

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

QMA vs. QMA(2) ?

Page 22: The complexity of the  Separable Hamiltonian  Problem

Thank you!

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

Page 23: The complexity of the  Separable Hamiltonian  Problem

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.

Page 24: The complexity of the  Separable Hamiltonian  Problem

Adapting Kitaev’s Hamiltonian for QMA(2) 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.

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