The complexity of the Separable Hamiltonian Problem

André Chailloux, Université Paris 7 and UC BerkeleyOr 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.

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

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QMA(2)

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

Similar to interrogation of suspects:

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?”

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

Main results

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.

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

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.

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.

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:

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.

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.

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

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.

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.

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:

Harrow-Montanaro’s construction

SWAP

SWAP

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time

Adapting Kitaev’s Hamiltonian for QMA(2)

SWAP

SWAP

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⊗

|𝜂 ⟩ (∑𝑡=0

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The history state is a tensor product state:

Adapting Kitaev’s Hamiltonian for QMA(2)

There is a delicate issue in the argument:

SWAP

SWAP

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Non-local operator!

Causes H to be non-local!

C-SWAP=.

Adapting Kitaev’s Hamiltonian for QMA(2): Local vs. Sparse

Control-Swap over multiple qubits is sparse.

Local & Sparse Hamiltonian common properties:

Local

Sparse

Compact description

Simulatable

Hamiltonian in QMA

Separable Hamiltonian in QMA(2)

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.

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).

Open problems

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

QMA vs. QMA(2) ?

Thank you!

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

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.

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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