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 BerkeleyOr Sattath, the Hebrew University
QIP 2012
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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)
¿𝝍 𝑨⟩
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⊗
The same as QMA, but with 2 provers, that do not share entanglement.
Similar to interrogation of suspects:
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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?”
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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.
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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.
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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.
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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:
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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.
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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.
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Part IISeparable Sparse Hamiltonian is QMA(2)-Complete.
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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.
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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.
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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:
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Harrow-Montanaro’s construction
SWAP
SWAP
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time
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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:
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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!
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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)
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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.
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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).
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Open problems
Can this help resolve whether Pure N- Representability is QMA(2)-Complete or not?
QMA vs. QMA(2) ?
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Thank you!
We would especially like to thank Fernando Brandão for suggesting the soundness proof technique.
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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.
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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.
SWAPSWAP