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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:
|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
Theorem 2: Separable Sparse Hamiltonian
Local Hamiltonian problem:
Given 𝐻 = 𝑖 𝐻𝑖 , (𝐻𝑖 acts on k qubits)
is sparse
each row has at
is Hamiltonian
there a state𝐻
energyif <a
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 HamiltonianQMA.
First try: the prover provides the witness, and the
verifier checks that it is not entangled. We don’t know
Theorem 1: Separable Local HamiltonianQMA.
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
 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
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
 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
What happens if we use Kitaev’s Hamiltonian
for a QMA(2) circuit, and allow only separable
 Even if 𝜓 = 𝜓𝐴 ⊗ |𝜓𝐵 〉, then |𝜓𝑡 〉 is
typically not separable.
 Even if ∀𝑡 |𝜓𝑡 〉 is separable , |𝜂𝜓 〉 is typically
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:
The history state is a tensor product state:
𝑡 ⊗ 𝑈𝑡 … 𝑈1 𝜓
⊗ |𝜓〉
There is a delicate issue in the argument:
Causes H to be
Control-Swap over multiple qubits is
 Local & Sparse
Hamiltonian common
properties: 1
Compact description
Hamiltonian in QMA
Local Sparse
Separable Hamiltonian in QMA(2)
The instance that we constructed is local,
except one term which is sparse.
Theorem 2: Separable Sparse Hamiltonian is
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
Similar to CLDM, but with the additional
requirement that the state is pure (i.e. not a mixed
 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
Why not: Separable Local Hamiltonian is
If we use the local implementation of C-SWAP, the
history state becomes entangled.
Only Seems like a technicality.