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Quantum NP

Quantum NP. Dorit Aharonov & Tomer Naveh Presented by Alex Rapaport. Introduction. The goal is to define something like NP and NP completeness in the quantum world. Kitaev defines quantum analog of NP and a complete problem analog to SAT. MA. Definition:. x. V. y. MA.

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Quantum NP

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  1. Quantum NP Dorit Aharonov & Tomer Naveh Presented by Alex Rapaport

  2. Introduction • The goal is to define something like NP and NP completeness in the quantum world. • Kitaev defines quantum analog of NP and a complete problem analog to SAT.

  3. MA • Definition:

  4. x V y MA • Can be viewed as a game between the prover Merlin and the verifier Artur. • Quantum analog of NP is derived from MA.

  5. QMA • Definition: x V y

  6. V QCMA • Definition: x y

  7. QCMA • In QCMA the witness is classical. • In QMA the witness is quantum state. • Left is trivial and rightis because V can measure the witness first turning it into classical.

  8. Amplification • We can gain more power by changing the 2/3 and the 1/3 parameters and define a general MA(c,s) or QMA(c,s) • Theorem:

  9. Amplification • The idea of the proof is to take polynomial number of witnesses and run the verifier on each. Taking the majority • Merlin can’t cheat by entangling the witnesses. M A J O R I T Y v … … … v

  10. Complexity • Theorem: • BPP – Can be solved in polynomial time with bounded small error probability. • BQP – same with quantum machine. • PP – Can be solved in polynomial time with error probability less than half (maybe exponentially close).

  11. 5-Local Hamiltonian • Input: H1,…, Hr a set of Hermitian positive semi-definite matrices operating on 5 qubits each with norm ||Hi|| 1. • Each Hi operates on 5 qubits out of total n. • Two real numbers a<b (not exponentially close). • Output: Is the smallest eigenvalue of H=H1+…+Hr smaller then a (YES) or larger than b (NO).

  12. 5-Local Hamiltonial V … … V … … + … V … …

  13. 3-SAT Connection • 3-SAT Can be reduced to 3-local Hamiltonial. • Let f = C1^…^Cr be a 3-SAT formula, with variables v1,…,vn. • Each Ci has exactly one unsatisfying assignment zi. Ci(zi)=0. • For every Ci define Hi = . Operating on qubits numbered as the variables of Ci. • For any satisfying assignment z of f if exists (H1+…+Hr)z = 0. • We take a = 0 and b = 1.

  14. Local Hamiltonial in QMA • Theorem: • k-Local Hamiltonial problem is in QMA for any k = O(log(n)). • The witness is the eigenvector that gets measured after applying randomly chosen H.

  15. QMA Completeness • 5-local Hamiltonian is QMA Complete. • For any L in QMA there exists a Quantum circuit Q with two-qubit gates U1,…,UTfor verification with exponentially close to one probability of right answer. • From this we will build an instance of local Hamiltonian problem given input x.

  16. Reduction • We imagine the matrixes as operating on the history of computation by Q given as superposition: • Hin is the matrix that checks that the input is really x. • Hout is the matrix that checks that the output is 1 at time t. • Hprop checks that the computation from t-1 to t is corect.

  17. Reduction Xi V t Hin(i) X1 V t Hout

  18. Reduction • We will take a = 1/T10 and b = 1/4(T+1)3

  19. Improving to 5-local • We proved the completeness for logT+2-local. • To turn it into 5-local we will represent the time in unary representation of T qubits |11…100..0> and replace by operators that operate on 3 qubits.

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