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

5 QMA Definition: V x y

6 QCMA Definition: V x y

7 QCMA In QCMA the witness is classical. In QMA the witness is quantum state. Left is trivial and right is 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. v v ……… MAJORITYMAJORITY

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: H 1,…, H r a set of Hermitian positive semi-definite matrices operating on 5 qubits each with norm ||H i || 1. Each H i operates on 5 qubits out of total n. Two real numbers a<b (not exponentially close). Output: Is the smallest eigenvalue of H=H 1 +…+H r 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 = C 1 ^…^C r be a 3-SAT formula, with variables v 1,…,v n. Each C i has exactly one unsatisfying assignment z i. C i (z i )=0. For every C i define H i =. Operating on qubits numbered as the variables of C i. For any satisfying assignment z of f if exists (H 1 +…+H r )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 U 1,…,U T for 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: H in is the matrix that checks that the input is really x. H out is the matrix that checks that the output is 1 at time t. H prop checks that the computation from t-1 to t is corect.

17 Reduction XiVtXiVt H in (i) X1VtX1Vt H out

18 Reduction We will take a = 1/T 10 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.