2 nd Lecture: QMA & The local Hamiltonian problem

1. 2nd Lecture: QMA & The local Hamiltonian problem

2. …. U5 U4 U3 U2 U1 Classical and Quantum Computation Turing Machine Circuits Cn ≈ time ● Input: ●Gates ● Measure B C A A Quantum Turing Machine… [Deutch’85] Q Quantum circuits [Yao’89] time Running time: number of gates L.

3. NP A Computational complexity map BQP: Class of problems solvable in polynomial time by quantum computers QNP BPP: Class of problems solvable in polynomial time by classical computers All physically realizable computational models can be simulated in poly time by a Turing machine” (Extended CTT) BQP BPP factoring P 3 3 Widely believed: QC violates ECTT BQP is strictly larger than BPP, Quantum Systems can in principle physically implement BQP

4. Dawn of Quantum Hamiltonian Complexity… K-SAT NP-Completeness theory Quantum SAT? Quantum NP? Quantum hardness? Cook-Levin’71: k-SAT is NP-complete • Quantum Cook-Levin: • Kitaev’98: Local Hamiltonian problem is quantum NP-complete. • Since then: QNP-hardness for a variety of physical systems. 9

5. Computer Science CSP is a special case! Condensed Matter Physics Major CS problem: Constraint Satisfaction Problem (CSP) Major CMP problem: The Local Hamiltonian Problem: Given: CSP formula Objectives: Min. # of Violations Optimal assignment Approximations Given: Local Hamiltonian Objective: Ground value, ground state(s) On the board: Recall what’s a Hamiltonian (Hermitian, energies, eigenstates) Define local Hamiltonian How a kSAT can be viewed as a special case of k-Local Ham. The # number of violations  groundenergy.

6.        Example for CSPs: Spin glass Which  spin distributionminimizes red green (1 violation.) Want to be different Want to be the same violation  Energy Penalty: Project on unsatisfying values of x The lowest energy state (ground state) of the spin glass is the solution to our optimization problem.

7.        More generally: Quantum constraints Now the terms need not be diagonal in the comp. basis Energy Penalty: Project on an unsatisfying subspace The lowest energy state (ground state) of the system is the solution to our quantum optimization problem. #violations  Energy

8. Computer Science CSP is a special case! Condensed Matter Physics        Multiparticle Entanglement Major CS problem: Constraint Satisfaction Problem (CSP) Major CMP problem: The Local Hamiltonian (LH) Problem: Given: CSP formula Objectives: Min. # of Violations Optimal assignment Approximations Given: Local Hamiltonian Objective: Ground state(s) Strings – not interesting from a comp complexity perspective Apply the computer science questions to the extremely difficult physics situations

9. Quantum NP (QMA) Quantum NP (QMA) NP Q Verifier Verifier X in L: Exists Ψ s.t. Pr(Q accepts)>2/3 X not in L: for all Ψ, Pr(Q accepts)<1/3 (Completeness & soundness) On the board: Amplification when |c-s|>1/poly Mention QCMA

10. The Local Hamiltonian problem Given: Local Hamiltonian H on n qubits, Terms are Projections, or PSD b-a>1/poly(n) Objective: Is min. eigenvalue of H<a or >b? The Quantum-Cook-Levin Theorem[Kitaev’98] The local Hamiltonian problem is QMA complete Quantum NP (QMA) X in L: Exists Ψ s.t. Pr(Q accepts)>2/3 X not in L: for all Ψ, Pr(Q accepts)<1/3 Q Verifier

11. The Easier direction: LH is in QMA The Local Hamiltonian problem (LH): Given: Local Hamiltonian H on n qubits, Terms are Projections, or PSD b-a>1/poly(n) Objective: Is min. eigenvalue of H<a or >b? Quantum NP (QMA) X in L: Exists Ψ s.t. Pr(Q accepts)>c X not in L: for all Ψ, Pr(Q accepts)<s c-s>1/poly Q Verifier Claim: Local Hamiltonian is in QMA On the board: the proof