3 rd Lecture: QMA & The local Hamiltonian problem (CNT’D)

1. 3rd Lecture: QMA & The local Hamiltonian problem (CNT’D)

2. Recap: What did we do Yesterday…

3. …. U5 U4 U3 a + b U2 U1 The Quantum Circuit model 0 0 1 ● Input: ●Gates ● Measure 1 Qubits, states, measurements, gates Quantum circuits time Running time: number of gates L.

4. NP A Computational complexity map BQP: Class of problems solvable in polynomial time by quantum computers 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 factoring BPP P 4 4 Widely believed: QC violates ECTT BQP is strictly larger than BPP, Quantum Systems can in principle physically implement BQP

5. Quantum algorithms using Fourier sampling Prime factors of Integers N=PQ Deutsch Josza [‘92] N Bernstein Vazirani[‘93] Shor’94 Simon [‘94] Q P

6. Interference between exponentially many paths : : Classical computation: moving from one configuration to the next. Quantum: moving to superpositions of configurations. Can work in parallel with a huge number of computational paths which “communicate” between themselves Using interference we can make sure that paths leading to wrong answers will cancel each other

7. Quantum Computational Hardness? K-SAT NP-Completeness theory Quantum SAT? Quantum NP? Quantum hardness? Cook-Levin’71: k-SAT is NP-complete 9

8.        Classical CSPs as Local Hamiltonians: Spin glass Which  spin distribution minimizes red green (1 violation.) Want to be different Want to be the same The groundstate of H is the solution of optimization problem.

9. 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) Which groundstates have efficient description?

10. Towards proving LH is QMA complete The Local Hamiltonian problem (LH): Given: Local Hamiltonian H on n qubits, Terms are Projections b-a>1/poly(n) Objective: Is min. eigenvalue of H>b or <a? 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 We proved the easy direction: LH is in QMA with c-s>1/poly, by having the verifier pick a random constraint and measure it, and accepting if the state happens to be in its groundspace.

11. Today: QMA hardness of LH & Kitaev’sCiruit-to-Hamiltonian Construction.

12. The interesting direction: LH is QMA hard 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>b or <a? 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 X in L: Exists Ψ s.t. Pr(Q accepts)>1-1/exp(n) X not in L: for all Ψ, Pr(Q accepts)<1/exp(n) Q Verifier We now want to show how every quantum verification circuit V Can be mapped into a local Hamiltonian whose ground-energy will Indicate whether there exists a state which V accepts with good prob, or V rejects all states with good probability

13. Why not mimic Cook-Levin’s proof? : Time steps Cook-Levin: K-SAT is NP complete. Q-Cook-Levin: LH is QMA complete [Kitaev’99] Q Verifier Verifier Computation is local Problem!

14. Entanglement, Ex. II: Inner Product Estimation Two distributions over n bit strings. Are they equal or their supports do not intersect? need exp(n) many samples. Can estimate <P|Q> efficiently (by measuring the left qubit) On the board – Compute the prob above. Extend to show how to locally test propagation of one step

15. : Using entanglement for local tests Time steps Given: Local Hamiltonian H on n qubits , a,bs.t. b-a>1/poly(n) Objective: Is min. eigenvalue of H <a or >b Computation is local

16. Circuit to Hamiltonian construction: • Universality of Adiabatic evolution [A’KempeLandauLloydRegevVanDam’04] • (QMA hardness often goes together w/ universality. see David’s lecture) • 2. Hardness of the Physics “Density functional theory” [SchuchVerstraete’09] • 3. Creation of Hamiltonians with “adversarially” highly entangled Gstates • [Irani’09, GottesmanHastings’09, A’HarrowLandauNagajSzegedyVazirani’14] • Creation of approximate quantum codes with local constraints [NirkheVaziraniYuen’18] • …. A B WHY? Dynamics to Statics

17. The reduction: LH is QMA hard

18. Applications of Circuit to Hamiltonian construction: • Universality of Adiabatic evolution [A’KempeLandauLloydRegevVanDam’04] • (QMA hardness often goes together w/ universality. see David’s lecture) • 2. Hardness of the Physics “Density functional theory” [SchuchVerstraete’09] • 3. Creation of Hamiltonians with “adversarially” highly entangled Gstates • [Irani’09, GottesmanHastings’09, A’HarrowLandauNagajSzegedyVazirani’14] • Creation of approximate quantum codes with local constraints [NirkheVaziraniYuen’18] • …. A B WHY? Dynamics to Statics

19.  Quantum Hamiltonian complexity Computationally Hard Easy/ tractable QMA hardness: Many results… 1D systems are QMA hard [AharonovGottesmanIraniKempe’07, 1D Trans Invariant systems can be hard computationally [GottesmanIrani’09] Useful object (also for Q simulations): gadgets [KempeKitaevRegev’04,OliveiraTerhal’05] A class of Hamiltonians is QMA hard  No classical description of GSs (unless QMA=NP). QMA hardness often goes together with BQP universality… See David’s lecture A B