Computational Lens on Quantum Physics

1. Quantum physics through theComputational lens DoritAharonov, The Hebrew University

2. FOCS’96, Radisson, Burlington, Vermont

3. Hardness Computational economicscomputational biology…Computational quantum physics Approximation Reductions Robustness Pseudo- randomness Communication Error correction Interaction

4. Computational complexity in physics1. NP  quantum states2. interactive proofs  testing quantum mechanics3. Error correction black holes

5. Setting the language: Quantum computation

6. The computational power of physical systems Uniform Circuits B C A A Q Extended Church TuringThesis: ≈ ≈

7. Crash course in Quantum mechanics: quantum states 1 n quantum bits – require 2n complex numbers. Multiparticle Entanglement

8. …. U5 U4 U3 U2 U1 Input: Quantum Circuits Dynamics: Measurement  output Hadamard NOT CNOT Hadamard + classical gates are quantum universal Complexity measure: number of gates.

9. Quantum algorithms Prime factors of Integers N=PQ N Shor’94 Q P HarrowHassidimLloyd’08 UnitGroup ChildsCleveDeottoFarhi GuttmanSpielman’02 kitaevLarsenWang’02 A’JonesLandau’04 Eisentrager HallgrenKitaevSong’14

10. Example 1: Bell’s game They win if: > 0.85!Pr(success) with EPR

11. Example 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)

12. Example III: Quantum error correcting codes Groundstates: Eigenstates of H with Lowest eigenvalues (energies) å å = + H B A P V P V

13. 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) factoring BQP BPP P 13 13 QECC: Shor,Steane’95 Fault tolerant QC: A’BenOr96 KnillLaflammeZurek’96 Kitaev’96 Widely believed: QC violates ECTT BQP is strictly larger than BPP, Quantum Systems can in principle physically implement BQP

14. What does quantum complexity tell us about physics?

15. 1.NP &The complexity of groundstates

16. New quantum algorithms, Cryptographic protocols We do not understand Quantum States… Simulating quantum Systems, Understanding Physical phenomena P, BPP Easy/ tractable Computationally Hard Area laws Quantum NP A B

17. NP & Quantum NP Verifier BQP Verifier input “hint” input 3-SAT Is the input formula satisfiable? Quantum SAT? Quantum Cook-Levin? Cook-Levin’71: 3-sat is NP complete

18. K-SAT  Local Hamiltonians        Classical CSPs CSP is a special case! Local Hamiltonians Multiparticle Entanglement Major CS problem: Constraint Satisfaction Problem (CSP) Major CMP problem: The Local Hamiltonian Problem: Given: CSP formula Objectives: Given: Local Hamiltonian Objective: Ground energy Min. # of Violations Optimal assignment Approximations In Quantum NP

19. Quantum Cook Levin : Time steps Theorem: Approxgroundvalue of a local Hamiltonian is QNP complete [Kitaev98(based on Feynman82)] Computation is local Given: Local Hamiltonian H on n qubits , a,bs.t. b-a>1/poly(n) Objective: Is min. eigenvalue of H <a or >b

20. Complexity of groundstates of Local Hamiltonians        A class of Hamiltonians is quantum NP complete  Do not expect a classical description of GSs (unless QNP=NP) Exponentially Long relaxation times to GSs

21. 1D systems Classically: 1D constraint systems are in P Quantumly: Heuristics: DMRG works. Quantum 1D is Quantum NP hard! [A’KempeGottesmanIrani’07] Even trans. Invariant! [GottesmanIrani’10] If gapped: Quantum 1D in P [LandauVaziraniVidick’13] Reductions: Hamiltonian Road map Area law

22. 2.Interactive Experiments Inspired by GoldwasserMicaliRakoff’85 Motivated by conversations with Oded Goldreich and Madhu Sudan

23. A Physical Experiment A physical theory F=ma Quantum Mechanics

24. Is Quantum Mechanics (QM) Falsifiable? Question 1: Fundamental: Is QM Falsifiable? Question 2: Experimental: Can Experimentalists Test Their systems, claimed to be quantum computers? Question 3: Cryptographic: How can we safely Delagate computations to an untrusted company claiming to have a Q comp.?

25. Why did Shor’s algorithm for factoring N=PQ succeed in getting arround the “usual type of experiments” pitfall? Interactive proofs [Goldwasser, Micali, Rackoff’85] N X P,Q . . . All powerful prover, but untrusted BQP Prover BPP verifier With interaction, A computationally weak Verifier Can get convinced of highly complex claims Without knowing how to prove them!!!

26. The power of interaction

27. Verifying quantum evolutions [A’EbanBenOr’08, BroadbentKashefiFitzimons’09, Broadbent’15,A’BenOrMahadev’16] Verifier: BPP . . BQPProver . + O(1) qubits Theorem: A BQP prover can prove any quantum circuit to a BPP+O(1) qubits verifier! dim Hilbert space 2 dim Hilbert space

28. Verifying quantum evolutions Open: Can this be done with one classical verifier? Possible with two entangled provers [ReichardtUngerVazirani’12] Open: Relations to qPCP? Cryptography: Q. obfuscation? Can interactive experiments be used elsewhere? Other verification of quantum supremacy: Boson sampling[AaronsonArkhipov’13] QC with commuting gates[BremnerJoszaScheperd’10, BremnerMontaneroSchepard’15]

29. 3.Quantum gravity & quantum information processing?

30. AdS/CFT [Maldacena’97]quantum gravity AlmheiriDongHarlow’14: CFT as a code subspace Complexity of quantum states = Length of wormhole [Susskind’15] Testing unitarity of blackhole evolution [Hayden & Preskill’07]

32. Thanks Avi, and happy birthday!!