Dorit Aharonov Hebrew Univ. & UC Berkeley

1. Adiabatic Quantum Computation Dorit Aharonov Hebrew Univ. & UC Berkeley

2. Ground State Solutions Which  spin distributionminimizes the number of red edges with similar spins and green edges withopposite spins?        (1 violation.) 1) A combinatorial minimization problem. 2) A lowest energy question for magnetic materials. The ground state of the magnet is the solution toour optimization problem.

3. Properties of Adiabatic Computation • Language of Hamiltonians. • New approach to designing quantum • algorithms • Equivalent in power to quantum ckts. • Natural fault-tolerance properties • Laid back approach!

4. …. U5 U4 U3 U2 U1 The Conventional Model of Quantum Computers Input Output: measure Quantum Computing of “Classical” functions “Quantum states”

5. Ground States Schrodinger’s Equation: The Hamiltonian (A Hermitian Matrix) Eigenvectors (eigenstates) Eigenvalues (Energies) Ground state:Eigenvector with lowest eigenvalue

6. Given: f: {0,1}n N, f(x) for x=x1,…..xn, Objective: find xmin which minimizes f Classical Optimization in terms of Quantum states are the eigenvectors f(x) are the eigenvalues The answer = state with minimal eigenvalue

7. 2. Closest Lattice Vector v v2 v1 0 1. Graph Isomorphism Special Quantum States[AharonovTa-Shma’02] As well as Factoring, Discrete Log… [A’TaShma’02]

8. Apply a Hamiltonian with the desired ground state AND…. ? Adiabatic Computation A method to help the system reach a desired groundstate

9. Adiabatic Evolution H(T) H(0) • Adiabatic theorem:[BornFock ’28, Kato ’51] • Ground state of H(0)ground state of H(T)

10. HT H0 Adiabatic Systems as Computation Devices Input Output • Algorithm: • HT Hamiltonian with ground state |Y(T)i • H0 Hamiltonian with known ground state |Y(0)I • Slowly transform H0 into HT Efficient: T< nc i.e.

11. Physics:Periodic Hamiltonians, n∞ γ > const or γ0 Remark 1: Non Negligible Spectral Gaps Adiabatic computation: Tailored Hamiltonians , n∞ The interesting line is Allow it to go to zero if sufficiently slowly.

12. HT H0 Remark 2: Connection to Simulated Annealing Adiabatic Rapidly mixing Computation Markov Chains Hamiltonian  Transition rate matrix Groundstate  Limiting Distribution Spectral gap  Spectral gap for rapid mixing Quantum Simulated Annealing

13. Remark 3:Adiabatic Optimization [FGGS’00]Adiabatic Computation [ADKLLR’03] Without increasing the physical resources: General Local HT Final state is the groundstate of a local Hamiltonian Diagonal HT Final state is a basis state

14. Adiabatic Computation The set of computations that can be performed by Quantum systems, evolving adiabatically under the action local Hamiltonians with non negligible spectral gaps. A Natural Model of Computation What is thecomputational power of Adiabatic Computers ? What are the possible dynamics of Adiabatic systems ?

15. Overview 1 AdiabaticComputation 2 Previous Results AdiabaticOptimization 3 Main Result: Adiabatic Computers Can perform any Quantum Computation 4Adding Geometry: True even if the adiabatic computation is on 2 dim grid, nearest neighbor interactions Implications and Open Questions

16. 2.Examples: Adiabatic Optimization

17. [FarhiGoldstoneGutmanSipser’00]. Given: f: {0,1}n N, f(x) for x=x1,…..xn, Objective: find xmin which minimizes f Adiabatic Algorithms for Optimization f(x) is number of unsatisfied clauses Energy Penalty: Project on Unsatisfying values of x

18. HT H0 [FarhiGoldstoneGutmanSipser’00]. Adiabatic Algorithms for Optimization (Cont’d) • 20 bits: promising simulation [Farhi et al.’00,’01…] • Mounting evidence that γ(s) is exponentially small in worst case [vanDamVazirani’01, Reichhardt’03]. • Quadratic speed up: Adiabatic algorithm to solve NP in √2n. Classical NP algorithm: 2n [RolandCerf’01,vanDamMoscaVazirani’01]

19. E(x) E(x) w(x) w(x) 0 0 Tunneling: Simulated Annealing vs Adiabatic Optimization[FGGRV’03] n n Adiabatic optimization is Exponentially faster than simulated annealing! But finding 0 is easy….

20. 3.How to Implement any Quantum Algorithm Adiabatically

21. All of Quantum Computation can be done adiabatically! Spectral gaps, Eigenstates Unitary gates Result [A’TaShma’02,A’02,A’vanDamKempeLandauLloydRegev’03] Condensed matter & Mathematical Physics • Implication for Quantumcomputation: • Equivalence: New Language, new tools ! • New vantage point to tackle the challenges of quantum computation: • 1. Designing new algorithms: change of langauge, new tools. • 2. Adiabatic Computation is resilient to certain types of errors • [ChildsFarhiPreskill’01]  Possible applications for • fault tolerance. (2-dim architecture) • Implications for Physics: • Understanding ground states, Adiabatic Dynamics from • an information perspective.

22. H(T) H(0) …. U5 U4 U3 U2 U1 What’s the Problem? Want to construct adiabatic computation with γ(t)>1/Lc from which we can deduce the answer. Local unitary gates First try: Make the ground state of H(T). Problem: To specify such aHamiltonian we need to know !

23. : Time steps Key Idea Kitaev’99, based on Feynman: Instead of , use a local Hamiltonian H(T) whose ground state is the History. Classical computation: Correct History can be checked locally.

24. Time steps Key Idea Kitaev’99, based on Feynman: Instead of , use a local Hamiltonian H(T) whose ground state is the History. Classical computation: Correct History can be checked locally.

25. The Hamiltonian H(s) HT: ● Test correct propagation: Energy penalty Local interaction: H0: ● Test that input is 0

26. 4.Adding Geometry: Adiabatic Computation on a Two-D Lattice

27. Particles on a 2-d Lattice Wanted: Evolution of the form Problem: Not enough interaction between clock and computer to have terms like: Solution: Relax notion of computation/clock particles. Each particle will have both types of degrees of freedom. States will no longer be tensor products but will encode time in their geometric shape. To do this we use a like evolution.

28. * * * * * * * * * * * * * * * * * * 0 0 1 1 The 2-Dim Lattice Construction Six states particles: Unborn First Phase Second Phase Dead R * * * * * * * 0 1 0 0 0 1 * * * * * * n

29. 0 0 0 0 0 0 0 0 0 0 The Hamiltonian As before:Check correct propagation by checking each move; Each move involves only two particles. Except:Moves may seem correct locally but are not. Space of legal states is no longer invariant. Solution:Add penalty for all “forbidden” shapes: 0 0 0 0 0 0 0 Fortunately, can be checked by checking nearest neighbors: Hclock=∑

30. Saw how to implement any Q algorithm adiabatically. • Algorithm Design: New language: Ground states, spectral gaps. • What states can we reach? What statesare ground states of local Hamiltonians? To Summarize Methods from Mathematical Physics? Fault Tolerance:Adiabatic comp. is naturally robust. Adiabatic Fault Tolerance? • Ground states: • All states are ground states of local Hamiltonians, • Adiabatic dynamics are general.

31. Slow down, you move too fast……