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

2. 2 Ground State Solutions        Which  spin distribution minimizes the number of red edges with similar spins and green edges with opposite 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 to our optimization problem.

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

4. 4 The Conventional Model of Quantum Computers Input Input U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2 Output:measure Quantum Computing of “Classical” functions “Quantum states”

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

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

7. 7 Special Quantum States [AharonovTa-Shma’02] 1. Graph Isomorphism 2. Closest Lattice Vector 0 v2v2 v1v1 v As well as Factoring, Discrete Log… [A’TaShma’02]

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

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

10. 10 Adiabatic Systems as Computation Devices Input Output Algorithm: H T Hamiltonian with ground state |  (T) i H T Hamiltonian with ground state |  (T) i H 0 Hamiltonian with known ground state |  (0) I H 0 Hamiltonian with known ground state |  (0) I Slowly transform H 0 into H T Slowly transform H 0 into H T Efficient: T< n c i.e. H0H0H0H0 HTHTHTHT

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

12. 12 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 H0H0H0H0 HTHTHTHT

13. 13 Remark 3: Adiabatic Optimization [FGGS’00]  Adiabatic Computation [ADKLLR’03] Without increasing the physical resources: Diagonal H T Final state is a basis state General Local H T Final state is the groundstate of a local Hamiltonian

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

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

16. 16 2. Examples: Adiabatic Optimization

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

18. 18 Adiabatic Algorithms for Optimization (Cont’d) [FarhiGoldstoneGutmanSipser’00]. H0H0H0H0 HTHTHTHT 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 √2 n. Classical NP algorithm: 2 n [RolandCerf’01,vanDamMoscaVazirani’01]

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

20. 20 3. How to Implement any Quantum Algorithm Adiabatically

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

22. 22 Want to construct adiabatic computation with γ(t)>1/L c from which we can deduce the answer. H(0)H(T) First try: Make the ground state of H(T). Problem: To specify such a Hamiltonian we need to know ! What’s the Problem? Local unitary gates U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2

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

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

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

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

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

28. 28 * The 2-Dim Lattice Construction Six states particles: Six states particles: Unborn First Phase Second Phase Dead 1 0 1 01 001 R n 0 0 ***** ******* *** *** *** *** ******

29. 29 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: H clock = ∑ 0 0 0000 Fortunately, can be checked by checking nearest neighbors: 0000 0 0 0 00 0 0

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

31. 31 Slow down, down, you you move move too too fast…… fast……