Adiabatic Quantum Computation An alternative approach to a quantum computer

Transcription

1 An alternative approach to a quantum computer ETH Zürich May

2 Table of Contents 1 Introduction 2 3 4

3 Literature: Introduction E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, D. Preda A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem. e-print quant-ph/ ; Science 292, Jèrèmie Roland and Nicolas J. Cerf Quantum search by local adiabatic evolution Physical Review A, Volume 65,

4 Idea Introduction Alternative approach to quantum computation. Encode problem in a constructed Hamiltonian.

5 Idea Introduction Alternative approach to quantum computation. Encode problem in a constructed Hamiltonian. Encode solution in ground state of this Hamiltonian.

6 Spin glass Introduction Figure: Three frustrated spins with ferromagnetic and anti-ferromagnetic coupling.

7 Spin glass Introduction Figure: Three frustrated spins with ferromagnetic and anti-ferromagnetic coupling. Frustrations lead to many local minima.

8 Spin glass Introduction Figure: Three frustrated spins with ferromagnetic and anti-ferromagnetic coupling. Frustrations lead to many local minima. Minima are separated by large potential walls.

9 Spin glass Introduction Figure: Three frustrated spins with ferromagnetic and anti-ferromagnetic coupling. Frustrations lead to many local minima. Minima are separated by large potential walls. Ground state not reachable by cooling.

10 Theorem A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum.

11 Theorem A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum. A system in the ground state remains in the ground state.

12 Theorem A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum. A system in the ground state remains in the ground state. The perturbation does not have to be small.

13 Theorem A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum. A system in the ground state remains in the ground state. The perturbation does not have to be small. Can switch Hamiltonian: H(t) = (1 t T )H 0 + t T H P

14 Runtime of an adiabatic algorithm Figure: Eigenvalues of the time-dependent Hamiltonian. E 0 and E 1 avoid crossing each other.

15 Runtime of an adiabatic algorithm For probability 1 ɛ 2 of remaining in the ground state: g min = min 0 t T [E 1(t) E 0 (t)] (1) D max = max 0 t T E 1; t dh dt E 0; t (2) Condition for the adiabatic Theorem: D max g 2 min ɛ (3)

16 Introduction String of n bits z 1, z 2...z n satisfying a set of clauses of the form z i + z j + z k = 1. Determining a string satisfying all clauses involves checking all 2 n assignments, and is a NP-complete problem.

17 Define energy for a clause: h C (z ic, z jc, z kc ) = { 0 if z ic + z jc + z kc = 1 1 if z ic + z jc + z kc 1 (4) Define total energy: h = C H C (5)

18 Define energy for a clause: h C (z ic, z jc, z kc ) = { 0 if z ic + z jc + z kc = 1 1 if z ic + z jc + z kc 1 (4) Define total energy: h = C H C (5) The energy h 0 and h(z 1, z 2,...z n ) = 0 only if the string satisfies all clauses.

19 Problem Hamiltonian Use spin- 1 2 qubits labeled by z 1 where z 1 = 0, 1. 0 = ( ) 1 and 1 = 0 ( ) 0 1 (6)

20 Problem Hamiltonian Use spin- 1 2 qubits labeled by z 1 where z 1 = 0, 1. 0 = ( ) 1 and 1 = 0 ( ) 0 1 (6) Define operator corresponding to clause C H P,C ( z 1... z n ) = h C (z ic, z jc, z kc ) z 1... z n (7)

21 Problem Hamiltonian Use spin- 1 2 qubits labeled by z 1 where z 1 = 0, 1. 0 = ( ) 1 and 1 = 0 ( ) 0 1 (6) Define operator corresponding to clause C H P,C ( z 1... z n ) = h C (z ic, z jc, z kc ) z 1... z n (7) Problem Hamiltonian is given by H P = C H P,C (8)

22 Initial Hamiltonian Use x basis for initial state x i = 0 = 1 2 ( 1 1 ) and x i = 1 = 1 ( ) (9) Define operator H (i) B x i = x = 1 (1 σ(i) x ) x i = x = x x i = x (10) 2

23 Initial Hamiltonian is given by H P = n i=1 d i H (i) B (11) where d i is the number of clauses involving bit i. The ground state is given by x 1 = 0... x n = 0 = 1 2 n/2 ( z 1 = 0 + z 1 = 1 )...( z n = 0 + z n = 1 ) (12)

24 Runtime for probability 1/8 Figure: Median time to achieve success probability of 1/8 for different sized problems.

25 Introduction Alternative, non-gate based approach to quantum computation.

26 Introduction Alternative, non-gate based approach to quantum computation. Encode solution in ground state of a Hamiltonian.

27 Introduction Alternative, non-gate based approach to quantum computation. Encode solution in ground state of a Hamiltonian. Adiabatic theorem provides way to reach the ground state.