Adiabatic Quantum Computation An alternative approach to a ... · A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem. e-print quant-ph/0104129;

IntroductionAdiabatic Theorem

Exact CoverConclusion

Adiabatic Quantum ComputationAn alternative approach to a quantum computer

Lukas Gerster

ETH Zurich

May 2. 2014

Lukas Gerster Adiabatic Quantum Computation



Table of Contents

1 Introduction

2 Adiabatic Theorem

3 Exact Cover

4 Conclusion




Literature:

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, D.PredaA Quantum Adiabatic Evolution Algorithm Applied to RandomInstances of an NP-Complete Problem.e-print quant-ph/0104129; Science 292, 472 2001.

Jeremie Roland and Nicolas J. CerfQuantum search by local adiabatic evolutionPhysical Review A, Volume 65, 042308




Idea

Alternative approach to quantum computation.

Encode problem in a constructed Hamiltonian.

Encode solution in ground state of this Hamiltonian.




Idea

Alternative approach to quantum computation.

Encode problem in a constructed Hamiltonian.

Encode solution in ground state of this Hamiltonian.




Spin glass

Figure: Three frustrated spins with ferromagnetic and anti-ferromagneticcoupling.

Frustrations lead to many local minima.

Minima are separated by large potential walls.

Ground state not reachable by cooling.




Spin glass








Spin glass








Spin glass








Adiabatic Theorem

Theorem

A physical system remains in its instantaneous eigenstate if a givenperturbation is acting on it slowly enough and if there is a gapbetween 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− tT )H0 +

tTHP




Adiabatic Theorem

Theorem





tTHP




Adiabatic Theorem

Theorem





tTHP




Adiabatic Theorem

Theorem





tTHP




Runtime of an adiabatic algorithm

Figure: Eigenvalues of the time-dependent Hamiltonian. E0 and E1

avoid crossing each other.




Runtime of an adiabatic algorithm

For probability 1− ε2 of remaining in the ground state:

gmin = min0≤t≤T

[E1(t)− E0(t)] (1)

Dmax = max0≤t≤T

|〈E1; t|dH

dt|E0; t〉| (2)

Condition for the adiabatic Theorem:

Dmax

g2min

≤ ε (3)




Exact Cover

String of n bits z1, z2...zn satisfying a set of clauses of the formzi + zj + zk = 1.Determining a string satisfying all clauses involves checking all 2n

assignments, and is a NP-complete problem.




Define energy for a clause:

hC(ziC , zjC , zkC ) =

{0 if ziC + zjC + zkC = 1

1 if ziC + zjC + zkC 6= 1(4)

Define total energy:

h =∑C

HC (5)

The energy h ≥ 0 and h(z1, z2, ...zn) = 0 only if the string satisfiesall clauses.




Define energy for a clause:

hC(ziC , zjC , zkC ) =

{0 if ziC + zjC + zkC = 1

1 if ziC + zjC + zkC 6= 1(4)

Define total energy:

h =∑C

HC (5)

The energy h ≥ 0 and h(z1, z2, ...zn) = 0 only if the string satisfiesall clauses.




Problem Hamiltonian

Use spin-12 qubits labeled by |z1〉 where z1 = 0, 1.

|0〉 =(10

)and |1〉 =

(01

)(6)

Define operator corresponding to clause C

HP,C(|z1〉...|zn〉) = hC(ziC , zjC , zkC )|z1〉...|zn〉 (7)

Problem Hamiltonian is given by

HP =∑C

HP,C (8)




Problem Hamiltonian


|0〉 =(10

)and |1〉 =

(01

)(6)




HP =∑C

HP,C (8)




Problem Hamiltonian


|0〉 =(10

)and |1〉 =

(01

)(6)




HP =∑C

HP,C (8)




Initial Hamiltonian

Use x basis for initial state

|xi = 0〉 = 1√2

(11

)and |xi = 1〉 = 1√

2

(1−1

)(9)

Define operator

H(i)B |xi = x〉 = 1

2(1− σ(i)x )|xi = x〉 = x|xi = x〉 (10)




Initial Hamiltonian is given by

HP =

n∑i=1

diH(i)B (11)

where di is the number of clauses involving bit i.The ground state is given by

|x1 = 0〉...|xn = 0〉 = 1

2n/2(|z1 = 0〉+|z1 = 1〉)...(|zn = 0〉+|zn = 1〉)

(12)




Runtime for probability 1/8

Figure: Median time to achieve success probability of 1/8 for differentsized problems.




Conclusion

Alternative, non-gate based approach to quantumcomputation.

Encode solution in ground state of a Hamiltonian.

Adiabatic theorem provides way to reach the ground state.




Conclusion







Conclusion





Documents

Adiabatic Quantum Computation An alternative approach to a ... · A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem. e-print quant-ph/0104129;