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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
IntroductionAdiabatic Theorem
Exact CoverConclusion
Table of Contents
1 Introduction
2 Adiabatic Theorem
3 Exact Cover
4 Conclusion
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
Idea
Alternative approach to quantum computation.
Encode problem in a constructed Hamiltonian.
Encode solution in ground state of this Hamiltonian.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
Idea
Alternative approach to quantum computation.
Encode problem in a constructed Hamiltonian.
Encode solution in ground state of this Hamiltonian.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
Runtime of an adiabatic algorithm
Figure: Eigenvalues of the time-dependent Hamiltonian. E0 and E1
avoid crossing each other.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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)
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
Runtime for probability 1/8
Figure: Median time to achieve success probability of 1/8 for differentsized problems.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
IntroductionAdiabatic Theorem
Exact CoverConclusion
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.
Lukas Gerster Adiabatic Quantum Computation
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