adiabatic quantum computation an alternative approach to a ... · a quantum adiabatic evolution...
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![Page 1: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/1.jpg)
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
![Page 2: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/2.jpg)
IntroductionAdiabatic Theorem
Exact CoverConclusion
Table of Contents
1 Introduction
2 Adiabatic Theorem
3 Exact Cover
4 Conclusion
Lukas Gerster Adiabatic Quantum Computation
![Page 3: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/3.jpg)
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
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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
![Page 5: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/5.jpg)
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
![Page 6: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/6.jpg)
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
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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
![Page 8: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/8.jpg)
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
![Page 9: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/9.jpg)
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
![Page 10: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/10.jpg)
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
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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
![Page 12: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/12.jpg)
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
![Page 13: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/13.jpg)
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
![Page 14: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/14.jpg)
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
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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
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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
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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
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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
![Page 19: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/19.jpg)
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
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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
![Page 21: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/21.jpg)
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
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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
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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
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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
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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
![Page 26: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/26.jpg)
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
![Page 27: 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;](https://reader034.vdocument.in/reader034/viewer/2022050312/5f743753d2660a1d306fb24b/html5/thumbnails/27.jpg)
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