Inhomogeneous quantum annealing schedules
Methods and apparatus for performing quantum annealing using a quantum system. In one aspect, a method includes controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises applying an inhomogeneous driving field to the quantum system to drive the quantum system across a quantum phase transition.
1. A method for performing algorithmic quantum annealing using a quantum system comprising a spin chain, comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying inhomogeneous driving fields to the quantum system to drive a state of the quantum system from an initial state across a quantum phase transition to a final state, wherein clusters of spins included in the spin chain are driven separately from one another by each respective inhomogeneous driving field.
2. The method of claim 1 , wherein controlling the quantum system further comprises:
driving, by the applied inhomogeneous driving field, the quantum system across a quantum phase transition.
3. The method of claim 1 , wherein the method further comprises performing one or more measurements of the energy of the quantum system to determine the solution to the computational task.
4. The method of claim 1 , wherein the computational task comprises an optimization task.
5. The method of claim 1 , and wherein the inhomogeneous driving fields comprise transverse fields that can be locally modulated for each spin in the spin chain.
6. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying an inhomogeneous driving field to the quantum system, wherein applying the inhomogeneous driving field to the quantum system causes quantum fluctuations and the quantum system responds to the quantum fluctuations within multiple clusters of the quantum system, wherein the inhomogeneous driving field is given by:
g ( n,t )=ω 0 g 0 ( n ) v 0 ( t )+Σ k=1 M ω k g k (∥ n−n k ∥−v k ( n,t ) t )
wherein g 0 represents a time-independent global magnetic field which has a spatial structure, v 0 (t) represents a critical-front velocity which is spatially uniform and has non-linear dependence to time, g k represents the non-separable spatiotemporal modulation of the drive field, and M represents the total number of clusters.
7. The method of claim 6 , wherein the inhomogeneous driving field is characterized by set of effective hyper-parameters describing the spatial structure and velocities of a critical front, the hyper-parameters including (i) a shape in space, (ii) horizontal velocity, and (iii) vertical velocity.
8. The method of claim 7 , wherein the (i) horizontal velocity, (ii) vertical velocity, and (iii) inhomogeneity slope are given by
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9. The method of claim 8 , wherein the inhomogeneous driving field comprises space and time separated inhomogeneity.
10. The method of claim 9 , wherein the inhomogeneous driving field is given by g(n,t)=g 0 (n)v 0 (t).
11. The method of claim 8 , wherein the inhomogeneous driving field comprises a periodic spatial inhomogeneity combined with spatially independent time-local inhomogeneity.
12. The method of claim 11 , wherein the inhomogeneous driving field comprises a single standing wave.
13. The method of claim 12 , wherein the (i) horizontal velocity, (ii) vertical velocity, and (iii) inhomogeneity slope are given by
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14. The method of claim 8 , wherein the inhomogeneous driving field comprises spatiotemporal inhomogeneities.
15. The method of claim 14 , wherein the inhomogeneous driving field comprises a locally linear function.
16. The method of claim 14 , wherein the inhomogeneous driving field is given by g(n,t)=g c {1+Σ k=2 M ω k tanh[θ k (∥n−n k ∥−v k (n)t)]}.
17. The method of claim 16 , wherein the activation function tanh[θ k (∥n−n k ∥−v k (n)t)] is linearized in each cluster near the quantum critical point.
18. A method for performing algorithmic quantum annealing using a quantum system, comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying an inhomogeneous driving field to the quantum system and applying a causality control strategy.
19. A method for performing algorithmic quantum annealing using a quantum system, comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying an inhomogeneous driving field to the quantum system, wherein the inhomogeneous driving field comprises one or more critical fronts, and wherein the velocities of the one or more critical fronts are below a threshold value that conserves effective inhomogeneity.
20. A method for performing algorithmic quantum annealing using a quantum system, comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying an inhomogeneous driving field to the quantum system, wherein applying the inhomogeneous driving field comprises applying an inhomogeneous driving field that creates an effective gap between low energy states and higher energy states of the quantum system.
21. A quantum annealing device comprising:
a quantum system comprising a spin chain;
one or more control devices that operate on the quantum system; and
one or more classical processors,
wherein the one or more classical processors and control devices are configured to perform operations comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein
the energy spectrum of the problem Hamiltonian encodes a solution to a computational task, and
controlling the quantum system comprises applying inhomogeneous driving fields to the quantum system to drive a state of the quantum system from an initial state across a quantum phase transition to a final state, wherein clusters of spins included in the spin chain are driven separately from one another by each respective inhomogeneous driving field.
22. A method for performing algorithmic quantum annealing using a quantum system comprising a spin chain, comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying inhomogeneous driving fields to the quantum system; and
driving, by the inhomogeneous driving fields, the quantum system across a quantum phase transition, wherein clusters of spins included in the spin chain are driven separately from one another by each respective inhomogeneous driving field.
23. The method of claim 22 , wherein the inhomogeneous driving field comprises a transverse field that can be locally modulated for each spin in the spin chain.
24. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein the energy spectrum of the problem Hamiltonian encodes a solution to a computational task and wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system;
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition; and
performing one or more measurements of the energy of the quantum system to determine the solution to the computational task.
25. The method of claim 24 , wherein the computational task comprises an optimization task.
26. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system, wherein applying the inhomogeneous driving field to the quantum system causes quantum fluctuations and the quantum system responds to the quantum fluctuations within multiple clusters of the quantum system; and
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition, wherein the inhomogeneous driving field is given by:
g ( n,t )=ω 0 g 0 ( n ) v 0 ( t )+Σ k=1 M ω k g k (∥ n−n k ∥−v k ( n,t ) t )
wherein g 0 represents a time-independent global magnetic field which has a spatial structure, v 0 (t) represents a critical-front velocity which is spatially uniform and has non-linear dependence to time, g k represents the non-separable spatiotemporal modulation of the drive field, and M represents the total number of clusters.
27. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system, wherein the inhomogeneous driving field is characterized by set of effective hyper-parameters describing the spatial structure and velocities of a critical front, the hyper-parameters including (i) a shape in space, (ii) horizontal velocity, and (iii) vertical velocity; and
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition.
28. The method of claim 27 , wherein the (i) horizontal velocity, (ii) vertical velocity, and (iii) inhomogeneity slope are given by
v
v
(
n
,
t
)
=
-
∂
g
(
n
,
t
)
/
∂
t
,
v
k
h
(
n
,
t
)
=
∂
n
(
g
f
i
x
,
t
)
/
∂
t
,
and
α
(
n
,
t
)
=
∂
g
(
n
,
t
)
∂
n
.
29. The method of claim 28 , wherein the inhomogeneous driving field comprises space and time separated inhomogeneity.
30. The method of claim 29 , wherein the inhomogeneous driving field is given by g(n,t)=g 0 (n)v 0 (t).
31. The method of claim 29 , wherein the inhomogeneous driving field comprises a periodic spatial inhomogeneity combined with spatially independent time-local inhomogeneity.
32. The method of claim 31 , wherein the inhomogeneous driving field comprises a single standing wave.
33. The method of claim 32 , wherein the (i) horizontal velocity, (ii) vertical velocity, and (iii) inhomogeneity slope are given by
v
h
(
n
,
t
)
=
v
v
(
t
)
=
a
sin
(
kn
+
φ
)
v
.
o
(
t
)
-
ag
fix
v
.
o
(
t
)
kv
o
2
(
t
)
1
-
[
g
fix
/
av
o
(
t
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]
2
α
(
n
,
t
)
=
ak
cos
(
kn
+
φ
)
v
.
o
(
t
)
.
34. The method of claim 28 , wherein the inhomogeneous driving field comprises spatiotemporal inhomogeneities.
35. The method of claim 34 , wherein the inhomogeneous driving field comprises a locally linear function.
36. The method of claim 34 , wherein the inhomogeneous driving field is given by g(n,t)=g c {1+Σ k=2 M ω k tanh[θ(∥n−n k ∥−v k (n)t)]}.
37. The method of claim 36 , wherein the activation function tanh[θ k (∥n−n k ∥−v k (n)t)] is linearized in each cluster near the quantum critical point.
38. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system;
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition; and
applying a causality control strategy.
39. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system, wherein the inhomogeneous driving field comprises one or more critical fronts, and wherein the velocities of the one or more critical fronts are below a threshold value that conserves effective inhomogeneity; and
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition.
40. A method for performing algorithmic quantum annealing using a quantum system, the method comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying an inhomogeneous driving field to the quantum system, wherein applying the inhomogeneous driving field comprises applying an inhomogeneous driving field that creates an effective gap between low energy states and higher energy states of the quantum system; and
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition.
41. A quantum annealing device comprising:
a quantum system comprising a spin chain;
one or more control devices that operate on the quantum system; and
one or more classical processors,
wherein the one or more classical processors and control devices are configured to perform operations comprising:
controlling the quantum system such that a total Hamiltonian characterizing the quantum system evolves from an initial quantum Hamiltonian to a problem quantum Hamiltonian, wherein controlling the quantum system comprises:
applying inhomogeneous driving fields to the quantum system; and
driving, by the inhomogeneous driving field, the quantum system across a quantum phase transition, wherein clusters of spins included in the spin chain are driven separately from one another by each respective inhomogeneous driving field.