Constructing and programming quantum hardware for quantum annealing processes

1. A method, comprising:

configuring a first pair of logical superconducting units for use in computation to each have quantum states that can be spanned over Pauli operator σ sj i , where i=x, y, or z;

configuring a first pair of control superconducting units for use in assisting the computation without being a computational unit to each have quantum states that can be spanned over Pauli operator σ ck i , where i=x, y, or z;

coupling a first logical superconducting unit and a second logical superconducting unit by applying a magnetic field along a z direction;

coupling a first control superconducting unit and a second control superconducting unit by applying a magnetic field along a z direction;,

coupling the first logical superconducting unit and the second control superconducting unit by applying a magnetic field along the z direction; and

coupling the second logical superconducting unit and the first control superconducting unit by applying a magnetic field along the z direction.

2. The method of claim 1 , wherein:

when applying the magnetic field to the first logical superconducting unit and the second logical superconducting unit, a quantum operator σ s1 z of the first logical superconducting unit is coupled with a quantum operator σ s2 z of the second logical superconducting unit in a first coupling represented by σ s1 z σ s2 z ;

when applying the magnetic field to the first control superconducting unit and the second control superconducting unit, a quantum operator σ c1 z of the first control superconducting unit is coupled with a quantum operator σ c2 z of the second control superconducting unit in a second coupling represented by σ c1 z σ c2 z ;

when applying the magnetic field to the first logical superconducting unit and the second control superconducting unit, a quantum operator σ s1 z of the first logical superconducting unit is coupled with a quantum operator σ s2 z of the second control superconducting unit in a third coupling represented by σ s1 x σ s2 z ; and

when applying the magnetic field to the second logical superconducting unit and the first control superconducting unit, the magnetic field couples a quantum operator σ s2 z of the first control superconducting unit with a quantum operator σ c1 z of the second logical superconducting unit in a fourth coupling represented by σ s2 x σ c1 z .

3. The method of claim 1 , wherein the logical superconducting units and superconducting units comprise superconducting qubits having binary quantum states.

4. The method of claim 1 , wherein the logical superconducting units have the same precision as the control superconducting units.

5. The method of claim 1 , wherein the control superconducting units have less precision than the logical superconducting units.

6. The method of claim 1 , wherein the superconducting units and the couplers are configured such that a quantum Hamiltonian characterizing the superconducting units and the couplers is:

H

tot

=

I

⁡

(

t

)

⁢

∑

i

N

⁢

σ

i

x

+

G

⁡

(

t

)

⁢

(

∑

i

=

1

N

G

⁢

h

i

G

⁢

σ

i

z

+

∑

i

,

j

=

1

N

G

⁢

J

ij

G

⁢

σ

i

z

⁢

σ

j

z

+

∑

i

=

1

N

G

⁢

∑

j

=

1

N

A

⁢

J

ij

GA

⁢

σ

i

z

⁢

σ

j

z

)

+

P

⁡

(

t

)

⁢

(

-

∑

i

N

⁢

h

i

⁢

σ

i

z

+

∑

ij

N

⁢

J

ij

⁢

σ

i

z

⁢

σ

j

z

)

where i represents the i th pair of superconducting unit, h i and J ij have real values that are associated with coupling strength between the first and second superconducting units, I(t), G(t), and P(t) are time-dependent control parameters.