Constructing and programming quantum hardware for quantum annealing processes

1. A method comprising:

configuring a first logical superconducting unit and a second logical superconducting unit that form 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 control superconducting unit and a second control superconducting unit that form 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 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 control superconducting unit, a quantum operator σ s1 z of the first logical superconducting unit is coupled with a quantum operator σ c2 z of the second control superconducting unit in a first 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 second coupling represented by σ s2 x σ c1 z .

3. The method of claim 1 , wherein the first and second logical superconducting units and first and second control 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 couplings 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 an i th pair of superconducting units, 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.

7. An apparatus comprising:

a first logical superconducting unit and a second logical superconducting unit that form 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;

a first control superconducting unit and a second control superconducting unit that form 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;

a first coupler between the first logical superconducting unit and the second control superconducting unit, wherein the first coupler couples the first logical superconducting unit and the second control superconducting unit when a magnetic field is applied along the z direction; and

a second coupler the second logical superconducting unit and the first control superconducting unit, wherein the second coupler couples the second logical superconducting unit and the first control superconducting unit when a magnetic field is applied along the z direction.

8. The apparatus of claim 7 , wherein:

the first coupler couples a quantum operator σ s1 z of the first logical superconducting unit with a quantum operator σ c2 z of the second control superconducting unit in a first coupling represented by σ s1 x σ c2 z ; and

the second coupler 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 second coupling represented by σ s2 x σ c1 z .

9. The apparatus of claim 7 , wherein the first and second logical superconducting units and first and second control superconducting units comprise superconducting qubits having binary quantum states.

10. The apparatus of claim 7 , wherein the logical superconducting units have the same precision as the control superconducting units.

11. The apparatus of claim 7 , wherein the control superconducting units have less precision than the logical superconducting units.

12. The apparatus of claim 7 , 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 an i th pair of superconducting units, 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.