Methods and apparatus for performing phase operations

1. A method, performed by a quantum computing device, for duplicating a phased plus state, the method comprising:

performing a phase operation on a first qubit included in the quantum computing device and a second qubit included in the quantum computing device, wherein the first qubit is prepared in an arbitrary initial state and the second qubit is prepared in a plus state, comprising:

performing a first NOT operation on a third qubit included in the quantum computing device, wherein the third qubit is prepared in the phased plus state;

computing a controlled adder operation on the first, second and third qubits, comprising encoding a result of the controlled adder operation in a fourth qubit included in the quantum computing device;

performing a square of the phase operation on the fourth qubit;

uncomputing the controlled adder operation on the first, second and third qubits;

performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control;

performing a CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control; and

performing a second NOT operation on the third qubit.

2. The method of claim 1 , wherein after performing the CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control, the second qubit is in the phased plus state and the phase operation has been performed on the first qubit.

3. The method of claim 1 , wherein after performing the second NOT operation on the third qubit the third qubit is in a phased plus state.

4. The method of claim 1 , wherein computing the controlled adder operation comprises:

performing a multi target CNOT on the first, second and third qubits, wherein the first qubit acts as the control;

computing a logical AND operation between the second and third qubits;

encoding a result of the logical AND computation in the fourth qubit; and

performing a CNOT operation between the first qubit and the fourth qubit, wherein the first qubit acts as the control.

5. The method of claim 1 , wherein uncomputing the controlled adder operation comprises:

performing a CNOT operation on the first qubit and the fourth qubit, wherein the first qubit acts as the control; and

un-computing a logical AND operation between the second and third qubits; and

wherein performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control comprises: performing a multi target CNOT operation on the first, second and third qubits, wherein the first qubit acts as the control.

6. The method of claim 1 , wherein:

uncomputing the controlled adder operation on the first, second and third qubits;

performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control;

performing a CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control, comprises:

performing a CNOT operation on the first qubit and the fourth qubit, wherein the first qubit acts as the control;

un-computing a logical AND operation between the second and third qubits; and

performing a CNOT operation on the second qubit and the third qubit, wherein the second qubit acts as the control; and

performing a multi target CNOT operation on the first, second and third qubits, wherein the first qubit acts as the control.

7. The method of claim 2 , further comprising providing the second qubit in the phased plus state for use in a gate teleportation operation.

8. The method of claim 7 , wherein the gate teleportation operation comprises one of:

performing a second phase operation on a fifth and sixth qubit;

preparing a sixth qubit in a same state as a seventh qubit when performing a phase operation on a fifth qubit; or

performing the phase operation on a fifth qubit.

9. The method of claim 8 , wherein performing the phase operation on a fifth qubit comprises:

applying a CNOT operation between the second qubit prepared in the phased plus state and the fifth qubit prepared in an arbitrary state, wherein the fifth qubit acts as the control;

measuring the second qubit; and

applying a squared phase operation to the fifth qubit if a generated measurement result from measuring the second qubit indicates that the second qubit is ON.

10. The method of claim 1 , wherein performing a square of the phase operation on the fourth qubit comprises preparing a sixth qubit in a same state as a seventh qubit when performing the square of the phase operation on a fifth qubit, optionally further comprising iteratively performing squares of the phase operation.

11. The method of claim 1 , wherein the phase operation comprises a single qubit operation of the form

Z

θ

=

(

1

0

0

e

i

⁢

π

⁢

θ

)

where θ specifies the phase operation, and wherein the square of the phase operation is given by Z 2θ .

12. The method of claim 1 , wherein a phased plus state comprises the phase operation applied to a plus state |+ =(|0 +|1 )/√{square root over (2)}, and/or wherein the phase operation comprises a √{square root over (T)}=Z π/8 operation.

13. The method of claim 1 , further comprising, for a system requiring N phase operations to be performed on multiple respective qubits:

grouping qubits that require a phase operation into O(sqrt(N)) groups of size O(sqrt(N));

preparing a full-total qubit register of size O(log(N));

for each group:

computing a Hamming weight of the qubits in the group;

adding a computed group-total into the full-total register;

uncompute the Hamming weight of the qubits in the group;

performing phase operations on the full-total register;

for each group:

computing a Hamming weight of the qubits in the group;

subtracting a computed group-total out of the full-total register; and

uncomputing the Hamming weight of the qubits in the group,

optionally further comprising clearing and discarding the full-total register.

14. An apparatus comprising a quantum computing device in data communication with one or more classical processors, wherein the quantum computing device is configured to perform operations comprising:

performing a phase operation on a first qubit included in the quantum computing device and a second qubit included in the quantum computing device, wherein the first qubit is prepared in an arbitrary initial state and the second qubit is prepared in a plus state, comprising:

performing a first NOT operation on a third qubit included in the quantum computing device, wherein the third qubit is prepared in a phased plus state;

computing a controlled adder operation on the first, second and third qubits, comprising encoding a result of the controlled adder operation in a fourth qubit included in the quantum computing device;

performing a square of the phase operation on the fourth qubit;

uncomputing the controlled adder operation on the first, second and third qubits;

performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control;

performing a CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control; and

performing a second NOT operation on the third qubit.

15. The apparatus of claim 14 , wherein the quantum computing device comprises:

a register of qubits;

a plurality of control lines coupled to the register of qubits; and

a plurality of control circuits coupled to the plurality of control lines.

16. The apparatus of claim 14 , wherein after performing the CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control, the second qubit is in the phased plus state and the phase operation has been performed on the first qubit.

17. The apparatus of claim 14 , wherein after performing the second NOT operation on the third qubit the third qubit is in a phased plus state.

18. The apparatus of claim 14 , wherein computing the controlled adder operation comprises:

performing a multi target CNOT on the first, second and third qubits, wherein the first qubit acts as the control;

computing a logical AND operation between the second and third qubits;

encoding a result of the logical AND computation in the fourth qubit; and

performing a CNOT operation between the first qubit and the fourth qubit, wherein the first qubit acts as the control.

19. The apparatus of claim 14 , wherein uncomputing the controlled adder operation comprises:

performing a CNOT operation on the first qubit and the fourth qubit, wherein the first qubit acts as the control; and

un-computing a logical AND operation between the second and third qubits; and

wherein performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control comprises: performing a multi target CNOT operation on the first, second and third qubits, wherein the first qubit acts as the control.

20. The apparatus of claim 14 , wherein:

uncomputing the controlled adder operation on the first, second and third qubits;

performing a CNOT operation between the first qubit and the third qubit, wherein the first qubit acts as the control;

performing a CNOT operation between the second qubit and the third qubit, wherein the second qubit acts as the control, comprises:

performing a CNOT operation on the first qubit and the fourth qubit, wherein the first qubit acts as the control;

un-computing a logical AND operation between the second and third qubits; and

performing a CNOT operation on the second qubit and the third qubit, wherein the second qubit acts as the control; and

performing a multi target CNOT operation on the first, second and third qubits, wherein the first qubit acts as the control.