Methods and apparatus for performing phase operations
Methods, systems, and apparatus for performing phase operations. In one aspect, a method for performing a same phase operation on a first and second qubit using a third qubit prepared in a phased plus state includes: performing a first NOT operation on the third qubit; computing a controlled adder operation on the first, second and third qubit, comprising encoding the result of the controlled adder operation in a fourth qubit; performing a square of the phase operation on the fourth qubit; uncomputing the controlled adder operation on the first, second and third qubit; 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.
1. A method, performed by a quantum computing device, for applying a target phase to a first qubit and a second qubit by performing a phase operation on a first and second qubit using a third qubit prepared in a phased plus state, wherein the phased plus state comprises a plus state that has been phased by an angle equivalent to the target phase, comprising:
performing a first NOT operation on the third qubit;
performing a first multi target CNOT gate on the first, second and third qubits, wherein the first and second qubits comprise target qubits and the third qubit acts as a control;
performing a first CNOT gate on the third qubit, wherein the first and second qubits act as controls;
performing a square phase operation on the third qubit,
performing a second CNOT operation on the third qubit, wherein the first and second qubits act as controls;
performing a second multi target CNOT gate on the first, second and third qubits, wherein the first and second qubits comprise target qubits and the third qubit acts as a control;
performing a third CNOT gate on the third qubit, wherein the first qubit acts as a control;
performing a fourth CNOT gate on 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 performing the second NOT operation on the third qubit returns the third qubit to the phased plus state.
3. The method of claim 1 , wherein the first and second qubits are initially prepared in arbitrary initial states.
4. The method of claim 1 , wherein:
the first qubit is prepared in an arbitrary initial state, the second qubit is prepared in a plus state, and the third qubit is prepared in a phased plus state; and
after performing the second NOT operation on the third qubit, the second qubit is in a phased plus state, the third qubit is in a phased plus state, and the phase operation has been performed on the first qubit.
5. The method of claim 4 , further comprising providing the second qubit in the phased plus state for use in a gate teleportation operation.
6. The method of claim 5 , wherein the gate teleportation operation comprises one of:
performing a second phase operation on a fifth and sixth qubits;
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.
7. The method of claim 6 , 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.
8. The method of claim 4 , wherein performing the square phase operation on the third qubit comprises preparing a sixth qubit in a same state as a seventh qubit when performing the square phase operation on a fifth qubit.
9. The method of claim 8 , further comprising iteratively performing squares of the phase operation.
10. The method of claim 1 , wherein the phase operation comprises a single qubit operation of the form
Z
θ
=
(
1
0
0
e
i
πθ
)
where θ specifies the phase operation, and wherein the square of the phase operation is given by Z 2θ .
11. The method of claim 1 , wherein the phased plus state comprises the phase operation applied to a plus state ├|+ =(|0 +|1 )/√2, and/or wherein the phase operation comprises a
T
=
Z
π
8
operation.
12. 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.
13. An apparatus ( 100 ) comprising quantum hardware ( 102 ) in data communication with one or more classical processors ( 104 ), wherein the quantum hardware is configured to perform operations for applying a target phase to a first qubit and a second qubit by performing a phase operation on a first and second qubit using a third qubit prepared in a phased plus state, wherein the phased plus state comprises a plus state that has been phased by an angle equivalent to the target phase, the operations comprising:
performing a first NOT operation on the third qubit;
performing a first multi target CNOT gate on the first, second and third qubits, wherein the first and second qubits comprise target qubits and the third qubit acts as a control;
performing a first CNOT gate on the third qubit, wherein the first and second qubits act as controls;
performing a square phase operation on the third qubit,
performing a second CNOT operation on the third qubit, wherein the first and second qubits act as controls;
performing a second multi target CNOT gate on the first, second and third qubits, wherein the first and second qubits comprise target qubits and the third qubit acts as a control;
performing a third CNOT gate on the third qubit, wherein the first qubit acts as a control;
performing a fourth CNOT gate on the third qubit, wherein the second qubit acts as the control; and
performing a second NOT operation on the third qubit.
14. The apparatus of claim 13 , wherein the quantum hardware 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.
15. The apparatus of claim 13 , wherein:
the first qubit is prepared in an arbitrary initial state, the second qubit is prepared in a plus state, and the third qubit is prepared in a phased plus state; and
after performing the second NOT operation on the third qubit, the second qubit is in a phased plus state, the third qubit is in a phased plus state, and the phase operation has been performed on the first qubit.
16. The apparatus of claim 15 , further comprising providing the second qubit in the phased plus state for use in a gate teleportation operation.
17. The apparatus of claim 13 , wherein performing the second NOT operation on the third qubit returns the third qubit to the phased plus state.
18. The apparatus of claim 13 , wherein the first and second qubits are initially prepared in arbitrary initial states.
19. The apparatus of claim 13 , wherein the phase operation comprises a single qubit operation of the form
Z
θ
=
(
1
0
0
e
i
πθ
)
where θ specifies the phase operation, and wherein the square of the phase operation is given by Z 2θ .
20. The apparatus of claim 13 , wherein the phased plus state comprises the phase operation applied to a plus state ├|+ =(|0 +|1 )/√2, and/or wherein the phase operation comprises a
T
=
Z
π
8
operation.