Patent US10242321

Efficient synthesis of repeat-until-success circuits in clifford + T basis
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US10242321

1. A method of defining a quantum circuit, comprising: (4) (4)

establishing a rational cyclotomic approximation of a target single-qubit unitary to a requested precision;

expanding the rational cyclotomic approximation into a multi-qubit unitary that implements the target single-qubit unitary upon successful measurement;

decomposing the multi-qubit unitary into a Clifford+T circuit; and

defining a Repeat-Until-Success (RUS) circuit based on the decomposition.

2. The method of claim 1, further comprising: (1) (2)

establishing the rational cyclotomic approximation of the target single-qubit unitary so as to solve a required norm equation; and

defining the multiple-qubit unitary based on the rational cyclotomic approximation.

4. The method of claim 2, wherein the target unitary is an axial rotation and is approximated by z*/z wherein z is a cyclotomic integer; and (1) (1)

modifying z by randomly selecting at least one value of r∈ custom character

[√{square root over (2)}] such that the norm equation is solvable for z replaced by rz.

5. The method of claim 4, wherein a single-qubit matrix defined by z and r is of the form (2) (0)

6. The method of claim 5, wherein at least one value of r is selected so as to boost the one-round probability of success p(r)=|rz|²/2^Lof the RUS circuit. (1) (0)

8. The method of claim 5, further comprising defining the multi-qubit unitary based on the single-qubit matrix. (1) (0)

9. The method of claim 8, further comprising expanding the single-qubit matrix corresponding to the target single-qubit unitary into a multi-qubit unitary, defining the multi-qubit unitary as a Clifford+T unitary by identifying the entangling two-qubit gates required for implementation of the RUS circuit. (2) (0)

10. The method of claim 9, further comprising decomposing the single-qubit matrix corresponding to the target single-qubit unitary into a product of axial rotations and implementing the multi-qubit unitary corresponding to each axial rotation as a Clifford+T circuit obtained from a product g₃c′g₄, wherein g₁, g₂, g₃, g₄are Clifford gates, c is a T-code, and c′ is a decoration of c, and defining the two-qubit unitary as Λ(g₅)(Id⊗g₁)Lift(c′)(Id⊗g₂)Λ(g₆). (1) (0)

12. The method of claim 9, further comprising decomposing the single-qubit matrix corresponding to the target single-qubit unitary, wherein the target single-qubit unitary performs a rotation around an axis in the x-y plane; (0) (1)

3. The method of claim 1, wherein the multi-qubit unitary is a multi-qubit unitary defined with respect to at least one ancillary qubit and a primary qubit. (1) (0)

15. The method of claim 1, further comprising receiving a rotation angle θ and the precision, and the cyclotomic approximation is associated with a phase factor e^iθ. (2) (0)

20. A compiler for defining a quantum circuit, comprising: (0) (2)

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Patent US10242321

Efficient synthesis of repeat-until-success circuits in clifford + T basis

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