Method and system for approximate quantum circuit synthesis using quaternion algebra

A quantum circuit synthesizer system, comprising:

• a processor;
• and
• at least one memory coupled to the processor and having stored thereon processor-executable instructions for a quantum computer synthesis procedure that comprises:
• receiving a target unitary described by a target angle and target precision;
• determining a corresponding quaternion approximation of the target unitary;
• and
• synthesizing a quantum circuit corresponding to the quaternion approximation, the circuit being over a single qubit gate set, the single qubit gate set being realizable by a target quantum computer architecture, wherein the determining the corresponding quaternion approximation of the target unitary comprises:
• selecting a first algebraic integer for use as a first quaternion element in the quaternion approximation, the first algebraic integer being selected so that a distance threshold to a Rx, Ry, or Rz rotation is satisfied.

A method, comprising:

• by a quantum computer synthesis tool adapted for use in a quantum computer design and implementation process and implemented by one or more computing devices:
• inputting a program describing a desired computation to be performed in a target quantum computer architecture;
• generating a gate set adapted for implementation on the target quantum
• computer architecture based on the program, wherein the generating includes generating a single qubit circuit for one or more respective unitaries used to perform the desired computation,
• wherein the generating the single qubit circuit comprises:
• identifying a target unitary, and
• determining a corresponding quaternion approximation of the target unitary using a process in which one or more of the elements in the
• corresponding quaternion approximation are randomly selected, wherein the determining the corresponding quaternion approximation of the target unitary comprises:
• randomly selecting one or more values for use in the corresponding quaternion approximation from a constrained body of values that guarantees that a norm equation is solvable;
• and
• solving the norm equation to determine at least another one of the elements in the corresponding quaternion approximation.

A method, comprising:

• by a quantum computer synthesis tool adapted for use in a quantum computer design and implementation process and implemented by one or more computing devices:
• inputting a program describing a desired computation to be performed in a target quantum computer architecture;
• generating a gate set adapted for implementation on the target quantum
• computer architecture based on the program, wherein the generating includes generating a single qubit circuit for one or more respective unitaries used to perform the desired computation,
• wherein the generating the single qubit circuit comprises:
• identifying a target unitary, and
• determining a corresponding quaternion approximation of the target unitary using a process in which one or more of the elements in the
• corresponding quaternion approximation are randomly selected, wherein the determining the corresponding quaternion approximation of the target unitary comprises:
• randomly sampling points from subsets of a ring of integers;
• and
• using the randomly selected sampling points as quaternion elements in the quaternion approximation.

A method, comprising:

• by a quantum computer synthesis tool adapted for use in a quantum computer design and implementation process and implemented by one or more computing devices:
• inputting a program describing a desired computation to be performed in a target quantum computer architecture;
• generating a gate set adapted for implementation on the target quantum
• computer architecture based on the program, wherein the generating includes generating a single qubit circuit for one or more respective unitaries used to perform the desired computation,
• wherein the generating the single qubit circuit comprises:
• identifying a target unitary, and
• determining a corresponding quaternion approximation of the target unitary using a process in which one or more of the elements in the corresponding quaternion approximation are randomly selected, wherein the determining the corresponding quaternion approximation of the target comprises:
• partitioning a convex body of complex numbers into convex subsets, the partitioning being
• performed such that each subset has a lattice point corresponding to K;
• randomly selecting one of the subsets;
• selecting a lattice point corresponding to K from the randomly selected subset;
• and
• using the selected lattice point as an element in the quaternion approximation.

One or more computer-readable memory or storage devices storing computer-executable instructions which when executed by a computer cause the computer to perform a quantum computer synthesis procedure comprising:

• receiving a target unitary described by a target angle and target precision;
• determining a corresponding quaternion approximation of the target unitary;
• synthesizing the corresponding quaternion approximation to produce a single qubit circuit for the target unitary, the single qubit circuit being realizable by a quantum computer architecture,
• wherein the determining the corresponding quaternion approximation of the target unitary uses a Closest Vector Problem (CVP) technique to select values for the corresponding quaternion approximation.