# Linear Pauli Rotations

This function performs a rotation on a series of $$m$$ target qubits, where the rotation angle is a linear function of an $$n$$-qubit control register, as follows:

$\left|x\right\rangle _{n}\left|q\right\rangle _{m}\rightarrow\left|x\right\rangle _{n}\prod_{k=1}^{m}\left(\cos\left(\frac{a_{k}}{2}x+\frac{b_{k}}{2}\right)- i\sin\left(\frac{a_{k}}{2}x+\frac{b_{k}}{2}\right)P_{k}\right)\left|q_{k}\right\rangle$

where $$\left|x\right\rangle$$ is the control register, $$\left|q\right\rangle$$ is the target register, each $$P_{k}$$ is one of the three Pauli matrices $$X$$, $$Y$$, or $$Z$$, and $$a_{k}$$, $$b_{k}$$ are the user given slopes and offsets, respectively.

For example, the operation of a linear $$Y$$ rotation on a zero-input qubit is

$\left|x\right\rangle _{n}\left|0\right\rangle \rightarrow\left|x\right\rangle _{n}\left( \cos\left(\frac{a}{2}x+\frac{b}{2}\right)\left|0\right\rangle +\sin\left(\frac{a}{2}x+\frac{b}{2}\right)\left|1\right\rangle \right)$

Such a rotation can be realized as a series of controlled rotations as follows:

$\left[R_{y}\left(2^{n-1}a\right)\right]^{x_{n-1}}\cdots \left[R_{y}\left(2^{1}a\right)\right]^{x_{1}} \left[R_{y}\left(2^{0}a\right)\right]^{x_{0}}R_{y}\left(b\right)$

Function: linear_pauli_rotations

Arguments:

• bases: CArray[int] - List of Pauli Enums.

• slopes: CArray[float] - Rotation slopes for each of the given Pauli bases.

• offsets: CArray[float] - Rotation offsets for each of the given Pauli bases.

• x: QArray[QBit] - Quantum state to apply the rotation based on its value.

• q: QArray[QBit] - List of indicator qubits for each of the given Pauli bases.

Notice that bases, slopes, offset and q should be of the same size.

## Example: Three Y Rotations Controlled by a 6-qubit State

This example generates a quantum program with a $$6$$-qubit control state and $$3$$ target qubits, acted upon by Y rotations with different slopes and offsets.

from classiq import (
Output,
Pauli,
QArray,
QBit,
allocate,
create_model,
linear_pauli_rotations,
qfunc,
)

NUM_STATE_QUBITS = 6
BASES = [Pauli.Y.value] * 3
OFFSETS = [0.1, 0.3, 0.33]
SLOPES = [2.1, 1, 7.0]

@qfunc
def main(x: Output[QArray[QBit]], ind: Output[QArray[QBit]]):
allocate(NUM_STATE_QUBITS, x)
allocate(len(BASES), ind)
linear_pauli_rotations(BASES, SLOPES, OFFSETS, x, ind)

qmod = create_model(main)

from classiq import synthesize, write_qmod

write_qmod(qmod, "linear_pauli_rotations")
qprog = synthesize(qmod)