# State Preparation

Most quantum applications start with preparing a state in a quantum register. For example, in finance the state may represent the price distribution of some assets. In chemistry, it may be an initial guess for the ground state of a molecule, and in a quantum machine learning, a feature vector to analyze.

The state preparation functions creates a quantum program that outputs either a probability distribution $$p_{i}$$ or a real amplitudes vector $$a_{i}$$ in the computational basis, with $$i$$ denoting the corresponding basis state. The amplitudes take the form of list of float numbers. The probabilities are a list of positive numbers. This is the resulting wave function for probability:

$\left|\psi\right\rangle = \sum_{i}\sqrt{p_{i}} \left|i\right\rangle,$

and this is for amplitude:

$\left|\psi\right\rangle = \sum_{i}a_{i} \left|i\right\rangle.$

In general, state preparation is hard. Only a very small portion of the Hilbert space can be prepared efficiently (in $$O(poly(n))$$ steps) on a quantum program. Therefore, in practice, an approximation is often used to lower the complexity. The approximation is specified by an error bound, using the $$L_2$$ norm.

The higher the specified error tolerance, the smaller the output quantum program. For exact state preparation, specify an error bound of $$0$$.

The state preparation algorithm can be tuned depending on whether the probability distribution is sparse or dense. The synthesis engine will automatically select the parameterization based on the given constraints and optimization level.

Function: prepare_state

Parameters:

• probabilities: CArray[CReal] - Probabilities to load. Should be non-negative and sum to 1.
• bound: CReal - Approximation Error Bound, in the $$L_2$$ metric (with respect to the given probabilies vector).
• out: Output[QArray[QBit]]

Function: inplace_prepare_state

Parameters:

• probabilities: CArray[CReal]
• bound: CReal
• out: QArray[QBit] - Should of size exactly $$\log_2$$(probabilities.len)

The inplace_prepare_state works the same, but for a given allocated QArray.

Function: prepare_amplitudes

Parameters:

• amplitudes: CArray[CReal] - Amplitudes of the loaded state. Each should be real and the vector norm should be equal to 1.
• bound: CReal - Approximation Error Bound, in the $$L_2$$ metric (with respect to the given amplitudes vector).
• out: Output[QArray[QBit]]

Function: inplace_prepare_amplitudes

Parameters:

• amplitudes: CArray[CReal] - Amplitudes of the loaded state. Each should be real and the vector norm should be equal to 1.
• bound: CReal - Approximation Error Bound, in the $$L_2$$ metric (with respect to the given amplitudes vector).
• out: QArray[QBit] - Should of size exactly $$\log_2$$(amplitudes.len)

The inplace_prepare_amplitudes works the same, but for a given allocated QArray.

This example generates a quantum program whose output state probabilities are an approximation to the PMF given. That is, the probability of measuring the state $$|000⟩$$ is $$0.05$$, $$|001⟩$$ is $$0.11$$,... , and the probability to measure $$|111⟩$$ is $$0.06$$.

from classiq import Output, QArray, QBit, allocate, create_model, prepare_state, qfunc

@qfunc
def main(x: Output[QArray[QBit]]):
probabilities = [0.05, 0.11, 0.13, 0.23, 0.27, 0.12, 0.03, 0.06]
prepare_state(probabilities=probabilities, bound=0.01, out=x)

qmod = create_model(main)

from classiq import synthesize, write_qmod

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


Print the resulting probabilities:

import numpy as np

from classiq import execute

res = execute(qprog).result()[0].value

probs = np.zeros(8)
for sample in res.parsed_counts:
probs[int(sample.state["x"])] = sample.shots / res.num_shots
print("Resulting probabilities:", probs)

Resulting probabilities: [0.045 0.104 0.134 0.227 0.289 0.112 0.034 0.055]


## Example 2 - Preparating Amplitudes

This example loads a normalized linear space between -1 to 1. The load state has an accuracy of 99 present under the L2 norm.

from classiq import (
Output,
QArray,
QBit,
allocate,
create_model,
prepare_amplitudes,
qfunc,
)
from classiq.execution import ClassiqBackendPreferences, ExecutionPreferences

@qfunc
def main(x: Output[QArray[QBit]]):
amps = np.linspace(-1, 1, 8)
amps = amps / np.linalg.norm(amps)
prepare_amplitudes(amplitudes=amps.tolist(), bound=0, out=x)

backend_preferences = ClassiqBackendPreferences(
backend_name="aer_simulator_statevector"
)
execution_preferences = ExecutionPreferences(
num_shots=1, backend_preferences=backend_preferences
)
qmod = create_model(main, execution_preferences=execution_preferences)

from classiq import synthesize, write_qmod

write_qmod(qmod, "prepare_amplitudes", decimal_precision=15)
qprog = synthesize(qmod)


Print the resulting amplitudes:

import numpy as np

from classiq import execute

res = execute(qprog).result()[0].value

amps = np.zeros(8, dtype=complex)
for sample in res.parsed_state_vector:
amps[int(sample.state["x"])] = sample.amplitude

# remove global phase
global_phase = np.angle(amps[0])
amps = np.real(amps / np.exp(1j * global_phase))

print("Resulting amplitudes:", amps)

Resulting amplitudes: [ 0.54006172  0.38575837  0.23145502  0.07715167 -0.07715167 -0.23145502
-0.38575837 -0.54006172]
`