Grover Operator

The Grover operator is a unitary used in amplitude estimation and amplitude amplification algorithms [1]. The Grover operator is given by

$Q = Q(A,\chi) = -AS_0A^{-1}S_\chi$

where $$A$$ is a state preparation operator,

$A|0 \rangle= |\psi \rangle$

$$S_\chi$$ marks good states and is called an oracle, $$S_\chi\lvert x \rangle = \begin{cases} -\lvert x \rangle & \text{if } \chi(x) = 1 \ \phantom{-} \lvert x \rangle & \text{if } \chi(x) = 0 \end{cases}$$ and $$S_0$$ is a reflection about the zero state.

$S_0 = I - 2|0\rangle\langle0|$

Function: grover_operator

Arguments:

• oracle: QCallable[QArray[QBit]] - Oracle representing $$S_{\chi}$$, accepting quantum state to apply on.
• space_transform: QCallable[QArray[QBit]] - State preparation operator $$A$$, accepting quantum state to apply on.
• packed_vars: QArray[QBit] - Packed form of the variable to apply the grover operator on.

Example

The following example implements a grover search algorithm using the grover operator for a specific oracle, with a uniform superposition over the search space. The circuit starts with a uniform superposition on the search space, followed by 2 applications of the grover operator.

from classiq import (
Constraints,
Output,
QArray,
QBit,
QCallable,
QNum,
allocate,
bind,
create_model,
grover_operator,
phase_oracle,
power,
qfunc,
)
from classiq.qmod.symbolic import logical_and

VAR_SIZE = 2

@qfunc
def my_predicate(x: QNum, y: QNum, res: QBit) -> None:
res ^= logical_and((x + y < 9), ((x * y) % 4 == 1))

@qfunc
def main(x: Output[QNum[VAR_SIZE, False, 0]], y: Output[QNum[VAR_SIZE, False, 0]]):
packed_vars = QArray("packed_vars")
allocate(2 * VAR_SIZE, packed_vars)

power(
2,
lambda: grover_operator(
lambda vars: phase_oracle(
predicate=lambda _vars, _res: my_predicate(
_vars[0:VAR_SIZE], _vars[VAR_SIZE : _vars.len], _res
),
target=vars,
),
packed_vars,
),
)
bind(packed_vars, [x, y])

constraints = Constraints(max_width=15)
qmod_grover = create_model(main, constraints=constraints)

from classiq import synthesize, write_qmod

write_qmod(qmod_grover, "grover_operator")
qprog = synthesize(qmod_grover)


And the next is a verification of the amplification of the solutions to the oracle:

from classiq import execute

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

[{'x': 3.0, 'y': 3.0}: 479,
{'x': 1.0, 'y': 1.0}: 473,
{'x': 1.0, 'y': 0.0}: 5,
{'x': 1.0, 'y': 3.0}: 5,
{'x': 2.0, 'y': 1.0}: 5,
{'x': 3.0, 'y': 2.0}: 5,
{'x': 2.0, 'y': 0.0}: 4,
{'x': 0.0, 'y': 2.0}: 4,
{'x': 3.0, 'y': 1.0}: 4,
{'x': 0.0, 'y': 0.0}: 4,
{'x': 0.0, 'y': 3.0}: 3,
{'x': 1.0, 'y': 2.0}: 3,
{'x': 2.0, 'y': 3.0}: 2,
{'x': 0.0, 'y': 1.0}: 2,
{'x': 2.0, 'y': 2.0}: 1,
{'x': 3.0, 'y': 0.0}: 1]

from itertools import product

for x, y in product(range(2**VAR_SIZE), range(2**VAR_SIZE)):
print(x, y, (x + y < 9) and ((x * y) % 4 == 1))

0 0 False
0 1 False
0 2 False
0 3 False
1 0 False
1 1 True
1 2 False
1 3 False
2 0 False
2 1 False
2 2 False
2 3 False
3 0 False
3 1 False
3 2 False
3 3 True


References

[1] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, “Quantum Amplitude Amplification and Estimation,” arXiv:quant-ph/0005055, vol. 305, pp. 53–74, 2002, doi: 10.1090/conm/305/05215.