Grover Operator¶

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

The grover operator can be given as:

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

where \(A\) is state preparation operator,

\[ A|0 \rangle= |\psi \rangle \]

\(S_x\) marks good states,

and \(S_0\), called diffuser, flips around the zero state,

\[ S_0 = I - 2|0\rangle\langle0| \]

By default, the state preparation used is the uniform superposition by applying hadamard on all inputs.

\[ A = H^{\bigotimes n} \]

Syntax¶

Function: GroverOperator

Parameters:

-

oracle: ArithmeticOracle

state_preparation: Optional[StatePreparation]

{
  "function": "GroverOperator",
  "function_params": {
    "oracle": {
      "expression": "(a + b + (c & 6)) % 4 | 4 & c  == 4",
      "definitions": {
        "a": {
          "size": 2
        },
        "b": {
          "size": 2
        },
        "c": {
          "size": 3
        }
      },
      "uncomputation_method": "optimized",
      "qubit_count": 16
    },
    "state_preparation": null
  }
}

Example 1:¶

SDK

from classiq import Model, RegisterUserInput
from classiq.builtin_functions import ArithmeticOracle, GroverOperator

oracle_params = ArithmeticOracle(
    expression="(a + b + (c & 6)) % 4 | 4 & c  == 4",
    definitions=dict(
        a=RegisterUserInput(size=2),
        b=RegisterUserInput(size=2),
        c=RegisterUserInput(size=3),
    ),
    uncomputation_method="optimized",
    qubit_count=16,
)
grover_params = GroverOperator(oracle=oracle_params)

model = Model()
model.GroverOperator(grover_params)

circuit = model.synthesize()

The example above generates grover operator for a specific oracle. The functional blocks include an arithmetic oracle, state preparation, a diffuser and finally the global phase.

zoom out circuit

An explanation of execution of a Grover Operator can be found in the Amplitude Amplification section.

Grover in the IDE¶

In the screenshot below the IDE view when the Grover option is selected form the application suite is shown.

grover-ide

Details on executing a Grover model with Amplitude Amplification can be found in the Amplitude Amplification section.

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.