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Arithmetic oracle

A grover oracle is a black-box which sets a minus phase to specific states.

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The arithmetic oracle function takes an arithmetic expression and implements it as a grover oracle. Any arithmetic expression that outputs a boolean, i.e. the result is '1' if the expression is true and '0' if false, can be used by initializing the result qubit with the \(\ket{-}\) state[1]. An example for such expression is an expression that uses logical comparator ('==','<','>', etc.) and binary connectives (and, or, etc.). The oracle can later be incorporated into a grover operator.

Syntax

Function: ArithmeticOracle

Parameters:

  • expression: str
  • definitions: Dict[str, Union[int, float, FixPointNumber, RegisterUserInput]
  • uncomputation_method: ['naive', 'optimized']
  • qubit_count: Optional[PositiveInt]
  • simplify: bool
  • max_fraction_places: Optional[PositiveInt]
{
  "function": "ArithmeticOracle",
  "function_params": {
    "expression": "(a + b) == 3 and c ^ a == 2",
    "definitions": {
      "a": {
        "size": 2
      },
      "b": {
        "size": 1
      },
      "c": {
        "size": 3
      }
    },
    "uncomputation_method": "optimized"
  }
}

Register Names

The names of the input and output registers are determined by the definitions field. Each defined variable has a corresponding input and output register with the same name.

Example 1:

{
    "logic_flow": [
        {
            "function": "ArithmeticOracle",
            "function_params": {
                "expression": "(a + b) == 3 and c ^ a == 2",
                "definitions": {
                    "a": {
                        "size": 2
                    },
                    "b": {
                        "size": 1
                    },
                    "c": {
                        "size": 3
                    }
                },
                "uncomputation_method": "optimized"
            }
        }
    ]
}

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

params = ArithmeticOracle(
    expression="(a + b) == 3 and c ^ a == 2",
    definitions=dict(
        a=RegisterUserInput(size=2),
        b=RegisterUserInput(size=1),
        c=RegisterUserInput(size=3),
    ),
    uncomputation_method="optimized",
)
model = Model()
model.ArithmeticOracle(params)
circuit = model.synthesize()

This examples marks all states which make the expression \(\verb!(a + b) == 3 and c ^ a == 2!\) true. The qubit \(q_0\) is initialized with \(\ket{-}\) state.

img_4.png

The Classiq executor allows you to find a solution to the oracle using Grover's algorithm. An explanation of execution of a Grover Operator can be found in the Amplitude Amplification section.

References

[1] Nielsen, M., & Chuang, I. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511976667, page 249