Addition

The "Addition" operation, denoted '+', is implemented according to the following truth table. Here \(a\) and \(b\) denote numbers, \(i\) bit index, and \(c_{in / out}\) the incoming and outgoing carries of that step, respectively.

\(a_i\)	\(b_i\)	\(c_{in}\)	\({(a + b)}_i\)	\(c_{out}\)
0	0	0	0	0
0	1	0	1	0
1	0	0	1	0
1	1	0	0	1
0	0	1	1	0
0	1	1	0	1
1	0	1	0	1
1	1	1	1	1

Note that integer and fixed-point numbers are represented in a 2-complement method during function evaluation. The binary number is extended in the case of a register size miss-match. For example, the positive signed number \((110)_2=6\) is expressed as \((00110)_2\) when operating with a 5-qubit register. Similarly, the negative signed number \((110)_2=-2\) is expressed as \((11110)_2\).

Examples

• 5 + 3 = 8 , 0101 + 0011 = 1000
• 5 + -3 = 2, 0101 + 1101 = 0010
• -5 + -3 = -8, 1011 + 1101 = 1000

Several adder algorithms, differing in depth, number of gates used, and number of auxiliaries, are implemented [1][2].

Syntax

Function: Adder

Parameters:

• left_arg: Union[float, int, FixPointNumber, RegisterUserInput]
• right_arg: Union[float, int, FixPointNumber, RegisterUserInput]
• output_size: Optional[PositiveInt]
• output_name: Optional[str]
• inplace: bool = True

{
  "function": "Adder",
  "function_params": {
    "left_arg": 3,
    "right_arg": {
      "size": 3
    },
    "inplace": false
  }
}

Example 1: Two Register Addition

Textual ModelSDK

{
        "constraints": {
          "max_width": 7,
          "max_depth": 100
        },
        "logic_flow": [
            {
                "function": "Adder",
                "function_params": {
                    "left_arg": {"size": 3},
                    "right_arg": {"size": 3}
                }
            }
        ]
    }

from classiq import ModelDesigner
from classiq.builtin_functions import Adder
from classiq.interface.generator.arith.arithmetic import RegisterUserInput

params = Adder(left_arg=RegisterUserInput(size=3), right_arg=RegisterUserInput(size=3))
model_designer = ModelDesigner()
model_designer.Adder(params)
circuit = model_designer.synthesize()

This code example generates a circuit that adds 2 arguments. Both "left_arg" and "right_arg" are defined to be quantum registers of size 3.

Generated Circuit

Example 2: Float and Register Addition

Textual ModelSDK

{
    "constraints": {
      "max_width": 5,
      "max_depth": 100
    },
    "logic_flow": [
        {
            "function": "Adder",
            "function_params": {
                "left_arg": 3.5,
                "right_arg": {
                    "size": 3
                }
            }
        }
    ]
}

from classiq import ModelDesigner
from classiq.builtin_functions import Adder
from classiq.interface.generator.arith.arithmetic import RegisterUserInput

params = Adder(left_arg=3.5, right_arg=RegisterUserInput(size=3))
model_designer = ModelDesigner()
model_designer.Adder(params)
circuit = model_designer.synthesize()

This code example generates a circuit that adds 2 arguments. Here "left_arg" is a fixed-point number \((11.1)_2\) and "right_arg" a quantum register of size 3.

Generated Circuit

References

[1]S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, “A new quantum ripple-carry addition circuit,” arXiv:quant-ph/0410184, Oct. 2004, Accessed: Aug. 09, 2021. [Online]. Available: http://arxiv.org/abs/quant-ph/0410184

[2]T. G. Draper, “Addition on a Quantum Computer,” arXiv:quant-ph/0008033, Aug. 2000, Accessed: Aug. 09, 2021. [Online]. Available: http://arxiv.org/abs/quant-ph/0008033