# Using the Numerics library

## Overview

The Numerics library consists of three components

**Basic integer arithmetic**with integer adders and comparators**High-level integer functionality**that is built on top of the basic functionality; it includes multiplication, division, inversion, etc. for signed and unsigned integers.**Fixed-point arithmetic functionality**with fixed-point initialization, addition, multiplication, reciprocal, polynomial evaluation, and measurement.

All of these components can be accessed using a single `open`

statement:

```
open Microsoft.Quantum.Arithmetic;
```

## Types

The numerics library supports the following types

: A qubit array`LittleEndian`

`qArr : Qubit[]`

that represents an integer where`qArr[0]`

denotes the least significant bit.: Same as`SignedLittleEndian`

`LittleEndian`

except that it represents a signed integer stored in two's complement.: Represents a real number consisting of a qubit array`FixedPoint`

`qArr2 : Qubit[]`

and a binary point position`pos`

, which counts the number of binary digits to the left of the binary point.`qArr2`

is stored in the same way as`SignedLittleEndian`

.

## Operations

For each of the three types above, a variety of operations is available:

`LittleEndian`

- Addition
- Comparison
- Multiplication
- Squaring
- Division (with remainder)

`SignedLittleEndian`

- Addition
- Comparison
- Inversion modulo 2's complement
- Multiplication
- Squaring

`FixedPoint`

- Preparation / initialization to a classical values
- Addition (classical constant or other quantum fixed-point)
- Comparison
- Multiplication
- Squaring
- Polynomial evaluation with specialization for even and odd functions
- Reciprocal (1/x)
- Measurement (classical Double)

For more information and detailed documentation for each of these operations, see the Q# library reference docs at docs.microsoft.com.

## Sample: Integer addition

As a basic example, consider the operation $$ \ket x\ket y\mapsto \ket x\ket{x+y} $$ that is, an operation that takes an n-qubit integer $x$ and an n- or (n+1)-qubit register $y$ as input, the latter of which it maps to the sum $(x+y)$. Note that the sum is computed modulo $2^n$ if $y$ is stored in an $n$-bit register.

Using the Quantum Development Kit, this operation can be applied as follows:

```
operation TestMyAddition(xValue : Int, yValue : Int, n : Int) : Unit {
use (xQubits, yQubits) = (Qubit[n], Qubit[n]);
let x = LittleEndian(xQubits); // define bit order
let y = LittleEndian(yQubits);
ApplyXorInPlace(xValue, x); // initialize values
ApplyXorInPlace(yValue, y);
AddI(x, y); // perform addition x+y into y
// ... (use the result)
}
```

## Sample: Evaluating smooth functions

To evaluate smooth functions such as $\sin(x)$ on a quantum computer, where $x$ is a quantum `FixedPoint`

number,
the Quantum Development Kit numerics library provides the operations `EvaluatePolynomialFxP`

and `Evaluate[Even/Odd]PolynomialFxP`

.

The first, `EvaluatePolynomialFxP`

, allows to evaluate a polynomial of the form
$$
P(x) = a_0 + a_1x + a_2x^2 + \cdots + a_dx^d,
$$
where $d$ denotes the *degree*. To do so, all that is needed are the polynomial coefficients `[a_0,..., a_d]`

(of type `Double[]`

),
the input `x : FixedPoint`

and the output `y : FixedPoint`

(initially zero):

```
EvaluatePolynomialFxP([1.0, 2.0], x, y);
```

The result, $P(x)=1+2x$, will be stored in `yFxP`

.

The second, `EvaluateEvenPolynomialFxP`

, and the third, `EvaluateOddPolynomialFxP`

, are specializations
for the cases of even and odd functions, respectively. That is, for an even/odd function $f(x)$ and
$$
P_{even}(x)=a_0 + a_1 x^2 + a_2 x^4 + \cdots + a_d x^{2d},
$$
$f(x)$ is approximated well by $P_{even}(x)$ or $P_{odd}(x) := x\cdot P_{even}(x)$, respectively.
In Q#, these two cases can be handled as follows:

```
EvaluateEvenPolynomialFxP([1.0, 2.0], x, y);
```

which evaluates $P_{even}(x) = 1 + 2x^2$, and

```
EvaluateOddPolynomialFxP([1.0, 2.0], x, y);
```

which evaluates $P_{odd}(x) = x + 2x^3$.

## More samples

You can find more samples in the main samples repository.

To get started, clone the repo and open the `Numerics`

subfolder:

```
git clone https://github.com/Microsoft/Quantum.git
cd Quantum/samples/numerics
```

Then, `cd`

into one of the sample folders and run the sample via

```
dotnet run
```