Using the Numerics library


The Numerics library consists of three components

  1. Basic integer arithmetic with integer adders and comparators
  2. High-level integer functionality that is built on top of the basic functionality; it includes multiplication, division, inversion, etc. for signed and unsigned integers.
  3. 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;


The numerics library supports the following types

  1. LittleEndian: A qubit array qArr : Qubit[] that represents an integer where qArr[0] denotes the least significant bit.
  2. SignedLittleEndian: Same as LittleEndian except that it represents a signed integer stored in two's complement.
  3. FixedPoint: Represents a real number consisting of a qubit array 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.


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

  1. LittleEndian

    • Addition
    • Comparison
    • Multiplication
    • Squaring
    • Division (with remainder)
  2. SignedLittleEndian

    • Addition
    • Comparison
    • Inversion modulo 2's complement
    • Multiplication
    • Squaring
  3. 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

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 {
    using ((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
cd Quantum/Numerics

Then, cd into one of the sample folders and run the sample via

dotnet run