Struggling With The Quantum Fourier Transform?

The Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform is the most popular routine in Quantum Computing.Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing You see it in textbooks, tutorials, and lecture slides. It is introduced as the workhorse behind Shor's AlgorithmShor’s Algorithm is a quantum algorithm for factoring large integers efficiently—something classical computers can only do very slowly. It works by using quantum parallelism and the Quantum Fourier Transform to find the period of a modular exponentiation function, which reveals the factors. Its efficiency threatens current cryptographic systems like RSA that rely on the hardness of factoring.
Learn more about Shor's Algorithm and Quantum Phase Estimation.Quantum Phase Estimation (QPE) is a quantum algorithm that determines the phase in an eigenvalue equation for a given unitary U and its eigenvector . It does this by encoding into the amplitudes of qubits using controlled applications of U and then extracting via the inverse quantum Fourier transform. QPE is a core subroutine in many quantum algorithms, such as Shor’s factoring algorithm and quantum simulations.
Learn more about Quantum Phase Estimation

This post is accompanied by a PDF file summarizing the key points.

At its core the idea is straightforward. The routine is not mysterious. Yet somehow, the moment you look it up, clarity evaporates.

Textbooks present it as sums of items filled with complex exponentials. They emphasize that it is a unitaryA **unitary operator** is a linear operator ( U ) on a complex vector space that satisfies ( U^\dagger U = UU^\dagger = I ), meaning it preserves inner products. In simpler terms, it preserves the **length** and **angle** between vectors—so it represents a **reversible, norm-preserving transformation**. In quantum mechanics, unitary operators describe the evolution of isolated systems because they conserve probability.
Learn more about Unitary Operator but rarely explain what this means in practice. Physics sources describe it as abasis change into momentum eigenstates. That is correct if you already think like a physicist, but for most readers it only adds another layer of confusion.

Let's take a look at the standard reference work on Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing. What do Nielsen and ChuangNielsen, M.A., 2010, Cambridge university press, say?

One such transformation is the discrete Fourier transform. In the usual mathematical notation, the discrete Fourier transform takes as input a vector of complex numbers, where the length of the vector is a fixed parameter. It outputs the transformed data, a vector of complex numbers , defined by

The quantum Fourier transform is exactly the same transformation, although the conventional notation for the quantum Fourier transform is somewhat different. The quantum Fourier transform on an orthonormal basis is defined to be a linear operator with the following action on the basis states,

Equivalently, the action on an arbitrary state may be written

where the amplitudes are the discrete Fourier transform of the amplitudes . It is not obvious from the definition, but this transformation is a unitary transformation, and thus can be implemented as the dynamics for a quantum computer.

This type of explanation is not an isolated case. Grab your favorite book on quantum computing and have a look yourself.

Okay, those are textbooks for students. We might think the situation is certainly different when it comes to online materials. So, let's have a look at IBM'sIBM, 2025, , material.

From the analogy with the Discrete Fourier Transform, the Quantum Fourier Transform acts on a Quantum State for qubits and maps it to the Quantum State .

The definition of the Quantum Fourier Transform is

where .

Or, written in the unitary matrix representation:

All we get is a wall of symbols. You come away believing the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform is something arcane when in fact it is not.

? captures the essence of the (classical) Fourier Transform.The Fourier Transform converts a signal from the time domain into the frequency domain, showing which frequencies make up the signal and with what amplitudes. It expresses any time-based function as a sum of sine and cosine waves. In essence, it decomposes complex signals into their basic frequency components.
Learn more about Fourier Transform Depending on the perspective you apply, you'll see a composition of waves either as a tangled waveform or as a few sharp spikes.

In the time domain a signal often appears complicated. Several waves combine into a curve that looks chaotic. You cannot easily see which components are hidden inside.

The Fourier TransformThe Fourier Transform converts a signal from the time domain into the frequency domain, showing which frequencies make up the signal and with what amplitudes. It expresses any time-based function as a sum of sine and cosine waves. In essence, it decomposes complex signals into their basic frequency components.
Learn more about Fourier Transform changes this perspective. Instead of describing the signal over time, it reveals the frequencies that are present and their relative strength. This is their Amplitude.In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude The same information remains, but expressed in a different basis. What seemed tangled now looks simple.

This shift of view is powerful. A signal in the time basis conceals its inner structure. Once expressed in the frequency basis the structure becomes clear. Nothing has been altered. Only the lens has changed.

Every Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State lives in a complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number Vector Space.A vector space is a set of objects called **vectors** that can be **added together** and **scaled by numbers** (called scalars) while still staying in the same set. The addition and scaling follow specific algebraic rules (like associativity, distributivity, and the existence of a zero vector). Examples include geometric vectors in 2D or 3D space, or functions that can be added and multiplied by numbers.
Learn more about Vector Space For a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit this space is two-dimensional. The two basic directions are called the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis and are written as is a basis state.
Learn more about and is a basis state.
Learn more about . You can think of them as the vertical axis in the diagram, with is a basis state.
Learn more about at the top and is a basis state.
Learn more about at the bottom.

The Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State of a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is a combination of these two directions. This is the SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition. Mathematically it looks like . The symbols and are Complex Numbers.A complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number The realA real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve .
Learn more about Real Number part of these numbers represents the Amplitude,In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude which determines the probability of measuringIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in that state (by its absolute square). And the imaginary part denotes the Quantum Phase,A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase which encodes an angle on the complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number plane.

The Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase is subtle. You cannot observe it directly with a MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement. What you do measure are probabilities given by and . The phase shows its importance only when states interfere or when multiple qubits interact. In those cases the relative phase controls how amplitudes combine and whether outcomes reinforce or cancel.

Seen differently, the Quantum State VectorA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State points to the surface of a sphere (the Bloch SphereThe Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere). The vector's latitude sets the balance between is a basis state.
Learn more about and is a basis state.
Learn more about . Its longitude sets the phase. Together these two angles describe the full state of the qubit.

Essentially, the qubit is not just zero or one. It is a vector in a complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number Vector Space.A vector space is a set of objects called **vectors** that can be **added together** and **scaled by numbers** (called scalars) while still staying in the same set. The addition and scaling follow specific algebraic rules (like associativity, distributivity, and the existence of a zero vector). Examples include geometric vectors in 2D or 3D space, or functions that can be added and multiplied by numbers.
Learn more about Vector Space with both magnitude and phase.

All the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform does is to take the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State and rewrite it in a new basis. It transforms a Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State that resides in the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis into phaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase information as shown in ?

A QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis appears as a mixture of is a basis state.
Learn more about and is a basis state.
Learn more about with certain probabilities. That is what you see if you measureIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit directly. The outcome is either zero or one, and the distribution of many measurements reveals the probabilities.

When you apply the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform its Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State is expressed differently. Instead of describing probabilities in the computational view, the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform reorganizes the information into Quantum Phases.A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase These phasesA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase form structured patterns across the Basis States.A basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State But the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform does not change the meaning of the underlying Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State. It only changes how you represent it, much like moving from a time domain to a frequency domain description of a signal.

The simplest way to understand the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform is to see it in action. The Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform is almost trivial for a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit because it is exactly the Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator. It takes the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis states is a basis state.
Learn more about and is a basis state.
Learn more about and rewrites them in the Fourier Basis.The Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis

If the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is in is a basis state.
Learn more about , applying the Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator produces . Instead of pointing straight up in the Bloch Sphere,The Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit now lies on the equator, residing on the -axis. This is the Fourier BasisThe Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis version of zero. Like all points on the equator, this is a Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State with equal weight on both Basis States.A basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State

If the qubit is in is a basis state.
Learn more about , the Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator produces . This state also lies on the equator of the Bloch Sphere,The Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere but the relative Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase between is a basis state.
Learn more about and is a basis state.
Learn more about is different. That Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase is what distinguishes from .

This simple example demonstrates what the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform does. While it does not erase the difference between is a basis state.
Learn more about and is a basis state.
Learn more about , it translates them into new Quantum StatesA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State where the difference shows up as a change of Quantum Phase.A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase The Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator makes the role of relative Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase explicit.

With two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform shows its structure more clearly. In the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis there are four possible states: , , , and . Each one corresponds to a distinct configuration of the two Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit either up or down on the Bloch Sphere.The Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere

The two-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform, depicted in Figure 5, maps each of these Quantum StatesA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State into a new state in the Fourier Basis.The Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis Now the relative Quantum PhasesA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase between the QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit become even clearer. turns into , becomes , where the essential feature is the quarter-turn phase shift. Similarly, and transform into and . The difference among these outputs lies in the relative Quantum PhasesA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase that distinguish them from one another.

So the Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform on two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit does what the Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator did for one. It arranges the results in clear, evenly spaced intervals in the Fourier Basis.The Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis

When you scale this up to multiple QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit, the idea generalizes.

Put simply, the -QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform maps the Basis StatesA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State of the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis into evenly spaced intervals in the Fourier Basis.The Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis The more qubits we use, the more intricate and fine-grained the phase patterns become.

This exponential growth in structure is why Quantum Fourier TransformThe Quantum Fourier Transform (QFT) is the quantum analog of the discrete Fourier transform, mapping quantum states from the computational basis to the frequency domain using quantum superposition and interference. It transforms amplitudes so that periodic patterns in the input become phase differences in the output. The QFT is exponentially faster than the classical Fourier transform, making it central to algorithms like Shor’s factoring algorithm.
Learn more about Quantum Fourier Transform is such a powerful tool. It can encode periodicity at scales far beyond what is visible in the raw Computational Basis.The computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis That is the resource algorithms like Shor's AlgorithmShor’s Algorithm is a quantum algorithm for factoring large integers efficiently—something classical computers can only do very slowly. It works by using quantum parallelism and the Quantum Fourier Transform to find the period of a modular exponentiation function, which reveals the factors. Its efficiency threatens current cryptographic systems like RSA that rely on the hardness of factoring.
Learn more about Shor's Algorithm use to extract answers to problems that are intractable for classical machines.