Quantum Data Encoding

A widespread misunderstanding in Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing is that QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit behave like strange memory cells. The picture is simple: take your data, load it into Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit let theQuantum AlgorithmA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms.
Learn more about Quantum Algorithm run, and then read outIn 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 answer.

That picture collapses as soon as you look closer. QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit are not blank containers waiting for information. How you put data into them already determines what transformationsA **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 are possible and which ones are ruled out. The encoding step is not passive. Instead, it reshapes the landscape of what the Quantum AlgorithmA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms.
Learn more about Quantum Algorithm can even do.

This is where the contrast with classical practice becomes sharp. A data scientist moving from Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming.
Learn more about Machine Learning might think of encoding as a preprocessing step, like scaling or normalizing features. In Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing, it isn't preparation. Data encoding is the opening move of the Quantum AlgorithmA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms.
Learn more about Quantum Algorithm itself. The physics forces your hand: differentencodings give rise to different computational possibilities, and some pathways close off entirely depending on the choice you make.

And this is only the beginning. Once you see encoding in this way, you realize there is no single recipe to follow. What looks best is inseparable from the context and the problem at hand.

Why Can’t We Just Load Data Into Qubits?

In classical computing, memory is a neutral canvas. Binary DigitsA bit (short for “binary digit”) is the smallest unit of data in computing, representing a value of either 0 or 1. It’s the fundamental building block of all digital information. Multiple bits combine to form larger units like bytes (8 bits) and encode more complex data such as numbers, text, or images.
Learn more about Binary Digit orVectorsA vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector are written directly, no questions asked. In Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing things are a little different.

• Data Structure Mismatch: 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 don’t work this way. A QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
  Learn more about Quantum Bit isn't a container. It's a WavefunctionIn quantum computing, a wavefunction describes the complete quantum state of a system. Essentially it encodes all possible outcomes and their probabilities. It’s a mathematical object (usually a complex-valued vector) that evolves according to the Schrödinger equation. Measuring the system collapses the wavefunction to one outcome, destroying the superposition of states.
  Learn more about Wavefunction in a Hilbert SpaceA Hilbert space is a complete vector space equipped with an inner product, which allows for measuring angles and lengths between vectors. "Complete" means that every Cauchy sequence of vectors converges to a vector within the space. It generalizes the idea of Euclidean space to possibly infinite dimensions and forms the foundation for quantum mechanics and functional analysis.
  Learn more about Hilbert Space. That means you don't just store numbers. You have to map them into Amplitudes,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 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 or 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 You don't start with a dataset and load it. You start with physics and reshape the dataset to fit.
• Required Normalization: Classical data can take any scale. Store a or a and the machine doesn't care. 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 are stricter. 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 must satisfy the normalization condition:. This means raw values can't be used directly. They must be rescaled, which can warp relationships in the data. Get the normalization wrong, and you don't just lose accuracy. You change the problem definition.
• Cost of State Preparation: Loading data classically is trivial: per item. In quantum computing, preparing an -dimensional amplitude state generally costs Quantum Gates.A quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
  Learn more about Quantum Gate That's before a single algorithmic step is run. For many practical datasets, the overhead swamps any theoretical speedup. Encoding isn't just a design choice. It's the tax you pay up front.
• Encoding Determines Computation: In classical computing, how you store the data doesn't restrict the algorithms you can apply. Arrays, lists, or hash maps all lead to the same computations. Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
  Learn more about Quantum Computing flips this logic. The encoding defines the set of computations you can perform. Pick the wrong encoding, and the Quantum AlgorithmA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms.
  Learn more about Quantum Algorithm you wanted to run may be inaccessible. Encoding isn't preparation. It's part of your strategy.

So, encoding is not a simple loading step. It is the first algorithmic choice in any quantum workflow, and it dictates what is possible. There is no universal method. Instead, four main approaches have emerged, each reshaping the physics in a different way and each carrying its own trade-offs.

The first is Basis EncodingBasis encoding represents classical data by mapping each possible data value to a distinct computational basis state of qubits. For example, an ( n )-bit binary string ( x ) is encoded as the quantum state (|x\rangle) among the (2^n) basis states. It’s a direct, one-to-one encoding—no superposition or amplitude manipulation is used.
Learn more about Basis Encoding. In this method, you map each classical bitA bit (short for “binary digit”) is the smallest unit of data in computing, representing a value of either 0 or 1. It’s the fundamental building block of all digital information. Multiple bits combine to form larger units like bytes (8 bits) and encode more complex data such as numbers, text, or images.
Learn more about Binary Digit directly to a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit A value of zero remains in is a basis state.
Learn more about , while a value of one flips to is a basis state.
Learn more about with a Not GateIn quantum computing, the NOT gate (also called the **X gate**) flips the state of a qubit: it changes to and to . Mathematically, it’s represented by the Pauli-X matrix . On the Bloch sphere, it corresponds to a rotation around the X-axis.
Learn more about Not Gate. The overall dataset is simply the Tensor ProductA tensor product combines two vector spaces (or matrices) into a new, larger space that encodes all possible pairwise combinations of their elements. If one space has dimension *m* and the other *n*, the tensor product space has dimension *m × n*. In matrix terms, it generalizes the outer product, producing a block matrix that represents how elements from one space interact with elements of another.
Learn more about Tensor Product of these single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit states. The advantage lies in its simplicity. It is transparent, easy to teach, and requires minimal Circuit DepthCircuit depth in quantum computing is the number of layers of quantum gates that must be applied sequentially, where gates acting on different qubits in parallel count as one layer. It measures how long a quantum computation takes, assuming gates in the same layer happen simultaneously. Lower depth is crucial because qubits lose coherence over time, so deep circuits are more error-prone.
Learn more about Circuit Depth. Yet it consumes one QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit per feature and scales poorly. Use it only for small binary or categorical datasets, toy problems, or demonstrations.

The second is Amplitude EncodingAmplitude encoding represents classical data as the amplitudes of a quantum state's basis vectors. If you have a normalized data vector , it’s encoded into an -qubit state . This allows data values to be stored in only qubits, but preparing the quantum state can be computationally expensive.
Learn more about Amplitude Encoding. Here, you normalize a vector so that and then load it into the AmplitudeIn 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 of a quantum state. With Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit you can represent features. This is extremely efficient. However, preparing such 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 requires many Quantum Gates,A quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate Quantum CircuitsA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit become deepCircuit depth in quantum computing is the number of layers of quantum gates that must be applied sequentially, where gates acting on different qubits in parallel count as one layer. It measures how long a quantum computation takes, assuming gates in the same layer happen simultaneously. Lower depth is crucial because qubits lose coherence over time, so deep circuits are more error-prone.
Learn more about Circuit Depth, and NoiseNoise
Learn more about Noise quickly destroys them. It is elegant in theory but problematic on current hardware. Yet, looking forward to error-corrected Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Amplitude EncodingAmplitude encoding represents classical data as the amplitudes of a quantum state's basis vectors. If you have a normalized data vector , it’s encoded into an -qubit state . This allows data values to be stored in only qubits, but preparing the quantum state can be computationally expensive.
Learn more about Amplitude Encoding likely becomes even more important.

The third approach is Angle EncodingAngle encoding is a method of loading classical data into a quantum state by mapping data values to rotation angles of qubits (e.g., using quantum gates like Rx Gate, Ry Gate, or Rz Gate) Each feature of the data is represented as the angle of the quantum state vector’s rotation, which changes its probability amplitudes. This allows continuous classical values to be embedded in qQuantum states for use in quantum algorithms or quantum circuits.
Learn more about Angle Encoding. Each feature is mapped to a rotation angle and applied through Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate such as Ry GateThe **Ry gate** is a single-qubit quantum gate that rotates a qubit’s state around the **Y-axis** of the Bloch sphere by a specified angle . Mathematically, it’s represented as . It changes the probabilities (amplitudes) of measuring 0 or 1 without altering their relative phase.
Learn more about Ry Gate, Rx GateThe **Rx gate** is a single-qubit quantum gate that performs a rotation around the **x-axis** of the Bloch sphere by a specified angle . Mathematically, it’s represented as , where is the Pauli-X matrix. It changes the qubit’s state by mixing the amplitudes of and while keeping their probabilities dependent on the rotation angle.
Learn more about Rx Gate, or Rz GateThe Rz gate is a single-qubit rotation gate that rotates the qubit’s state around the Z-axis of the Bloch sphere by an angle . Its matrix form is . It changes the relative phase between and without altering their probabilities.
Learn more about Rz Gate. 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 is the product of these single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit rotations, with optional entangling gatesEntanglement is a quantum phenomenon where two or more particles become correlated so that measuring one instantly determines the state of the other, no matter how far apart they are. This correlation arises because their quantum states are linked as a single system, not as independent parts. It doesn’t allow faster-than-light communication but shows that quantum systems can share information in ways classical physics can’t explain.
Learn more about Entanglement like Controlled Not GateA controlled-NOT (CNOT) gate is a two-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Quantum GateA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate where the first QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit (control) determines whether the second QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit (target) is flipped on the -axis. If the control QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is in state is a basis state.
Learn more about , the target Qubit'sA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit 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 inverted . If the control is is a basis state.
Learn more about , the target is unchanged. It’s essential for creating entanglement, since applying aCNOT to a superposed control qubit links the states of both qubits.
Learn more about Controlled Not Gate to introduce correlations. This scheme producesshallow circuitsCircuit depth in quantum computing is the number of layers of quantum gates that must be applied sequentially, where gates acting on different qubits in parallel count as one layer. It measures how long a quantum computation takes, assuming gates in the same layer happen simultaneously. Lower depth is crucial because qubits lose coherence over time, so deep circuits are more error-prone.
Learn more about Circuit Depth, handles real-valued features naturally, and integrates well with hybrid algorithmsA Variational Quantum Algorithm is a hybrid quantum–classical algorithm in which a quantum circuit is paramterized by a classical routine. This means, it usually computes the values for rotation angles used inside this parameterized quantum circuit during a classical pre-processing step. Additionally, the measurement results are interpreted during a classical post-processing.
Learn more about Variational Quantum Algorithm. Its weaknesses are limited compression and low ExpressivenessExpressiveness in quantum computing refers to how effectively a quantum model (like a variational circuit) can represent complex functions or quantum states. A more expressive model can capture richer correlations and solve more complex problems, but may also be harder to train. It’s roughly analogous to the “capacity” or “model complexity” in classical machine learning.
Learn more about Expressiveness without Entanglement.Entanglement is a quantum phenomenon where two or more particles become correlated so that measuring one instantly determines the state of the other, no matter how far apart they are. This correlation arises because their quantum states are linked as a single system, not as independent parts. It doesn’t allow faster-than-light communication but shows that quantum systems can share information in ways classical physics can’t explain.
Learn more about Entanglement Despite this, it is the most practical option on Noisy Intermediate-Scale QuantumNoisy Intermediate-Scale Quantum refers to the current generation of quantum devices that have enough qubits to run non-trivial algorithms but are still small and error-prone, limiting their reliability and scalability.
Learn more about Noisy Intermediate-Scale Quantum hardware and the default for near-term Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from quantum computing with traditional machine learning to solve complex problems more efficiently than classical approaches.
Learn more about Quantum Machine Learning.

Finally, there is Block EncodingBlock encoding is a quantum computing method for representing a classical or quantum matrix (A) as a submatrix of a larger unitary matrix (U). Specifically, (U) is built so that (A = \alpha (\langle 0^k | \otimes I) U (|0^k\rangle \otimes I)) for some scaling factor (\alpha) and number of ancilla qubits (k). This lets quantum algorithms access (A) efficiently using unitary operations, enabling tasks like Hamiltonian simulation or matrix inversion.
Learn more about Block Encoding. Instead of embedding Vectors,A vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector you embed a matrix into the top-left block of a larger unitary matrixUnitary evolution is the rule that describes how a closed quantum system changes over time according to the Schrödinger equation. The system’s state vector is transformed by a *unitary operator*, meaning probabilities (the total norm) are exactly preserved. This ensures quantum evolution is reversible and information is never lost.
Learn more about Unitary Evolution. Ancilla QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit provide the scaffolding, and by post-selecting them in is a basis state.
Learn more about , you can extract the action of . This technique opens access to powerful algorithms such as Harrow–Hassidim–Lloyd Algorithm,The HHL algorithm (Harrow–Hassidim–Lloyd) is a quantum algorithm for solving systems of linear equations exponentially faster than the best known classical methods, under certain conditions. It encodes the solution vector as a quantum state using phase estimation and controlled rotations based on the eigenvalues of . However, it only provides a quantum state proportional to , not its classical components, and its speedup depends on properties like sparsity and condition number of .
Learn more about Harrow–Hassidim–Lloyd Algorithm Hamiltonian Simulation,Hamiltonian simulation is the process of using a quantum computer to mimic the time evolution of a quantum system governed by a Hamiltonian , typically by approximating . It allows prediction of how a quantum state changes over time without physically realizing the system. This is fundamental to quantum algorithms for chemistry, materials science, and physics because it efficiently reproduces complex quantum dynamics that are intractable for classical computers.
Learn more about Hamiltonian Simulation and Quantum Singular Value Transformation.Quantum Singular Value Transformation (QSVT) is a quantum algorithmic framework that applies a polynomial function to the singular values of a given matrix, using a sequence of controlled unitary operations. It generalizes many quantum algorithms (like amplitude amplification and HHL) under a single formalism. In essence, QSVT lets you manipulate matrix spectra directly on a quantum computer without explicitly diagonalizing the matrix.
Learn more about Quantum Singular Value Transformation The price is high resource demand and complexity, which makes it unsuitable for today's devices. It remains a forward-looking tool for Linear AlgebraLinear algebra is the branch of mathematics that studies vectors, vector spaces, and linear transformations between them. It provides tools for solving systems of linear equations and understanding geometric operations like rotations, projections, and scaling. Its core objects—matrices and vectors—form the basis for much of modern computation, physics, and machine learning.
Learn more about Linear Algebra and physics simulations.

Each encoding strikes a different balance between resources, efficiency, and computational scope. Basis EncodingBasis encoding represents classical data by mapping each possible data value to a distinct computational basis state of qubits. For example, an ( n )-bit binary string ( x ) is encoded as the quantum state (|x\rangle) among the (2^n) basis states. It’s a direct, one-to-one encoding—no superposition or amplitude manipulation is used.
Learn more about Basis Encoding is simple but wasteful. Amplitude EncodingAmplitude encoding represents classical data as the amplitudes of a quantum state's basis vectors. If you have a normalized data vector , it’s encoded into an -qubit state . This allows data values to be stored in only qubits, but preparing the quantum state can be computationally expensive.
Learn more about Amplitude Encoding is compact but costly to prepare. Angle EncodingAngle encoding is a method of loading classical data into a quantum state by mapping data values to rotation angles of qubits (e.g., using quantum gates like Rx Gate, Ry Gate, or Rz Gate) Each feature of the data is represented as the angle of the quantum state vector’s rotation, which changes its probability amplitudes. This allows continuous classical values to be embedded in qQuantum states for use in quantum algorithms or quantum circuits.
Learn more about Angle Encoding is hardware-friendly yet limited in compression. Block EncodingBlock encoding is a quantum computing method for representing a classical or quantum matrix (A) as a submatrix of a larger unitary matrix (U). Specifically, (U) is built so that (A = \alpha (\langle 0^k | \otimes I) U (|0^k\rangle \otimes I)) for some scaling factor (\alpha) and number of ancilla qubits (k). This lets quantum algorithms access (A) efficiently using unitary operations, enabling tasks like Hamiltonian simulation or matrix inversion.
Learn more about Block Encoding is powerful but resource-intensive. None is universally superior.

The lesson is clear. Encoding is not preprocessing. It is the first constraint that shapes what is computable. Each method trades off QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit count, Circuit Depth,Circuit depth in quantum computing is the number of layers of quantum gates that must be applied sequentially, where gates acting on different qubits in parallel count as one layer. It measures how long a quantum computation takes, assuming gates in the same layer happen simultaneously. Lower depth is crucial because qubits lose coherence over time, so deep circuits are more error-prone.
Learn more about Circuit Depth and ExpressivenessExpressiveness in quantum computing refers to how effectively a quantum model (like a variational circuit) can represent complex functions or quantum states. A more expressive model can capture richer correlations and solve more complex problems, but may also be harder to train. It’s roughly analogous to the “capacity” or “model complexity” in classical machine learning.
Learn more about Expressiveness in a different way. The right choice depends on your data, your hardware, and your goals. For teaching and exploration, start with Basis EncodingBasis encoding represents classical data by mapping each possible data value to a distinct computational basis state of qubits. For example, an ( n )-bit binary string ( x ) is encoded as the quantum state (|x\rangle) among the (2^n) basis states. It’s a direct, one-to-one encoding—no superposition or amplitude manipulation is used.
Learn more about Basis Encoding. On real devices with modest datasets, use Angle EncodingAngle encoding is a method of loading classical data into a quantum state by mapping data values to rotation angles of qubits (e.g., using quantum gates like Rx Gate, Ry Gate, or Rz Gate) Each feature of the data is represented as the angle of the quantum state vector’s rotation, which changes its probability amplitudes. This allows continuous classical values to be embedded in qQuantum states for use in quantum algorithms or quantum circuits.
Learn more about Angle Encoding. For theoretical work, experiment with Amplitude EncodingAmplitude encoding represents classical data as the amplitudes of a quantum state's basis vectors. If you have a normalized data vector , it’s encoded into an -qubit state . This allows data values to be stored in only qubits, but preparing the quantum state can be computationally expensive.
Learn more about Amplitude Encoding. For advanced operator-based algorithms, turn to Block EncodingBlock encoding is a quantum computing method for representing a classical or quantum matrix (A) as a submatrix of a larger unitary matrix (U). Specifically, (U) is built so that (A = \alpha (\langle 0^k | \otimes I) U (|0^k\rangle \otimes I)) for some scaling factor (\alpha) and number of ancilla qubits (k). This lets quantum algorithms access (A) efficiently using unitary operations, enabling tasks like Hamiltonian simulation or matrix inversion.
Learn more about Block Encoding.