Encoding Patterns

Basis Encoding

Basis encoding represents classical data by mapping each possible data value to a distinct computational basis state of qubits. For example, an -bit binary string is encoded as the quantum state among the basis states. It's a direct, one-to-one encoding. No superposition or amplitude manipulation is used.

by Frank Zickert

January 4, 2026

We hear that a handful of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit can exist in a space with exponentially many possibilities. Isn't that tempting? However, if you take a closer look at most Quantum Circuits,A 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 you will find that they use one QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit for one classical Bit.A 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 But why would they do that?

? denotes a short program that prepares such a Quantum CircuitA 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 with Qiskit.Qiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit It encodes the 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 sequence x=10110. Of course, we can us to any 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 sequence we want. The number of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit corresponds to the number of BitsA 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 in the sequence.

basis_encoding.py

import json

import pytest

from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister

from qiskit.quantum_info import Statevector

from qiskit.primitives import StatevectorSampler

# our target classical bit string

x = "10110"

# number of qubits equals number of bits

m = len(x)

# the quantum circuit with m-sized quantum and classical registers

qr = QuantumRegister(m, "qr")

cr = ClassicalRegister(m, "cr")

qc = QuantumCircuit(qr, cr)

# Prepare |x> by flipping qubits where the bit is '1'

# reversed because qubit 0 is x_0

for i, b in enumerate(reversed(x)):

    if b == "1":

        qc.x(i)

# measure each qubit to a classical bit

qc.measure(range(m), range(m))

# simulate the state |x>

sampler = StatevectorSampler()

counts = sampler.run([qc], shots=1000).result()[0].data.cr.get_counts()

# compute the probability

total = sum(counts.values()) or 1

probs = {k: v / total for k, v in counts.items()}

# Verify the bitstring appears

assert x in counts, f"Expected bitstring '{x}' not found in results: {counts}"

# Verify it appears with probability of 1

assert probs[x] == 1.0, f"Bitstring '{x}' appeared only {probs[x]*100:.1f}% of the time"

print(json.dumps({"counts": counts, "probabilities": probs}, indent=4))

Listing 1 Bit string encoding

We the quantum and classical registers and build a circuit with QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit and classical Bits.A 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 The classical BitsA 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 are used to store the 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 results.

The circuit starts in the state , in which all BitsA 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 have the value . We through the BitsA 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 of and apply an X-GateThe X-Gate flips a qubit's probability amplitudes between its two basis states, effectively exchanging their roles. It doesn’t just turn into , but rotates any quantum state halfway around the x-axis of the Bloch sphere. This means it inverts the qubit's orientation while preserving the overall shape of its state on the sphere.
Learn more about X-Gate to each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit whose corresponding 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 has the value . This is the core of this coding pattern, which we refer to as Basis Encoding.Basis encoding represents classical data by mapping each possible data value to a distinct computational basis state of qubits. For example, an -bit binary string is encoded as the quantum state among the basis states. It's a direct, one-to-one encoding. No superposition or amplitude manipulation is used.
Learn more about Basis Encoding

For a Qubit,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit there are two 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 called is a basis state.
Learn more about and is a basis state.
Learn more about . They are Orthogonal.Orthogonal means independent. In geometry, two lines or vectors are orthogonal if they meet at a angle. In math or data science, it means two things don’t influence each other, like variables or directions that are completely uncorrelated. In quantum computing, orthogonal states are quantum states that are completely distinct. Their inner product is zero, meaning they can be perfectly told apart when measured. For example, the basis states and are orthogonal. This property ensures they can represent different, non-overlapping pieces of information in a quantum system.
Learn more about Orthogonal In Quantum Computing,Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing OrthogonalOrthogonal means independent. In geometry, two lines or vectors are orthogonal if they meet at a angle. In math or data science, it means two things don’t influence each other, like variables or directions that are completely uncorrelated. In quantum computing, orthogonal states are quantum states that are completely distinct. Their inner product is zero, meaning they can be perfectly told apart when measured. For example, the basis states and are orthogonal. This property ensures they can represent different, non-overlapping pieces of information in a quantum system.
Learn more about Orthogonal states are 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 that do not overlap at all, which means that their Inner ProductAn inner product is a mathematical operation that takes two vectors and returns a single number measuring how similar or aligned they are. In Euclidean space, it’s the sum of the products of corresponding components (e.g., ). It generalizes the dot product and defines geometric concepts likelength and angle in vector spaces.
Learn more about Inner Product is zero. As a result, they can be perfectly distinguished from each other during a Measurement.In 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

For Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit the JointJoint probability is the likelihood that two (or more) events happen at the same time. For example, if is it rains and is you carry an umbrella, the joint probability is the chance that both rain and carrying an umbrella occur together. Mathematically, .
Learn more about Joint Probability Basis StateA 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 is a product of single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit 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 So, if is a 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 sequence with the BitsA 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 , the corresponding product state is

where the symbol means that we place the individual QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit states side by side to form the JointJoint probability is the likelihood that two (or more) events happen at the same time. For example, if is it rains and is you carry an umbrella, the joint probability is the chance that both rain and carrying an umbrella occur together. Mathematically, .
Learn more about Joint Probability state. If is equal to , then is equal to is a basis state.
Learn more about . If is equal to , then is equal to is a basis state.
Learn more about .

The encoding map is

which reads as follows. Take a 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 string with Bits.A 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 Map it into the dimensional 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 by returning the basis VectorA 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 that matches the 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 pattern. The two sets are the set of all 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 strings and 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 space.

To prepare from the all zeros state, you can apply the following 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

which says apply the Pauli X gate on those QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit whose 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 is one and do nothing on the others. The exponent here is just a switch. If equals zero then is the Identity MatrixAn identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts as the multiplicative identity in matrix algebra, meaning ( A \times I = I \times A = A ) for any compatible matrix ( A ). Essentially, multiplying by it leaves a matrix unchanged.
Learn more about Identity Matrix (that does nothing). If equals one then is the X gate that flips 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 from to . Acting on this 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 gives .

This operator maps to the Basis StateA 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 .

Simply put, the operators we apply in our Quantum CircuitA 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 act as a composite operator themselves. We refer to this composite operator as . When we apply this operator to a Quantum SystemA quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
Learn more about Quantum System in which all QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit are in the state , we obtain 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 encodes , i.e., .

The in the source code uses the reverse order of the 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 sequence because QiskitQiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit follows the little-endian convention. The QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit corresponds to the least significant 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 , which appears on the right in the binary sequence. Python,Python is a high-level, interpreted programming language known for its simple syntax and readability. It supports multiple programming paradigms, including procedural, object-oriented, and functional programming. Its extensive standard library and large ecosystem make it useful for tasks ranging from web development to data science and automation.
Learn more about Python on the other hand, traverses a string from left (position ) to right. Reversing the string ensures that the 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 is applied to the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit . If the same convention is used throughout for preparation and Measurement,In 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 input and output 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 strings will match.

Each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is into the corresponding classical Bit.A 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 This converts 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 back into classical data.

In Qiskit,Qiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit we always measure 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 whose results coincide with these OrthogonalOrthogonal means independent. In geometry, two lines or vectors are orthogonal if they meet at a angle. In math or data science, it means two things don’t influence each other, like variables or directions that are completely uncorrelated. In quantum computing, orthogonal states are quantum states that are completely distinct. Their inner product is zero, meaning they can be perfectly told apart when measured. For example, the basis states and are orthogonal. This property ensures they can represent different, non-overlapping pieces of information in a quantum system.
Learn more about Orthogonal 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 and . The probability of seeing the output when the state is is

Here, is the Inner ProductAn inner product is a mathematical operation that takes two vectors and returns a single number measuring how similar or aligned they are. In Euclidean space, it’s the sum of the products of corresponding components (e.g., ). It generalizes the dot product and defines geometric concepts likelength and angle in vector spaces.
Learn more about Inner Product between the two basis 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 It is equal to one if is equal to , and otherwise equal to zero. The symbol is called the Kronecker delta and encodes this rule. It is either or .

If you prefer to think in terms of integers, define

This is the usual binary value of the string when is the least significant Bit.A 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 Then you can write , which means that the integer label refers to the corresponding basis Vector.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

We a StatevectorSampler and execute the circuit several times. We check the preparation by sampling the 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 results, just as we would on real hardware, with the exception of errors and Noise.Noise
Learn more about Noise

The result of the execution is a dictionary with measured values. We these values to empirical probabilities. If the Basis StateA 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 is prepared correctly, all 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 results in each 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 correspond to the original 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 sequence .

{

  "counts": { "10110": 1000 },

  "probabilities": { "10110": 1.0 }

}

This means that the Basis-encoding provides certainty during readout.

That's the long version of a short story: We simply apply X gates to every 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 that is . We flip the states of the corresponding QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit from is a basis state.
Learn more about to is a basis state.
Learn more about . By measuring the Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit we obtain the original 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 sequence. Each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit gives us the value of the corresponding 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 in the 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 sequence.

No clever packaging. No Superposition.Superposition 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 Just the exact pattern you already had. We treat QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit as or . Just like classical Bits.A 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

Doesn't that feel like buying a concert hall and using it for just one note? Doesn't everyone say that a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is not just or ? Isn't it in multiple states at the same time?

Can't we use these properties to compress classical BitsA 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 into far fewer Qubits?A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Isn't that exactly where the Quantum AdvantageQuantum advantage is the point where a quantum computer performs a specific task faster or more efficiently than the best possible classical computer. It doesn’t mean quantum computers are universally better—just that they outperform classical ones for that task. The first demonstrations (e.g., Google’s 2019 Sycamore experiment) showed speedups for highly specialized problems, not yet for practical applications.
Learn more about Quantum Advantage comes from?

The Tempting but Incomplete Intuition

The natural intuition is simple. A QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit feels like a richer Bit.A 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 If it can be in many states, then maybe we can pack more classical information into fewer Qubits...?A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit

Of course, a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is not in multiple states at the same time. If you haven't already internalized this yet, repeat after me: a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is not in multiple states at the same time. But it is in a complex linear combination of its 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 is a basis state.
Learn more about and . Even if this is something different than being in multiple states at the same time, we can use this structure for data compression.

Let's take one of the other EncodingEncoding is the process of converting information from one form into another, usually so it can be stored, transmitted, or processed more efficiently. For example, text can be encoded into binary for computers to handle, or sounds into digital signals for transmission. The key idea is that encoding changes the representation, not the meaning, of the data.
Learn more about Encoding patterns. Consider Angle Encoding,Angle 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 quantum states for use in quantum algorithms or quantum circuits.
Learn more about Angle Encoding for example. Conceptually, you can map an entire 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 sequence to the angle of a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit and then read back this angle statistically.

You can take a 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 sequence of length , convert it to an integer , set , prepare , and then sample many times to estimate the probability of measuring , which is equal to . With enough MeasurementsIn 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 and suitable 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 bases, one can estimate and then derive .

What this example is really showing

However, this is not compression in the information-theoretical sense. The reason for this is that a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit cannot reliably store and recover multiple classical BitsA 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 from a Measurement.In 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

We cannot directly observe the angle of rotation. What makes recovery seemingly possible here is that we are allowed to

measure multiple times,
estimate a real parameter (a probability) that contains more information than a single Bit,A 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 and
decode the angle of rotation from the 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 probability.

In other words, the additional capacity arises from repeated sampling and not from a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit storing multiple BitsA 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 at once.

If you want to retrieve classical data with certainty in a single Measurement,In 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 every possible message must be encoded in a state that can be perfectly distinguished from every other message state. It must be measurable.

In Quantum Computing,Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing perfect distinguishability requires orthogonality. And as we have already learned, orthogonality means that the Inner ProductAn inner product is a mathematical operation that takes two vectors and returns a single number measuring how similar or aligned they are. In Euclidean space, it’s the sum of the products of corresponding components (e.g., ). It generalizes the dot product and defines geometric concepts likelength and angle in vector spaces.
Learn more about Inner Product between two states is zero, which guarantees that an appropriate 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 can distinguish reliably.

Let's count. 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 have at most mutually OrthogonalOrthogonal means independent. In geometry, two lines or vectors are orthogonal if they meet at a angle. In math or data science, it means two things don’t influence each other, like variables or directions that are completely uncorrelated. In quantum computing, orthogonal states are quantum states that are completely distinct. Their inner product is zero, meaning they can be perfectly told apart when measured. For example, the basis states and are orthogonal. This property ensures they can represent different, non-overlapping pieces of information in a quantum system.
Learn more about Orthogonal states. If you want to encode all possible -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 strings and recover them with certainty in one pass, then must hold, which gives .

Another way to recognize the same limit is the following information limit. When measuring 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 extract at most BitsA 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 of classical information, even if you try very clever EncodingsEncoding is the process of converting information from one form into another, usually so it can be stored, transmitted, or processed more efficiently. For example, text can be encoded into binary for computers to handle, or sounds into digital signals for transmission. The key idea is that encoding changes the representation, not the meaning, of the data.
Learn more about Encoding and Measurements.In 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 An ensemble of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit cannot transmit more than classical BitsA 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 in a single use.

This limitation is unavoidable. If you use fewer Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit you must accept probabilistic or lossy recovery.

Basis-encoding is not wasteful. It is a deliberate lossless EmbeddingAn embedding is a numerical representation of data (like words, images, or documents) in a continuous vector space where similar items are close together. It captures semantic or structural relationships by encoding features into dense vectors. Embeddings are used in machine learning to make non-numeric data comparable and computable.
Learn more about Embedding in which OrthogonalOrthogonal means independent. In geometry, two lines or vectors are orthogonal if they meet at a angle. In math or data science, it means two things don’t influence each other, like variables or directions that are completely uncorrelated. In quantum computing, orthogonal states are quantum states that are completely distinct. Their inner product is zero, meaning they can be perfectly told apart when measured. For example, the basis states and are orthogonal. This property ensures they can represent different, non-overlapping pieces of information in a quantum system.
Learn more about Orthogonal states are intentionally chosen to ensure that the 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 is reliable. We cannot use the properties of Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing to compress classical BitsA 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 into far fewer QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit while retaining the ability to read the data with certainty.

Fortunately, however, this is not where the Quantum AdvantageQuantum advantage is the point where a quantum computer performs a specific task faster or more efficiently than the best possible classical computer. It doesn’t mean quantum computers are universally better—just that they outperform classical ones for that task. The first demonstrations (e.g., Google’s 2019 Sycamore experiment) showed speedups for highly specialized problems, not yet for practical applications.
Learn more about Quantum Advantage lies. Rather, advantage results from the parallel generation and transformation of SuperpositionsSuperposition 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 and 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 of many such strings and the subsequent arrangement of InterferencesInterference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference so that the 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 reveals a global property.

So, the exponential state space is not used for denser classical storage, but for computation. Basis-encoding is the stable method for loading and unloading specific classical data into and out of this quantum computation.