Quantum Circuit

What's good for a quantum computer can't be bad for humans, can it?

Quantum circuits are more than just abstract math. They bridge theory and hardware by turning dense unitary matrices into structured recipes that real devices can execute. From the simplest Bell state to complex algorithms, circuits provide an essential layer of abstraction that makes quantum computation both understandable and practical.

by Frank Zickert
September 17, 2025
Quantum Circuit

A quantum circuit is a structured sequence of A **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 applied to A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit. Nielsen, M.A., 2010, Cambridge university press, .

When you think of a circuit, you may picture wires, resistors, and capacitors. A quantum circuit works very differently. Instead of currents, it uses A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit, and instead of resistors, it employs A **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. Yet both quantum and electrical circuits function as abstract models. They allow you to analyze how complex hardware systems are built and how they operate.

Do Quantum Circuits Look Alien to You?

Let's make them familiar
•5 min
When you first see a quantum circuit, it can feel a bit like staring at an alien language. Strange wires run across the page. Boxes with cryptic symbols sit on top. And the whole thing seems detached from the way we normally think about software. But here’s the twist: quantum circuits aren't alien at all. They're just the way we use to describe quantum computations. And once you know the building blocks, you’ll see they’re not only familiar, they’re surprisingly intuitive.

Yet, treating quantum circuits as classical logic analogues is misleading . Although both types of circuits serve the same purpose, their internal mechanisms differ completely. quantum circuits capture 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 and 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, Preskill, J., 2018, Quantum, Vol. 2, pp. 79.

The A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit is the simplest nontrivial 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 that reveals what sets quantum computing apart from classical logic. Think of it as the hello world of quantum computing.

The circuit begins with two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in the state . Applying a The 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 to the first A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit places it into a 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 of is a basis state.
Learn more about and is a basis state.
Learn more about . A The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator then links the A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit For the first A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is in the A 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 a basis state.
Learn more about , the second flips from is a basis state.
Learn more about to is a basis state.
Learn more about and vice versa. If the first is is a basis state.
Learn more about , the second remains unchanged.

You can represent this 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 in different ways. Textbooks often start with mathematics. The circuit corresponds to the equation:

Figure 1 Equation of the bell state quantum cicruit

Figure 1 denotes the euqation of the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit.

A 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 like this is a sequence of matrix multiplications that reads from right to left.

  1. We apply , where is the The 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     acting on the first A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
    Learn more about Quantum Bit     and is the An 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     acting on the second.
    • The symbol denotes the A 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      .
    • is a A **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      .
    • is the An 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      .
    • The A 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       is a A **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       acting on two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
      Learn more about Quantum Bit       simultaneously. It applies a The 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       to the first while leaving the second unchanged.
    Figure 2 The tensor product of H and I

    Figure 2 denotes the A 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     .

  2. We apply the The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
    Learn more about Controlled-NOT Operator     whose matrix is depicted in ?. Since this operator is an entanglement gate, we cannot write it as a A 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 two single-A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
    Learn more about Quantum Bit     matrices.
    Figure 3 The matrix of the CNOT operator

The second representation of a 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 is a circuit diagram shown in ?.

Figure 4 Quantum circuit diagram of the Bell state

In this diagram, each A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit appears as a horizontal line. You read the it from left to right. A single qubit gate is a block on its line. A A **controlled operator** is a quantum operation that acts on a target qubit (or system) **only if** one or more control qubits are in a specific state (usually (|1\rangle)). It’s represented as ( C(U) ), where (U) is the operator applied conditionally. For example, the **CNOT gate** is a controlled-(X) operator: it flips the target qubit only when the control qubit is (|1\rangle).
Learn more about Controlled Operator marks the control with a dot on the control line and shows the target as the gate block on the target line.

You can create the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit with a short Python program using 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 depicted in ?.

bell-state.py
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from qiskit import QuantumCircuit, QuantumRegister
 
qc = QuantumCircuit(2)
qc.h(0) # Hadamard on the first qubit
qc.cx(0, 1) # CNOT with control=0, target=1
Listing 1 Circuit that creates a Bell state
    In this circuit, we
  1. the QuantumCircuit class from 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    ,
  2. define a with two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
    Learn more about Quantum Bit    ,
  3. apply the operator on the first qubit at position 0,
  4. and apply the (cx) with the first qubit as the control qubit and the second qubit (at position 1) as the target.

On paper, the logic is straightforward. Every 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 corresponds to a A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation, and the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit can be described as the product of a The 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 and a The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator applied to the state , which entangles the two A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit.

Plain Matrix multiplication combines two matrices by taking the dot product of rows from the first matrix with columns from the second. Each entry in the result is the sum of elementwise products between a row of the first and a column of the second. It only works when the number of columns in the first matrix equals the number of rows in the second.
Learn more about Matrix Multiplication explains the outcome because each 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 has a concrete matrix representation: the The 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 is a A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation, while the The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator is a A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation. When used in sequence, these matrices multiply to form one composite A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation. The result is why the equation states that the series of operations is equal to . In this sense, the entire 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 can be represented as one large matrix as depicted in ?.

Figure 5 Equation of the bell state quantum cicruit

Figure 5 denotes the euqation of the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit.

When this matrix acts on the initial state vector , it produces a new state vector. That vector describes the state of the entire system after the circuit has been applied as shown in ?.

Figure 6 Apply the circuit onto the basis state |00>

The result is the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State. In this A 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, information is no longer stored locally in individual A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit but in the system as a whole. This representation illustrates the principles of Non-locality in quantum computing refers to the way entangled quantum states exhibit correlations that can’t be explained by local hidden variables—meaning changes to one qubit’s state can instantaneously affect another, even at a distance. This doesn’t allow faster-than-light communication but shows that quantum systems share information in a fundamentally non-classical way. It’s what enables quantum phenomena like teleportation and certain computational speedups.
Learn more about Non Locality.

At first glance, the equivalence appears airtight. The A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit looks like nothing more than the multiplication of the The 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 and The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator matrices. Following this reasoning, every 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 is a A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation, and every A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation could in principle be expressed as a 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 might then wonder why the distinction matters at all. Why not ignore the circuit representation and write down the final matrices directly?

The mathematical representation of a quantum computation offers only the theorist's perspective. Real quantum hardware does not manipulate large matrices directly. Instead, it relies on basic building blocks such as single-qubit rotations and a restricted set of two-qubit operators. The most common example is the The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator, which can only act between specific pairs of connected A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit.

These physical constraints mean that even though the mathematics allows clean decompositions of A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation, Shende, V.V., 2004, , . Kjaergaard, M., 2019, , .

If we were given only a large matrix, the practical challenge would be to break it down into a sequence of 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 the hardware can execute reliably. This decomposition step is unavoidable because the hardware understands only the primitive operations available to it.

However, in the literature, you will rarely see a 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 expressed using only native 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. The reason is that, much like an electrical circuit, a 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 fulfills another crucial role. It serves as an abstraction layer that humans can understand. This is similar to higher-level electrical circuits that make use of pre-built components such as microprocessors or sensors.

A circuit uses components
Figure 7 A circuit uses components

A 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 also acts as a practical tool for decomposition because it conveys a mathematical transformation in a way that humans can follow step by step. This quality makes it an effective bridge between abstract Linear 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 implementable hardware instructions.

In real work, you almost never manipulate A unitary transformation is a mathematical operation that preserves the length (or norm) of vectors, meaning it doesn’t change probabilities or total energy in quantum systems. Formally, a matrix is unitary if , where is its conjugate transpose. In essence, it represents a rotation or change of basis in complex vector space that keeps inner products unchanged.
Learn more about Unitary Transformation directly. Instead, you construct 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 that then will be decomposed, mapped, and optimized for the hardware in use. Through decomposition, what begins as an overwhelmingly dense mathematical object becomes a structured and manageable recipe for computation. The A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State illustrates this principle on a small scale, but the same reasoning extends to more complex algorithms.

At the level of circuits, there are many established recipes beyond the A **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State circuit. These recipes help you decide which 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 to apply and in which sequence, giving you a practical framework for building computations on real hardware.