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

A quantum circuit is a structured sequence of Unitary OperatorsAn Unitary operator is a reversible quantum transformation. applied to QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.. It executes a complete computation, starting with the initial state and ending with the final measurementNielsen, 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 QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states., and instead of resistors, it employs Unitary OperatorsAn Unitary operator is a reversible quantum transformation.. 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

September 19•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 SuperpositionSuperposition is... and EntanglementEntanglement is..., which invalidates the intuition of classical logic-gate wiringPreskill, J., 2018, Quantum, Vol. 2, pp. 79.

The Bell StateBell State circuit is the simplest nontrivial Quantum CircuitQuantum 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 QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. in the state . Applying a Hadamard OperatorHadamard Operator to the first QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. places it into a SuperpositionSuperposition is... of is a basis state. and is a basis state.. A Controlled-NOT OperatorCnot Operator then links the Qubits.A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. For the first QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. is in the stateQuantum State is... is a basis state., the second flips from is a basis state. to is a basis state. and vice versa. If the first is is a basis state., the second remains unchanged.

You can represent this Quantum CircuitQuantum 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 Bell StateBell State circuit.

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

We apply , where is the Hadamard OperatorHadamard Operator acting on the first QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. and is the Identity MatrixIdentity Matrix acting on the second.
- The symbol denotes the Tensor ProductTensor Product.
- is a unitary matrixAn Unitary operator is a reversible quantum transformation..
- is the Identity MatrixIdentity Matrix.
- The Tensor ProductTensor Product is a unitary matrixAn Unitary operator is a reversible quantum transformation. acting on two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. simultaneously. It applies a Hadamard OperatorHadamard Operator to the first while leaving the second unchanged.
Figure 2 The tensor product of H and I
Figure 2 denotes the Tensor ProductTensor Product .
We apply the Controlled-NOT OperatorCnot Operator whose matrix is depicted in ?. Since this operator is an entanglement gate, we cannot write it as a Tensor ProductTensor Product of two single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. matrices.
Figure 3 The matrix of the CNOT operator

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

Figure 4 Quantum circuit diagram of the Bell state

In this diagram, each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. appears as a horizontal line. You read the it from left to right. A single qubit gate is a block on its line. A Controlled OperatorControlled 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 Bell StateBell State circuit with a short Python program using QiskitQiskit depicted in ?.

bell-state.py

from qiskit import QuantumCircuit

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

the QuantumCircuit class from QiskitQiskit,
define a with two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.,
apply the operator on the first qubit at position 0,
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 Quantum GateQuantum Gate corresponds to a Unitary MatrixUnitary Transformation, and the Bell StateBell State circuit can be described as the product of a Hadamard OperatorHadamard Operator and a Controlled-NOT OperatorCnot Operator applied to the state , which entangles the two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states..

Plain Matrix MultiplicationMatrix Multiplication explains the outcome because each Quantum GateQuantum Gate has a concrete matrix representation: the Hadamard OperatorHadamard Operator is a unitary matrixUnitary Transformation, while the Controlled-NOT OperatorCnot Operator is a unitary matrixUnitary Transformation. When used in sequence, these matrices multiply to form one composite unitary matrixUnitary Transformation. The result is why the equation states that the series of operations is equal to . In this sense, the entire Quantum CircuitQuantum 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 Bell StateBell 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 Bell StateBell State. In this Quantum StateQuantum State is..., information is no longer stored locally in individual QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. but in the system as a whole. This representation illustrates the principles of Non LocalityNon Locality.

At first glance, the equivalence appears airtight. The Bell StateBell State circuit looks like nothing more than the multiplication of the Hadamard OperatorHadamard Operator and Controlled-NOT OperatorCnot Operator matrices. Following this reasoning, every Quantum CircuitQuantum Circuit is a Unitary MatrixUnitary Transformation, and every Unitary MatrixUnitary Transformation could in principle be expressed as a Quantum CircuitQuantum 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 Controlled-NOT OperatorCnot Operator, which can only act between specific pairs of connected QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states..

These physical constraints mean that even though the mathematics allows clean decompositions of Unitary MatricesUnitary Transformation, the hardware cannot always realize them in the same formShende, V.V., 2004, , . Limited QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. connectivity and the presence of NoiseNoise further complicate what can actually be implemented on a Quantum ComputerQuantum ComputerKjaergaard, M., 2019, , .

If we were given only a large matrix, the practical challenge would be to break it down into a sequence of Quantum GatesQuantum 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 Quantum CircuitQuantum Circuit expressed using only native Quantum GatesQuantum Gate. The reason is that, much like an electrical circuit, a Quantum CircuitQuantum 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 Quantum CircuitQuantum 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 AlgebraLinear Algebra and implementable hardware instructions.

In real work, you almost never manipulate Unitary MatricesUnitary Transformation directly. Instead, you construct Quantum CircuitsQuantum 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 Bell StateBell 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 Bell StateBell State circuit. These recipes help you decide which Quantum GatesQuantum Gate to apply and in which sequence, giving you a practical framework for building computations on real hardware.