what is Quantum Circuit in quantum computing. explain with examples

 A Quantum Circuit is the fundamental blueprint used in quantum computing to represent a computation.¹ Just as a classical circuit diagram shows how bits flow through logic gates (like AND, OR, NOT) to produce a result, a quantum circuit shows how qubits flow through quantum gates to manipulate information.²

Here is a breakdown of what it is, its components, and a step-by-step example.

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1. The Core Components

A quantum circuit looks like a music score where "time" flows from left to right.³

• Qubits (The Wires): Horizontal lines represent individual qubits.⁴ Each line tracks the state of a qubit as it moves through time.⁵

• Quantum Gates (The Operations): Symbols placed on the wires that change the state of the qubits.⁶ Unlike classical gates, quantum gates are reversible (except for measurement).⁷

  • H (Hadamard): Puts a qubit into Superposition (a state of being 0 and 1 simultaneously).⁸

  • X (Pauli-X): The quantum "NOT" gate.⁹ Flips ¹⁰$\lvert 0 \rangle$ to ¹¹$\lvert 1 \rangle$.¹²

  • CNOT (Controlled-NOT):¹³ Entangles two qubits.¹⁴ It flips the second qubit (Target) only if the first qubit (Control) is ¹⁵$\lvert 1 \rangle$.¹⁶

• Measurement: Represented by a meter symbol.¹⁷ This collapses the fragile quantum state into a classical bit (0 or 1) that we can read.¹⁸

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2. Example: Creating "Entanglement" (The Bell State)

The most famous simple quantum circuit is the Bell State generator. It takes two independent qubits and links them so perfectly that measuring one instantly determines the state of the other, no matter how far apart they are.¹⁹

The Circuit Diagram

Visually, the circuit looks like this:

$$\lvert 0 \rangle_A \quad \xrightarrow{\quad H \quad} \quad \bullet \quad \xrightarrow{\quad \text{Measure} \quad}$$

$$\lvert 0 \rangle_B \quad \xrightarrow{\quad \phantom{H} \quad} \quad \oplus \quad \xrightarrow{\quad \text{Measure} \quad}$$

(Note: The ²⁰$\bullet$ and ²¹$\oplus$ connected by a vertical line represent the CNOT gate.)²²

Step-by-Step Explanation

Step 1: Initialization

We start with two qubits, let's call them Qubit A and Qubit B, both initialized to the state $\lvert 0 \rangle$ (zero).

• Current State: $\lvert 00 \rangle$

Step 2: Superposition (The H Gate)

We apply a Hadamard (H) Gate to Qubit A.23

• This places Qubit A into a superposition, meaning it has a 50% chance of being ²⁴$\lvert 0 \rangle$ and a 50% chance of being ²⁵$\lvert 1 \rangle$.²⁶

• Qubit B is untouched.

• Current State: A mix where the system is in states $\lvert 00 \rangle$ and $\lvert 10 \rangle$ simultaneously.

Step 3: Entanglement (The CNOT Gate)

We apply a CNOT Gate.27 We use Qubit A as the Control (28$\bullet$) and Qubit B as the Target (29$\oplus$).30

• The CNOT gate looks at Qubit A.

• If Qubit A is $\lvert 0 \rangle$, it does nothing to Qubit B.

• If Qubit A is $\lvert 1 \rangle$, it flips Qubit B from $\lvert 0 \rangle$ to $\lvert 1 \rangle$.

Because Qubit A is in a superposition of both 0 and 1, the CNOT gate acts on both possibilities at once:

1. In the reality where A is 0, B stays 0. (Result: $\lvert 00 \rangle$)

2. In the reality where A is 1, B flips to 1. (Result: $\lvert 11 \rangle$)

Step 4: The Result (Bell State)

The two qubits are now entangled.31 The system is in a superposition of 32$\lvert 00 \rangle$ and 33$\lvert 11 \rangle$.34

• There is no $\lvert 01 \rangle$ or $\lvert 10 \rangle$ possibility anymore.

• If you measure Qubit A and see 0, Qubit B must be 0.

• If you measure Qubit A and see 1, Qubit B must be 1.

Summary Table: Classical vs. Quantum Circuit

Feature	Classical Circuit	Quantum Circuit
Information Unit	Bit (0 or 1)	Qubit ($\alpha\lvert 0 \rangle + \beta\lvert 1 \rangle$)
Wires	Physical wires carrying voltage	Abstract timeline of a particle
Logic Gates	AND, OR, NOT, NAND	Hadamard, CNOT, Pauli-X, Z
Reversibility	mostly Irreversible (information is lost)	Reversible (information is preserved)
Outcome	Deterministic (Same input = Same output)	Probabilistic (Outcome has probability)

Would you like me to explain how to write this Bell State circuit in code using Python and Qiskit?

... What are Bell States ...

This video is relevant because it visually demonstrates the specific Bell State circuit example discussed above, explaining how the Hadamard and CNOT gates work together to create entanglement.