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1Introduction toQuantumComputingExploring the Future of Computational PowerDr. P. ArunaAssociate Professor/CS, PMIST

2What is QuantumComputing? uses the principles of quantum mechanics to processinformationComparison with Classical Computing: Classical bits: 0 or 1 Quantum bits (qubits): Can be 0, 1, or both at the sametime (superposition). A qubit can exist in multiple states simultaneously due tosuperposition and entanglement.

3Key Principles of QuantumComputing Superposition: Qubits can exist in multiple statessimultaneously. Example: A spinning coin represents superposition; it’s notjust heads or tails until you observe it. Entanglement: Qubits can be linked together, so the stateof one qubit can depend on the state of another, no matterhow far apart they are. Example: If one qubit is measured and found to be 0, theentangled qubit will be 1.

4How Quantum ComputingWorksQuantum Circuits: A sequence of quantum gates applied to qubits.Quantum Gates: Operations that change the state of qubits, similar to logicgates in classical computing. building blocks of quantum circuits. Example: Hadamard gate creates superposition; CNOT gatecreates entanglement.

5Quantum Gates Hadamard Gate (H): Creates superposition, transforming a qubitfrom a definite state 0 or 1 into a superposition of both.∣ ⟩ ∣ ⟩ Controlled-NOT (CNOT) Gate: A two-qubit gate that flips the secondqubit (target) if the first qubit (control) is in state 1 . It is used to∣ ⟩create entanglement. Pauli-X Gate: Similar to the classical NOT gate, flips the state of aqubit Phase Gate (S and T Gates): Changes the phase of a qubit,important for interference in quantum algorithms.

6Structure of a QuantumCircuit Qubit Initialization: Qubits start in a known state,typically 0 |0rangle 0 .∣ ⟩ ∣ ⟩ Gate Operations: A series of quantum gates are applied toqubits, creating superposition, entanglement, andinterference. Measurement: At the end of the computation, qubits aremeasured. Measurement collapses the qubits’ states intoclassical 0s and 1s, producing a result that reflects theprobabilities encoded by the quantum circuit.

7Simple Quantum Circuit Consider a quantum circuit that creates a Bell State (amaximally entangled state of two qubits): Step 1: Apply a Hadamard gate to the first qubit: This creates a superposition of 0 and 1 .∣ ⟩ ∣ ⟩ Step 2: Apply a CNOT gate with the first qubit as the controland the second qubit as the target: Now the two qubits are entangled.

8 CNOT Operation: If the control qubit is 0 , the target qubit remains unchanged.∣ ⟩ If the control qubit is 1 , the target qubit is flipped (changed∣ ⟩from 0 to 1 or vice versa).∣ ⟩ ∣ ⟩ Applying CNOT to the Initial State: Now let's apply the CNOT gate to the state 00 + 10 :∣ ⟩ ∣ ⟩ For 00∣ ⟩: Control is 0 → Target remains 0 .∣ ⟩ ∣ ⟩ CNOT acts as: CNOT( 00 )= 00∣ ⟩ ∣ ⟩ For 10∣ ⟩: Control is 1 → Target flips from 0 to 1 .∣ ⟩ ∣ ⟩ ∣ ⟩ CNOT acts as: CNOT( 10 )= 11∣ ⟩ ∣ ⟩

9Why Quantum Computing isImportant Speed: Can solve complex problems much faster than classicalcomputers. Applications: Cryptography: Break current encryption methods (RSA, whichdepend on the difficulty of factoring large numbers). Drug discovery: Simulate molecular interactions. Googledemonstrated quantum supremacy by simulating a simplechemical reaction (hydrogen molecules interacting). Companieslike Pfizer and Boehringer Ingelheim are also exploringquantum computing to accelerate drug discovery. Classical computers struggle to model and simulate moleculesbecause their complexity grows exponentially with size. Quantumcomputers, however, can model molecules and their interactions atthe quantum level

10Why Quantum Computing isImportant Artificial Intelligence and Machine Learning potential to accelerate machine learning by optimizing the training processfor models that involve large datasets Healthcare and Personalized Medicine could personalize medical treatments by analyzing massive amounts ofgenetic and molecular data could accelerate the analysis of DNA and RNA data, leading to moreeffective personalized treatments for diseases like cancer or geneticdisorders (Genomics) Climate Modeling process the massive amount of data involved in weather patterns moreefficiently, leading to more accurate and timely forecasts (WeatherForecasting) can simulate how different mitigation strategies (like reducing carbonemissions) will impact the climate, helping policymakers create effectiveaction plans (Climate Change Mitigation)

11Current State of QuantumComputing Companies Involved: IBM, Google, Microsoft, Intel, Rigetti, etc. Progress: IBM Quantum leader in quantum computing and offers access to quantum computers via thecloud through IBM Quantum Experience also developed the Qiskit open-source quantum computing framework built several quantum processors, including a 127-qubit processor called Eagle IBM Quantum Network, a global community of companies, universities, and labsworking on quantum computing Google Quantum AI In 2019, Google claimed to have achieved quantum supremacy with theirSycamore processor, a 54-qubit quantum computer, performed a calculation in200 seconds that would take the fastest classical computer 10,000 years. Volkswagen used a quantum algorithm to optimize traffic flow inBeijing, reducing congestion

12Google's Sycamore quantumcomputer

13Challenges in QuantumComputing Decoherence: Qubits are highly sensitive to theirenvironment, which can cause loss of information Error Correction: Developing robust quantum errorcorrection methods is still a significant challenge Hardware Scalability: Scaling quantum computers tomillions of qubits is needed for practical applications.

14The Future of QuantumComputing Potential Developments: More robust quantum algorithms. Advances in quantum networking. Integration with classical computing systems. Impact on Society: Revolutionary changes in technology, medicine, and finance.

15Implementation - QC Qiskit (for IBM), Cirq (for Google), and Q# (for Microsoft).Quantum programming involves controlling qubits andgates and managing superposition, entanglement, andprobabilistic outcomes

16Key quantum algorithms andtheir use cases solve complex problems more efficiently than classicalalgorithms Shor's Algorithm - Integer factorization of large numbers Grover's Algorithm - Unstructured database search Quantum Machine Learning (QML) Algorithms - Machinelearning tasks such as classification, clustering, orregression, but using quantum systems

17Mathematical Notation Superposition: A qubit can be in a state of superposition,represented mathematically as: ∣ψ =α 0 +β 1⟩ ∣ ⟩ ∣ ⟩ where αand β are complex numbers representing the probabilityamplitudes of the qubit being 0 or 1. symbol ψ(psi) - represent wave functions and quantumstatesBra-Ket Notation ∣ψ is a⟩ ket representing the state of the system ⟨ψ is a∣ bra representing the dual vector associated withthe ket

18Mathematical Notation ϕ (phi) - phases or angles χ (khi) - other types of wave functions Bra-Ket Notation Overview Ket: The ψ is called a∣ ⟩ ket and represents a quantumstate (in this case, the state of a qubit). The vertical bar ∣and the right angle bracket denote that it is a vector⟩(direction & Magnitude) in a complex vector space. Bra: In contrast, a bra is denoted as ψ . The bra notation⟨ ∣represents the dual vector associated with the ket.Together, bra and ket notations are used to express innerproducts.

19Mathematical Notation The ψ in ∣ψ⟩ is just a label or name for the specificquantum state. It can represent any quantum state, suchas: ∣0 for the ground state,⟩ ∣1 for the excited state,⟩ or a superposition of these states like ∣ψ =α 0 +β 1⟩ ∣ ⟩ ∣ ⟩

20Quantum States(Superposition) Qubit States: A qubit is the basic unit of quantuminformation, similar to a classical bit, but it can be in morethan just two states. It can be in a combination of twostates, which we write as: ψ =α 0 +β 1∣ ⟩ ∣ ⟩ ∣ ⟩ Here: ∣ψ : This is the state of the qubit we want to teleport.⟩ ∣0 and 1⟩ ∣ ⟩: These are the two basic states of the qubit, like the 'off'and 'on' states of a classical bit. α and β: These are numbers (specifically, complex numbers) that tellus how much of each state is in the qubit. They determine theprobability of measuring the qubit in each state.

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