Quantum computing! quantum gates! Fisica dell Energia!
What is Quantum Computing?! Calculation based on the laws of Quantum Mechanics.! Uses Quantum Mechanical Phenomena to perform operations on data.! Operations done at an atomic/sub-atomic level.!
Linear algebra:! Quantum computing depends heavily on linear algebra.! Some of the Quantum Mechanical concepts come from the mathematical formalism, not experiments.!
Dirac Notation:! Dirac notation is used for Quantum Computing.! States of a Quantum system are represented by Ket vectors (Column Matrix).! Example: 0, 1! Other notation: Bra notation-complex conjugate of Ket vectors(row Matrix).!
Data representation:! Quantum Bit (Qubit or q-bit) is used.! Qubit, just like classical bit, is a memory element, but can hold not only the states 0 and 1 but also linear superposition of both states, α 1 0 +α 2 1.! This superposition makes Quantum Computing fundamentally different.!
Classical bit Vs Qubit:! Classical bit: {0, 1}! Qubit: {0, 1, superposed states of 0 and 1}!
Quantum properties used:! Superposition! Entanglement! Decoherence!
Superposition:! Property to exist in multiple states.! In a quantum system, if a particle can be in states A and B, then it can also be in the state α 1 A + α 2 B ; α 1 and α 2 are complex numbers.!
Entanglement:! Most important property in quantum information.! States that two or more particles can be linked, and if linked, can change properties of particle(s) changing the linked one.! Two particles can be linked and changed each other without interaction.!
Decoherence:! The biggest problem.! States that if a coherent (superposed) state interacts with the environment, it falls into a classical state without superposition.! So quantum computer to work with superposed states, it has to be completely isolated from the rest of the universe (not observing the state, not measuring it,...)!
Uncertainty Principle:! Quantum systems are so small.! It is impossible to measure all properties of a Quantum system without disturbing it.! As a result there is no way of accurately predicting all the properties of a particle in a Quantum System.!
Physical representation of qubits:! A single atom that is in either Ground or Excited state Ground state representing 0 Excited state representing 1
Physical representation of qubits:!
More about qubits:! By superposition principle, a Qubit can be forced to be in a superposed state.! i.e. ψ = α 1 0 + α 2 1 Qubit in superposed state occupies all the states between 0 and 1 simultaneously, but collapses into 0 or 1 when observed physically.! A qubit can thus encode an infinite amount of information.!
Qubits in Superposed state:!
How do we operate on q-bits?! Logic gates?!
Classical gates! NOT! OR! AND! They are logically irreversible! A process is said to be logically reversible if the transition function that maps old computational states to new ones is a one-to-one function; i.e. the output logical states uniquely defines the input logical states of the computational operation!!
Reversible computing! Feynman gate! Fredkin gate! Toffoli gate! Peres gate!
Feynman gate! When A = 0 then Q = B, when A = 1 then Q = B.! Every linear reversible function can be built by composing only 2*2 Feynman gates and inverters! With B=0 Feynman gate is used as a fan-out gate.!
Fredkin gate! Invented by Ed. Fredkin.! The Fredkin gate is a computational circuit suitable for reversible computing.! It is universal, which means that any logical or arithmetic operation can be constructed entirely of Fredkin gates!
Fredkin gate!
Minimal Full Adder Using Fredkin gates!
Toffoli gate! Invented by Tommaso Toffoli! It is a universal reversible logic gate! It is also known as the "controlled-controlled-not" gate!
Toffoli gate!
Peres Gate!
Full adder with Peres gates!
Goals of reversible logic synthesis! Minimize the garbage! Minimize the width of the circuit (the number of additional inputs)! Minimize the total number of gates! Minimize the delay!
Operations on qubits: quantum gates! Quantum logic gates are used. They are logically reversible.! Quantum logic gates are represented by Unitary Matrices-U U=UU =I.! States are also represented by matrices as:!
Let s start with operation on single bit! Hadamard Gate! acts on a single qubit.! transforms 0 to ( 0 + 1 )/ 2! And 1 to ( 0-1 )/ 2!
Hadamard Gate! H! acts on a single qubit.! Transforms: 0 to ( 0 + 1 )/ 2! 1 to ( 0-1 )/ 2! H! H! 0 à 0! 1 à 1! Two consecutive applications leads to the identity!
Pauli-X gate:! acts on a single qubit.! Quantum equivalent of NOT gate.! Transforms 1 to 0 and 0 to 1
Pauli-Y gate:! acts on a single qubit.! Transforms 1 to -i 0 and 0 to i 1
Pauli-Z gate:! acts on a single qubit.! Transforms 1 to - 1 and 0 remains unchanged.!
Phase shift gate:! acts on a single qubit.! Transforms 1 to e iθ 1 and 0 remains unchanged.! Modifies (rotates) the phase of quantum state by θ.!
Two inputs q-bits! Unitary operator is a 4x4 matrix!
Unitary matrix for XOR!
Combining XOR! q-bits swapping!
Advantages:! Could process massive amount of complex data.! Ability to solve scientific and commercial problems.! Process data in a much faster speed.! Capability to convey more accurate answers.! More can be computed in less time.!
Disadvantages and Problems:! Security and Privacy Issues:! Ability to crack down passwords.! Capability to break every level of encryption.! Problem of Decoherence, the need of a noise free environment.! Complex hardware schemes like superconductors.! Not suitable for word processing and email.!
References! Quantum Computing - Basic Concepts, Sendash Pangambam!