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, α1|0〉+α2|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 α1|A〉 + α2|B〉 ; α1 and α2 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〉 = α1|0〉+ α2|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!

They are logically irreversible!

AND!OR!NOT!

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:!

Hadamard Gate!

•  acts on a single qubit.!

•  transforms |0〉 to (|0〉 +|1〉)/√2!

•  And  |1〉 to (|0〉 -|1〉)/√2!

Let’s start with operation on single bit!

Hadamard Gate!

acts on a single qubit.!

Transforms:  |0〉 to (|0〉 +|1〉)/√2!

|1〉 to (|0〉 -|1〉)/√2!

H!

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 eiθ |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!