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Page 1: Quantum  computation  and  quantum  information

Igor Ilijašević

QUANTUM COMPUTATION AND

QUANTUM INFORMATION

GROVER'S ALGORITHM

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COMPLEX NUMBERS BASICS

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BASICS

• Dirac (bra–ket) notation φ|ψ⟨ ⟩• Introduced in 1939 by Paul Dirac

• Interpreted as the probability amplitude for the state ψ to collapse into the state φ

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BASICS

• - vectors must have same dimensions

• - complex number not a vector - inner product

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BASICS

• - Hermitian matrix (measurable)

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BASICS

• - Identity (unit) matrix

• - Unitary matrix

• - Eigenvector (state of the system) ( - Eigenvalue (result (is a number)))

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BASICS

• - orthonormal basis vector

• Identity operator

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BASICS

• Expressing a linear operator as a matrix

• with respect to

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BASICS

• Qubit• Has two states |0⟩ and |1⟩ - Computational basis states

• Can also be in states other than |0⟩ or |1⟩• Can also form linear combinations of states – Superpositions

• α and β are complex numbers

• We can determine whether a qubit is in the state 0 or 1, but we cannot determine its quantum state (α and β)

• We can get the result 0 with probability or 1 with probability , where + = 1

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MORE BASICS• Bloch sphere

• Since + = 1

• A single qubit when measured gives us the following probabilities

• The state is often denoted as

• The state is often denoted as

• What if we need more than 1 (qu)bit, say for example 2?

• Classical bits 4 states: 00, 01, 10, 11

• Qubits 4 computational basis states: , , ,

• But can also be in superpositions of these 4 states - amplitude

• + +

• Bell state – EPR pair

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QUANTUM GATES AND CIRCUIT SYMBOLS

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QUANTUM GATES AND CIRCUIT SYMBOLS

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QUANTUM GATES AND CIRCUIT SYMBOLS

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QUANTUM GATES• All quantum gates are reversible

• Quantum gates can be easily represented using matrix form• Matrix must be unitary

• U†U = I

• That is the only constraint!

• Quantum NOT gate acts linearly

• Hadamard gate

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QUANTUM GATES

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QUANTUM GATES• There are as much single qubit gates as there are 2x2 unitary matrices

• Arbitrary single qubit gate can be decomposed as a product of rotations

• Rotation about the axis

• Multi-quantum gates

• CNOT

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QUANTUM PARALLELISM

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WALSH–HADAMARD TRANSFORM

• Example:• Performing a function on bit input and 1 bit output

• Prepare qubit states as

• Apply the Hadamard transform to the first bits

• Implement the quantum circuit for

• As a result we get

• Superposition over all states

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GROVER'S ALGORITHM

• Quantum algorithm• Probabilistic

• The probability of failure can be decreased by repeating the algorithm

• Deutsch–Jozsa algorithm is a deterministic quantum algorithm

• Searching an unsorted database with N entries in time using space

• May be more accurate to describe it as "inverting a function"

• “Only” a quadratic speedup compared to other quantum algorithms (exponential speedup)

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GROVER'S ALGORITHM• We have N entities

• Database entries

• We need an N-dimensional state space H which can be provided by qubits

• Choose an observable, Ω, acting on H, with N distinct eigenvalues whose values are all known

• Each of the eigenstates of Ω encode one of the entries in the database

• We are provided with a unitary operator (quantum oracle) which acts as a subroutine that compares database entities

• We need to identify the eigenstate or the eigenvalue that acts specially upon

• Grover diffusion operator

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GROVER'S ALGORITHM

1. Perform the following "Grover iteration" r(N) times (asymptotically )

1. Apply

2. Apply

2. Perform the measurement Ω which will give the result with probability approaching 1 for N 1≫

3. Get from

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GROVER'S ALGORITHM

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GOOGLE QUANTUM COMPUTING PLAYGROUND EXAMPLES

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GOOGLE QUANTUM COMPUTING PLAYGROUND GROVER'S ALGORITHM IMPLEMENTATION

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REFERENCES

• “Quantum Computation and Quantum Information, 10th Anniversary Edition”, Michael A. Nielsen & Isaac L. Chuang

• Google Quantum Computing Playground, http://qcplayground.withgoogle.com/

• http://twistedoakstudios.com/blog/Post2644_grovers-quantum-search-algorithm

• http://en.wikipedia.org/wiki/Grover's_algorithm

• http://www.quantiki.org/wiki/Main_Page

• https://www.youtube.com/watch?v=T2DXrs0OpHU

• https://www.youtube.com/watch?v=Xmq_FJd1oUQ

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