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Igor Ilijašević
QUANTUM COMPUTATION AND
QUANTUM INFORMATION
GROVER'S ALGORITHM
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Igor Ilijašević 2 / 26
Igor Ilijašević 3 / 26
COMPLEX NUMBERS BASICS
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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Igor Ilijašević
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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Igor Ilijašević
QUANTUM GATES AND CIRCUIT SYMBOLS
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Igor Ilijašević
QUANTUM GATES AND CIRCUIT SYMBOLS
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Igor Ilijašević
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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Igor Ilijašević
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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Igor Ilijašević
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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Igor Ilijašević
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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