Evolutionary Computation Online Course
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Section: Evolutionary Computation in Coded Communications: an Implementation of Viterbi Algorithm
Lesson: The quantum Grover’s algorithm
Evolutionary Computation.
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Introduction
Quantum Computing hopefully is the future of computing systems. It still on its first steps. The development of some quantum algorithms gives the quantum computing a boost on its importance. These algorithms (such as Shor’s and Grover’s algorithms) proved to have superior performance over classical algorithms. The recent findings, that quantum error correction can be used, showed that the decoherence problem can be solved and hence the quantum computers can be realized. The quantum algorithms are based on the use of special gates applied on one, two or more qubits (quantum bits). The classical computer uses different gates (NOT, AND, NAND, OR and XOR). Quantum gates are in many aspects different from classical gates where all gates must be reversible. This makes the quantum gates act as 2 nx2 ntransformation operators, where we have ninput qubits and noutput qubits.
To understand the quantum bits and gates we describe the group of amplitudes that describes the state of a quantum register as a vector. A qubit with state|0〉 , which is guaranteed to read logic 0 when measured, is represented by the vector(10) , and a qubit with state|1〉 which is guaranteed to read logic 1 when measured is represented by the vector(01). An arbitrary qubit state is then represented by the vector(αβ) as:
whereα andβ are complex numbers and|α|2+|β|2=1 .
One important quantum gate is the Hadamard gate given by:
When the input is|0〉 , Hadamard gate changes the state of the qubit to:
that is,|ϕ〉=12|0〉+12|1〉. So when reading the qubit at the end, we have exactly 50% chance of seeing a 0, and an equal chance of seeing a 1. Generalizing the above example, if an n-qubits register originally contains the value|0n〉 , it can be transformed using the Hadamard gate to the superpositional state:
where we would see each of the 2 nbinary numbers xwith equal probability when we observe the register. Other gates operate similar to Hadmard gate with different matrices, where Pauli gates are given by:
and phase gates:
The quantum computers use quantum gates to produce results in a faster and more efficient way than the classical computers. Implementation of quantum computers is still in its very beginning state, therefore, in this lecture, we need not to worry about the implementation issues. In addition to entanglement, the strength of quantum computing comes from the parallelism feature of the quantum computers and the fact that the quantum state is a superposition state. Using classical bits an nbit vector can represent one of 2nsymbols, but in quantum bits an nqubits vector can represent the 2nsymbols simultaneously.
Quantum Algorithms are usually organized in the following three steps:
Step 1.Initialize the quantum states.
Step 2.Apply the oracle quantum core as many times as needed.
Step 3.Measure the output states (results).
In many classical algorithms especially those used for searching mechanisms, speed and complexity are the main limiting factors in their implementation. Viterbi decoding algorithm is an important algorithm that is used to decode the received data when using Convolutional or Turbo codes at the transmitter. These codes are superior in their performance over many other types of codes. In the following we devise a quantum algorithm to implement the Viterbi algorithm (VA) .
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