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# Notes

Last updated Mar 17, 2022

Quantum information and quantum computations is a new, rapidly developing branch of physics that has arisen from quantum mechanics, mathematical physics and classical information theory. Significant interest in this area is explained by the great prospects that will open upon the implementation of its ideas, capturing almost all areas of human activity related to the transfer, storage and processing of information. The purpose of this course is to show the basic ideas of quantum informatics, as well as the physical laws and basic mathematical principles. Much attention is paid to such phenomena as quantum entanglement, quantum parallelism, and quantum interference. It is these phenomena that underlie most of the known quantum protocols and algorithms, which are devoted to individual sections of this course. In particular, from the course, students will learn about quantum teleportation, quantum algorithms, quantum error correction and other topics related to the quantum computations theory. As a result of the course, the students will be able to master the modern mathematical apparatus of quantum mechanics used in quantum computations, master the ideas that underlie the most important quantum logic algorithms and protocols for transmitting and processing quantum information, and learn how to solve problems on these topics.

## # Learning Outcomes

The student who completed this course should:

• know
• fundamental concepts of quantum mechanics and the quantum information theory;
• The most important protocols for the transfer and processing of quantum information;
• The most important quantum logic algorithms;
• The basic protocols of the classical and quantum error correction theory.
• be able to
• work with classical and quantum circuits;
• solve problems in the quantum information theory.
• master
• mathematical apparatus of quantum mechanics used in the quantum information theory.

# # Notes

## # Week 1: Statistical aspects of quantum mechanics

### # Physical implementations of qubit

• A Qubit is a unit of the amount of information in quantum computation.
• It has two basis states, denoted as $\ket{0}$ and $\ket{1}$, and it may exist in a Superposition of these basis states.
• David DiVincenzo formulated a criteria that a physical system must meet in order to be able to implement a Qubit. This is known as the DiVincenzo criteria.
• According to this criteria:
• A quantum physical system on which a qubit is realized must have two distinguished orthogonal (basis) quantum states.
• It is possible to prepare a physical system in one of these two quantum states.
• There is a procedure for measuring qubit (macroscopic distinguishability of the basis states).
• One can create a universal set of quantum logic elements (gates) for the qubit.
• The decoherence time longer than the operating time of quantum logic elements.
• We want the time to decoherence to be long enough so that we can perform useful operations on the qubit.
• In addition to these criteria, there’s an additional criteria called scalability.
• It should be possible to create many qubits that could efficiently interact with each other in a controlled manner and would be protected from the influence on decoherence.
• The simplest physical system on the basis of which a qubit can be implemented is a photon.
• The vertical polarization can be compared to state $\ket{1}$ and the horizontal polarization can be compared to state $\ket{0}$.
• The photon is relatively stable with respect to the noise in the environment, i.e. it has great decoherence time.
• However, there are some limitations with this system as well.
• It is very difficult to prepare a pulse of light that will be guaranteed to contain only one photon.
• It is even more difficult to prepare such single-photon pulses at certain specific points in time.
• Moreover, the preparing time will increase exponentially as we scale the system.
• Additionally, it is very difficult to force photons to interact with each other, i.e. use logic gates.
• Another physical implementation of the qubit can be created on the basis of a two-level atom.
• The ground state $\ket{g}$ of the atom can be taken as state $\ket{0}$ and the excited state $\ket{e}$ as $\ket{1}$.
• Here, any single qubit operation can be performed with a good accuracy.
• Generally speaking, the advantages and disadvantages on an atomic system are strongly dependent on its specific implementations.
• Another qubit implementation is a qubit based on a particle with spin $1/2$.
• As the basis states, the positive and negative projections of the spin along a given direction are chosen: $$\ket{0} = \ket{\uparrow} \qquad \ket{1} = \ket{\downarrow}.$$

### # Qubit as a quantum unit of information

• If a certain quantity can take $m$ possible values, then its description requires $(\log_2 m)$ bits.
• Bloch sphere is often used to find out the information capacity of a qubit.
• In the case when a qubit is realized using polarization of a photon, the Bloch sphere is called the Poincare sphere.
• The number of quantum states in which a qubit may exist is infinite.
• One measurement over a qubit will give one of the two basis states.

### # Pure and mixed states of quantum systems

• The Pure state of a quantum system is described using a ket Vector (or a bra vector), which is an element of the Hilbert Space.
• The superposition of pure states is also a pure state.
• The Mixed state cannot be described using a single ket vector. It is a statistical ensemble of pure states.
• We can use Density matrix to describe both pure and mixed states.
• Density matrix for Pure state: $\hat{\rho} = \ket{\psi}\bra{\psi}$.
• Density matrix for Mixed state: $$\hat{\rho} = \sum_{i = 1}^{n} w_i \ket{\psi_i}\bra{\psi_i}.$$