# Study quantum basics

- This is a list of topics which are unsorted at this point, but which give a basis to start your research in quantum domain. This is a way to help me organize thoughts on topics, saying what I know and what I am willing to understand
- Last Updated: May 20, 2024
- Published: Mar 15, 2023

## Table of Contents

# Quantum computing: answers

## Existing quantum computers

**Which physical realizations of quantum computers exist?**Current physical realizations include- superconducting circuits,
- trapped ions,
- neutral atoms,
- photonics,
- topological qubits using anyons.

- How many qubits are there? How many do we need?
**What is error correction?**Quantum error correction is a set of techniques to protect quantum information from errors induced by decoherence and other imperfections in quantum hardware. It involves encoding logical qubits using multiple physical qubits and implementing error-detecting and error-correcting codes. For this reason currently we need hundreds of physical qubits to realize a single logical qubit.

## Qudit

**What is a qubit?**A qubit is the fundamental unit of quantum information, representing a quantum version of classical bits. It can exist in a superposition of states, allowing for parallel processing of information. Mathematically it could be written as follows: $$|\psi\rangle= \alpha|0\rangle +e^{\text{i}\phi} \beta|1\rangle$$ with normalization: \(\alpha^2 + \beta^2 =1\).**How qubit compares to a classical bit?**Unlike classical bits that can only exist in states 0 or 1, qubits can exist in a superposition of states, enabling quantum parallelism. Qubits are subject to entanglement, allowing correlations between qubits that classical bits cannot exhibit.**What is a qudit? In which Hilbert space it leaves?**A qudit is a generalization of a qubit, representing a quantum system with more than two levels. $$|\alpha\rangle= \alpha_0|0\rangle+\alpha_1|1\rangle+…+ \alpha_{d-1}|d-1\rangle$$ where once again coefficients satisfy normalization. This object leaves in a Hilbert space of dimension d, \(|\alpha\rangle\in \mathbb{C}^d\).

## Bloch sphere

**What is a Bloch sphere? Where do I find my qubit on a Bloch sphere?**The Bloch sphere is a geometric representation of the state space of a qubit. It is a unitary sphere. A qubit state is represented by a point on the surface of the Bloch sphere.In a sense an infinite plane is wrapped or mapped onto a sphere, this is called stereographic projection. How to perform this is nicely described here.**Main axis of a Bloch sphere**The main axis are Pauli matrices eigenvectors.If qubit is oriented along Z axis, we find it either in 0 or 1 state. On equator qubit is in equal superposition between two levels, with an arbitrary phase.**How to represent mixed states?**If the system is not in a pure state, so we are working with a density matrix. In this case the qubit will be anywhere within a Bloch sphere. From wikipedia:

Mixed states are represented by an interior point. Thus, the purity of a state can be visualized as the degree to which the point is close to the surface of the sphere. The completely mixed state of a single qubit is represented by the center of the sphere.

**How to represent a qudit on a Bloch sphere?**If you are working with a qudit, you can still use a sphere. Although representation will be a constellation of stars. Once again a nice reference is this blog.

## Basic gates

**Rotation under Pauli matrices**Single-qubit gates include rotations under Pauli matrices (X, Y, Z). A rotation by an angle θ about the X-axis is represented by the matrix exp(-iθX/2), and similarly for Y and Z.**How to perform an arbitrary rotation**Any arbitrary single-qubit rotation can be performed using a combination of rotations under Pauli matrices. For example, the single-qubit gate Rx(θ) can be decomposed into rotations around the X, Y, and Z axes.**Single-qubit gates**Common single-qubit gates include Hadamard (H), phase (S), and pi/8 (T) gates, each performing specific rotations on the Bloch sphere.**Two-qubit gates**Examples of two-qubit gates include the CNOT gate, which performs a controlled-X operation based on the state of another qubit.

## Universal quantum computer

**DiVincenzo criteria**DiVincenzo criteria are a set of requirements for building a practical quantum computer. Criteria include a scalable physical qubit array, universal gate set, efficient qubit initialization, long qubit coherence times, and a faithful qubit measurement.**Set of universal gates**A universal gate set is a set of quantum gates that can be combined to approximate any unitary transformation. Examples include the Hadamard, phase, and CNOT gates.

## Quantum advantage

**How much information can I save on a single qubit?**A single qubit can store an infinite amount of classical information when in a superposition of states. In practice, when reading out a single**Where the advantage comes from?**Quantum advantage arises from the ability of quantum systems to process information in parallel through superposition and to exhibit entanglement, enabling correlations not possible with classical systems.**Lloyd’s theorem**Lloyd’s theorem states that a quantum system with n qubits can simulate 2^n classical bits, illustrating the exponential information processing capacity of quantum systems.**Which algorithms are supposed to have quantum advantage?**Quantum algorithms such as Shor’s algorithm (for factoring) and Grover’s algorithm (for search) are believed to provide exponential speedup compared to their best-known classical counterparts.

## Preparation of a qubit in an atomic system?

**Select a two-level system within Rb87 sublevels.**In rubidium-87 (Rb87), the ground state hyperfine levels can be used as a two-level system for qubit manipulation.**Which couplings will allow single-qubit gates?**Applying external electromagnetic fields, such as microwave or radiofrequency fields, can induce transitions between sublevels, allowing for single-qubit operations.**What is Hamiltonian and evolution operator for this system****How rotation of a qubit is performed?**Rotations of a qubit are achieved by applying specific frequency components of an external field, causing transitions between sublevels.

## Measurement of a qubit

**Stern-Gerlach technique**The Stern-Gerlach technique involves passing a qubit through an inhomogeneous magnetic field, causing it to align along one of the field’s axes. The resulting measurement provides information about the qubit’s state.**Basis rotation**Basis rotation is a transformation applied to a qubit’s state before measurement to simplify the measurement process. It involves rotating the Bloch vector to align with the measurement basis.**How much information from each measurement?**Each measurement provides a single classical bit of information about the qubit’s state. The outcome is probabilistic, reflecting the quantum nature of the measurement process.**Step-by-step measurement of a qubit**The step-by-step measurement involves preparing the qubit state, applying basis rotations if necessary, and performing the measurement. The result is a classical outcome corresponding to a specific qubit state.