Principles of quantum computing

• Prerequisites: Basic notations of quantum state, Bloch sphere

In general, "computing" is supported by underlying mechanisms that rely on some types of physical systems. Extremely, we can argue that single-light bulb system is a 1-bit classical computer. Technically, a system composed of numerous lightbulbs can perform n-bit computation. Since this system should be sophisticately and effectively managed, we can think that transistor is a great alternative for lightbulb. Similarly, quantum computing also should be supported by a physical system. Compared to the classical computing, physical system that supports quantum computing should show quantum behavior. Some quantum mechanical systems including atoms, nuclear spins are utilized to design quantum mechanical system that enables quantum computing.
David P. DiVincenzo 's paper explained about some conditions should be met to implement quantum computation, and this paper will also be reviewed at another article.

By the way, quantum computations exploit postulates of quantum mechanics in terms of:

Quantum state preparation

Quantum state can be represented in a Dirac notation, i.e. $|\psi\rangle \in \mathcal{H}$. n-qubit quantum state can represent complex vector in $\mathbb{C}^{2^n}$ space.
Density matrix is an equivalent formalism to represent such quantum state: $\rho = |\psi\rangle\langle\psi|$.
For an ensemble of pure states $|\psi_i\rangle$ with probability $p_i$, its corresponding density matrix $\rho = \sum_i p_i|\psi_i\rangle\langle\psi_i|$. Purity of a quantum state represented by a density matrix can be measured by calculating $Tr(\rho)$, sinle $Tr(\rho) = 1$ if and only if $\rho$ is a pure state.

Quantum state dynamics

Evolution of the quantum state is governed by schrodinger equation, i.e. $i\bar{h} {\partial|\psi\rangle \over \partial t} = H|\psi\rangle$.
H is called the hamiltonian of the system, and especially for circuit model, we assume time-independant hamiltonain H to describe time evolution of the system. Later, we'll discuss about time-dependant hamiltonain cases for quantum annealing and its time-independant approximation to make connections with the concepts of QAOA(Quantum Approximation Optimization Algorithm).
By the way, hamiltonian H is hermitian matrix, thus always spectral decomposable. H can be decomposed into $H = \sum_ip_i|\psi_i\rangle\langle\psi_i|$ where eigenvalue $p_i$ corresponds to the energy of the system, while eigenstate $|\psi_i\rangle$ represents configuration of the system. Under time-independant hamiltonain H, solving the partial derivative equation about time yields a unitary evolution of quantum state: i.e. $|\psi(t_2)\rangle=U(t_2,t_2)|\psi(t_1)\rangle$ where $U^\dag U=I.$

In terms of density matrix, unitary evolution $\{p_i, |\psi_i\rangle\} \rarr \{p_i, U|\psi_i\rangle\}$ result in a new density matrix $\rho' = \sum_ip_iU|\psi_i\rangle\langle\psi_i|U^\dagger = U\rho U^\dagger$.

Quantum state measurement

Measurement should be done with set of projection operators $M = \{M_1, M_2, ..., M_n\}$. In order to check wheter a set of projection operator $\{M_1, ..., M_n\}$ is valid, it is enough to verify that $\sum M_i^{\dagger}M_i = I$( $\because$ sum of the probabilities = 1). An observable $O \in M$ is also hermitian matrix, thus eigendecomposable into $O=\sum\lambda_mP_m$. Expectation value of an observable $O$, i.e. $\langle O\rangle = \sum \lambda_m P(\lambda_m) = \sum \lambda_m \langle\psi|P_m|\psi\rangle=\langle\psi|\sum\lambda_mP_m|\psi\rangle=\langle\psi|O|\psi\rangle$, which is strainghforward to the concept of quantum mechanics. By the cyclic property of Trace operator, $\langle O\rangle = \langle\psi|O|\psi\rangle = Tr(O|\psi\rangle\langle\psi|)$.

Density matrix after measurement $M_m$ is given as $\rho_m = \frac{M_m\rho M_m^\dagger}{Tr(M_m^\dagger M_m\rho)}$. For entangled composite systems, since we cannot factor out the state vectors, partial trace gives a density matrix for each qubits: $\rho_A = Tr_B(\rho_{AB}), \rho_B = Tr_A(\rho_{AB})$

Example) $Tr_B(|a_1\rangle\langle a_2|\otimes|b_1\rangle\langle b_2|) = |a_1\rangle\langle a_2|Tr(|b_1\rangle\langle b_2|)=|a_1\rangle\langle a_2|\langle b_2|b_1\rangle$

Next article, we will discuss about variational quantum circuits and parameter update rules.