Chapter 2: Quantum computing

To conduct computations, quantum computing is a sort of computing that makes use of the collective qualities of quantum states, such as superposition, interference, and entanglement. Quantum computers are the electronic devices that are capable of carrying out quantum calculations. One of the subfields that falls under the umbrella of quantum information science is the study of quantum computing.

The quantum circuit model, the quantum Turing machine, the adiabatic quantum computer, the one-way quantum computer, and a variety of quantum cellular automata are some of the several kinds of quantum computers (also known as quantum computing systems). The quantum circuit, which is based on the quantum bit, also known as the "qubit," which is somewhat comparable to the bit used in conventional computing, is the model that is employed the most often. A qubit may exist in a quantum state of either one or zero, or it can exist in a superposition of both states at the same time. However, when it is measured, it is either 0 or 1, and the likelihood of either occurrence relies on the quantum state of the qubit just before measurement.

In order to construct a physical quantum computer, researchers are concentrating their efforts on developing technologies such as transmons, ion traps, and topological quantum computers. These technologies are designed to produce high-quality qubits. On the other hand, every issue that can be solved by a quantum computer may also be solved by a conventional computer given enough time, at least in theory. To put it another way, the Church-Turing thesis is followed by quantum computers. This indicates that while quantum computers do not provide any additional advantages over classical computers in terms of computability, quantum algorithms for specific problems have significantly lower time complexities than corresponding known classical algorithms. This is the case even though quantum computers do not provide any additional advantages. Notably, it is thought that quantum computers can swiftly answer some problems that no conventional computer could solve in any conceivable period of time. This accomplishment is known as "quantum supremacy," and it is something that no classical computer could do. In the context of quantum computers, the field of research known as quantum complexity theory examines the degree of difficulty that issues provide to quantum computers.

In 1980, scientist Paul Benioff suggested a quantum mechanical model of the Turing machine, which is considered to be the beginning of quantum computing.

The most widely accepted model of quantum computing portrays the process of computing as consisting of a network of quantum logic gates.

A memory consisting of

bits of information has

possible states.

A vector representing all memory states thus has

entries (one for each state).

This vector is considered to be a probability vector, since it reflects the fact that the memory will most likely be discovered in a certain state.

According to the traditional point of view, just one of the entries would have a value of one (meaning that there is a one hundred percent chance of being in this condition), while the other entries would all have a value of zero.

The concept of probability vectors may be applied to that of density operators in quantum mechanics. In most cases, the quantum state vector formalism is presented first because it is theoretically easier, and also because it may be used instead of the density matrix formalism for pure states, which is applicable in situations in which the whole quantum system is known.

Let's start by thinking about a basic memory that just has one bit in it. This memory might be located in either the zero state or the one state. These are the two possible states. We may use the Dirac notation to express the state of this memory in order to ensure that

A quantum memory may then be found in any quantum superposition

of the two classical states

and

:

The coefficients

and

are complex numbers.

It is stated that the quantum memory has one qubit worth of information embedded in it.

The state

is not itself a probability vector but can be connected with a probability vector via a measurement operation.

If the quantum memory is measured to determine whether the state is

or

(this is known as a computational basis measurement), the zero state would be observed with probability

and the one state with probability

.

The numbers

and

are called probability amplitudes.

The state of this one-qubit quantum memory may be modified by applying quantum logic gates, which is comparable to the manner in which the state of conventional memory can be changed by using classical logic gates. The NOT gate, which may be represented as a matrix, is an essential gate for both conventional and quantum processing.

Mathematically, Modeling the application of a logic gate like this to a quantum state vector may be done by matrix multiplication.

Thus

and

.

In two significant ways, the mathematics of single-qubit gates may be extended to work on multi-qubit quantum memory. This opens up new possibilities for the field of quantum computing. One technique to do this is to choose a qubit at random and then apply the gate that you picked to the qubit you want to change while keeping the rest of the memory untouched. There is also the option of applying the gate to its target only under the condition that another portion of the memory is already in the required state. One more illustration may be used to compare and contrast these two options. The following are the states that may be stored in a two-qubit quantum memory:

Following is a matrix that may be used to represent the CNOT gate, which can be used::

As a direct result of this definition using mathematical terms,

,

,

, and

.

To put it another way, the CNOT applies a NOT gate (

from before) to the second qubit if and only if the first qubit is in the state

.

If the first qubit is

, There is no change made to either qubit.

In a nutshell, a quantum computer may be conceptualized as a network that consists of quantum logic gates and measurements. Nevertheless, any measurement may be postponed until the completion of the quantum computation; however, this postponement may come with a computational cost, which is why the majority of quantum circuits portray a network that solely consists of quantum logic gates and does not include any measurements.

Any computation using quantum mechanics (which is, in the preceding formalization, any unitary matrix over

qubits) can be represented as a network of quantum logic gates from a fairly small family of gates.

A universal gate set refers to a certain gate family of one's choosing that makes this construction possible, as a computer that is capable of running such circuits is considered to be a universal quantum computer.

One such set often has all of the single-qubit gates in addition to the CNOT gate described above.

This indicates that any quantum computing may be carried out by performing a series of gates that operate on a single qubit in conjunction with gates that operate on CNOT qubits.

Despite the fact that this gate set has no end,, In accordance with the Solovay-Kitaev theorem, it is possible to substitute it with a finite gate set.

The quantum circuit model is often where researchers concentrate their efforts when attempting to make headway in the search for quantum algorithms; however, there are exceptions, such as the quantum adiabatic algorithm. Quantum algorithms may be generally sorted into several categories based on the kind of speedups they produce in comparison to their conventional counterparts. Certain oracle problems, such as Simon's problem and the Bernstein-Vazirani problem, do give provable speedups; however, this is in the quantum query model, which is a restricted model where lower bounds are much easier to prove; however, this does not necessarily translate to speedups for practical problems. Simon's problem and the Bernstein-Vazirani problem are examples of such problems.

Other problems, such as the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations, all have quantum algorithms that appear to give super-polynomial speedups and are BQP-complete. These problems include: Due to the fact that these issues are BQP-complete, the existence of an equally fast classical method for them would suggest that no quantum algorithm delivers a super-polynomial speedup, which is something that is thought to be very improbable. This is an alternative approach to the search issue.

One important use of quantum computing is in the development of new methods for breaking encryption protocols that are already in use. Integer factorization, which is the foundation of the security of public key cryptographic systems, is thought to be computationally infeasible with a typical computer for large integers if those integers are the product of few prime numbers. This is because large integers are more difficult to factor (e.g., products of two 300-digit primes). In contrast, a quantum computer might answer this issue in a time-efficient manner by using Shor's method to identify its components. Because of this capabilities, a quantum computer would be...