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Now that the concepts of quantum mechanics are known, it’s time to talk about how to compute! The first concept that the quantum learner must acknowledge is understanding the fundamental differentiation between the classical technology that is currently utilized in the world and the quantum technology that is to come. Classical or digital technology is the paradigm that we are familiar with using commonly to solve many kinds of computations calculated today. They are both tools used to “compute” solutions to problems and can certainly be used in tandem, yet the significant differences lie in the methods or operations that each of these problem solving tools use. Though they may use the same word of "computer" in their phrasing, digital and quantum technology are solving computational problems of vastly differing complexity.
In the most fundamental sense, a computer is something that processes an input to produce output.
- Bits vs Qubits
On digital computers, the input is taken in bits of information using classical mechanics, switches, and circuits in order to process the information to output. In a mathematical context, a bit is described in binary descriptions on digital computers, as an information value being either of value 0 or 1. This value of 0 or 1 can symbolize many things but in short was a symbolic construct to pass through a classical computer which can only process information in binary ways by default. So the input will always process as a value of 0 or 1 (or Black or white, Red or Blue, Alive or Dead.) and also output in a singular binary answer of values 0 or 1. From the invention of switches and circuits, this was revolutionary in the past to be able to compute & control an outcome directly because it gave us the ability to manipulate 2 outcomes of information. It was only later on that we created additional logic to add complexity to further manipulate 2 bits of information (things like the NOT operation, OR conditional, as well as the AND which were the basic components of an electrical circuit which later turned digital) in ways that further served the desired outcome.
Binary methods include turning on and off a light switch or opening & closing a door. So many ways to create information and technologies are based on this binary principle, largely because it is just simpler to compute with. It was this ability to manipulate this off/on inputs & outputs that allowed the eventual ability to create a world of binary applications, all stitched together with various off/on bits and constructed logic operations to create all the TVs, computers, cell phones,cars, electrical devices, and pretty every “modern” technology that we enjoy today, all from the concept of 0’s and 1’s.
Conversely, on quantum computers, the input is taken in qubits of information using quantum mechanics in order to process the information to output. In a mathematical context, a qubit on the other hand is described in vector descriptions(the shorthand for a matrix) for quantum computers, as an information value being a range from|0> or |1 > . Similarly to digital counterpart, the states |0> and |1> can symbolize many things but it is the customary construct to pass through a quantum computer in order to make up one logical qubit.
This value of 0 or 1 can symbolize many things (and many systems!) but in short is a symbolic construct to pass through a quantum computer to process many outputs. These inputs are processed as values on the bloch sphere (depicted nearby). So the input will always process as a value of 0 and 1 (or Black and white, Red and Blue, Alive and Dead.) and also output in a multitude of answers for values 0 and 1. This is why quantum computers are known to output probabilistic answers. From this point we convert the probabilistic output stored as a binary output for a human-readable answer. From the discovery of Shor’s algorithm to the invention of D-Wave’s quantum device, it has been revolutionary over the past few years to observe the research & results coming out constantly within this field. It is an exciting time to be in the field indeed.**The understanding between digital technology that is currently utilized in the world and quantum technology will be differentiated. Digital technology solves different kinds of problems than quantum technology. They are both tools for different problems and can certainly be used in tandem. Yet significant differences are there though they may use the same word of "computer" in their phrasing since they are indeed solving computational problems of differing complexity.
The initial state of each qubit is |0〉. Then, the Hadamard gate is applied to each qubit, preparing for the quantum parallelism.
- highlighted by Shwetha Jayaraj at page 28 on BasicQuantumAlgorithms.pdf
Quantum parallelism is the simultaneous execution of an algorithm with more than one input to a single quantum processor. To execute the same task on a classical computer would require an exponential number of classical processors
- highlighted by Shwetha Jayaraj at page 28 on BasicQuantumAlgorithms.pdf
Due to the linearity of linear algebra, U is applied to all kets of the sum abov n simultaneously. Therefore, it is possible to perform 2simultaneous computations on a n-qubit quantum computer. Note, however, that the result of these computations is a superpositio state and, after a measurement, the final output is a single n-bit string.
- highlighted by Shwetha Jayaraj at page 29 on BasicQuantumAlgorithms.pdf
We can refine the standard model by adding an extra register for draft calculations. Sinc unitary gates are invertible, the whole calculation process without measurement is reversible This means that no information is erased. The number of output qubits must be equal to the number of input qubits.
The basic memory unit of a classical computer is the bit, which assumes 0 or 1. Usually, th bit is implemented using two distinct voltages, following the convention that null or low voltag represents bit 0 and high voltage represents bit 1. To determine whether the output is bit 0 or 1 at the end of the computation, it is necessary to measure the voltage The basic memory unit of a quantum computer is the qubit, which also assumes, at the end of the computation, 0 or 1. The qubit is implemented using an electric current in a small super conductor, following the convention that clockwise current represents 0 and counter-clockwise current represents 1, or the other way around. The difference from the classical device happen during the computation since the qubit admits the simultaneous coexistence of 0 and 1. During the computation, that is, before the measurement, the state of a qubit is represented by norm-1 two-dimensional vector and the states of a qubit corresponding to 0 and 1 are |0〉 an |1〉. The definition of state is a vector of norm 1 in a complex vector space endowed with the 1 inner product presented in the previous Section.The state can be thought of as the “value” of the qubit before the measurement. Quantum coexistence is represented mathematically by a linear combination of orthonormal vectors as follows |ψ〉 = α|0〉 + β|1〉, where α and β are complex numbers that obey the constrain 2 2 |α|+ |β|= 1. The state of the qubit is vector |ψ〉 of norm 1 with the entries α and β. The complex numbers α and β are the amplitudes of the state |ψ〉
- highlighted by Shwetha Jayaraj at page 9 on BasicQuantumAlgorithms.pdf
The state of a qubit can be characterized by two angles θ and φ as follow θθiφ |ψ〉 = cos |0〉 + esin |1〉 |0〉 + esin 2 2 where 0 ≤ θ ≤ π and 0 ≤ φ < 2π. This notation shows that there is a one-to-one correspondence between the set of states of a qubit and points on the surface of a sphere of radius 1, called Bloc sphere.
- highlighted by Shwetha Jayaraj at page 11 on BasicQuantumAlgorithms.pdf