An Introduction to Quantum: Classical Computing - RPIQuantumComputing/QuantumCircuits GitHub Wiki

Introduction to Classical Information and Classical States

Before going deeper into quantum computing, it will be beneficial to have sufficient knowledge regarding classical computing. More particularly, classical information is dictated by one main principle: classical systems (mediums which store information) can be in a finite number of states. A prime example of a classical medium is a bit: a bit is a classical, small unit of information, which has two states: zero and one. Another example of a classical medium is a six-sided die-its states are one, two, three, four, five and six. A nucleobase in a strand of DNA has four states: A, C, G, and T. Electric fans have four states: low, medium, high, and off. Below is a diagram representing the two possible states a bit can have: zero and one. A bit is a prime example of a classical system.

Above is a clear example of two classical states regarding a bit: zero and one. People who regularly code may also visualize these states as those seen with boolean states in low-level languages: in C++, when printing out a boolean, false is printed out as '0', and true is printed out as '1.'

Across all of these examples, it is important to note one thing: the number of states all aforementioned mediums can have is finite. There are two perspectives to consider when interpreting classical information: a situation in which we know with certainty what state our classical system is in and a situation in which we are only able to assign probabilities to different states. For example, if I roll a six-sided die, look at what side is face-up, and see that it is a three, I can say with certainty that the die has a state of three. On the other hand, if I roll a six-sided die while looking in another direction, I only know that the die has a one-sixth chance of being in all six states. This would be written in the form of a probability vector: for the die, the probability vector= (1/6, 1/6, 1/6, 1/6, 1/6, 1/6). These probabilities correspond to the set of all possible states for the die, which is equal to {1, 2, 3, 4, 5, 6}. When writing probability vectors, it is important to note that all elements in a probability vector must be nonnegative real numbers and that the sum of the elements must be equal to one. While considering a system in a probabilistic state, we may want to unequivocally determine which state the system is in. To do this, we will attempt to "measure" the system being considered. By measuring the system, we gain knowledge of the system's state. For example, say you flip a coin but close your eyes to conceal the result. To someone else who is looking at the coin after it has been flipped, it is in one particular unambiguous state: heads or tails. However, you must "measure" the system to gain knowledge regarding what state it is in.

Pictured above is a six-sided die. As described previously, a die is a simple example of a medium with a finite number of classical states: zero, one, two, three, four, five, and six.

As previously noted, column vectors store the probabilities regarding the various possible classical states. For example, if we have a bit, X, and if we know that this bit has a state of zero with probability 3/4 and a state of one with probability 1/4, we could represent the bit's probability(column) vector like so:

Introduction to Deterministic Operations (Classical Information, for a single system)

Deterministic Operations are those in which each classical state is transformed by some function $f(a)$, where $a$ represents every possible classical state within the system. For example, if we are talking about a bit, whose states are either zero or one, a few deterministic operations can occur. First, a constant deterministic operation can occur. A constant deterministic operation transforms the system's state into one particular state among the set of states, regardless of what state it is currently in. For example, consider a bit. Under a constant deterministic operation, whether the bit is in its zero state or its one state, it will be transformed to either the same state or the other state, depending on the operation. For example, a constant deterministic operation, which transforms the bit into its zero state, will transform the bit so that it is in its zero state, regardless of what state it is currently in. If the bit is in its zero state, after the operation, it will remain in the zero state. If the bit is in state one, it will be transformed to state zero. There can be another deterministic operation in which regardless of what state the bit(or any other classical system) is in, it will be transformed so the bit is in state one. The not deterministic operation transforms the bit such that it is in the opposite state from that which it was previously in. For example, if it was in $\ket{0}$, it will now be in $\ket{1}$. An identity deterministic operation will maintain the state which the bit is currently in. Many times, it is convenient to represent Deterministic Operations through matrix-vector multiplication. Corresponding to the order that they were described in above, the matrix representation of each Deterministic Operation f(a) mentioned above is put below. With the matrices, the order is as follows: rows first, and columns second. When multiplying a matrix by a vector, it is like grabbing the vector, placing it at the top of one of the matrices, where the column you visualize accedes with the observed classical state. The output received is a probability vector based on the column the input was placed in. For example, for the first operation, in which both results come out to be state zero, both columns in the matrix have a probability vector corresponding to the zero state. This fundamental idea applies to the rest of the deterministic operations as well.

##Dirac Notation(Second Part) Previously, Dirac Notation was introduced as vector notation given a column probabilistic vector. Additionally, Dirac Notation can be applied to row vectors as well. For example, a row vector that is (1,0) will correspond to <0| and (0,1) will correspond to <1|.

Introduction to Probabilistic Operations and stochastic matrices

Probabilistic operations also fall under the realm of operations performed on classical systems. Probabilistic operations build on the idea of changing the state of the classical system. However, instead of transforming the state with absolute certainty with a certain operation, the probabilistic operations transform a classical system into various possible states with a certain probability. These operations can be represented with matrices; these matrices are matrices whose columns form probability vectors. One of the most intuitive ways to think about probabilistic operations is that they are just like deterministic operations, except for the fact that each potential operation happens under a certain chance.

Introduction to Quantum Information

In many ways, quantum information is similar to classical information. However, there are some differences. Firstly, the entries of a quantum state vector are complex vectors. Since the entries are complex numbers, the sum of all the vector's elements squared will be equal to one. The simple difference being that quantum information stores complex numbers and classical information stores nonnegative real numbers is conducive for the differences between quantum and classical information. Example states are (1,0), (0,1), (1/rad2, 1rad/2), (1/rad2, -1/rad2), and ((1+2i)/3. Squaring and summing these elements results in a sum of 1. The state (1/rad2, 1/rad2) is known as the plus state, and the state (1/rad2, -1/rad2) is known as the minus state. Therefore, it is appropriate to regard the state (1/rad2, 1/rad2) as the |+> state, and it is appropriate to regard the state (1/rad2, -1/rad2) as the |-> state. Furthermore, it is common to use the notation ∣∣ψ⟩ to represent an arbitrary vector that may not be a standard basis vector.

Pictured above is a column vector showing various quantum states.

Quantum States of other systems

At first, it seems like certain systems may have only simple, classical state sets. However, it is possible to analyze and interpret quantum states of systems having arbitrary classical state sets. For example, an electrical fan switch may have the states: (1/2, 0, -i/2, 1/rad2).