## Quantum Computations: an Analysis and Comparative Study

The Quantum Computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.
Quantum computers will harness the power of atoms and molecules to perform memory and processing tasks. Quantum computers have the potential to perform certain calculations significantly faster than any silicon-based computer.Paul Benioff is credited with first applying quantum theory to computers in 1981. Benioff theorized about creating a quantum Turing machine. Most digital computers, like the one you are using to read this article, are based on the Turing Theory.

Turing in the 1930s, is a theoretical device that consists of tape of unlimited length that is divided into little squares. Each square can either hold a symbol (1 or 0) or be left blank. A read-write device reads these symbols and blanks, which gives the machine its instructions to perform a certain program. Well, in a quantum Turing machine, the difference is that the tape exists in a quantum state, as does the read-write head. This means that the symbols on the tape can be either 0 or 1 or a superposition of 0 and 1; in other words the symbols are both 0 and 1 (and all points in between) at the same time. While a normal Turing machine can only perform one calculation at a time, a quantum Turing machine can perform many calculations at once.

Today’s computers, like a Turing machine, work by manipulating bits that exist in one of two states: a 0 or a 1. Quantum computers aren’t limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today’s most powerful supercomputers.

WORKING(based on physical model)—

Quantum computers also utilize another aspect of quantum mechanics known as entanglement. One problem with the idea of quantum computers is that if you try to look at the subatomic particles, you could bump them, and thereby change their value. If you look at a qubit in superposition to determine its value, the qubit will assume the value of either 0 or 1, but not both (effectively turning your spiffy quantum computer into a mundane digital computer). Entanglement provides a potential answer. In quantum physics, if you apply an outside force to two atoms, it can cause them to become entangled, and the second atom can take on the properties of the first atom. So if left alone, an atom will spin in all directions. The instant it is disturbed it chooses one spin, or one value; and at the same time, the second entangled atom will choose an opposite spin, or value. This allows us to know the value of the qubits without actually looking at them.
A quantum computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or, crucially, any quantum superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8. In general a quantum computer with n qubits can be in an arbitrary superposition of up to 2n different states simultaneously (this compares to a normal computer that can only be in one of these 2n states at any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm.An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: “down” and “up” (typically written and , or and ). But in fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an effective spin-1/2 system.

Bits vs. qubits

Qubits are made up of controlled particles and the means of control (e.g. devices that trap particles and switch them from one state to another). Consider first a classical computer that operates on a three-bit register. The state of the computer at any time is a probability distribution over the 23 = 8 different three-bit strings 000, 001, …, 111. If it is a deterministic computer, then it is in exactly one of these states with probability 1. The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (a,b,c,d,e,f,g,h), called a wave function. However, instead of adding to one, the sum of the squares of the coefficient magnitudes, | a | 2 + | b | 2 + … + | h | 2, must equal one. Moreover, the coefficients are complex numbers. Since states are represented by complex wavefunctions, two states being added together will undergo interference. This is a key difference between quantum computing and probabilistic classical computing.
If you measure the three qubits, then you will observe a three-bit string. The probability of measuring a string will equal the squared magnitude of that string’s coefficients (using our example, probability that we read state as 000 = | a | 2, probability that we read state as 001 = | b | 2, etc..). Thus a measurement of the quantum state with coefficients (a,b,…,h) gives the classical probability distribution ( | a | 2, | b | 2,…, | h | 2). We say that the quantum state “collapses” to a classical state.

Operation
While a classical three-bit state and a quantum three-qubit state are both eight-dimensional vectors, they are manipulated quite differently for classical or quantum computation. For computing in either case, the system must be initialized, for example into the all-zeros string, , corresponding to the vector (1,0,0,0,0,0,0,0). In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations (they preserve that the sum of the squares add up to one, the Euclidean or L2 norm). Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we sample from the probability distribution on the three-bit register to obtain one definite three-bit string, say 000. Quantum mechanically, we measure the three-qubit state, which is equivalent to collapsing the quantum state down to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients for the quantum state, as described above) followed by sampling from that distribution. Note that this destroys the original quantum state. Many algorithms will only give the correct answer with a certain probability, however by repeatedly initializing, running and measuring the quantum computer, the probability of getting the correct answer can be increased.

Quantum computers also utilize another aspect of quantum mechanics known as entanglement. One problem with the idea of quantum computers is that if you try to look at the subatomic particles, you could bump them, and thereby change their value. If you look at a qubit in superposition to determine its value, the qubit will assume the value of either 0 or 1, but not both. To make a practical quantum computer, scientists have to devise ways of making measurements indirectly to preserve the system’s integrity. Entanglement provides a potential answer. In quantum physics, if you apply an outside force to two atoms, it can cause them to become entangled, and the second atom can take on the properties of the first atom. So if left alone, an atom will spin in all directions. The instant it is disturbed it chooses one spin, or one value; and at the same time, the second entangled atom will choose an opposite spin, or value. This allows scientists to know the value of the qubits without actually looking at them.

WORKING (BASED ON CLASICAL MACHINE MODEL)-
Here is a simple model for a quantum computer based on a classical computer instructing a machine to manipulate a set of spins. Consider the following model for the operation of a quantum computer: Several thousand spin- particles (or two-level systems) are initially in some well defined state, such as spin-down. A classical machine takes single spins or pairs of spins and entangles them (performing an elementary one-bit operation or the two-bit XOR gate); see Fig. These stages are repeated on different pairs of spins according to the instructions of a conventional computer program. Since the spins are entangled, we must not look at the spins at intermediate stages: We must keep the quantum superposition intact. Furthermore, nothing else may interfere with the spins which could destroy their orientation or interrupt their unitary evolution. Once this well-defined cycle of manipulation is complete the orientations of the spins are measured This set of measured orientations is the output of the computation.

Initially all particles are spin-down. Stage a) a classical machine takes a single or pair of spins and in stage b) it performs a selected one-bit or two-bit operation; in stage c) the “entangled” particles are returned to their original locations. These three stages are repeated many times in accord with the instructions given by an ordinary classical computer. When this cycle is complete stage d) consists of measuring the state of the particles (leaving them in some particular bit-string); this bit-string is the result of the computation.Given this paradigm for a quantum computer, what might its high-level language (its computer code) look like? The most serious difficulty that must be dealt with is that the quantum information is manipulated by a conventional computer in a completely blind manner—without any access to the values of this quantum information. This means that the program cannot utilize “shortcuts” conditional on the value of a quantum variable (or register or bit). For example, loops must be iterated through exactly the same number of times independent of the values of the quantum variables. Similarly, conditional branches around large pieces of code must be broken down into repeated conditions for each step. In addition, each instruction performed upon the quantum bits must be logically reversible. Thus, ordinary assignments of a value to a variable, such as
| a = n, are not legal and must instead be performed as increments on an initially zeroed variable, such as | a = | a + n.