Quantum Computing PDF

Quantum Computing Contents  Introduction to Quantum Computing Superposition, Entanglement Interference and Coherence/de-coherence  Representation of a qubit-Bloch sphere (Qualitative) Pure and mixed states Polarization, von Neumann Entropy  Quantum logic gates: Hadamard Gate,...

Quantum Computing Contents  Introduction to Quantum Computing Superposition, Entanglement Interference and Coherence/de-coherence  Representation of a qubit-Bloch sphere (Qualitative) Pure and mixed states Polarization, von Neumann Entropy  Quantum logic gates: Hadamard Gate, Pauli Gates, C - NOT Gate and Toffoli gates, The Stern–Gerlach experiment. What is a quantum computer?  A quantum computer is a machine that performs calculations based on the laws of quantum mechanics >>>>>>>> the study of particles at the sub-atomic level.   “I think I can safely say that nobody understands quantum mechanics” - Feynman  1982 - Feynman proposed the idea of creating machines based on the laws of quantum mechanics instead of the laws of classical physics.  1985 - David Deutsch developed the quantum turing machine, showing that quantum circuits are universal.  Three-gate quantum circuit and its corresponding fully connected quantum neural network (QNN)  Qubit A qubit (or quantum bit) is the quantum mechanical analogue of a classical bit. In classical computing, the information is encoded in bits, where each bit can have the value zero or one. In quantum computing the information is encoded in qubits. A qubit is a two-level quantum system where the two basis qubit states are usually written as ∣0 ⟩, ∣1 ⟩ or (unlike a classical bit) in a linear combination of both states ∣0 ⟩ +∣1 ⟩. The name of this phenomenon is known as superposition. Examples for qubits : Single photons, neutral atoms, spin half electrons, Quantum dots, superconductors, Nuclear spins etc A physical implementation of a qubit could use the two energy levels of an atom. An excited state representing |1> and a ground state representing |0>. Light pulse of frequency  for Excited time interval t State Nucleu Ground s State Electr State on State |0> |1> What is a qubit?  Relationships among data - Entanglement Entanglement is the ability of quantum systems to exhibit correlations between states within a superposition. Imagine two qubits, each in the state |0> + |1> (a superposition of the 0 and 1.) We can entangle the two qubits such that the measurement of one qubit is always correlated to the measurement of the other qubit.  Principle of Superposition Suppose |0> and |1> are two allowed quantum states of a system; then the system can exist in any linear superposition of these states   | 0    |1  where  and  are complex numbers "macroscopic quantum superposition" If this was observable in a macroscopic object + live dead |1  |0 But you don’t see such states in everyday objects: "Schrodinger's cat paradox" (Schrodinger, 1935) Quantum Entanglement (Schrodinger, 1935) Multiple quantum systems can exist in entangled superposition states in which the state of an individual system has no well-defined physical meaning. Example for Separable state  | 0 a | 1 b Qubit "a" is in |0> state and Qubit "b" is in |1> state. This is a "separable state" or "product state". 1 Example for an Entangled state    | 0 a | 1 b  | 1 a | 0 b  2 Quantum Entanglement 1    | 0 a | 1 b  | 1 a | 0 b  2 if this were the state of two macroscopic + objects (dead, live) and (live, dead) Quantum Computing with one qubit Consider one qubit with energy eigenstates |0> and |1>. We will need to be able to put it into superposition states:   | 0    |1  - probability amplitudes a and b can be complex numbers 2 2 - The state must be normalized to unity so    1 - an overall phase factor has no effect, so we can choose a to be real - then define  cos( / 2)  ei sin( / 2) 2 2 2 2     cos( / 2)  e sin( / 2) i cos 2 ( / 2)  sin 2 ( / 2) 1 can always write a superposition state in the form:    | 0    | 1   cos( / 2) | 0   ei sin( / 2) | 1   The Bloch Sphere Superposition States are Points on the Bloch Sphere The poles represent the classical bit; let us use the notation |0⟩ and |1⟩. Quantum bits cover the whole sphere. Thus, there is      cos  | 0  ei sin   | 1  much more information involved in the quantum bits,  2  2 and the Bloch sphere depicts that. z |0 >  y  sphere with radius R=1 …..this is the “Bloch Sphere” x |1 > Superposition States as Points on the Unit Sphere z Example:  = 0 |0>      cos  | 0  e sin   | 1  i  2  2  0  0 cos  | 0  ei sin   | 1   2  2 | 0   0 y x Superposition States as Points on the Unit Sphere z Example:  =  |0>      cos  | 0  ei sin   | 1   2  2     cos  | 0  ei 0 sin   | 1   2  2 | 1    y x |1> Superposition States as Points on the Unit Sphere z Example:  = /2, = 0 |0>      cos  | 0  ei sin   | 1   2  2   i0   cos  | 0  e sin   | 1   4  4 | 0   |1     / 2 2 | 0   |1  y 2 x |1> Superposition States as Points on the Unit Sphere z Example:  = /2, = /2 |0>      cos  | 0  ei sin   | 1   2  2    i   cos  | 0  e sin   | 1  2  4  4 | 0  i | 1     / 2 2 | 0   |1  y 2   / 2 | 0  i | 1  2 x |1> There are an infinite number of states on the Bloch sphere,..... but we can choose a "digital" subset for computing o  0 1  1 0 1 0  1 0  1 x  x  2 2 2 0 i1 2 0 i 1 0 i1 y   y  2 2 Note: one classical bit has 21 possible states (0 and 1). One of these qubits has of 1 order ~22 accessible states 2-Qubit Representation A two-qubit system combines the quantum states of two qubits into a single quantum state, representing all possible combinations of the individual qubit states. This system lives in a 4-dimensional Hilbert space. Basis States Each qubit can be in one of the two basis states ∣0⟩ or ∣1⟩. The combined system has four basis states: ∣00⟩,∣01⟩,∣10⟩,∣11⟩ These are the computational basis states, formed by the tensor product of the individual qubits Mathematical Representation For two qubits ∣q1⟩ and ∣q2⟩, the combined state is given by the tensor product: ∣ψ⟩=∣q1⟩⊗∣q2⟩ For example: If ∣q1⟩=∣0⟩ and ∣q2⟩=∣1⟩, ∣ψ⟩=∣0⟩⊗∣1⟩ The vector representation of ∣01⟩ is: ∣01⟩= ​ Qubit Representation One Qubit Two Qubit Three Qubit Operations on Qubits - Reversible Logic Due to the nature of quantum physics, the destruction of information in a gate will cause heat to be evolved which can destroy the superposition of qubits. Ex. Input Output The AND Gate In these 3 cases, A B C information is 0 0 0 being destroyed A 0 1 0 C B 1 0 0 1 1 1 This type of gate cannot be used. We must use Quantum Gates. Quantum Gates  Quantum Gates are similar to classical gates, but do not have a degenerate output. i.e. their original input state can be derived from their output state, uniquely. They must be reversible. This means that a deterministic computation can be performed on a quantum computer only if it is reversible. Luckily, it has been shown that any deterministic computation can be made reversible.(Charles Bennet, 1973) Quantum Gates - Hadamard Simplest gate involves one qubit and is called a Hadamard Gate (also known as a square-root of NOT gate.) Used to put qubits into superposition. H H State State | State 0> + |1> |1> |0> Note: Two Hadamard gates used in succession can be used as a NOT gate Quantum Gates - Controlled NOT A gate which operates on two qubits is called a Controlled-NOT (CN) Gate. If the bit on the control line is 1, invert the bit on the target line. Input Output A - Target A’ A B A’ B’ 0 0 0 0 0 1 1 1 B - Control B’ 1 0 1 0 1 1 0 1 Note: The CN gate has a similar behavior to the XOR gate with some extra information to make it reversible. Example Operation - Multiplication By 2  We can build a reversible logic circuit to calculate multiplication by 2 using CN gates arranged in the following manner: Input Output Carry Ones Carry Ones Bit Bit Bit Bit 0 0 0 0 0 1 1 0 0 Carry Bit Ones Bit H Quantum Gates - Controlled Controlled NOT (CCN) A gate which operates on three qubits is called a Controlled Controlled NOT (CCN) Gate. Iff the bits on both of the control lines is 1,then the target bit is inverted. Input Output A B C A’ B’ C’ A - Target A’ 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 B - Control 1 B’ 0 1 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 C - Control 2 C’ 1 1 0 1 1 0 1 1 1 0 1 1  A Universal Quantum Computer  The CCN gate has been shown to be a universal reversible logic gate as it can be used as a NAND gate. A - Target A’ Input Output A B C A’ B’ C’ 0 0 0 0 0 0 B - Control 1 B’ 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 C - Control 2 C’ 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 When our target input is 1, our target output is a result of a NAND of B and C. Quantum Gate  Stern-Gerlac Experiment A decisive (and very famous) early experiment indicating the qubit structure was conceived by Stern in 1921 and performed with Gerlach in 1922 in Frankfurt In this experiment, hot atoms were ‘beamed’ from an oven through a magnetic field which caused the atoms to be deflected, and then the position of each atom was recorded, as illustrated in Figure The original experiment was done with silver atoms, which have a complicated structure The same basic effect is observed with hydrogen atoms. If the atoms had retained their | + Z > orientation, then the output would be expected to