Chapter 2. Basic Principles of Quantum mechanics In this chapter we introduce basic principles of the quantum mechanics. Quantum computers are based on the principles of the quantum mechanics. In the classical computation we are familiar with bits and we use Boolean algebra. Please read the following analysis: In physics there are some models describe physical events in the macroscopic universe and microscopic universe. For instance, working principle of an engine based on laws of thermodynamics. If this sentence make sense then you realized the second law! Research and development in the physics, leads to the production to the high-technological devices, computers, scientific instruments, electronics technology, communication technology, as well as scientific papers and patents. I hope at the end of this course your point of view to technological developments will improve and you realize existence of a bridge between theory and practice. Quantum computation is based on SUPERPOSITION and COMPOSITION of the state. In the classical computation we cannot obtain superposition of two binary number (bit) 1 and 0. In quantum computation it is possible to obtain mixture of two quantum bit (qubits): i.e spin up and spin down electron. The representations of bit and qubit are: The representation is known as DIRAC notation which is called a KET vector. In the classical system we can not express a bit as decomposition of the bits. The ket vectors also represented by: The quantum theory is the "theory of everything". Note that one can complete all of the calculations of the quantum computation by using simple matrix and vector algebra! In this lecture we will discuss: Four postulates of QM States: Dirac notation (Bra-Ket notation), Pauli Spin Matrices Transformation of the states: Pauli Gates Measurement Composite systems & Entanglement States: Dirac notation (Bra-Ket notation), Pauli Spin Matrices A quantum mechanical system completely described by a state vector. A state vector include all information about the quantum system. In classical computation all computations based on a bit in the base 2 number system. In quantum computation theory the computation based on state (qubits). As you see that this open a new way to the computation theory! A stationary state corresponds to a quantum state with a single definite energy. A superposition state is mixture of stationary states. (For more details come to class) Consider two level quantum system (for example spin up and down state of the electron). Wave function of the system may be written as the superposition of the spins:
The wave should be normalized. Then BRA of the wave function can be written as Normalization yields: Note that are orthogonal vectors, then The expression reads as follows. The state is with probability and with a probability. Transformation of the states: Pauli Gates Time evolution of a quantum system in the Hilbert space is described by a unitary transformation, generally by using unitary matrices. Unitary matrix is defined as follows: If transpose conjugate of a matrix U is equal to the inverse of matrix then the matrix is known as unitary matrix. Note that transpose conjugate of U can be abbreviated as U + and satisfy the relation U + U=I (unit matrix). Examples: The matrices are unitary and first 3 matrices are known as Pauli Matrices. Transformation of the state can be realized as change of the physical situation of the quantum particle from one state to other. Pauli Matrices are mathematical representations of the transformation of spin state of the particles. Technically the spin of the particles can be changed by applying magnetic field. Consider a particle of zero (down) spin. By applying magnetic field we can change the direction of the spin to. This transformation mathematically represented as: Explicitly Generally, transformation of the state with a unitary matrix can be expressed as i.e: Means, the quantum particle has a spin up and down with probability a and b respectively. The state of the particle can be changed by applying an external field. New spin state of the particle is again up and down, but in this case the probabilities are b and a respectively. One of the important transformation can be done by the matrix:, is known as Hadamard Matrix. This transformation is:
Before going further, let us practically construct some simple quantum gates namely PAULI GATES. As it is mentioned before this gates physically realized by using material (ions, for instance) whose spins controlled by magnetic field. Similar devices also constructed by using semi transparent mirrors and polarization properties of the photos. The devices also constructed by using an appropriate crystal and again polarization properties of the photons. Pauli X-Gate X Pauli Y-Gate Y Pauli Z-Gate Z Exercise: Show that XY=iZ, YZ=iX, ZX=iY and X 2 =Y 2 =Z 2 =I. One can also construct a gate using the transformation matrix:. This gate is known as Walsh-Hadamard gate that is one ofthe important gate of the quantum computers. Walsh Hadamard H-Gate
H Measurement Measurement of a qubit is another postulates of a quantum mechanics and play an important role in the quantum computation. Consiter a state. We want to measure a or/and b in order to determine output of the calculation. Unfortunately quantum mechanics does not allow determination of a and b. However, quantum we can obtain an indirect information about the state. The measurement always disturb the system then only one of the output can be determined with a probalities: As an example for the given state the probabilities. The postulate can be stated as follows: Quantum measurements can be described by measuring transformation of the state by the operator, mathematically Consider operators.probability of measurement of the output is and the state vector Then the probability of measuring the state are. Composite systems & Entanglement Mathematical Supplement As in the digital computation system, using simple quantum logic gates we can design more complicated circuits. For multi uncoupled qubits (state), the state of the composite system is given by tensor product or direct product of the matrix representation. Consider tensor product of two qubit system: These two qubit states can be represented by matrices: Then tensor product of two matrices are given by:
The quantum computation requires this multiplication. Sometimes we might want to compute effect of a gate conditionally. The mathematical operation composing gates conditionally is the direct sum of the corresponding unitary matrices. Direct sum of two matrices can be described as: We will turn our attention to the composition of multi qubit systems in the next chapter. For a useful quantum computation device we have to construct multi-qubit quantum states. "The state space of a composite physical system is the tensor product of the state spaces of the component physical systems." Suppose that there are two states then tensor product produce multiple state (superposition of the states) such that For quantum systems tensor product captures the essence of superposition. Properties of tensor product will be introduced in the class!! Quantum entanglement Let us ask a question: how can we determine individual 1-qubit state of a 2-qubit quantum register: We recall the last postulate (composition) and define the states:, the tensor product If we ask the same question for the decomposition:, interestingly we can see that no individual 1-qubit state exist to form this decomposition. At first sight we can think that quantum mechanics does not allow this decomposition! If we decide to measure the first qubit then either will be measured with the probabilities respectively. In the computational basis if we measure qubit then without touching or measuring other qubit it is quantum state can be determined as. This shows that there is a mysterious connection between states!measurements of one qubit on the side effect another qubit. This shows that aforementioned entangled state can also be written as:. This effect seems to be in total contradiction with Einstain's theory of speed of light! Finally we define the entanglement state: If a multi qubit state can not be factored into individually one qubit state then the state is called entangled state. Entanglement is an important phenomenon in multi-qubit quantum memory registers and many quantum algorithms.