7 Summary

In this course you have learnt the fundamentals of quantum computing.The key points are as follows.

1. Quantum computers may be able solve problems more quickly than classical computers if problem solving algorithms which have exponential run-times on a classical computer can be written to have polynomial run-times on a quantum computer.

2. For a given square matrix, , it is possible to solve the equation where are column vectors known as eigenvectors and is a scalar called an eigenvalue. In quantum mechanics, an operator is a mathematical entity which converts one function into another function. Given an operator , the eigenvalue equation for that operator is . Here, the eigenvalue may be a complex number, and is function known as an eigenfunction. There may be more than one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation.

3. A general spin state (known as a ket) can be written as a linear combination of a spin-up state and a spin-down state (known as basis vectors), thus where and are complex numbers. For an atom in any spin state, , the probability of the outcome of a measurement indicating spin-up is and for spin-down is . Since these are the only possible outcomes the corresponding probabilities must sum to one, therefore

4. Matrices can be used as an alternative representation of spin states to simplify calculations. and are represented by the following column vectors:

   

   Any vector in spin space may be written as a linear combination of and . This means that becomes:

   

   In this way any spin state of a spin-½ particle can be represented as a two-element matrix, which is called a spinor.

5. For two electrons, the two-particle spin state can have an overall spin function which is either symmetric or antisymmetric to exchange of electrons. There is a set of triplet states and a singlet state, as follows:

   

   where the first arrow in each ket refers to particle 1 and the second to particle 2. The triplet states are symmetric and the singlet state is antisymmetric under particle exchange. Two-particle states which cannot be factorised (i.e. and ) are known as entangled states and exhibit entanglement. The other states (i.e. and ) are not entangled.

6. Quantum computing is based on units of information called qubits (quantum bits, and pronounced kew-bits), which obey the laws of quantum mechanics. A qubit is the quantum analogue of a classical bit. The classical bit values 0 and 1 are replaced by the orthonormal basis states of the quantum-mechanical qubit and . The basis states are given the name logical states , since they correspond to the classical bits upon which the logic gates operate. The key difference between qubits and classical bits is that qubits can exist in a superposition of the and states, which means qubits can be prepared in the superposition state .

7. The quantum NOT gate is denoted by the operator, and, in the basis of the logical qubits and , is represented by the matrix:

   

   which is also the Pauli-X operator, . If the input to a quantum NOT gate is then the output is .

8. The Hadamard gate is a single-qubit gate defined by the matrix:

   

   If the input to a Hadamard gate is then the output is . A Hadamard gate allows the transformation of the logical qubit state into a superposition state.

9. A straightforward way to define the two-qubit states is to build the two-qubit basis states from product states . There are four possible product states of the usual single-qubit basis states: . A general two-qubit state, therefore can be expressed in terms of the product states as

   

   where is normalised in the usual way:

   

10. The quantum CNOT gate acts on two qubits, a control qubit and a target qubit. It is represented by an operator which acts on a target qubit depending on the state of a control qubit . If the input to a CNOT gate is then the output is . The CNOT gate can entangle a pair of disentangled qubits and can also disentangle a pair of entangled qubits.

11. Single-qubit gates and two-qubit gates can be combined in a structured sequence to give a quantum circuit that executes a particular algorithm. Measurements can also be made in quantum circuits when one of the eigenstates of the measurement operator will be obtained with a certain probability.

12. Real-world quantum computing has been impemented on platforms including the transmon qubit (as in the IBM quantum computers), the NMR qubit (based on technology used in MRI scanners), and using ultra-cold atom technology.