3.1 Two qubit operators

1. Now, let $q$ be a system with two qubits, i.e., a 2-vector of qubits.

2. $q$ is now a linear combo of 4 basis values, $ | 00\!\!>$, $ | 01\!\!>$, $ | 10\!\!>$, $ | 11\!\!>$.

3. $q = a_0 | 00\!\!> + a_1 | 01\!\!> + a_2 | 10\!\!> + a_3 | 11\!\!> $

4. where $a_i$ are complex and $ \sum | a_i | ^2 = 1$.

5. $q$ exists in all 4 states simultaneously.

6. If $q$ is a vector with n component qubits, then it exists in $2^n$ states simultaneously.

7. This is part of the reason that quantum computation is powerful.

8. A measurement operator applied to $q$ will rotate it to a basis {00, 01, 10, 11}, so that it will be observed in one of those four cases, with probabilities $ | a_i | ^2$.

9. You operate on $q$ by multiplying it by a 4x4 matrix operator.

10. The matrices are all invertible (except for measurement matrices), and all leave $ | q | = 1$.

11. You set the initial value of $q$ by setting its two qubits each to 0 or 1.

12. How this is done depends on the particular hw.

13. I.e., initially, $q_1 = \begin{pmatrix}a_1 | 0\!\!> \\b_1 | 1\!\!> \end{pmatrix}$ and $q_2 = \begin{pmatrix}a_2 | 0\!\!> \\b_2 | 1\!\!> \end{pmatrix}$, and so

    $$q = \begin{pmatrix} q_1 \\ q_2 \end{pmatrix} = \begin{pmatrix} a_1 a_2 | 00 \!\!> \\ a_1 b_2 | 01 \!\!> \\ a_2 b_1 | 10 \!\!> \\ b_1 b_2 | 11 \!\!> \end{pmatrix}$$.

14. The combined state is the tensor product of the individual qubits.

15. In this case, you could separate out the individual qubits again.

16. However, sometimes after operating on the combo (i.e., multiplying by a matrix), you cannot any more separate out the result into a tensor product of individual qubits.

17. For $n$ qubits, the tensor product is a vector with $2^n$ elements, one element for each possible value of each qubit.

18. Each element of the tensor product has a complex weight.

19. You transform a state by multiplying it by a matrix.

20. The matrix is invertible.

21. The transformation doesn't destroy information.

22. When you measure a state, it collapses into one of the basis states. (aka component states)

23. You don't need to bring in consciousness etc. The collapse happens because the measurement causes the state to interact with the outside world.

24. The probability of collapsing into a particular state is the squared magnitude of its complex weight.

25. For some sets of weights, particularly after a transformation, the combined state cannot be separated into a tensor product of individual qubits. In this case, the individual qubits are entangled.

26. That is the next part of why quantum computation is powerful.

27. Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.

28. However that does not let you communicate.

29. The current limitations are that IBM does only a few qubits and that the operation is noisy.