exercises.md

TKET Exercises

1. Build a circuit to prepare the 3 qubit GHZ state $| \psi \rangle_{GHZ}$. Verify your circuit with Circuit.get_statevector() or AerStateBackend .

$$ \begin{equation*} | \psi \rangle^{(3)}_{GHZ} = \frac{1}{\sqrt{2}} \big( |000\rangle + |111\rangle \big) \end{equation*} $$

Can you define a function build_ghz_circ which builds a circuit to prepare an $n$ qubit GHZ state?

$$ \begin{equation*} | \psi \rangle^{(n)}_{GHZ} = \frac{1}{\sqrt{2}} \big( |0\rangle^{\otimes n} + |1\rangle^{\otimes n} \big) \end{equation*} $$

2. Take a look at the Quantum Fourier transform as implemented in the intro notebook. Can you define a function build_qft_circ(n_qubits) that builds a circuit for the Quantum Fourier Transform for $n$ qubits?

(a) Define a custom rebase to the ${\text{H}, \text{Rx}, \text{CU1}}$ gateset. See This page for the gate definitions. Test out this rebase pass on the circuits you have made so far.

(b) Define a SequencePass which optimises a Circuit using FullPeepholeOptimise before applying the rebase defined above. Test out this rebase pass on the circuits you have made so far.

4. Shown below is a circuit that performs the SWAP test. This circuit can be used to calculate the inner product between the states prepared in the registers $p$ and $q$. In the circuit the registers $p$ and $q$ contain 1 qubit however this test works for registers of any size.

If we measure qubit $a$ the probability measuring the state $| 0 \rangle$ state is related to the value of the inner product as follows

$$ \begin{equation} P \big(a=|0\rangle \big) = \frac{1}{2}\big( 1 + |\langle p | q\rangle|^{2} \big) \end{equation} $$

Use the AerBackend to calculate the inner product $\langle p | q\rangle$ of the states $|p \rangle$ and $|q \rangle$ (defined below). How does your answer change when the number of shots is increased?

$$ \begin{align} |p \rangle &= \frac{1}{\sqrt{2}} \big( |00\rangle - i |11\rangle \big) \\ |q \rangle &= \frac{1}{\sqrt{2}} \big( |01\rangle - |10\rangle \big) \end{align} $$

Note that here $|p \rangle$ and $|q \rangle$ are two qubit states so you will need to perform the swap test on a pair of 2 qubit registers. Therefore the circuit will involve 5 qubits in total if we add the single measurement qubit.

5. Write a pytket CustomPass that iterates through a Circuit and eliminates parameterised gates with small angles. Can you make a method where we can set a threshold for a gate to be eliminated?

Lengthier exercise (not part of the workshop)

6. (BONUS) Write your Backend that can run simulations of a pytket Circuit for a gateset of your choice. You can use this notebook tutorial as a guide.