Exercises are from QUANTUM COMPUTING: A GENTLE INTRODUCTION, by Eleanor Rieffel and Wolfgang Polak, published by The MIT Press.

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 12.1}$

Show that a single-qubit gate followed by a single-qubit error is equivalent to a (possibly different) single-qubit error followed by the same gate.

$\mathbf{Exercise\ 12.2}$

Why do we consider the preparation of the cat state in Figure 12.6 to be fault-tolerant even though it includes two $C_{not}$ gates from the qubit on which the $Z$ measurement is made.

$\mathbf{Exercise\ 12.3}$

What effect does applying $P_{\pi/4}$ to each of the qubits in the Steane seven qubit encoding have?

$\mathbf{Exercise\ 12.4}$

Show that the transversal circuit shown in Figure 12.8 does not implement the Toffoli gate. Consider the effect of the circuit on $\ket{\tilde 1}\otimes\ket{\tilde 0}\otimes\ket{\tilde 0}$.

$\mathbf{Exercise\ 12.5}$

Design a fault-tolerant version of the Toffoli gate for the Steane code.