Nielsen and Chuang -- Chapter 4
(4.1) Find the points on the Bloch sphere which correspond to the normalized eigenvectors of the different Paul matrices.
Recall that, from Exercise 2.11,
Similarly,
Finally,
First, we solve for
So, for
Similarly, for the second eigenvector,
Therefore, for the first eigenvector,
And for the second we have,
Similarly, we find the Bloch vectors
(4.2) Let and be a matrix that satisfies . Show that
From the power series definition of
We can use exercise 4.2 to write the above equations more conveniently.
(4.3) Show that, up to a global phase, the gate satisfies .
Note that
Now, using the definition of
(4.4) Express the Hadamard gate as a product of and rotations and for some .
We’ve discussed a procedure for expressing
We know that
We’ll use that
Note that the above is simply showing that the anti-commutator of
Hence,
which gives the Hadamard transform with phase
(4.5) Prove that , and use this to verify the following equation
Evidently,
because
Therefore, using Exercise 4.2 (Nielsen & Chuang), if we let
(4.7) Show that and use this to prove that .
From above, we have that distinct Pauli matrices anti-commute. Furthermore, the Pauli matrices are hermitian and unitary
So,
using that
(4.12) Give , and for the Hadamard gate.
Using Lemma 4.12 (Nielsen & Chuang) above we can solve, assuming
So, the proof of Corollary 4.2 in Nielsen & Chuang tells us to set
and
(4.16)
For the first circuit, we consider action on the computational basis.
Now, given that
Similarly, for the second circuit we have
(4.17)
Construct a CNOT gate from one controlled-
and two Hadamard gates, specifying the control and target qubits.
Recall that, in terms of the computational basis, the action of the CNOT is given by
We construct our algorithm by first making the observation that
(1) If
(2) If
In summary, we have the circuit, beginning with state
(1) Apply
(2) Controlled-
(3) Apply
From the next exercise, we’ll see that it didn’t actually matter whether we used the first or second qubit as control/target:
(4.18)
We simply prove the statement for the computational basis.
(1)
(2)
(3)
(4)
(4.19) The CNOT gate is a simple permutation whose action on a density matrix is to rearrange the elements in the matrix. Write out this action explicitly in the computational basis.
Now,
Hence, the permutation matrix acting on the computational basis as
satisfies this permutation.
(4.20)
Consider action on
Similarly, for the circuit on the RHS, action on
Now, using that