# Solutions: Quantum Computation and Quantum Information by Nielsen and Chuang

## Chapter 1

### 1.1

$$
\begin{eqnarray}
\frac{\binom{\frac{2^n}{2}}{2}}{\binom{2^n}{2}}\times2 &=& \frac{2^{n-1}}{2^n} \times 2 \\\

&=&\frac{2^{n-1}\times\left(2^{n-1}-1\right)\times\frac{1}{2}}{2^{n-1}\times\left(2^n-1\right)}\times 2 \\\

&=&\frac{2^{n-1}-1}{2^n-1}<\frac{1}{2}
\end{eqnarray}
$$
A probabilistic classical computer can solve Deutsch’s problem with two evaluations with some probability of error $<\frac{1}{2}$.

### 1.2

If states are distinguishable, you can determine which state you send and then, employing an appropreate designer of Hamiltonian, build a second system in the same state.

Conversely, if you can prepare many identical copies of a qubit, then it is possible to measure the mean value of noncommuting observables.