We hope you’ve enjoyed tinkering with the computational problems we presented in this book! Before closing, we’ll briefly introduce a few subjects we didn’t have space to go into previously, and provide pointers on where to go to learn more about these and other topics in quantum computing. We won’t go into too much depth here; the aim here is rather to link what you’ve learned so far to material reaching beyond the scope of this book. Let the intuition you’ve built here be only the first step in your exploration of quantum programming!

The $|x\rangle $ notation that we use throughout the book to refer to states in a quantum register is called *bra-ket notation* or, sometimes *Dirac notation*, in honor of the 20th-century physicist of the same name. Throughout the quantum computing literature, this—rather than circle notation—is the notation used to represent quantum states. In Chapter 2 we hinted at the equivalence between these two notations, but it’s worth saying a little more to set you on your way. A general superposition within a single-qubit register can be expressed in Dirac notation as $\alpha |0\rangle +\beta |1\rangle $, where $\alpha $ and $\beta $ are the states’ *amplitudes*, represented as complex numbers satisfying the equation ${\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}=1$. The magnitude and relative phase of each value in the circle notation we’ve been using are given by the *modulus* and *argument* of the complex numbers $\alpha $

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