Abstract
The chapter introduces the core ideas of quantum computing and quantum information theory. Most importantly, the concept of the qubit as the basic indivisible unit of quantum information is covered in detail, along with the circuit approach to modeling quantum computations. The chapter introduces the most common quantum gates—both single- and multi-qubit ones—and draws parallels to the quantum mechanical concepts covered earlier in the book.
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Notes
- 1.
Some quantum computing literature refers to classical bits as cbits, to emphasize their classical nature. We shall not do that in this book though—we will assume that a term bit always refers to the classical concept.
- 2.
It is sometimes said that a qubit in a superposition “is both 0 and 1 at the same time”. This is a rather simplistic and inaccurate description, but one that is commonly used in popular science articles, as it manages to convey the weirdness of conceptualizing quantum states. As we discussed earlier, it would be more appropriate to say that a qubit is neither 0 and 1, and only acquires a definite basis state upon measurement using a specific observable.
- 3.
An alternative name often used in literature is the Hadamard basis, named like this after a French mathematician, Jacques Hadamard
- 4.
We are explicitly ignoring the topic of quantum hardware errors here.
- 5.
- 6.
- 7.
The reader may point out that we already achieved that when measuring a default in the X-basis in the previous section. This is a correct observation—however it is the computational basis that is really the standard way of expressing quantum algorithms, and indeed in this particular example a measurement is a standard Z-basis measurement.
- 8.
We are obviously taking a simplified approach here. Quantum random number generation is a major pillar of cryptography and a deeply complex field. Readers willing to dive deeper into the related challenges and statistical consequences are referred to specialized studies in the topic such as [3].
- 9.
When considering the basis states only, quantum computer behaves like a classical computer.
- 10.
While the \(\mathbf {S}\) gate can be created from two \(\mathbf {T}\) gates, Nielsen and Chuang include the \(\mathbf {S}\) gate in their universal set due to its important role in quantum error correction protocols. This is, however, beyond the scope of this book.
References
Bernhardt, C. (2019). Quantum computing for everyone. MIT Press.
DiVincenzo, D. P., & IBM. (2000). The physical implementation of quantum computation. Protein Science, 48, 771–783.
Kollmitzer, C., Schauer, S., Rass, S., & Rainer, B. (2020). Quantum random number generation. Springer Nature Switzerland.
Nielsen, M. A., & Chuang, I. (2010). Quantum computation and quantum information: 10th anniversary edition. Cambridge University Press.
Rieffel, E. G., & Polak, W. H. (2014). Quantum computing: A gentle introduction. MIT Press.
Sutor, R. S. (2019). Dancing with Qubits: How quantum computing works and how it can change the world. Packt Publishing.
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Wojcieszyn, F. (2022). Quantum Computing. In: Introduction to Quantum Computing with Q# and QDK. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-99379-5_4
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DOI: https://doi.org/10.1007/978-3-030-99379-5_4
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