By Mikio Nakahara, Robabeh Rahimi, Akira Saitoh

ISBN-10: 9812814477

ISBN-13: 9789812814470

This ebook offers a finished evaluate of the mathematical features of quantum computing. it is going to be important for graduate scholars and researchers attracted to quantum computing from diversified parts of physics, arithmetic, informatics and machine technology. The lecture notes during this quantity are written in a self-contained sort, and for that reason are obtainable for graduate scholars and researchers with even much less history within the issues.

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**Sample text**

This method cannot be applicable to qubits, however, due to no-cloning theorem. We have to somehow think out the way to overcome this theorem. 2. Three-qubit bit-flip code: the simplest example It is instructive to introduce a simple example of quantum error correcting codes (QECC). We closely follow Steane30 here. 1. Bit-ﬂip QECC Suppose Alice wants to send a qubit or a series of qubits to Bob through a noisy quantum channel. Let |ψ = a|0 +b|1 be the state she wants to send. If she is to transmit a serise of qubits, she sends them one by one and the following argument applies to each of the qubits.

7. 1. Introduction It has been shown in the previous chapter that interactions between a quantum system with environment cause undesirable changes in the state of the quantum system. In the case of qubits, they appear as bit-ﬂip and phaseﬂip errors, for example. To reduce such errors, we must implement some sort of error correcting mechanism in the algorithm. Before we introduce quantum error correcting codes, we have a brief look at the simplest version of error correcting code in classical bits.

Although a qubit may take inﬁnitely many diﬀerent states, it should be kept in mind that we can extract from it as the same amount of information as that of a classical bit. Information can be extracted only through measurements. When we measure a qubit, the state vector ‘collapses’ to the eigenvector that corresponds to the eigenvalue observed. Suppose a spin is in the state a|0 + b|1 . If we observe that the z-component of the spin is +1/2, the system immediately after the measurement is in |0 .

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