By Keisuke Fujii

This ebook provides a self-consistent assessment of quantum computation with topological quantum codes. The publication covers every little thing required to appreciate topological fault-tolerant quantum computation, starting from the definition of the skin code to topological quantum errors correction and topological fault-tolerant operations. The underlying uncomplicated options and robust instruments, equivalent to common quantum computation, quantum algorithms, stabilizer formalism, and measurement-based quantum computation, also are brought in a self-consistent approach. The interdisciplinary fields among quantum info and different fields of physics similar to condensed subject physics and statistical physics also are explored by way of the topological quantum codes. This ebook therefore offers the 1st entire description of the total photograph of topological quantum codes and quantum computation with them.

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**Additional resources for Quantum Computation with Topological Codes: From Qubit to Topological Fault-Tolerance**

**Sample text**

The logical |T state is measured in the X-basis transversally, which detects Z errors on the code state, projecting the code state on the local X-basis. The input state in the second lowest wire is entangled with the ancilla qubit in the lowest wire, where the distilled magic state is teleported. Let c be a 15-bit string specifying the location of the Z errors, E(c) ≡ i Zi(c)i .

Zi−2 Yi−1 Zi+1 , Zi−1 Yi+1 Zi+2 , . . , Yi . 26) By performing the phase gates S on the (i − 1)th and (i + 1)th qubits, we obtain a new stabilizer group . . , Zi−2 Xi−1 Zi+1 , Zi−1 Xi+1 Zi+2 , . . , Yi . 5 Graph States 35 Suppose three neighboring qubits (i−1), i, and (i+1) are measured in the Y -basis. This is equivalent to measuring the ith qubit in the Y -basis, and then measuring the (i − 1)th and (i + 1)th qubits of the post-measurement graph state in the X-basis, because there is a phase operation S acting on them as a product.

Thus the approximation of the Jones polynomial is a BQP-complete problem [15, 16]. The AJL algorithm for approximation of the Jone polynomial was extended for the Tutte polynomial in Ref. [18], where the Solovay–Kitaev algorithm for non-unitary linear operators was developed. 6 Quantum Noise Quantum coherence, one of the essential properties of quantum systems, is quite fragile against noise, due to interactions between the system and the environment. Suppose that the system S of interest interacts via a unitary operation U with the environment E, where the system and environment are initially uncorrelated.

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