Note that this gives us a finite generating set of gates. We need to add some additional gate outside the Clifford group to allow universal quantum computation a single gate will suffice, such as the single-qubit π/8 phase rotation diag 1, exp i π / 4. Unfortunately, the Clifford group by itself does not have much computational power: it can be efficiently simulated on a classical computer. Thus, for the 7-qubit code, the full logical Clifford group is accessible via transversal operations. Furthermore, for the 7-qubit code, transversal Hadamard performs a logical Hadamard, and the transversal π/4 rotation performs a logical − π/4 rotation. A particularly useful fact is that a transversal CNOT gate (i.e., CNOT acting between the ith qubit of one block of the QECC and the ith qubit of a second block for all i) acts as a logical CNOT gate on the encoded qubits for any CSS code. Some stabilizer codes have interesting symmetries under the action of certain Clifford group elements, and these symmetries result in transversal gate operations. The set of stabilizer codes is exactly the set of codes which can be created by a Clifford group encoder circuit using |0〉 ancilla states. Stabilizer codes have a special relationship to a finite subgroup C n of the unitary group U(2 n) frequently called the “Clifford group.” The Clifford group on n qubits is defined as the set of unitary operations which conjugate the Pauli group P n into itself C n can be generated by the Hadamard transform, the controlled-NOT (CNOT), and the single-qubit π/4 phase rotation diag(1, i). On a stabilizer code, therefore, logical Pauli operations can be performed via a transversal Pauli operation on the physical qubits. In particular, each coset S ⊥/ S corresponds to a different logical Pauli operator (with S itself corresponding to the identity). Indeed, the set S ⊥\ S of undetectable errors is a boon in this case, as it allows us to perform these gates. The Pauli group P k, however, can be performed transversally on any stabilizer code. Universal fault tolerance is known to be possible for any stabilizer code, but in most cases the more complicated type of construction is needed for all but a few gates. Gottesman, in Encyclopedia of Mathematical Physics, 2006 Fault-Tolerant Gates Finally, for a better understanding of the material presented in this chapter, a set of problems has been provided in the next section.ĭ. Entanglement-assisted codes were introduced in Section 9.10. Section 9.9 was devoted to surface codes. An important class of quantum codes, namely topological codes, was discussed in Section 9.8. In the same section, efficient encoding and decoding of subsystem codes was discussed as well. Subsystem codes were introduced in Section 9.7. Section 9.6 was devoted to nonbinary stabilizer codes, which generalizes the previous sections. The standard form is the basic representation for the efficient encoder and decoder implementations, which were discussed in Section 9.5. This representation was used in Section 9.4 to introduce the so-called standard form of stabilizer code. In Section 9.3, finite geometry interpretation of stabilizer codes was introduced. Their basic properties were discussed in Section 9.2 The encoded operations were introduced in the same section ( Section 9.2). The stabilizer codes were introduced in Section 9.1. This chapter was devoted to stabilizer codes and their relatives. Djordjevic, in Quantum Information Processing, Quantum Computing, and Quantum Error Correction (Second Edition), 2021 9.11 Summary
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