Third, the qubits of the quantum register are individually measured, wlog in the computational basis. Second, a sequence of unitary operations chosen from a fixed set, the so-called quantum gates, are applied to the quantum register. First, the quantum register is initialized in a fiducial state, e.g. Now, a quantum circuit proceeds in three stages. Specifically, for quantum circuits on pure states, substantial amounts of entanglement must necessarily be present in a quantum computation at some point in the evolution for otherwise, the quantum computation cannot yield a significant speedup over classical computation . Entanglement is a prominent feature that sets apart quantum from classical systems, and is intimately related to the quantum speedup. Such generic states are called ‘entangled’. As a simple counting argument shows, most of the states are not of product form | Ψ〉≠| ψ 1〉 1⊗| ψ 2〉 2⊗⋯⊗| ψ n〉 n. The dimension of this Hilbert space is rapidly growing with the number n of qubits. The state | Ψ〉 of the quantum register lives in a tensor product Hilbert space, constructed from the Hilbert spaces of the individual qubits, ( n factors). We make some suggestions for further reading in §5 and conclude in §6.Ģ.1The qubit exemplifies the unification of information theory with quantum mechanics namely, it inherits the two distinct basis states |0〉,|1〉 from the classical bit and the superposition principle from quantum mechanics.Ī number n of qubits stacked together form a quantum register, the logical structure on which quantum operations are performed in a quantum circuit. Section 4 moves from quantum error correction to fault-tolerant quantum computation. The phenomenon of error discretization is discussed. ![]() In §3, a class of quantum codes, the stabilizer codes , are developed from classical codes. In §2, we briefly review the circuit model of quantum computation and simple error channels. For example, it has been shown that long-range-correlated noise in space or time is not necessarily an obstacle to fault tolerance . In the years since a threshold theorem was first proved, its assumptions have been considerably relaxed. We hope that this degree of simplification is suited for the purpose of this study, which is to illustrate main ideas and concepts of quantum error correction without too much technicality. While they capture essential features of noise in a quantum-mechanical setting, the assumptions that go into them-such as locality and stochastic independence of error events-are not obeyed exactly in any physical system of interest. ![]() It shall be pointed out from the beginning that the decoherence models discussed in this study are phenomenological and thus overly simplistic. Woven together, and supplemented by further techniques, they provide the ground on which the celebrated threshold theorem can then be established. These are quantum codes, error discretization and transversal encoded gates. The purpose of this article was to explain three key ideas upon which the theory of fault-tolerant quantum computation is built. Put simply, a gate error below the threshold is effectively as good as no error at all. This is the content of the threshold theorem of fault-tolerant quantum computation , which is the capstone of the theory of quantum error correction. If the effect of decoherence per operation is below a certain limit-the error threshold-then arbitrarily long and accurate quantum computation is possible. All that is needed for quantum computation are gate operations of a sufficiently high but fixed quality. But does not the extra power harnessed in quantum mechanical ways of information processing rely on the assumption that quantum operations could be performed with perfect accuracy? Would not the slightest amount of decoherence ruin any quantum computation if it only takes long enough? ‘No’ says the theory of quantum fault tolerance . Assuming, of course, that a quantum computer could actually be built. The discovery of an efficient quantum algorithm to break the widespread Rivest–Shamir–Adleman cryptosystem showed that quantum computing has enormous potential for solving hard computational problems.
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