![]() A mathematical model of computation is an idealized abstraction. Mayers, in Advances in Cryptography: Proceedings of Crypto’96. Quantum Error Correction An Introduction to Quantum Computing Oxford Academic Abstract. ![]() Brassard, in Proceeding of IEEE International Conference on Computers, Systems and Signal Processing, (1984) pp. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear. Renner, Security of Quantum Key Distribution PhD thesis, ETH Zurich (2005), arXiv:quant-ph/0512258. Hayashi, A Group Theoretic Approach to Quantum Information (Kyoritsu-shuppan, Tokyo, 2014)ī.W. ![]() Robertson, The Theory of Groups and Quantum Mechanics (1931), reprinted by Dover (1950)) The spaces of multi-qubit states used to encode these logical states are called quantum error-correcting codes, and their ability to correct errors is. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. It is well known 22, 23, 24 that the probability of cor-rectly distinguishing a pair of quantum states is related to the distance induced by the trace-norm, kAk. Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. Hirzel, Leipzig 1928) (English translation by H. marily focus on encoding of states, which is the key for quantum information protection and correction in the Schroedinger’s picture. There are three steps for achieving this goal: (1) realising encoded logical qubits in a code capable of detecting and correcting errors, (2) realising operations on encoded qubits and error-correction cycles and (3) adding more ancillary qubits and improving the operation fidelity to achieve fault tolerance. Weyl, in Gruppentheorie und Quantenmechanik (Verlag von S. Gallager, in Information Theory and Reliable Communication (Wiley, New York, 1968) Weaver, in The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949) The same language, he said, ought to be applicable, in my opinion, to more general situations in particular, to a de Sitter universe like ours. We introduce stabilizer codes and their connection to classical. Chuang, in Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)Ĭ.E. Quantum error correction gives us a more general way of thinking about geometry in this code language, said Preskill, the Caltech physicist. The basic structure of a quantum code is laid out, and how errors are detected and corrected. We show that many previous ideas, such as the existence of a large number of "null states", a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, "state-dependent/state-specific" operator reconstruction, and the "simple entropy" approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.M.A. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of "non-isometric codes protected by computational complexity". inner-product preserving) encoding of the former into the latter. Scalable quantum computers require a far-reaching theory of fault-tolerant quantum computation. The encoder (defini- tion 1.1.2) is represented by an isometric quantum channel, mapping the logical states to states supported on a subspace of the physical. Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. To achieve large scale quantum computers and communication networks it is essential not only to overcome noise in stored quantum information, but also in general faulty quantum operations. O 19.2 CSS codes Restricting attention to isometric encoding motivates the notion of a quantum error - correcting code, which is simply a subspace.
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