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That these codes allow indeed for quantum computations of arbitrary length is the content of the quantum threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on ...
In this code, 5 physical qubits are used to encode the logical qubit. [2] With X {\displaystyle X} and Z {\displaystyle Z} being Pauli matrices and I {\displaystyle I} the Identity matrix , this code's generators are X Z Z X I , I X Z Z X , X I X Z Z , Z X I X Z {\displaystyle \langle XZZXI,IXZZX,XIXZZ,ZXIXZ\rangle } .
Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum logic is a fragment. Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an orthocomplemented lattice .
It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors.
Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space.
In addition to investigating fault tolerant quantum computation, the Eastin–Knill theorem is also useful for studying quantum gravity via the AdS/CFT correspondence and in condensed matter physics via quantum reference frame [2] or many-body theory. [3] The theorem is named after Bryan Eastin and Emanuel Knill, who published it in 2009. [1]
Physicist Leslie E. Ballentine gave the textbook a positive review, declaring it a good introduction to quantum foundations and ongoing research therein. [9] John C. Baez also gave the book a positive assessment, calling it "clear-headed" and finding that it contained "a lot of gems that I hadn't seen", such as the Wigner–Araki–Yanase theorem. [10]
Reviewer Corey S. Powell of The New York Times writes: . In the space of 221 dense, frequently thrilling and occasionally exasperating pages, ... tackles computer logic, thermodynamics, chaos theory, complexity, quantum mechanics, cosmology, consciousness, sex and the origin of life—throwing in, for good measure, a heartbreaking afterword that repaints the significance of all that has come ...