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# Quantum Computation Quantum Error Correcting Codes And Information Theory

## Contents

Hennrich and R. Ozeri and D. Nielsen, and B. Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip. weblink

A. R is called the recovery or decoding operation and serves to actually perform the correction of the state. A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer. The best practical quantum LDPC codes? http://bookstore.ams.org/tifr-7

## Quantum Error Correction Codes

References jQuantum development site Direct start of jQuantum from the web A local copy of jQuantum software Lecture notes and textbooks Preskill's notes on quantum computation. Using this or another verification procedure, we can check a non-fault-tolerant construction. If an error is modeled by a unitary transform U, which will act on a qubit | ψ ⟩ {\displaystyle |\psi \rangle } , then U {\displaystyle U} can be described However, in order to have a stabilizer code at all, the generators produced by the above procedure must commute.

1. We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error.
2. The salient point in these error-correction conditions is that the matrix element Cab does not depend on the encoded basis states i and j, which roughly speaking indicates that neither the
3. D.
4. Todd Brun, Igor Devetak, and Min-Hsiu Hsieh also constructed the entanglement-assisted stabilizer formalism as an extension of the standard stabilizer formalism that incorporates quantum entanglement shared between a sender and a
5. Shor, and N.

The most widely-used structure gives a class of codes known as stabilizer codes. A  + 1 eigenstate in the data therefore leaves us with ∣00…0⟩ + ∣11…1⟩ in the ancilla and a  − 1 eigenstate leaves us with ∣00…0⟩ + ∣11…1⟩. The task of protecting quantum information is more formidable and relies upon fascinating phenomena such as entanglement, which also enables communications with speeds greater than light, and suitable analogues of the 5 Qubit Code We use them to give a recovery operator independent definition of error-correcting codes.

For H2, we perform the same procedure, but each 1 is instead replaced by X. Quantum Error Correction For Beginners Please try the request again. Specifically: Definition 1 Let Hn be a 2n-dimensional Hilbert space (n qubits), and let C be a K-dimensional subspace of Hn. https://quantiki.org/wiki/quantum-error-correction-and-fault-tolerance-0 The decoder is sometimes also taken to map Hn into an unencoded Hilbert space HlogK isomorphic to C.

A. Surface Code The basic design principle of a fault-tolerant protocol is that an error in a single location --- either a faulty gate or noise on a quiescent qubit --- should not be Other approaches to quantum error correction Fault-tolerant quantum computation by anyons. Bernstein, G.

## Quantum Error Correction For Beginners

J. One of the first papers to suggest using LDPC codes, which have been very successful in non-quantum coding, for quantum error correction. Quantum Error Correction Codes The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation. Stabilizer Codes And Quantum Error Correction. Each measurement gives us one bit of the error syndrome, which we then decipher classically to determine the actual error.

Bennett, D. http://vealcine.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php A. Note that the square brackets specify that the code is a stabilizer code, and that the middle term k refers to the number of encoded qubits, and not the dimension 2k The syndrome measurement tells us as much as possible about the error that has happened, but nothing at all about the value that is stored in the logical qubit—as otherwise the Fault Tolerant Quantum Computation

Therefore, we must be careful and use some sort of technique to verify the cat state, for instance by checking if random pairs of qubits are the same. Knill, R. Extra reading: Arul Lakshminarayan's article on Borromean triangles and the GHZ state . check over here With the Shor code, a qubit state | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } will

Matthew Hastings is a theorist studying entanglement in quantum many-body systems. Quantum Code Reel Review So even if the error due to the noise was arbitrary, it can be expressed as a superposition of basis operations—the error basis (which is here given by the Pauli matrices The cat state contains as many qubits as the operator M to be measured, and we perform the controlled-X, -Y, or -Z operations transversally from the appropriate qubits of the cat

## Protecting quantum information from being lost is subject to the laws of both quantum physics and mathematics.

Vazirani, Strengths and weaknesses of quantum computing,'' quant-ph/9701001. Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct. The following circuit performs a π/8 rotation, given an ancilla state ∣ψπ/8⟩ = ∣0⟩ + exp(iπ/4)∣1⟩: Here P is the π/4 phase rotation diag(1, i), and X is the bit flip. Bit Flip Error Each row of the parity check matrix can be converted into a Pauli operator by replacing each 0 with an I operator and each 1 with a Z operator.

On the iterative decoding of sparse quantum codes. The Clifford group on n qubits is defined as the set of unitary operations which conjugate the Pauli group Pn into itself; Cn can be generated by the Hadamard transform, the Currently, he is also interested in speculative ideas such as analyzing the security of quantum key distribution in the presence of closed timelike curves and the effect of entanglement as a this content In general, a gate coupling pairs of qubits allows errors to spread in both directions across the coupling.

Nielsen, Quantum data processing and error correction,'' quant-ph/9604022. The solution is to use transversal gates whenever possible. A. Devabhaktuni, and J.

S. If U = σ z {\displaystyle U=\sigma _{z}} , a sign flip error occurs. We will start with these fundamental ideas, and proceed to study the subject up to its current state. Ben-Or, Fault-tolerant quantum computation with constant error,'' quant-ph/9611025.

His interests also include physics of information, quantum error correction, the transition from quantum to classical, as well as other subjects such as cosmology and dynamics of phase transformations. The conditions depend only on the behavior of the logical states. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely Horodecki, P.

re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—provided the error rate of individual quantum gates is below a certain threshold; as otherwise, the Girvin and R. For lower physical error rates, overhead requirements are more modest, particularly if we only attempt to optimize for calculations of a given size, but are still larger than one would like. Stabilizer Codes In order to better manipulate and discover quantum error-correcting codes, it is helpful to have a more detailed mathematical structure to work with.

For instance, P ∈ Pn can be represented by a pair of n-bit binary vectors (pX∣pZ) where pX is 1 for any location where P has an X or Y tensor factor and Prerequisites and Requirements Familiarity with Hilbert spaces or foundations of Quantum Mechanics. We get a similar result in the case where the noise is a general quantum operation on each qubit which differs from the identity by something of size O(ε). J.

Laflamme, and W.