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# Quantum Error Codes

## Contents

By using this site, you agree to the Terms of Use and Privacy Policy. 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 Reed, L. To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators: P 0 = | 000 ⟩ ⟨ 000 | + | weblink

To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators: P 0 = | 000 ⟩ ⟨ 000 | + | Classical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. 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. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

It’s possible to determine whether the first and second qubit have the same value, and whether the second and third qubit have the same value, without determining what that value is. In fact, his error-correction code was a response to skepticism about the feasibility of implementing his factoring algorithm. Definition 3 Let S ⊂ Pn be an Abelian subgroup of the Pauli group that does not contain  − 1 or  ± i, and let C(S) = {∣ψ⟩ s.t.

• Sloane ([2], [3]); these are also called additive codes.
• If we are instead interested in erasure errors, when the location of the error is known but not its precise nature, a distance d code corrects d − 1 erasure errors.
• Once you perform a measurement on the qubits, however, the superposition collapses, and the qubits take on definite values.
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• The syndrome measurement "forces" the qubit to "decide" for a certain specific "Pauli error" to "have happened", and the syndrome tells us which, so that we can let the same Pauli
• Then by comparing qubits within blocks of three, we can detect bit flip errors, and by comparing the signs of the three blocks, we can detect sign errors.
• The theory of quantum error-correcting codes has been developed to counteract noise introduced in this way.
• R.

We therefore have to resort to more complicated techniques. Knill, R. Therefore, if p is below the threshold pt, we can achieve an arbitrarily good error rate ε per logical gate or timestep using only poly(logε) resources, which is excellent theoretical scaling. Quantum Error Correction Book Nevertheless, in quantum computing there is another method, namely the three qubit bit flip code.

Girvin and R. Stabilizer Codes And Quantum Error Correction. One of the central problems in the theory of quantum error correction is to find codes which maximize the ratios (logK)/n and d/n, so they can encode as many qubits as Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing". https://arxiv.org/abs/quant-ph/9512032 R is called the recovery or decoding operation and serves to actually perform the correction of the state.

Note that the Gilbert-Varshamov bound simply states that codes at least this good exist; it does not suggest that better codes cannot exist. Quantum Code 7 Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Andersen, Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code , Nature Photonics 4 10 (2010)(this document online) External links Prospects Error-check breakthrough in quantum computing[permanent dead Therefore, given the ability to perform fault-tolerant Clifford group operations, fault-tolerant measurements, and to prepare the encoded ∣ψπ/8⟩ state, we have universal fault-tolerant quantum computation.

## Stabilizer Codes And Quantum Error Correction.

Monz, V. Optimized simulations of fault-tolerant protocols suggest the true threshold may be as high as 5%, but to tolerate this much error, existing protocols require enormous overhead, perhaps increasing the number of Quantum Error Correction For Beginners Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed. 5 Qubit Code A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors.

Nigg, L. have a peek at these guys M. Experimental realization There have been several experimental realizations of CSS-based codes. That these codes allow indeed for quantum computations of arbitrary length is the content of the threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct Steane Code

Through the transmission in a channel the relative sign between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } can become inverted. If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. Stabilizer codes have a special relationship to a finite subgroup Cn of the unitary group U(2n) frequently called the Clifford group. check over here S.

What is more, the outcome of this operation (the syndrome) tells us not only which physical qubit was affected, but also, in which of several possible ways it was affected. 5-qubit Quantum Error Correction Somaroo, "Experimental Quantum Error Correction," Phys. Unfortunately, the practical requirements for this result are not nearly so good.

## Unfortunately, it does not appear to be possible to perform universal quantum computations using just transversal gates.

Contents 1 The bit flip code 2 The sign flip code 3 The Shor code 4 General codes 5 Models 6 Experimental realization 7 See also 8 References 9 Bibliography 10 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. The phenomenon of degeneracy has no analogue for classical error correcting codes, and makes the study of quantum codes substantially more difficult than the study of classical error correction. Bit Flip Memory Error In particular, the task of determining what error has occured can be computationally difficult (NP-hard, in fact), and designing codes with efficient decoding algorithms is an important task in quantum correction

The code will be able to correct bit flip (X) errors as if it had a distance d1 and to correct phase (Z) errors as if it had a distance d2. We can imagine the various possible errors taking the subspace C into other subspaces of Hn, and we want those subspaces to be isomorphic to C, and to be distinguishable from This three qubits bit flip code can correct one error if at most one bit-flip-error occurred in the channel. this content The system returned: (22) Invalid argument The remote host or network may be down.

If U = σ z {\displaystyle U=\sigma _{z}} , a sign flip error occurs. Another useful representation is to map the single-qubit Pauli operators I, X, Y, Z to the finite field GF(4), which sets up a connection between stabilizer codes and a subset of Chwalla, M. Category:Introductory Tutorials Category:Quantum Error Correction Category:Handbook of Quantum InformationLast modified:Monday, October 26, 2015 - 17:56 Massachusetts Institute of Technology News Video Social Follow MIT MIT News RSS Follow MIT on Twitter

It may be possible to implement the researchers’ scheme without actually duplicating banks of qubits. L. The primary difference between a quantum state and a classical state is that a quantum state can be in a superposition of multiple different classical states. Wineland, "Realization of quantum error correction," Nature 432, 602-605 (2004), doi:10.1038/nature03074 ^ P.

This process is known as decoherence. Math. 1 (2001), no. 3, 325–332. In that case, let us consider tensor products of the Pauli matrices $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}, X=\begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ Define the Pauli group Pn as the group consisting of tensor products of I, X, Cerf, Ulrik L.

All of them share some basic features: they involve creation and verification of specialized ancilla states, and use transversal gates which interact the data block with the ancilla state. Price, W. Shor is also responsible for the theoretical result that put quantum computing on the map, an algorithm that would enable a quantum computer to factor large numbers exponentially faster than a D.

However, in a quantum channel, it is no longer possible, due to the no-cloning theorem, which forbids the creation of identical copies of an arbitrary unknown quantum state. Then the bit flip code from above can recover | ψ ⟩ {\displaystyle |\psi \rangle } by transforming into the Hadamard basis before and after transmission through E phase {\displaystyle E_{\text{phase}}} Zurek, T. 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⟩.

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