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# Quantum Error Correcting Codes

## Contents

J. So a single qubit can not be repeated three times as in the previous example, as any measurement of the qubit will change its wave function. The simplest way is to store the information multiple times, and—if these copies are later found to disagree—just take a majority vote; e.g. 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. http://caribtechsxm.com/quantum-error/quantum-computation-quantum-error-correcting-codes-and-information-theory.php

The procedure is transversal, so an error on a single qubit in the initial cat state or in a single gate during the interaction will only produce one error in the 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 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 By using this site, you agree to the Terms of Use and Privacy Policy. https://en.wikipedia.org/wiki/Quantum_error_correction

## Quantum Error Correction For Beginners

G. First we create special encoded ancilla states in a non-fault-tolerant way, but perform some sort of check on them (in addition to error correction) to make sure they are not too Mathematics of quantum computation, 287–320, Comput. Zurek, T.

Reed, L. Ozeri and D. 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 Quantum Error Correction Book Classical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state.

Generated Tue, 25 Oct 2016 00:18:59 GMT by s_wx1085 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection Stabilizer Codes And Quantum Error Correction. The first demonstration was with NMR qubits.[4] Subsequently, demonstrations have been made with linear optics,[5] trapped ions,[6][7] and superconducting (transmon) qubits.[8] Other error correcting codes have also been implemented, such as If U = σ x {\displaystyle U=\sigma _{x}} , a bit flip error occurs. https://arxiv.org/abs/quant-ph/9512032 T.

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. Quantum Code 7 D. But it is possible to spread the information of one qubit onto a highly entangled state of several (physical) qubits. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits.

## Stabilizer Codes And Quantum Error Correction.

Because the ancillas in Steane and Knill error correction are more complicated than the cat state, it is especially important to verify the ancillas before using them. visit The encoded $\left|\overline{0}\right\rangle$ for this code consists of the superposition of all even-weight classical codewords and the encoded $\left|\overline{1}\right\rangle$ is the superposition of all odd-weight classical codewords. Quantum Error Correction For Beginners The signs of states in a quantum superposition are important, so we need to be able to correct sign errors as well as bit flip errors. 5 Qubit Code The theory of fault-tolerant quantum computation tells us how to perform operations on states encoded in a quantum error-correcting code without compromising the code's ability to protect against errors.

Please try the request again. navigate to this website Then the Z generators from H1 will all commute with the X generators from H2 iff C2 ⊥  ⊆ C1 (or equivalently, C1 ⊥  ⊆ C2). 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 Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006]. Steane Code

E. 5-qubit Quantum Error Correction Suppose that the state of qubit 8 at time 5 has implications for the states of both qubit 8 and qubit 11 at time 6. B.

## Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors.

They are less general than arbitrary quantum codes, but have a number of useful properties that make them easier to work with than the general QECC. Blakestad, J. Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one. Bit Flip Memory Error Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred.

Comput. Fault-Tolerant Gates We will focus on stabilizer codes. Andrew Steane found a code which does the same with 7 instead of 9 qubits, see Steane code. http://caribtechsxm.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php In most codes, the effect is either a bit flip, or a sign (of the phase) flip, or both (corresponding to the Pauli matrices X, Z, and Y).

The computational complexity of the encoder is frequently a great deal lower than that of the decoder. In the case where M is a multi-qubit Pauli operator, this can be broken down into a sequence of controlled-X, controlled-Y, and controlled-Z operations. Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits. The decoder is sometimes also taken to map Hn into an unencoded Hilbert space HlogK isomorphic to C.

Fault-Tolerance Given a QECC, we can attempt to supplement it with protocols for performing fault-tolerant operations. It is possible to correct for both types of errors using one code, and the Shor code does just that. 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. Thus, it is sufficient in general to check that the error-correction conditions hold for a basis of errors.

Rev. Each measurement gives us one bit of the error syndrome, which we then decipher classically to determine the actual error. But we didn’t have constructive examples of getting here. For instance, a standard bound on classical error correction is the Hamming bound (or sphere-packing bound), but the analogous quantum Hamming bound k/n ≤ 1 − (t/n)log3 − h(t/n) for [[n, k, 2t + 1]] codes (when n is large) is

The ideal quantum error correction code would correct any errors in quantum data, and it would require measurement of only a few quantum bits, or qubits, at a time. The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation. A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. 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

R. Then S encodes k qubits and has distance d, where d is the smallest weight of an operator in S ⊥  \ S. Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip. This is just the simplest quantum code.

Those types of measurements, in a real system, can be very hard to do. Using the stabilizer formalism limits the available states but there is still a lot of interesting freedom. voting system? Back to Daniel Gottesman's home page - Quantum error correction sonnet February 25, 1999 Massachusetts Institute of Technology News Video Social Follow MIT MIT News RSS Follow MIT on Twitter Follow