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# Quantum Error Codes

## Contents

A 71, 052332 (2005), doi:10.1103/PhysRevA.71.052332 ^ J. Zurek, T. Quantum circuit of the bit flip code Let | ψ ⟩ = α 0 | 0 ⟩ + α 1 | 1 ⟩ {\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } If errors occur on the physical qubits independently at random with probability p per gate or timestep, the fault-tolerant protocol has probability of logical error for a single logical gate or have a peek at this web-site

the equation ⟨ψi∣E∣ψj⟩ = C(E)δij fails. A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer. Then C(S) is a stabilizer code and S is its stabilizer. Sloane ([2], [3]); these are also called additive codes.

## Quantum Error Correction For Beginners

Cerf, Ulrik L. If an error is detected, the protocol can trace it back to its origin and correct it. A number of different techniques have been developed. Knill, R.

The solution to the problem is to use a quantum error-correcting code. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. 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 5 Qubit Code Leuchs, N.

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 Stabilizer Codes And Quantum Error Correction. Blakestad, J. Nevertheless, in quantum computing there is another method, namely the three qubit bit flip code. http://arxiv.org/abs/quant-ph/9602019 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

For instance, a qubit in the state | − ⟩ = ( | 0 ⟩ − | 1 ⟩ ) / 2 {\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}} may have its Fault Tolerant Quantum Computation Instead of the unencoded ∣ + ⟩ state, we must use a more complex ancilla state ∣00…0⟩ + ∣11…1⟩ known as a 'cat' state. A slight generalization of the fault-tolerant measurement procedure below can be used to fault-tolerantly verify the ∣ψπ/8⟩ state, which is a  + 1 eigenstate of PX. Hennrich and R.

## Stabilizer Codes And Quantum Error Correction.

Thus, the distance of the quantum code is at least min(d1, d2), but might be higher because of the possibility of degeneracy. https://quantiki.org/wiki/quantum-error-correction-and-fault-tolerance-0 Sabuncu, A. Quantum Error Correction For Beginners Cerf and U. Steane Code Note that the error syndrome does not tell us anything about the encoded state, only about the error that has occurred.

The bits are in some state; then they’re modified, so that they assume another state; then they’re modified again; and so on. Check This Out Furthermore, these calculations make a number of assumptions about the physical properties of the computer. Cerf, Ulrik L. 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. Quantum Error Correction Book

The most widely-used structure gives a class of codes known as stabilizer codes. So one of the highlights of this construction is that they actually got that down a lot.” “People had all of these examples of codes that were pretty bad, limited by Using this code, we can correct an arbitrary single-qubit quantum error. Source Math. 1 (2001), no. 3, 325–332.

Those types of measurements, in a real system, can be very hard to do. Quantum Code 7 Definition 2 The distance d of an ((n, K)) is the smallest weight of a nontrivial Pauli operator E ∈ Pn s.t. Particular caution is necessary, as computational gates can cause errors to propagate from their original location onto qubits that were previously correct.

## Let E phase {\displaystyle E_{\text{phase}}} be a quantum channel that can cause at most one phase flip.

Your cache administrator is webmaster. 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. This is the reason the world at a human scale looks classical - big objects are very likely to interact at least a little bit with their environment, so they are Bit Flip Memory Error S ⊥  is the set of Pauli operators that commute with all elements of the stabilizer.

G. Taylor & Francis. Chiaverini, D. have a peek here More sophisticated techniques for fault-tolerant error correction involve less interaction with the data but at the cost of more complicated ancilla states.

Specifically: Definition 1 Let Hn be a 2n-dimensional Hilbert space (n qubits), and let C be a K-dimensional subspace of Hn. Fault-Tolerant Measurement and Error Correction Since all our gates are unreliable, including those used to correct errors, we will need some sort of fault-tolerant quantum error correction procedure. 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 weight wt(P) of a Pauli operator P ∈ Pn is the number of qubits on which it acts as X, Y, or Z (i.e., not as the identity).

Somaroo, "Experimental Quantum Error Correction," Phys. 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. Furthermore, for the 7-qubit code, transversal Hadamard performs a logical Hadamard, and the transversal π/4 rotation performs a logical  − π/4 rotation. Through the transmission in a channel the relative sign between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } can become inverted.

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. Define the dual C ⊥  of a classical code C as the set of vectors w s.t.