Implementation of a simple operator-quantum-error-correction scheme

Abstract

It is often assumed that the ancilla qubits required for encoding a qubit in quantum error correction (QEC) have to be in pure states, |00 . . . 0 for example. In this letter, we introduce an encoding scheme avoiding fully correlated errors, in which the ancillae may be in a uniformly mixed state. We demonstrate our scheme experimentally by making use of a three-qubit NMR quantum computer. Moreover, the encoded state has an interesting nature in terms of Quantum Discord, or purely quantum correlations between the data-qubit and the ancillae.A quantum computer is vulnerable against environmental noise and it must be protected by one way or another. Quantum error correction (QEC) is one of the most successful approaches to this end [1]. Despite this great success, QEC requires expensive resources, or ancillae that are usually assumed to be in pure states [2,3]. However, it is not yet proved that ancillae in uniformly mixed states are useless. We extend previous works [4] and show an encoding scheme robust against fully correlated noise in which all the ancillae can be in uniformly mixed states. The encoded state has an interesting nature in terms of Quantum Discord [5], or purely quantum correlations between the data-qubit and the ancillae. Our QEC scheme also provides an example of Deterministic Quantum Computation with 1-Qubit (DQC-1) [6,7].Suppose we have a single qubit in an arbitrary state ρ 1 , which we want to protect from noise. We introduce some additional qubits (ancillae) in order to protect the first qubit and suppose that all the qubits suffer from the same noise. Such a noise is called fully correlated and it may happen when the dimensions of the quantum computer are microscopic compared with the wavelength of external disturbances. Noiseless subsystem (NS) [8][9][10][11] and decoherence free subsystem (DFS) [12][13][14][15] are well known strategies to protect a system from such fully correlated noises [16,17]. These schemes, however, require ancillae in pure states and thus they are expensive.In the following, we show that it is indeed possible to devise a cheaper QEC scheme employing ancillae in the uniformly mixed state. Letbe the state of the qubit to be protected. Here σ 0 is a unit matrix of dimension 2, n = (n x , n y , n z ) is the Bloch vector, and σ i is the ith component of the Pauli matrices. We introduce two ancillae in uniformly mixed states, whose Bloch vectors are 0. The initial state of the three-qubit system is thus a tensor product state ρ 1 ⊗ (σ 0 /2) ⊗2 . The unitary encoding operator U E transforms the tensor product state to an entangled stateρ 3 . If the state of the system is againρ 3 even after the action of noises, a unitary recovery operator, U R = U † E , transforms ρ 3 back to the initial tensor product state ρ 1 ⊗ (σ 0 /2) ⊗2 and ρ 1 can be recovered after tracing over the ancilla states.It is highly counterintuitive that a QEC scheme works with ancillae in uniformly mixed states. The trick is that the uniformly mixed state (σ 0 /2) ⊗2 is rewritten aswhere n 2 and n ′ ...

Error correction schemes for fully correlated quantum channels protecting both quantum and classical information

“…Norm[Transpose[Q [3]]. X [7] Similarly, we can check the formula We can also check that for all n ≥ 3, P † n E(P n (σ ⊗ ρ)P † n )P n −σ ⊗ ρ = 0.…”

“…So, for such a fully correlated channel, if one would like to protect 2k-qubit data states, only one additional qubit is needed, and using two additional qubits to protect the 2k-qubit data state seems to be a waste of resources. In [7], it was shown that one can use 2 arbitrary qubits to protect 1 data qubit in a fully correlated channel on 3-qubits.…”

New construction of quantum error-avoiding codes via group representation of quantum stabilizer codes

“…The available code Q is used to correct a set of errors ⊂ E, and the dimension of Q is denoted nχ (1).…”

“…In quantum computation/communication systems, it is critical to maintain the coherence of a quantum system [1]. In practice, however, decoherence coming from the unwanted interaction between the system and the environment will collapse the state of the quantum communication/computation, thus making the information harder to correct [2].…”

Recursive encoding and decoding of the noiseless subsystem for qudits

“…The second QECC scheme is often called the "error-avoiding" coding due to this reason. Decoherence free subspace (DFS) and noiseless subsystem (NS) are two popular examples of the second kind [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”