Quantum Error Correction via Less Noisy Qubits

Abstract

Known quantum error correction schemes are typically able to take advantage of only a limited class of classical error-correcting codes. Entanglement-assisted quantum error correction is a partial solution which made it possible to exploit any classical linear codes over the binary or quaternary finite field. However, the known entanglement-assisted scheme requires noiseless qubits that help correct quantum errors on noisy qubits, which can be too severe an assumption. We prove that a more relaxed and realistic assumption is sufficient by presenting encoding and decoding operations assisted by qubits on which quantum errors of one particular kind may occur. As in entanglement assistance, our scheme can import any binary or quaternary linear codes. If the auxiliary qubits are noiseless, our codes become entanglement-assisted codes, and saturate the quantum Singleton bound when the underlying classical codes are maximum distance separable.

Quantum error correction assisted by two-way noisy communication

“…The case in which storage errors arise on the idle qudits has been discussed in Refs. 20 , 21 . However, as yet, there have been no efforts directed towards overcoming encoding errors, which is quite a complicated and difficult topic that is worth further exploration.…”

High-Rate Quantum Low-Density Parity-Check Codes Assisted by Reliable Qubits

“…. While it is straightforward to derive Theorem 2.2 from the results already presented in [36], for completeness, we give a formal proof in the remainder of this subsection.…”

“…The rest of this section is divided into two subsections. Section II-A presents the basics of the framework for quantum error correction given in [36] that is assisted by qubits on which only one particular kind of quantum error may occur. We briefly review in Section II-B the entanglement-assisted stabilizer formalism developed in [23] which uses completely noiseless qubits.…”

“…Lemma 2.3 ( [36]): Assume that there exits a linear code of length n and dimension k with a parity-check matrix H in standard form. Define…”

“…Very recently, a framework that significantly reduces the burden of providing extreme reliability has been proposed, where any binary or quaternary linear codes over F 2 or F 4 respectively can be fully exploited as long as we can provide auxiliary qubits that are only subject to a restricted quantum error model [36]. This framework takes advantage of the fact that while realizing completely noiseless qubits is a very difficult task, not every kind of quantum error is equally difficult to suppress through technical development on hardware.…”