1. IntroductionQuantum system is vulnerable to disturbance from the external environment, which leads todecoherence in the system. We have to overcome this difficulty to realize a working quanutumcomputer and a dependable quantum information processing. Quantum error correction (QEC)[1, 2, 3] is one of the most promising candidates for overcoming decoherence.QEC proposals to date are separated roughly into two classes, one employs extra ancilla qubitsfor error syndrome readout while the other, called operator quantum error correction (OQEC)employs a higher rank projection operator. However, the schemes have not been built in actualquantum computing or quantum information processing so far. This is dueto the following obstaclesinherent in these QECs; the syndrome must be read out by introducing ancilla qubits duringcomputing/information processing in the former case, while realization of a high-rank projectionoperator is physically challenging in the latter case.It was shown in [4] that for some quantum channels there exist different QECs in which nosyndrome measurements, no ancillas and no projection operators were required. Recovery operationand decoding operation are combined into a single unitary operation and the output state is a directproductof a decoded qubitstate and an ancilla state. Aqubit state isreproducedwithout recoveringthe codeword and, moreover, the projection operation is automatically built in our output state.The purpose of this paper is to extend the results in [4] to general quantum channels. We showthat for any quantum channel there is a unitary recovery operation, with which the output stateis a tensor product of decoded qubit state and an ancilla state. As a result, a decoding scheme canbe done by a unitary gate followed by a partial trace operation.The rest of the paper is organized as follows. We introduce the basic notions of OQEC andthen prove the main theorem in Section 2. We also give simple examples demonstrating our resultand a simplified proof of a theorem given in [2] illustrating that our recovery channel can be usedto do correction for many other channels related to ours. Section 3 is devoted to summary anddiscussions.
IntroductionQuantum system is vulnerable to disturbance from the external environment, which leads todecoherence in the system. We have to overcome this difficulty to realize a working quanutumcomputer and a dependable quantum information processing. Quantum error correction (QEC)[1, 2, 3] is one of the most promising candidates for overcoming decoherence.QEC proposals to date are separated roughly into two classes, one employs extra ancilla qubitsfor error syndrome readout while the other, called operator quantum error correction (OQEC)employs a higher rank projection operator. However, the schemes have not been built in actualquantum computing or quantum information processing so far. This is dueto the following obstaclesinherent in these QECs; the syndrome must be read out by introducing ancilla qubits duringcomputing/information processing in the former case, while realization of a high-rank projectionoperator is physically challenging in the latter case.It was shown in [4] that for some quantum channels there exist different QECs in which nosyndrome measurements, no ancillas and no projection operators were required. Recovery operationand decoding operation are combined into a single unitary operation and the output state is a directproductof a decoded qubitstate and an ancilla state. Aqubit state isreproducedwithout recoveringthe codeword and, moreover, the projection operation is automatically built in our output state.The purpose of this paper is to extend the results in [4] to general quantum channels. We showthat for any quantum channel there is a unitary recovery operation, with which the output stateis a tensor product of decoded qubit state and an ancilla state. As a result, a decoding scheme canbe done by a unitary gate followed by a partial trace operation.The rest of the paper is organized as follows. We introduce the basic notions of OQEC andthen prove the main theorem in Section 2. We also give simple examples demonstrating our resultand a simplified proof of a theorem given in [2] illustrating that our recovery channel can be usedto do correction for many other channels related to ours. Section 3 is devoted to summary anddiscussions.
It is known that one can do quantum error correction without syndrome measurement, which is often done in operator quantum error correction (OQEC). However, the physical realization could be challenging, especially when the recovery process involves high-rank projection operators and a superoperator. We use operator theory to improve OQEC so that the implementation can always be done by unitary gates followed by a partial trace operation. Examples are given to show that our error correction scheme outperforms the existing ones in various scenarios.
It is known that one can do quantum error correction without syndrome measurement, which is often done in operator quantum error correction (OQEC). However, the physical realization could be challenging, especially when the recovery process involves high-rank projection operators and a superoperator. We use operator theory to improve OQEC so that the implementation can always be done by unitary gates followed by a partial trace operation. Examples are given to show that our error correction scheme outperforms the existing ones in various scenarios.
We propose a quantum error correction without error detection. A quantum state ρ0 combined with an ancilla state σ is encoded unitarily and an error operator is applied on the encoded state. The recovery operation then produces a tensor product state ρ0⊗σ′. The decoding operation is combined with the recovery operation and the state ρ0 is directly reproduced without referring to the code word. A higher rank projection operator required for a conventional operator quantum error correction is not necessary to implement. Encoding and the recovery operations are implemented with unitary operators only, which makes quantum error correction much easier than any other proposals.
Most quantum error-correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $|0\ensuremath{\rangle}$ which can be used as ancilla qubits to prepare multiqubit logical states. In this Brief Report, we examine the consequences of relaxing this assumption and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates when the recovery operation would already be implemented. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation used to correct detected errors after decoding. This augmentation will be especially useful in quantum computing architectures that do not possess projective measurement, such as solid-state NMR quantum information processing.
Quantum error correction (QEC) is an essential concept for any quantum information processing device. Typically, QEC is designed with minimal assumptions about the noise process; this generic assumption exacts a high cost in efficiency and performance. In physical systems, errors are not likely to be arbitrary; rather we will have reasonable models for the structure of quantum decoherence. We may choose quantum error correcting codes and recovery operations that specifically target the most likely errors. We present a convex optimization method to determine the optimal (in terms of average entanglement fidelity) recovery operation for a given channel, encoding, and information source. This is solvable via a semidefinite program (SDP). We present computational algorithms to generate near-optimal recovery operations structured to begin with a projective syndrome measurement. These structured operations are more computationally scalable than the SDP required for computing the optimal; we can thus numerically analyze longer codes. Using Lagrange duality, we bound the performance of the structured recovery operations and show that they are nearly optimal in many relevant cases. We present two classes of channel-adapted quantum error correcting codes specifically designed for the amplitude damping channel. These have significantly higher rates with shorter block lengths than corresponding generic quantum error correcting codes. Both classes are stabilizer codes, and have good fidelity performance with stabilizer recovery operations. The encoding, syndrome measurement, and syndrome recovery operations can all be implemented with Clifford group operations.
We demonstrate that continuous-variable quantum error correction based on Gaussian ancilla states and Gaussian operations (for encoding, syndrome extraction, and recovery) can be very useful to suppress the effect of non-Gaussian error channels. For a certain class of stochastic error models, reminiscent of those typically considered in the qubit case, quantum error correction codes designed for single-channel errors may enhance the transfer fidelities even when errors occur in every channel employed for transmitting the encoded state. In fact, in this case, the error-correcting capability of the continuous-variable scheme turns out to be higher than that of its discrete-variable analogues.