Perfect Error Correction Research Articles

Quantum Error Correction in the Presence of Initial System-Environment Correlations

Quantum error correction is studied in a framework consisting of an open quantum system and its environment, jointly subjected to a unitary action, and an interaction-free reference system. It has been shown that coherent information between the initially correlated open system and reference system is conserved in the transmission stage of any quantum communication process, provided that the tripartite input is any pure Markov state and the overall evolution preserves its form. This conservation constitutes the necessary and sufficient condition for accomplishment of perfect error correction by a recovery channel even in the presence of initial system-environment correlations. Explicit expressions of the recovery operators and examples of the joint unitary evolution preserving the form of inputs are given for all classes of pure Markov states.

A high storage density strategy for digital information based on synthetic DNA.

DNA has been recognized as a promising natural medium for information storage. The expensive DNA synthesis process makes it an important challenge to utilize DNA nucleotides optimally and increase the storage density. Thus, a novel scheme is proposed for the storage of digital information in synthetic DNA with high storage density and perfect error correction capability. The proposed strategy introduces quaternary Huffman coding to compress the binary stream of an original file before it is converted into a DNA sequence. The proposed quaternary Huffman coding is based on the statistical properties of the source and can gain a very high compression ratio for files with a non-uniform probability distribution of the source. Consequently, the amount of information that each base can store increases, and the storage density is also improved. In addition, quaternary Hamming code with low redundancy is proposed to correct errors occurring in the synthesis and sequencing. We have successfully converted a total of 5.2KB of files into 3934 bits in DNA bases. The results of biological experiment indicate that the storage density of the proposed scheme is higher than that of state-of-the-art schemes.

Steane code single qubit Clifford gates

We apply single ‘logical’ qubit Clifford gates to states encoded in the [7,1,3] quantum error correction code in a nonequiprobable Pauli matrix error environment. The gates we analyze are the NOT, Phase, and Hadamard gates. We calculate accuracy measures of the applied gates by determining output state and logical gate fidelities. The latter is determined from the logical gate -matrix. In addition, we apply perfect and noisy quantum error correction to the output state of the Clifford gate. Perfect error correction allows the determination of whether errors that occurred during the gate are ‘correctable’. Noisy error correction gives an idea of what can be expected in a realistic quantum computation.

Non-fault-tolerantTgates for the [7,1,3] quantum error-correction code

We simulate the implementation of a $T$ gate, or $\frac{\ensuremath{\pi}}{8}$ gate, for a [7,1,3] encoded logical qubit in a nonequiprobable error environment. We demonstrate that the use of certain non-fault-tolerant methods in the implementation may nevertheless enable reliable quantum computation while reducing basic resource consumption. Reliability is determined by calculating gate fidelities for the one-qubit logical gate. Specifically, we show that despite using a non-fault-tolerant procedure in constructing a logical zero ancilla to implement the $T$ gate, the gate fidelity of the logical gate, after perfect error correction, has no first order error terms. Meaning, any errors that may have occurred during implementation are ``correctable'' and fault tolerance may still be achieved.

Towards a unified framework for approximate quantum error correction

Recent work on approximate quantum error correction (QEC) has opened up the possibility of constructing subspace codes that protect information with high fidelity in scenarios where perfect error correction is impossible. Motivated by this, we investigate the problem of approximate subsystem codes. Subsystem codes extend the standard formalism of subspace QEC to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Here, we demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, for certain classes of subsystem codes and noise processes, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel. This work generalizes our earlier approach [H.K. Ng and P. Mandayam, Phys. Rev. A 81 062342 (2010)] of using the transpose channel for approximate correction of subspace codes to the case of subsystem codes, and brings us closer to a unifying framework for approximate QEC.

Encoding an arbitrary state in a [7,1,3] quantum error correction code

We calculate the fidelity with which an arbitrary state can be encoded into a [7, 1, 3] Calderbank-Shor-Steane quantum error correction code in a non-equiprobable Pauli operator error environment with the goal of determining whether this encoding can be used for practical implementations of quantum computation. The determination of usability is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We also apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be implemented. Finally, fault tolerant noisy error correction is applied to the encoded states allowing us to compare noisy (realistic) and perfect error correction implementations. We find the encoding to be usable for the states \({|0\rangle, |1\rangle}\), and \({|\pm\rangle = |0\rangle\pm|1\rangle}\). These results have implications for when non-fault tolerant procedures may be used in practical quantum computation and whether quantum error correction must be applied at every step in a quantum protocol.

Fidelity of an encoded [7,1,3] logical zero

I calculate the fidelity of a $[7,1,3]$ Calderbank-Shor-Steane quantum error correction code logical zero state constructed in a nonequiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault-tolerant error correction to an arbitrary state of seven qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the nonequiprobable Pauli operator error environment, and their fidelity depends on the number of verifications done to ensure multiple errors will not propagate into the encoded quantum information. Surprisingly, performing these verifications may lower the fidelity of the constructed Shor states. The second encoding method is to simply implement the $[7,1,3]$ encoding gate sequence also in the nonequiprobable Pauli operator error environment. Perfect error correction is applied after both methods to determine the correctability of the implemented errors. I find that which method attains higher fidelity depends on which of the Pauli operators errors is dominant. Nevertheless, perfect error correction applied after the encoding suppresses errors to at least first order for both methods.

Constructions and noise threshold of topological subsystem codes

Topological subsystem codes proposed recently by Bombin are quantum error-correcting codes defined on a two-dimensional grid of qubits that permit reliable quantum information storage with a constant error threshold. These codes require only the measurement of two-qubit nearest-neighbor operators for error correction. In this paper, we demonstrate that topological subsystem codes (TSCs) can be viewed as generalizations of Kitaev's honeycomb model to 3-valent hypergraphs. This new connection provides a systematic way of constructing TSCs and analyzing their properties. We also derive a necessary and sufficient condition under which a syndrome measurement in a subsystem code can be reduced to measurements of the gauge group generators. Furthermore, we propose and implement some candidate decoding algorithms for one particular TSC assuming perfect error correction. Our Monte Carlo simulations indicate that this code, which we call the five-square code, has a threshold against depolarizing noise of at least 2%.

Optimized entanglement-assisted quantum error correction

Using convex optimization, we propose entanglement-assisted quantum error-correction procedures that are optimized for given noise channels. We demonstrate through numerical examples that such an optimized error-correction method achieves higher channel fidelities than existing methods. This improved performance, which leads to perfect error correction for a larger class of error channels, is interpreted in at least some cases by quantum teleportation, but for general channels this interpretation does not hold.

Checksum-Based Probabilistic Transient-Error Compensation for Linear Digital Systems

In this paper, a probabilistic compensation technique for minimizing the effect of transient errors effect is proposed. The focus is to develop a compensation technique for DSP applications in which exact error compensation is not necessary and end-to-end system level performance is degraded minimally as long as the impact of the ldquonoiserdquo injected into the system by the transient errors is minimized. The proposed technique, called checksum-based probabilistic compensation, uses real-number checksum codes for error detection and partial compensation. Traditional coding techniques need a code of distance three and relatively complex calculations for perfect error correction. Here, it is shown that a distance-two code can be used to perform probabilistic error compensation in linear systems with the objective of improving the signal-to-noise ratio in the presence of random transient errors. The goal is to have a technique with small power and area overhead and to perform compensation in real time with negligible latency. The proposed technique is comprehensive and can handle errors in the combinational circuitry and storage elements. Comparison against a system with no error correction shows that up to 13-dB SNR improvement is possible. The area, power, and timing overheads of the proposed technique are analyzed.

Strictly contractive quantum channels and physically realizable quantum computers

We study the robustness of quantum computers under the influence of errors modeled by strictly contractive channels. A channel T is defined to be strictly contractive if, for any pair of density operators \ensuremath{\rho}, \ensuremath{\sigma} in its domain, $\ensuremath{\Vert}T\ensuremath{\rho}\ensuremath{-}T\ensuremath{\sigma}{\ensuremath{\Vert}}_{1}l~k\ensuremath{\Vert}\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}{\ensuremath{\Vert}}_{1}$ for some $0l~kl1$ (here $\ensuremath{\Vert}\ensuremath{\cdot}{\ensuremath{\Vert}}_{1}$ denotes the trace norm). In other words, strictly contractive channels render the states of the computer less distinguishable in the sense of quantum detection theory. Starting from the premise that all experimental procedures can be carried out with finite precision, we argue that there exists a physically meaningful connection between strictly contractive channels and errors in physically realizable quantum computers. We show that, in the absence of error correction, sensitivity of quantum memories and computers to strictly contractive errors grows exponentially with storage time and computation time, respectively, and depends only on the constant k and the measurement precision. We prove that strict contractivity rules out the possibility of perfect error correction, and give an argument that approximate error correction, which covers previous work on fault-tolerant quantum computation as a special case, is possible.