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.
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 present a class of numerical algorithms which adapt a quantum error correction scheme to a channel model. Given an encoding and a channel model, it was previously shown that the quantum operation that maximizes the average entanglement fidelity may be calculated by a semidefinite program (SDP), which is a convex optimization. While optimal, this recovery operation is computationally difficult for long codes. Furthermore, the optimal recovery operation has no structure beyond the completely positive trace-preserving constraint. We derive methods to generate structured channel-adapted error recovery operations. Specifically, each recovery operation begins with a projective error syndrome measurement. The algorithms to compute the structured recovery operations are more scalable than the SDP and yield recovery operations with an intuitive physical form. Using Lagrange duality, we derive performance bounds to certify near-optimality.
Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set of errors, presumably a large subset of the physical noise process. In this paper, we examine the choice of recovery operation. Rather than seeking perfect correction on a subset of errors, we seek a recovery operation to maximize the entanglement fidelity for a given input state and noise model. In this way, the recovery operation is optimal for the given encoding and noise process. This optimization is shown to be calculable via a semidefinite program, a well-established form of convex optimization with efficient algorithms for its solution. The error recovery operation may also be interpreted as a combining operation following a quantum spreading channel, thus providing a quantum analogy to the classical diversity combining operation.
Quantum error correcting (QEC) plays a crucial role in protecting quantum information against decoherence and enabling scalable, reliable quantum computing. One of the most realistic and ubiquitous sources of noise affecting quantum hardware today is generalized amplitude-damping (GAD) noise. Conventional, deterministic QEC codes struggle to correct for GAD noise because of their inherent structure, leading to fidelity losses that scale linearly with the damping strength. In this work, we introduce the framework of probabilistic approximate quantum error correction (PAQEC), that combines the flexibility of approximate QEC with the potential of post-selected recovery, enabling high-fidelity, resource-efficient error correction. We construct a five-qubit permutation-invariant code that, under probabilistic recovery, achieves a fidelity loss quadratic in the damping strength, thus outperforming existing QEC codes. Formulating PAQEC as an optimization problem, we present a numerical technique based on Charnes-Cooper and semidefinite programming to identify the optimal recovery map for any PAQEC code. Our results establish PAQEC as a powerful tool for developing resource-efficient, high-fidelity quantum codes tailored to realistic noise, with promising implications for near-term quantum devices and future fault-tolerant architectures.
Quantum error correction (QEC) is crucial for faithful transmission of quantum information and successful realization of quantum computers. Typically, QEC is designed with minimal assumptions about the noise process, i.e. depolarizing noise that happens independently on each individual particle. In real physical systems, errors are unlikely to be arbitrary. Rather we have reasonable knowledge about the system to model its quantum decoherence process. In this talk, we examine QEC that are adapted to the physical noise models. We start from an overview of the concepts and general methods for channel-adapted QEC. We then examine a technique, called code concatenation, for constructing channel-adapted quantum codes. Applying to the amplitude damping channels, we demonstrate the power of the code concatenation method for constructing good codes for channel-adapted QEC.
The purpose of quantum error correction (QEC) is to preserve a quantum state despite the presence of decoherence. As we see throughout this book, the desired robustness exacts a cost in resources, most commonly the inclusion of redundant qubits. The following are reasonable engineering queries: How much will it cost to provide the desired robustness? For a fixed cost, what is the best performance I can achieve? In this chapter, we present some numerical tools to illuminate these kinds of questions. To understand the cost/performance trade-off quantitatively, we need a clear measure of performance and a model for permissible operations. With this in mind, we will revisit the concepts of fidelity and quantum operations. As it turns out, this quantitative approach can yield very well structured convex optimization problems. Using powerful numerical tools, we determine optimal encodings and recovery operations. Even if the optimal results are not directly implemented, the optimization tools can provide insight into practical solutions by providing the ultimate performance limits. Limitation of the independent arbitrary errors model As pointed out in Chapter 2, our rich history of classical error correction has provided a significant foundation for QEC methods. As such, the initial QEC breakthroughs involved importing classical coding concepts into a framework that made robust quantum codes (such as CSS codes or the stabilizer code formalism). We learned that we could create general purpose codes that made minimal assumptions about the structure of the decoherence process.
Quantum error correction (QEC) is essential for reliable quantum information processing. Targeting a particular error channel, both the encoding and the recovery channel can be optimized through a biconvex optimization to give a high-performance, noise-adapted QEC scheme. We solve the biconvex optimization by the technique of alternating semi-definite programming and identify a new four-qubit code for amplitude damping channel, one major noise in superconducting circuits and a good model for spontaneous emission and energy dissipation. We also construct analytical encoding and recovery channels that are close to the numerically optimized ones. We show that the new code notably outperforms the Leung-Nielsen-Chuang-Yamamoto four-qubit code in terms of the entanglement fidelity over an amplitude damping channel.