Entanglement-assisted Additive Qudit Stabilizer Codes

Abstract

Quantum systems governed by the laws of quantum mechanics are the most awaited technology of this century. Based on the shrinkage in the size of the devices over the years in classical systems, device sizes are expected to reach around the size of an atom and below. At this scale, quantum effects would be observed. As the current devices are engineered based on classical mechanics, the decline in Moore's law motivates us to build computing devices at atomic and subatomic scales designed using quantum mechanics in our pursuit of building more powerful computers compared to the current ones. Further, certain algorithms are known to be efficiently solved on quantum systems compared to their classical counterpart. Quantum computing, communication, and storage devices involve the processing, transmission, and storage of quantum information using qudits that are physically realized using NMR, photonics, superconducting circuits, ion traps, etc. As the environment is inherently quantum in nature, the state of the quantum system changes upon interacting with the environment due to decoherence; hence, quantum error correction codes are the need of the hour.

The primary motivation of my Ph.D. is to provide a framework for the stabilizer codes over qudits that use shared entanglement, similar to the qubit case, and obtain efficient codes based on the framework. The focus areas of my Ph.D. are as follows:

(i) We first generalize the Calderbank-Shor-Steane (CSS) coding framework to qudits to obtain two different codes, namely the generalized CSS codes and the well-known CSS codes. We explore the properties of these codes along with their code constructions and discuss the coding rate and minimum distance trade-off. The choice between the CSS and the generalized CSS code is based on whether the application requires better error correction capability or a higher coding rate, respectively.

(ii) We further provide the coding framework for the entanglement-assisted stabilizer codes over qudits of dimension p^k based on additive codes over F_{p^k}. Along with the code properties and the construction procedure, we also provide the encoding and the error correction procedure based on the Clifford operators. The entanglement-unassisted stabilizer codes are a special case of this framework with the number of shared entangled states to be zero.

(iii) For the entanglement-assisted CSS codes that are obtained from two classical codes, we provide the code construction and the encoding procedure, useful towards practical implementation.

(iv) Based on the proposed entanglement-assisted stabilizer framework, we provide the entanglement-assisted quantum Reed-Solomon codes over qudits that are non-degenerate codes saturating the quantum Singleton bound from the classical Reed-Solomon codes. We provide the construction, encoding, and error correction circuits for these codes using one and two qudit gates. We further propose a burst error correcting quantum code based on the proposed entanglement-assisted Reed-Solomon code.

(v) We introduce a field trace-based isomorphism to represent extension field elements based on primitive field elements. Using this trace-based isomorphic representation, we provide procedures to compute the product of two extension field elements and to compute the syndrome of a classical linear code over an extension field using primitive operations, which is efficient compared to the conventional procedure using the polynomial basis representation for the extension field. These procedures are very useful to implement multipliers for elements of F{2^k} and syndrome computation circuits for classical linear codes over F_{2^k} using the binary logic in digital integrated circuits.

(vi) Using the proposed entanglement-assisted stabilizer framework, we propose the entanglement-assisted quantum tensor product codes based on the classical tensor product codes. An exciting observation is that a non-zero rate entanglement-assisted tensor product code can be constructed from a classical tensor product code even when the entanglement-assisted CSS codes constructed from the component codes of the classical tensor product code yields zero rate quantum codes that are not useful. We view this as the coding analog of superadditivity. We provide the code construction, encoding, and error correction circuit for the code. We also propose two types of interleavers for the code using which the burst error correction capability of the code is enhanced.

(vii) Finally, we propose a reliable coded quantum network that is resilient to single node failure. We proposed the modified graph state code that encodes based on the network topology and provides a procedure that can recover the network from a single node loss if the network topology satisfies certain conditions. For a given number of network nodes n, we design the network links such that the modified graph state code obtained is rate-optimal and saturates the quantum Singleton bound when n is even.