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arXiv:1503.04334

Quantum Decoding with Venn Diagrams

Barros, C. M. F., Francisco de Assis, H. M. de Oliveira

Published Mar 14, 2015• 0 citations
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Abstract

The quantum error correction theory is as a rule formulated in a rather convoluted way, in comparison to classical algebraic theory. This work revisits the error correction in a noisy quantum channel so as to make it intelligible to engineers. An illustrative example is presented of a naive perfect quantum code (Hamming-like code) with five-qubits for transmitting a single qubit of information. Also the (9,1)-Shor codes is addressed.

DOI
10.48550/arxiv.1503.04334

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The quantum error correction theory is as a rule formulated in a rather convoluted way, in comparison to classical algebraic theory. This work revisits the error correction in a noisy quantum channel so as to make it intelligible to engineers. An illustrative example is presented of a naive perfect quantum code (Hamming-like code) with five-qubits for transmitting a single qubit of information. Also the (9,1)-Shor codes is addressed.
The limits of classical information theory are stretched at several points by the consideration of quantum mechanics. Entanglement in itself is a unique phenomenon, and its application has enabled solutions such as super-dense compression, teleportation or super-activation of quantum channels. Noisy classical channels can only be made error-free by adding redundancy, and even this can only be done for certain channels. However, in this article, we examine an equally unique possibility, namely, what properties can make the quantum channels error-free without adding redundancy just by using appropriate quantum transformations. We will show the quantum coding required for this and illustrate its efficiency through examples.
Quantum error correction draws many of its principles and constructions from classical coding theory, adapted to the unique aspects of quantum mechanics. Here we extend this correspondence to a block of logical qubits, each prepared in the same quantum code, by coupling the block through a classical error correcting code: syndrome qubits then carry error information from several logical qubits at once, and collective inference recovers the errors. Algebraically, the construction defines subgroups of the product stabilizer group corresponding to duals of classical codes, and we prove that a lookup table decoder corrects every error pattern respecting the correction radii of the two constituent codes. For classical algebraic code families, the number of syndrome qubits serving LLL logical qubits scales as O(log⁡2(L+1)){\cal O}(\log_2(L+1))O(log2​(L+1)) at the code-capacity level, though at the price of stabilizer weight growing with the block length. An entropy bound shows that check weights need not grow with LLL. Under a phenomenological noise model, the same lookup table identifies measurement errors by nearest-neighbor post-processing, and classical algebraic decoders locate exactly the logical qubits carrying errors even with noisy syndromes. More broadly, we argue that quantum error correction, reduced to its functional core, is source compression in the sense of Shannon, whose source and channel coding theorems bound the overhead rates of quantum post-selection tasks.
We investigate in this work a quantum error correction on a five-qubits graph state used for secret sharing through five noisy channels. We describe the procedure for the five, seven and nine qubits codes. It is known that the three codes always allow error recovery if only one among the sent qubits is disturbed in the transmitting channel. However, if two qubits and more are disturbed, then the correction will depend on the used code. We compare in this paper the three codes by computing the average fidelity between the sent secret and that measured by the receivers. We will treat the case where, at most, two qubits are affected in each one of five depolarizing channels.
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.