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Quantum Error Correction for Noisy Quantum Wires

Alastair Kay

Published Jul 22, 2015• 2 citations
OpenAlex

Abstract

Quantum wires can transport quantum states between distant locations in a solid state scenario without the need for intricate controls in the intervening distance. Fixed usage, time independent schemes permit the pre-manufacture and characterisation of the wires, minimising the resultant error. Nevertheless, noise inevitably enters the system and, even if it acts locally on a single spin at a single instant of time, the intrinsic dynamics of the system rapidly propagate it into a massively correlated error. By modifying the error correction procedure, we prove that in spite of this, standard error correcting codes (the Calderbank-Shor-Steane codes) can be used to protect states against sufficiently low rates of noise during transport through a perfect quantum wire.

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We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as $n$-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.

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"The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: 5.2 10. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor´s fault-tolerant method is used to measure the stabiliser´s generators."

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Quantum Steganography and Quantum Error-Correction

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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.

arXiv:1008.0425•Aug 2, 2010•1 citation

Quantum error correction assisted by two-way noisy communication

Zhuo Wang, Sixia Yu, Heng Fan +1 more

Pre-shared non-local entanglement dramatically simplifies and improves the performance of quantum error correction via entanglement-assisted quantum error-correcting codes (EAQECCs). However, even considering the noise in quantum communication only, the non-local sharing of a perfectly entangled pair is technically impossible unless additional resources are consumed, such as entanglement distillation, which actually compromises the efficiency of the codes. Here we propose an error-correcting protocol assisted by two-way noisy communication that is more easily realisable: all quantum communication is subjected to general noise and all entanglement is created locally without additional resources consumed. In our protocol the pre-shared noisy entangled pairs are purified simultaneously by the decoding process. For demonstration, we first present an easier implementation of the well-known EAQECC [[4, 1, 3; 1]]. Then, we construct for the first time a 1-error-correcting code of 4 physical qudits capable of encoding 1 qubit of information, which is impossible in standard quantum error correction, demonstrating that our protocol can also improve the encoding rate in some sense. A systematic construction of two-way-noisy-communication-assisted codes is also provided.

Scientific Reports•Nov 26, 2014•1 citation
Quantum wires can transport quantum states between distant locations in a solid state scenario without the need for intricate controls in the intervening distance. Fixed usage, time independent schemes permit the pre-manufacture and characterisation of the wires, minimising the resultant error. Nevertheless, noise inevitably enters the system and, even if it acts locally on a single spin at a single instant of time, the intrinsic dynamics of the system rapidly propagate it into a massively correlated error. By modifying the error correction procedure, we prove that in spite of this, standard error correcting codes (the Calderbank-Shor-Steane codes) can be used to protect states against sufficiently low rates of noise during transport through a perfect quantum wire.
Can robustness against experimental imperfections and noise be embedded into a quantum simulation? In this paper, we report on a special case in which this is possible. A spin chain can be engineered such that, in the absence of imperfections and noise, an unknown quantum state is transported from one end of the chain to the other, due only to the intrinsic dynamics of the system. We show that an encoding into a standard error-correcting code (a Calderbank-Shor-Steane code) can be embedded into this simulation task such that a modified error-correction procedure on readout can recover from sufficiently low rates of noise during transport.
We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as nnn-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.
"The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: 5.2 10. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor´s fault-tolerant method is used to measure the stabiliser´s generators."
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.
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.
Pre-shared non-local entanglement dramatically simplifies and improves the performance of quantum error correction via entanglement-assisted quantum error-correcting codes (EAQECCs). However, even considering the noise in quantum communication only, the non-local sharing of a perfectly entangled pair is technically impossible unless additional resources are consumed, such as entanglement distillation, which actually compromises the efficiency of the codes. Here we propose an error-correcting protocol assisted by two-way noisy communication that is more easily realisable: all quantum communication is subjected to general noise and all entanglement is created locally without additional resources consumed. In our protocol the pre-shared noisy entangled pairs are purified simultaneously by the decoding process. For demonstration, we first present an easier implementation of the well-known EAQECC [[4, 1, 3; 1]]. Then, we construct for the first time a 1-error-correcting code of 4 physical qudits capable of encoding 1 qubit of information, which is impossible in standard quantum error correction, demonstrating that our protocol can also improve the encoding rate in some sense. A systematic construction of two-way-noisy-communication-assisted codes is also provided.