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Information propagation through disordered quantum spin chains

Christian K. Burrell

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

One of the central aims of quantum information theory is to exploit quantum mechanical phenomena such as superposition and entanglement to build a quantum computer — a device capable of effciently performing computational tasks which are not feasible on a classical computer. In order to achieve this goal it will be necessary to connect together the different components of a quantum computer with channels which are capable of quickly and faithfully transmitting quantum states. It has been suggested that a good channel for this purpose is a chain of permanently coupled quantum spins and we begin by reviewing some of the protocols which have been previously developed to transfer quantum information along such a spin chain. We then examine the effects of noise on the propagation of quantum information along spin chains, using a toolkit of methods which include Lieb-Robinson bounds, the Jordan-Wigner transform and correlation functions. Several fundamentally different noise models are considered including a static (time-independent) on-site disorder model and two different models of fluctuating (time-dependent) on-site disorder. Each noise model has a different and distinctive effect on the propagation of information — static disorder leads to exponential localisation whilst dynamic disorder can lead to diffusive or even ballistic propagation of information. We finish by reviewing how a spin chain (whether noisy or noise-free) can be viewed as a depolarising channel and we make a detailed study of the geometrical structure of all possible depolarising channels with respect to various bases, concentrating on the Pauli, Gell-Mann and Heisenberg-Weyl bases. In particular, we show precisely when the set of all possible depolarising channels forms a simplex in compression space.

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Information propagation through quantum chains with fluctuating disorder

Christian K. Burrell, Jens Eisert, Tobias J. Osborne

We investigate the propagation of information through one-dimensional nearest-neighbor interacting quantum spin chains in the presence of external fields which fluctuate independently on each site. We study two fundamentally different models: (i) a model with general nearest-neighbor interactions in a field which fluctuates in both strength and direction and (ii) the $XX$ chain placed in a fluctuating field aligned in the $z$ direction. In both cases we find that information propagation is suppressed in a way which is quite different from the suppression observed when the $XX$ model is placed in a statically disordered field.

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One of the most essential processes in classical computation is that related to the information manipulation; each component or register of a computer needs to communicate to others by exchanging information encoded in bits and transforming it through logical operations. Hence the theoretical study of methods for information transfer and processing in classical information theory is of fundamental importance for telecommunications and computer science, along with study of errors and robustness of such proposals. When adding the quantum ingredient, there arises a whole new set of paradigms and devices, based on manipulations of \\emph{qubits}, the quantum analogues of conventional data bits. Such systems can show enormous advantage against their classical analogues, but at the same time present a whole new set of technical and conceptual challenges to overcome. The full and detailed understanding of quantum processes and studies of theoretical models and devices therefore provide the first logical steps to the future technological exploitation of these new machines. In this line, this thesis focuses on spin chains as such theoretical models, formed by series of coupled qubits that can be applied to a wide range of physical systems, and its several potential applications as quantum devices. In this work spin chains are presented as reliable devices for quantum communication with high transfer fidelities, entanglement generation and distribution over distant parties and protected storage of quantum information. Methods to design these tools to have some robustness against errors and noise are provided, giving optimism for future quantum technologies.

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The Lieb-Robinson correlation function for long disordered transverse-field Ising chains

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The Lieb-Robinson correlation function can be used to characterize the propagation of quantum information in Ising chains. Considerable work has been done to establish bounds on this correlation function in various circumstances. To actually calculate the value of the correlation function directly typically requires a state space which grows exponentially with system size, and so is intractable for all but relatively small systems. We employ a recently-developed method that enables direct calculation of the Lieb--Robinson correlation function and which scales linearly with system size. This enables the computation for systems with more than a thousand qubits, revealing the propagation of quantum information down the chain. We extend this technique to the problem of Ising chains with randomly disordered coupling strengths. Increasing disorder causes localization of the quantum correlations and halts propagation of quantum information down the chain. We see light-cone isocontours therefore becoming vertical. The decay of correlations down the chain is characterized by a correlation length which in the paramagnetic phase increases quadratically with mean coupling strength and inverse quadratically with disorder level. The correlation length peaks at the phase transition. We map our general expression for the correlation function in the Ising system onto the probability density in the single particle tight-binding Su-Schrieffer-Heeger model for a specific initial condition.

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Quantum state transfer in disordered spin chains: How much engineering is reasonable?

Analía Zwick, Gonzalo A. Álvarez, Joachim Stolze +1 more

The transmission of quantum states through spin chains is an important element in the implementation of quantum information technologies. Speed and fidelity of transfer are the main objectives which have to be achieved by the devices even in the presence of imperfections which are unavoidable in any manufacturing process. To reach these goals, several kinds of spin chains have been suggested, which differ in the degree of fine-tuning, or engineering, of the system parameters. In this work we present a systematic study of two important classes of such chains. In one class only the spin couplings at the ends of the chain have to be adjusted to a value different from the bulk coupling constant, while in the other class every coupling has to have a specific value. We demonstrate that configurations from the two different classes may perform similarly when subjected to the same kind of disorder in spite of the large difference in the engineering effort necessary to prepare the system. We identify the system features responsible for these similarities and we perform a detailed study of the transfer fidelity as a function of chain length and disorder strength, yielding empirical scaling laws for the fidelity which are similar for all kinds of chain and all disorder models. These results are helpful in identifying the optimal spin chain for a given quantum information transfer task. In particular, they help in judging whether it is worthwhile to engineer all couplings in the chain as compared to adjusting only the boundary couplings.

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Noise in Quantum Information Processing

Christopher T. Chubb

Quantum phenomena such as superposition and entanglement imbue quantum systems with information processing power in excess of their classical counterparts. These properties of quantum states are, however, highly fragile. As we enter the era of noisy intermediate-scale quantum (NISQ) devices, this vulnerability to noise is a major hurdle to the experimental realisation of quantum technologies. In this thesis we explore the role of noise in quantum information processing from two different perspectives. In Part I we consider noise from the perspective of quantum error correcting codes. Error correcting codes are often analysed with respect to simplified toy models of noise, such as iid depolarising noise. We consider generalising these techniques for analysing codes under more realistic noise models, including features such as biased or correlated errors. We also consider designing customised codes which not only take into account and exploit features of the underlying physical noise. Considering such tailored codes will be of particular importance for NISQ applications in which finite-size effects can be significant. In Part II we apply tools from information theory to study the finite-resource effects which arise in the trade-offs between resource costs and error rates for certain quantum information processing tasks. We start by considering classical communication over quantum channels, providing a refined analysis of the trade-off between communication rate and error in the regime of a finite number of channel uses. We then extend these techniques to the problem of resource interconversion in theories such as quantum entanglement and quantum thermodynamics, studying finite-size effects which arise in resource-error trade-offs. By studying this effect in detail, we also show how detrimental finite-size effects in devices such as thermal engines may be greatly suppressed by carefully engineering the underlying resource interconversion processes.

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Information propagation in isolated quantum systems

David J. Luitz, Yevgeny Bar Lev

How fast does quantum information spread in isolated quantum systems? The authors address this question using out-of-time order correlation functions and find that information spreads behind a power law front in disordered spin chains, with a characteristic exponent, which is related to the exponent of the power law growth of the entanglement entropy. The authors also observe that disorder inhibits information spreading and leads to unsaturated Lieb-Robinson bounds.

arXiv:1702.03929•Physical review. B./Physical review. B•Jul 12, 2017•174 citations
One of the central aims of quantum information theory is to exploit quantum mechanical phenomena such as superposition and entanglement to build a quantum computer — a device capable of effciently performing computational tasks which are not feasible on a classical computer. In order to achieve this goal it will be necessary to connect together the different components of a quantum computer with channels which are capable of quickly and faithfully transmitting quantum states. It has been suggested that a good channel for this purpose is a chain of permanently coupled quantum spins and we begin by reviewing some of the protocols which have been previously developed to transfer quantum information along such a spin chain. We then examine the effects of noise on the propagation of quantum information along spin chains, using a toolkit of methods which include Lieb-Robinson bounds, the Jordan-Wigner transform and correlation functions. Several fundamentally different noise models are considered including a static (time-independent) on-site disorder model and two different models of fluctuating (time-dependent) on-site disorder. Each noise model has a different and distinctive effect on the propagation of information — static disorder leads to exponential localisation whilst dynamic disorder can lead to diffusive or even ballistic propagation of information. We finish by reviewing how a spin chain (whether noisy or noise-free) can be viewed as a depolarising channel and we make a detailed study of the geometrical structure of all possible depolarising channels with respect to various bases, concentrating on the Pauli, Gell-Mann and Heisenberg-Weyl bases. In particular, we show precisely when the set of all possible depolarising channels forms a simplex in compression space.
We investigate the propagation of information through one-dimensional nearest-neighbor interacting quantum spin chains in the presence of external fields which fluctuate independently on each site. We study two fundamentally different models: (i) a model with general nearest-neighbor interactions in a field which fluctuates in both strength and direction and (ii) the XXXXXX chain placed in a fluctuating field aligned in the zzz direction. In both cases we find that information propagation is suppressed in a way which is quite different from the suppression observed when the XXXXXX model is placed in a statically disordered field.
One of the most essential processes in classical computation is that related to the information manipulation; each component or register of a computer needs to communicate to others by exchanging information encoded in bits and transforming it through logical operations. Hence the theoretical study of methods for information transfer and processing in classical information theory is of fundamental importance for telecommunications and computer science, along with study of errors and robustness of such proposals. When adding the quantum ingredient, there arises a whole new set of paradigms and devices, based on manipulations of \\emph{qubits}, the quantum analogues of conventional data bits. Such systems can show enormous advantage against their classical analogues, but at the same time present a whole new set of technical and conceptual challenges to overcome. The full and detailed understanding of quantum processes and studies of theoretical models and devices therefore provide the first logical steps to the future technological exploitation of these new machines. In this line, this thesis focuses on spin chains as such theoretical models, formed by series of coupled qubits that can be applied to a wide range of physical systems, and its several potential applications as quantum devices. In this work spin chains are presented as reliable devices for quantum communication with high transfer fidelities, entanglement generation and distribution over distant parties and protected storage of quantum information. Methods to design these tools to have some robustness against errors and noise are provided, giving optimism for future quantum technologies.
The Lieb-Robinson correlation function can be used to characterize the propagation of quantum information in Ising chains. Considerable work has been done to establish bounds on this correlation function in various circumstances. To actually calculate the value of the correlation function directly typically requires a state space which grows exponentially with system size, and so is intractable for all but relatively small systems. We employ a recently-developed method that enables direct calculation of the Lieb--Robinson correlation function and which scales linearly with system size. This enables the computation for systems with more than a thousand qubits, revealing the propagation of quantum information down the chain. We extend this technique to the problem of Ising chains with randomly disordered coupling strengths. Increasing disorder causes localization of the quantum correlations and halts propagation of quantum information down the chain. We see light-cone isocontours therefore becoming vertical. The decay of correlations down the chain is characterized by a correlation length which in the paramagnetic phase increases quadratically with mean coupling strength and inverse quadratically with disorder level. The correlation length peaks at the phase transition. We map our general expression for the correlation function in the Ising system onto the probability density in the single particle tight-binding Su-Schrieffer-Heeger model for a specific initial condition.
The transmission of quantum states through spin chains is an important element in the implementation of quantum information technologies. Speed and fidelity of transfer are the main objectives which have to be achieved by the devices even in the presence of imperfections which are unavoidable in any manufacturing process. To reach these goals, several kinds of spin chains have been suggested, which differ in the degree of fine-tuning, or engineering, of the system parameters. In this work we present a systematic study of two important classes of such chains. In one class only the spin couplings at the ends of the chain have to be adjusted to a value different from the bulk coupling constant, while in the other class every coupling has to have a specific value. We demonstrate that configurations from the two different classes may perform similarly when subjected to the same kind of disorder in spite of the large difference in the engineering effort necessary to prepare the system. We identify the system features responsible for these similarities and we perform a detailed study of the transfer fidelity as a function of chain length and disorder strength, yielding empirical scaling laws for the fidelity which are similar for all kinds of chain and all disorder models. These results are helpful in identifying the optimal spin chain for a given quantum information transfer task. In particular, they help in judging whether it is worthwhile to engineer all couplings in the chain as compared to adjusting only the boundary couplings.
Quantum phenomena such as superposition and entanglement imbue quantum systems with information processing power in excess of their classical counterparts. These properties of quantum states are, however, highly fragile. As we enter the era of noisy intermediate-scale quantum (NISQ) devices, this vulnerability to noise is a major hurdle to the experimental realisation of quantum technologies. In this thesis we explore the role of noise in quantum information processing from two different perspectives. In Part I we consider noise from the perspective of quantum error correcting codes. Error correcting codes are often analysed with respect to simplified toy models of noise, such as iid depolarising noise. We consider generalising these techniques for analysing codes under more realistic noise models, including features such as biased or correlated errors. We also consider designing customised codes which not only take into account and exploit features of the underlying physical noise. Considering such tailored codes will be of particular importance for NISQ applications in which finite-size effects can be significant. In Part II we apply tools from information theory to study the finite-resource effects which arise in the trade-offs between resource costs and error rates for certain quantum information processing tasks. We start by considering classical communication over quantum channels, providing a refined analysis of the trade-off between communication rate and error in the regime of a finite number of channel uses. We then extend these techniques to the problem of resource interconversion in theories such as quantum entanglement and quantum thermodynamics, studying finite-size effects which arise in resource-error trade-offs. By studying this effect in detail, we also show how detrimental finite-size effects in devices such as thermal engines may be greatly suppressed by carefully engineering the underlying resource interconversion processes.