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arXiv:1805.10292 · EPJ Quantum Technology

Finding critical states of enhanced memory capacity in attractive cold bosons

Gia Dvali, Marco Michel, Sebastian Zell

Published Mar 25, 2019• 24 citations
arXivHTMLPDF

Abstract

We discuss a class of quantum theories which exhibit a sharply increased memory storage capacity due to emergent gapless degrees of freedom. Their realization, both theoretical and experimental, is of great interest. On the one hand, such systems are motivated from a quantum information point of view. On the other hand, they can provide a framework for simulating systems with enhanced capacity of pattern storage, such as black holes and neural networks. In this paper, we develop an analytic method that enables us to find critical states with increased storage capabilities in a generic system of cold bosons with weak attractive interactions. The enhancement of memory capacity arises when the occupation number N of certain modes reaches a critical level. Such modes, via negative energy couplings, assist others in becoming effectively gapless. This leads to degenerate microstates labeled by the occupation numbers of the nearly-gapless modes. In the limit of large N , they become exactly gapless and their decoherence time diverges. In this way, a system becomes an ideal storer of quantum information. We demonstrate our method on a prototype model of N attractive cold bosons contained in a one-dimensional box with Dirichlet boundary conditions. Although we limit ourselves to a truncated system, we observe a rich structure of quantum phases with a critical point of enhanced memory capacity.

DOI
10.1140/epjqt/s40507-019-0071-1

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We discuss a class of quantum theories which exhibit a sharply increased memory storage capacity due to emergent gapless degrees of freedom. Their realization, both theoretical and experimental, is of great interest. On the one hand, such systems are motivated from a quantum information point of view. On the other hand, they can provide a framework for simulating systems with enhanced capacity of pattern storage, such as black holes and neural networks. In this paper, we develop an analytic method that enables us to find critical states with increased storage capabilities in a generic system of cold bosons with weak attractive interactions. The enhancement of memory capacity arises when the occupation number N of certain modes reaches a critical level. Such modes, via negative energy couplings, assist others in becoming effectively gapless. This leads to degenerate microstates labeled by the occupation numbers of the nearly-gapless modes. In the limit of large N , they become exactly gapless and their decoherence time diverges. In this way, a system becomes an ideal storer of quantum information. We demonstrate our method on a prototype model of N attractive cold bosons contained in a one-dimensional box with Dirichlet boundary conditions. Although we limit ourselves to a truncated system, we observe a rich structure of quantum phases with a critical point of enhanced memory capacity.
We establish a fundamental connection between quantum criticality of a many-body system, such as Bose-Einstein condensates, and its capacity of information-storage and processing. For deriving the effective theory of modes in the vicinity of the quantum critical point, we develop a new method by mapping a Bose-Einstein condensate of NNN-particles onto a sigma model with a continuous global (pseudo)symmetry that mixes bosons of different momenta. The Bogolyubov modes of the condensate are mapped onto the Goldstone modes of the sigma model, which become gapless at the critical point. These gapless Goldstone modes are the quantum carriers of information and entropy. Analyzing their effective theory, we observe information-processing properties strikingly similar to the ones predicted by the black hole portrait. The energy cost per qubit of information-storage vanishes in the large-NNN limit and the total information-storage capacity increases with NNN either exponentially or as a power law. The longevity of information-storage also increases with NNN, whereas the scrambling time in the over-critical regime is controlled by the Lyapunov exponent and scales logarithmically with NNN. This connection reveals that the origin of black hole information storage lies in the quantum criticality of the graviton Bose-gas, and that much simpler systems that can be manufactured in table-top experiments can exhibit very similar information-processing dynamics.
We implement a mechanism - originally proposed as a model for the large memory storage capacity of black holes - in quantum neural networks and show that an exponentially increased capacity of pattern storage and recognition is achieved in certain critically excited states, without involvement of synaptic plasticity. We consider a simple network of N interconnected quantum neurons with weak excitatory synaptic connections. We show that for frozen synaptic weights there exist the critical states of enhanced memory storage capacity. These states are achieved thanks to the high excitation levels of some of the neurons, which - despite of feeble synaptic connections - dramatically lower the response threshold of the remaining weaker-excited neurons. As a results, the latter neurons acquire a capacity to store an exponentially large number of patterns within a narrow energy gap. The stored patterns can be recognized and retrieved with perfect response under the influence of arbitrarily soft input stimuli. In sharp contrast, under the same stimuli the recall is absent in the ground-state of the system. The lesson is that the state with the highest micro-state entropy and memory storage capacity is not necessarily a local minimum of energy, but rather an excited critical state. The considered phenomenon has a smooth classical limit and can serve for achieving an enhanced memory storage capacity in classical brain networks.
An explicit microscopic realization of the phenomenon of holography is provided by a class of simple quantum theories of a bosonic field inhabiting a d-dimensional space and experiencing a momentum dependent attractive interaction. An exact mode counting reveals a family of holographic states. In each a set of gapless modes emerges with their number equal to the area of a (d-1)-dimensional sphere. These modes store an exponentially large number of patterns within a microscopic energy gap. The resulting micro-state entropy obeys the area-law reminiscent of a black hole entropy. We study the time-evolution of the stored patterns and observe the following phenomenon: Among the degenerate micro-states the ones with heavier loaded memories survive longer than those that store emptier patterns. Thus, a state gets stabilized by the burden of its own memory. From time to time the information pattern gets off-loaded from one holographic state into another but cannot escape the system. During this process the pattern becomes highly entangled and scrambled. We suggest that this phenomenon is universal in systems with enhanced memory storage capacity, such as black holes or critical neural networks. This universality sheds an interesting light on the puzzle of why, despite the evaporation, is a black hole forced to maintain information internally for a very long time.
Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here quantum generalizations of classical networks realizing associative memories---capable of retrieving patterns, or memories, from corrupted initial states---have been proposed. It is a challenging open problem to analyze quantum associative memories with an extensive number of patterns and to determine the maximal number of patterns the quantum networks can reliably store, i.e., their storage capacity. In this work, we propose and explore a general method for evaluating the maximal storage capacity of quantum neural network models. By generalizing what is known as Gardner's approach in the classical realm, we exploit the theory of classical spin glasses for deriving the optimal storage capacity of quantum networks with quenched pattern variables. As an example, we apply our method to an open-system quantum associative memory formed of interacting spin-1/2 particles realizing coupled artificial neurons. The system undergoes a Markovian time evolution resulting from a dissipative retrieval dynamics that competes with a coherent quantum dynamics. We map out the nonequilibrium phase diagram and study the effect of temperature and Hamiltonian dynamics on the storage capacity. Our method opens an avenue for a systematic characterization of the storage capacity of quantum associative memories.
Quantum critical phases are extended regions of phase space characterized by a diverging correlation length. By analogy, we define an \emph{information critical phase} as an extended region of a mixed state phase diagram where the Markov length, the characteristic length scale governing the decay of the conditional mutual information (CMI), diverges. We demonstrate that such a phase arises in decohered ZN\mathbb{Z}_{N}ZN​ toric codes by assessing both the CMI and the coherent information, the latter quantifying the robustness of the encoded logical qudits. For N>4N>4N>4, we find that the system hosts an information critical phase intervening between the decodable and non-decodable phases where the coherent information saturates to a fractional value in the thermodynamic limit, indicating that a finite fraction of logical information is still preserved. We show that the density matrix in this phase can be decomposed into a convex sum of Coulombic pure states, where gapped anyons reorganize into gapless photons. We further consider the ungauged ZN\mathbb{Z}_{N}ZN​ toric code and interpret its mixed state phase diagram in the language of strong-to-weak spontaneous symmetry breaking. We argue that in the dual model, the information critical phase arises because the spontaneously broken off-diagonal ZN\mathbb{Z}_{N}ZN​ symmetry gets enhanced to a U(1)U(1)U(1) symmetry, resulting in a novel superfluid phase whose gapless modes involve coherent excitations of both the system and the environment. Finally, we propose an optimal decoding protocol for the corrupted ZN\mathbb{Z}_{N}ZN​ toric code and show that its logical error rate saturates to a fractional value in the information critical phase. Our findings identify a gapless analog for mixed-state phases that still acts as a fractional topological quantum memory, thereby extending the conventional paradigm of quantum memory phases.
Motivated by the potential similarities between the underlying mechanisms of the enhanced memory storage capacity in black holes and in brain networks, we construct an artificial quantum neural network based on gravity-like synaptic connections and a symmetry structure that allows to describe the network in terms of geometry of a d-dimensional space. We show that the network possesses a critical state in which the gapless neurons emerge that appear to inhabit a (d-1)-dimensional surface, with their number given by the surface area. In the excitations of these neurons, the network can store and retrieve an exponentially large number of patterns within an arbitrarily narrow energy gap. The corresponding micro-state entropy of the brain network exhibits an area law. The neural network can be described in terms of a quantum field, via identifying the different neurons with the different momentum modes of the field, while identifying the synaptic connections among the neurons with the interactions among the corresponding momentum modes. Such a mapping allows to attribute a well-defined sense of geometry to an intrinsically non-local system, such as the neural network, and vice versa, it allows to represent the quantum field model as a neural network.