The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.

arXiv:1511.05738QuantumAug 11, 201727 citations

Quantum statistical complexity measure

Andre Tanus Cesario de Souza

Statistical complexity measures are distinctive tools that can detect significant physical phenomena. Many of these phenomena occur in such a way that they leave signatures that can be interpreted as changes in the state of the system, which can be read as statistical complexity through these measures, measured on a scale of order and disorder. We investigate the classical statistical complexity measure (CSCM) defined by López-Ruiz et al. and apply it to a simple model of discrete time evolution, characterized by finite Markov chains in dimension 2, for stochastic and bistochastic evolutions. Additionally, we introduce a quantum version for measuring statistical complexity: the quantum statistical complexity measure (QSCM), in the context of quantum information theory, and use it as a signaling function for quantum correlation transitions, measured on a scale of order and disorder. We consider the possibility that such transitions characterize intriguing physical phenomena, such as quantum phase transitions or sudden variations in correlation distributions. Our measure is applied to two analytically solvable Hamiltonian models: the one-dimensional quantum Ising model in the reduced single-particle state, and the Heisenberg chain XXZ of spin-1/2 in the state reduced by two particles. We analyze the measurement behavior in quantum phase transitions for finite system sizes and in the thermodynamic limit using the Bethe ansatz technique. We conclude that the quantum statistical complexity measure can serve as a signaling function of correlation transitions and distinguish quantum phase transitions, due to the abruptness that physical quantities experience at these transition points and the way in which the reduced states of these critical systems change within a scale of order and disorder.Statistical complexity measures are distinctive tools that can detect significant physical phenomena. Many of these phenomena occur in such a way that they leave signatures that can be interpreted as changes in the state of the system, which can be read as statistical complexity through these measures, measured on a scale of order and disorder. We investigate the classical statistical complexity measure (CSCM) defined by López-Ruiz et al. and apply it to a simple model of discrete time evolution, characterized by finite Markov chains in dimension 2, for stochastic and bistochastic evolutions. Additionally, we introduce a quantum version for measuring statistical complexity: the quantum statistical complexity measure (QSCM), in the context of quantum information theory, and use it as a signaling function for quantum correlation transitions, measured on a scale of order and disorder. We consider the possibility that such transitions characterize intriguing physical phenomena, such as quantum phase transitions or sudden variations in correlation distributions. Our measure is applied to two analytically solvable Hamiltonian models: the one-dimensional quantum Ising model in the reduced single-particle state, and the Heisenberg chain XXZ of spin-1/2 in the state reduced by two particles. We analyze the measurement behavior in quantum phase transitions for finite system sizes and in the thermodynamic limit using the Bethe ansatz technique. We conclude that the quantum statistical complexity measure can serve as a signaling function of correlation transitions and distinguish quantum phase transitions, due to the abruptness that physical quantities experience at these transition points and the way in which the reduced states of these critical systems change within a scale of order and disorder.

Dec 9, 20220 citations

New concept of statistical ensembles

M. I. Gorenstein

Journal of Physical StudiesJan 1, 20090 citations

Statistical complexity of quantum circuits

Kaifeng Bu, Dax Enshan Koh, Lu Li +2 more

In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely, the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a magic resource measure based on the $(p,q)$ group norm, which quantifies the amount of magic resource in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely, the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a magic resource measure based on the $(p,q)$ group norm, which quantifies the amount of magic resource in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.

arXiv:2101.06154Physical Review AJun 17, 202252 citations

(p,q)

group norm, which quantifies the amount of magic resource in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.