Quantum mechanics can reduce the complexity of classical models.

  title={Quantum mechanics can reduce the complexity of classical models.},
  author={Mile Gu and Karoline Wiesner and Elisabeth Rieper and Vlatko Vedral},
  journal={Nature communications},
Mathematical models are an essential component of quantitative science. They generate predictions about the future, based on information available in the present. In the spirit of simpler is better; should two models make identical predictions, the one that requires less input is preferred. Yet, for almost all stochastic processes, even the provably optimal classical models waste information. The amount of input information they demand exceeds the amount of predictive information they output… 
Optimal stochastic modeling with unitary quantum dynamics
A class of phase-enhanced quantum models are introduced, representing the most general means of simulating a stochastic process unitarily in causal order, and surpass previous state-of-art methods in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information.
Towards quantifying complexity with quantum mechanics
A new measure of complexity based on quantum, known asepsilon-machines, is proposed and applied to a simple system undergoing constant thermalization, which aligns more closely with the intuition of how complexity should behave.
The classical-quantum divergence of complexity in modelling spin chains
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
Equality conditions for internal entropies of certain classical and quantum models
Conditions when the internal entropies between classical models and specific quantum models coincide coincide are shown and it turns out that this situation appears very rarely.
Optimizing Quantum Models of Classical Channels: The Reverse Holevo Problem
This work determines when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation, and inverts Holevo's original question of quantifying the capacity of quantum channels with classical resources.
Using quantum theory to simplify input–output processes
It is shown that classical models cannot avoid inefficiency—storing past information that is unnecessary for correct future simulation, and construct quantum models that mitigate this waste, whenever it is physically possible to do so.
Matrix Product States for Quantum Stochastic Modeling.
This Letter associates each stochastic process with a suitable quantum state of a spin chain, and shows that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state.
Extreme Quantum Advantage when Simulating Strongly Coupled Classical Systems
It is found that the quantum advantage for strongly coupled spin systems---the Dyson-like one-dimensional Ising spin chain with variable interaction length--- scales with both interaction range and temperature, growing without bound as interaction increases.
Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes
Among the predictive hidden Markov models that describe a given stochastic process, the {\epsilon}-machine is strongly minimal in that it minimizes every Renyi-based memory measure, but there are those for which there does not exist any strongly minimal model.
Increasing complexity with quantum physics.
This work identifies correlations as a central concept connecting quantum information and complex systems science, and presents two examples for the power of correlations: using quantum resources to simulate the correlations of a stochastic process and to implement a classically impossible computational task.


Quantum computation and quantum information
  • T. Paul
  • Physics
    Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal
Toward a quantitative theory of self-generated complexity
Quantities are defined operationally which qualify as measures of complexity of patterns arising in physical situations, and are essentially Shannon information needed to specify not individual patterns, but either measure-theoretic or algebraic properties of ensembles of pattern arising ina priori translationally invariant situations.
Time's barbed arrow: irreversibility, crypticity, and stored information.
A closed-form expression for the excess entropy in terms of optimal causal predictors and retrodictors is presented, leading to two new system invariants: causal irreversibility-the asymmetry between the causal representations-and crypticity-the degree to which a process hides its state information.
Regularities unseen, randomness observed: levels of entropy convergence.
Several phenomenological approaches to applying information theoretic measures of randomness and memory to stochastic and deterministic processes are synthesized by using successive derivatives of the Shannon entropy growth curve to look at the relationships between a process's entropy convergence behavior and its underlying computational structure.
Lossless quantum data compression and variable-length coding
A general framework for variable-length quantum messages is developed in close analogy to the classical case and it is shown that it is possible to reduce the number of qbits passing through a quantum channel even below the von Neumann entropy by adding a classical side channel.
Computational Mechanics: Pattern and Prediction, Structure and Simplicity
It is shown that the causal-state representation—an ∈-machine—is the minimal one consistent with accurate prediction, and several results are established on ∉-machine optimality and uniqueness and on how∈-machines compare to alternative representations.
Statistical complexity of simple one-dimensional spin systems
We present exact results for two complementary measures of spatial structure generated by 1D spin systems with finite-range interactions. The first, excess entropy, measures the apparent spatial
Inferring statistical complexity.
A technique is presented that directly reconstructs minimal equations of motion from the recursive structure of measurement sequences, demonstrating a form of superuniversality that refers only to the entropy and complexity of a data stream.
Quantifying self-organization with optimal predictors.
This Letter proposes a new criterion, namely, an internally generated increase in the statistical complexity, the amount of information required for optimal prediction of the system's dynamics, for spatially extended dynamical systems.