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arXiv:1804.03242 · Journal of High Energy Physics

Quantum complexity and the virial theorem

Ning Bao, Junyu Liu

Published Aug 1, 2018• 9 citations
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Abstract

Abstract It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.

DOI
10.1007/jhep08(2018)144

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Abstract It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.
We give arguments for the existence of a thermodynamics of quantum complexity that includes a ``second law of complexity.'' To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of KKK qubits, and the positional entropy of a related classical system with 2K{2}^{K}2K degrees of freedom. We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system. We argue that the property of having less-than-maximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation. Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity resource as the accessible volume of spacetime behind a black hole horizon.
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to unnormalized density operators, which are subsequently "purified" into unnormalized vectors in Hilbert space. We discuss the optimality of our notion of Kolmogorov complexity in a broad sense.
We study the relations between the recently proposed machine-independent quantum complexity of P. Gacs [1] and the entropy of classical and quantum systems. On one hand, by restricting Gacs complexity to ergodic classical dynamical systems, we retrieve the equality between the Kolmogorov complexity rate and the Shannon entropy rate derived by A. A. Brudno [2]. On the other hand, using the quantum Shannon-McMillan theorem [3], we show that such an equality holds densely in the case of ergodic quantum spin chains.
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to positive operators, which are subsequently "purified". We discuss the optimality of our notion of Kolmogorov complexity in a broad sense and obtain a simple complexity bound.
The virial theorem, introduced by Clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics, is studied by making use of symplectic formalism as an approach, in the case of both the Hamiltonian and Lagrangian systems. The possibility of establishing virial-like theorems from one-parameter groups of non-strictly canonical transformations is analysed; the case of systems with a position-dependent mass is also discussed. Using the modern symplectic approach to quantum mechanics, we arrive at the quantum virial theorem in full analogy with the classical case.
We discuss, from a non-equilibrium thermodynamics perspective, the proposals that quantum computational complexity can be described via geometric approaches, i.e. complexity geometry. We review the basic concepts of complexity geometry and the principle of minimum complexity. We describe how to construct a partition function on special unitary group manifold, which may be extended to derive the Jarynzki identity from a Fokker-Planck equation with a sink term. In this way, we obtain, by using the Hamilton-Jacobi equation, the Jarzynski identity that directly relates to quantum computational complexity for what might be carried out to provide a non-equilibrium framework for describing quantum complexity. In this paper, we mainly utilize this framework to prove the second law of complexity and obtain a fluctuation-dissipation theorem based on the complexity version of the Jarzynski identity. Besides, we choose the transverse field Ising model as an example to test our results numerically.