Holographic fluctuations and the principle of minimal complexity

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 K qubits, and the positional entropy of a related classical system with 2 K 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-thanmaximal 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.

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“…More specifically, are they somehow the same? A similar suggestion in a slightly different context was recently proposed in [34].…”

Section: • Least Action and Least Computationsupporting

confidence: 78%

Second law of quantum complexity

2018

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“…), the precise value of the complexity will depend on the details of all of these choices. 41 The ambiguity in our reference state, i.e., the choice ω 0 , was already seen to modify the complexity in an interesting way in the preceding discussion.…”

Section: Ambiguities and Other Miscellaneous Complaintsmentioning

confidence: 71%

Circuit complexity in quantum field theory

Jefferson

Myers

2017

J. High Energ. Phys.

447

972

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Abstract:Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.

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“…where the second normal applies for both regulator surfaces next to the past and future singularities. For a constant r surface in the metric (2.1), the trace of the extrinsic curvature is given by 14) and as a result, we obtain…”

Section: Bulk Contributionmentioning

confidence: 99%

On the time dependence of holographic complexity

Carmi

Chapman

Marrochio

et al. 2017

J. High Energ. Phys.

289

554

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Abstract:We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. For either conjecture, we find that the late time limit for the rate of change of complexity is saturated at times of the order of the inverse temperature. Adding a charge to the eternal black holes washes out the early time behaviour, i.e. complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

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Holographic fluctuations and the principle of minimal complexity

Cited by 18 publications

References 66 publications

Second law of quantum complexity

Second law of quantum complexity

Circuit complexity in quantum field theory

On the time dependence of holographic complexity

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