Thoughts on Holographic Complexity and its Basis-dependence

Abstract

In this paper, we argue that holographic complexity should be a basis-dependent quantity. Computational complexity of a state is defined as a minimum number of gates required to obtain that state from the reference state. Due to this minimality, it satisfies the triangle inequality, and can be regarded as a (discrete version of) distance in the Hilbert space. However, we show a no-go theorem that any basis-independent distance cannot reproduce the behavior of the holographic complexity. Therefore, if holographic complexity is dual to a distance in the Hilbert space, it should be basis-dependent, i.e., it is not invariant under a change of the basis of the Hilbert space.

7 pages

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Cited by in corpus (28)

• Quantum Computation as Gravity
• Circuit complexity in interacting QFTs and RG flows
• Time Evolution of Complexity: A Critique of Three Methods
• Black holes, complexity and quantum chaos
• Complexity change under conformal transformations in AdS${}_3$/CFT${}_2$
• Complexity of Holographic Superconductors
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• Complexity and action for warped AdS black holes
• Time evolution of the complexity in chaotic systems: concrete examples
• Complexity growth rate during phase transitions
• Holographic Subregion Complexity in Einstein-Born-Infeld theory
• Holographic study of entanglement and complexity for mixed states
• Complexity of operators generated by quantum mechanical Hamiltonians
• Bulk reconstruction of metrics inside black holes by complexity
• Complexity growth in Gubser-Rocha models with momentum relaxation
• Circuit Complexity in ${\mathcal{Z}}_2$ $\mathcal{EEF}\mathcal{T}$
• Subregion complexity in holographic thermalization with dS boundary
• To be unitary-invariant or not?: a simple but non-trivial proposal for the complexity between states in quantum mechanics/field theory
• Subregion complexity and confinement-deconfinement transition in a holographic QCD model
• Complexity and scaling in quantum quench in $1+1$ dimensional fermionic field theories
• Properties of the (Un)Complexity of Subsystems
• Complexity growth for topological black holes by holographic method
• Weighting gates in circuit complexity and holography
• Effects of the acceleration on holographic complexity
• Causality Constraint on Circuit Complexity from $COSMOEFT$
• What kind of "complexity" is dual to holographic complexity?
• D5-brane on topological black holes
• Time dependent Interface in AdS Black Hole Spacetime