Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

TL;DR

This article surveys quantum computational complexity from its quantum-information foundations to its holographic applications, emphasizing Nielsen’s geometric framework and the extension to Gaussian states, QFT, and CFT. It then compares two leading holographic proposals (CV and CA) for complexity, tests them with shock waves and subregions, and discusses tensor-network perspectives. The work highlights how complexity captures growth, chaos, and thermal phenomena in black holes and quantum many-body systems, while exposing open problems in universality, scheme dependence, and connections to bulk dynamics. Overall, it provides a coherent, accessible bridge between quantum computation, field theory, and holography, outlining concrete avenues for further development.

Abstract

Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity can also serve to study a completely different physical problem - that of information processing inside black holes. Quantum computational complexity was suggested as a new entry in the holographic dictionary, which extends the connection between geometry and information and resolves the puzzle of why black hole interiors keep growing for a very long time. In this pedagogical review, we present the geometric approach to complexity advocated by Nielsen and show how it can be used to define complexity for generic quantum systems; in particular, we focus on Gaussian states in QFT, both pure and mixed, and on certain classes of CFT states. We then present the conjectured relation to gravitational quantities within the holographic correspondence and discuss several examples in which different versions of the conjectures have been tested. We highlight the relation between complexity, chaos and scrambling in chaotic systems. We conclude with a discussion of open problems and future directions. This article was written for the special issue of EPJ-C Frontiers in Holographic Duality.

Figures (22)

• Figure 1: Illustration of a circuit implementing the unitary transformation $\exp{\left(i \alpha \prod_i \sigma_z^i\right)}$.
• Figure 2: Illustration of a circuit representing time evolution according to a k-local (in this case 2-local) Hamiltonian.
• Figure 3: Illustration of the time dependence of complexity during chaotic Hamiltonian evolution. The complexity grows linearly until it reaches its maximal value which is exponential in the number of degrees of freedom, and is expected to decrease significantly around the quantum recurrence time which is doubly exponential in the number of degrees of freedom in the system, once the full unitary group has been explored.
• Figure 4: Illustration of the switchback effect. The perturbation $W$ is acted on by $U(t)$ on the left and $U^\dagger(t)$ on the right to create the precursor operator. Two qubit gates participating in the most efficient preparation of $U(t)$ are labeled $g_i$ and they appear as light-purple circles before applying them to $W$ (and as light-red circles after being applied). Estimating the complexity of the precursor operator at different times depend on delicate cancellations which can be seen after applying the gates. For example, the gate $g_2$ commutes with the perturbation and the previously applied gates and therefore does not contribute to the complexity.
• Figure 5: Illustration of the time dependence of complexity of the precursor. An initial exponential regime is followed by linear growth starting at the scrambling time $n_*$.