Computational pseudorandomness, the wormhole growth paradox, and constraints on the AdS/CFT duality

TL;DR

The paper investigates the wormhole growth paradox in AdS/CFT through a computational and cryptographic lens. It proposes that pseudorandom quantum states can arise within the CFT under scrambling dynamics and limited perturbations, leading to a tension: if wormhole volume equals boundary circuit complexity, either the AdS/CFT dictionary is exponentially complex or quantum gravity violates the quantum Extended Church-Turing thesis. A tailored PRS construction within the CFT domain is presented, along with black-box security arguments and a detailed discussion of wormhole volume being computable in a non-physical sense. The work highlights deep implications for the feasibility of the duality, suggesting that computational constraints may fundamentally shape holographic mappings and quantum gravity behavior.

Abstract

A fundamental issue in the AdS/CFT correspondence is the wormhole growth paradox. Susskind's conjectured resolution of the paradox was to equate the volume of the wormhole with the circuit complexity of its dual quantum state in the CFT. We study the ramifications of this conjecture from a complexity-theoretic perspective. Specifically we give evidence for the existence of computationally pseudorandom states in the CFT, and argue that wormhole volume is measureable in a non-physical but computational sense, by amalgamating the experiences of multiple observers in the wormhole. In other words the conjecture equates a quantity which is difficult to compute with one which is easy to compute. The pseudorandomness argument further implies that this is a necessary feature of any resolution of the wormhole growth paradox, not just of Susskind's Complexity=Volume conjecture. As a corollary we conclude that either the AdS/CFT dictionary map must be exponentially complex, or the quantum Extended Church-Turing thesis must be false in quantum gravity.

Figures (3)

• Figure 1: Representation of the tree $T(\sigma)$ for $k=3$. $R(\sigma)$ denotes the nodes in the second to last row of the tree, and $L(\sigma)$ denotes the leaves of the tree.
• Figure 2: Generalization of the proof to the case one can query both $\sigma$ and $\sigma^{-1}$ for $\ell=5$. By choosing whether or not to connect these trees at a random pair of points $a\in A,b\in B$, one changes a "yes" instance to a "no" instance. A hybrid argument then proves the lower bound on distinguishing these two cases.
• Figure 3: Visual representation of our integral \ref{['eq:integral']} for $k=3$. One connects the right $U$ indices to the left $U^\dagger$ indices via a permutation $\sigma$, and the left $U$ indices with the right $U^\dagger$ indices with a permutation $\tau$. The value of the term is $Wg(\sigma\tau^{-1},2^n)$ if all connected indices are identical and 0 otherwise. Here we illustrate two cases of $\sigma=\tau=Id$ and $\sigma=\tau=\pi$ for the permutation $\pi$ described above. One can easily check that the term on the left -- $\sigma=\tau=Id$ -- has value $0$ because the identifications of the connected indices produces a contradiction. On the other hand the term on the right has every variable identification repeated twice. For example $j_1=i_2'$ is repeated by both connecting the inner indices of the top left and bottom right node, as well as the outer indices of the second from top left and second from bottom right nodes. One can easily check that as the indices range over $\{0,1\}^n$ that this diagram has the maximum number of nonzero terms (i.e. number of settings of $i_1,i_2,\ldots$ in which all variable identification equalities are met).

Theorems & Definitions (8)

• Definition 4.1
• Conjecture 4.2: Complexity=Volume Conjecture susskind2014originalsusskind2016computational
• Definition 5.1
• Conjecture 5.2
• Claim 5.3
• Theorem 5.4
• proof
• Definition B.1