Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes, Part II

TL;DR

The paper investigates how gravity-like phenomena can be simulated in quantum devices by exploiting holographic duality, focusing on teleportation by size and the size-winding of operators in nearly-AdS$_2$ systems. It analyzes concrete models—SYK, random matrices, spin chains, and Brownian circuits—to demonstrate how size-winding signals bulk momentum and enables traversable-wormhole-like information transfer, with distinct high-temperature and low-temperature regimes. The work provides a detailed holographic interpretation, introduces a modified bulk protocol leveraging symmetry generators, and connects boundary operator growth to bulk geometry through entanglement wedges and quantum extremal surfaces. It also outlines near-term experimental benchmarks and the potential for lab-based demonstrations of quantum gravity phenomena in the NISQ era.

Abstract

In [1] we discussed how quantum gravity may be simulated using quantum devices and gave a specific proposal -- teleportation by size and the phenomenon of size-winding. Here we elaborate on what it means to do 'Quantum Gravity in the Lab' and how size-winding connects to bulk gravitational physics and traversable wormholes. Perfect size-winding is a remarkable, fine-grained property of the size wavefunction of an operator; we show from a bulk calculation that this property must hold for quantum systems with a nearly-AdS_2 bulk. We then examine in detail teleportation by size in three systems: the Sachdev-Ye-Kitaev model, random matrices, and spin chains, and discuss prospects for realizing these phenomena in near-term quantum devices.

Figures (14)

• Figure 1: Refinement of size-momentum duality to the level of wavefunctions. If the expansion of the time evolved thermal Pauli (or fermion) is $\rho_\beta^{1/2} P(t) =\sum c_P P$ (or $\rho_\beta^{1/2} \psi(t) =\sum c_u \Psi_u$), then one can define the winding size distribution $q(l) = \sum_{|P|=l} c_p^2$ (or $q(l) = \sum_{|u|=l} c_u^2$), in contrast to the conventional size distribution $\sum_{|P|=l} |c_p|^2$. We argue that the $q_l$ is the boundary analog of the bulk momentum wavefunction. The plot is the schematic drawing of the winding size distribution in the SYK model, near, but slightly before the scrambling time. This is the regime that the width of the distribution is of order $n$. One can observe that the Fourier transform of the winding size distribution mimics the behavior of the position of the infalling particle (measured, e.g., from the black hole horizon). Up middle. Magnitude and phase of winding size distribution. Bottom middle. The Fourier transform (or bulk location) is near the origin. Up left. Magnitude and phase of winding size distribution at a slightly earlier time. The size distribution is smaller and winds faster. Bottom left. The Fourier transform (or bulk location) is farther from the origin. Up right. The size distribution after acting by $e^{igV}$, with the proper value of $g$. Now the distribution is winding in the opposite direction. Bottom right. The Fourier transform shows that the particle is on the other side of the origin, a manifestation of the fact that the infalling particle has moved from one side of the horizon to the other side.
• Figure 2: A short summary teleportation by size, discussing different systems, different patterns of operator growth, and consequence of each growth pattern for signal transmission. Blue: Initial operator-size distribution. Red: Operator-size distribution of the time-evolved operator.
• Figure 3: Penrose diagram of wormholes. Left: Without the coupling, a message or particle inserted at early times on the left passes through the left horizon, and hits the singularity (the top line of the diagram). Right: In the presence of the left-right coupling, the message hits the negative energy shockwave (the thick blue line) created by the coupling. The effect of the collision is to rescue the message from behind the right horizon.
• Figure 4: The circuits considered in this paper, with $H_L = H_R^T$. Downward arrows indicate acting with the inverse of the time-evolution operator. In both protocols, the goal is to transmit information from the left to the right. The (a) state transfer protocol calls for us to discard the left message qubits ($A_L$) and replace them with our message $\Psi_{\mathrm{in}}$. The output state on the right then defines a channel applied to the input state. The (b) operator transfer protocol calls for the operator $O$ to be applied to $A_L$. Based on the choice of operator, the output state on the right is modified, similar to a perturbation-response experiment.
• Figure 5: Traversing the wormhole from the boundary point of view. (a) An operator $O$ inserted at negative time $t$ into the left boundary. (b) The (winding) size distribution of the thermal operator $O(t)\rho_\beta^{1/2}$, which is winding in the clockwise direction. (c) The size distribution after the application of $LR$ coupling. The coupling applies a linear phase to the size distribution of the thermal operator in part (b), unwinds it, and winds it in the opposite direction. In this way, we obtain a counter-clockwise size distribution corresponding to the thermal operator $\rho_\beta^{1/2}O(t)$. (d) As we saw in firstpaper, winding in the opposite direction corresponds to the operator inserted on the other boundary at a positive time. Thus, the coupling maps the operator $O$ on the left to operator $O^T$ on the right.