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Pre-Registered Finite-Size Predictions for Page-Curve Experiments on Near-Term Quantum Hardware

Ben Holland

Published Aug 9, 2026• 0 citations
Publisher

Abstract

The Page curve, the rise and fall of radiation entropy during unitary evaporation, is now measurable on quantum simulators: trapped-ion and superconducting platforms routinely prepare scrambled states of 10–50 qubits and estimate subsystem Rényi entropies by randomized measurements. Semiclassical gravity, including the quantum-extremal-surface and replica-wormhole machinery that derives the Page curve, is constructed in the G → 0 limit and makes no prediction at finite qubit number. We state a set of exact finite-N laws for a concrete evaporation protocol — the location and determinism of the fine-grained transition, its first-order (whole-island) character, a quantized entanglement barrier with a per-qubit mechanism identity, the barrier's monotone response to scrambling depth, and a one-particle "mode books" formula that predicts the transition time under partial (graybody) emission — together with the complete experimental design: register sizes, gate budgets (~84–348 two-qubit gates at 12–24 qubits), a one-line noise mitigation certified to 3× typical hardware error rates, a measurement plan in which no register larger than max(t, N_I − t + 1) qubits is ever measured, and a total shot budget of order 10⁵. Every prediction is stated with the condition that falsifies it. All numbers were fixed before any experiment; the reproduction package included with this deposit reproduces every claim, including the exact hardware-numbers table, in under a minute.

DOI
10.5281/zenodo.21862155

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A bstract If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, without resorting to typicality. Our argument is based on what we refer to as the “operator gas” approach, which allows one to understand the evolution of the microstate of the black hole from generic features of the Heisenberg evolution of operators. One key feature which leads to the Page curve is the possibility of dynamical processes where operators in the “gas” can “jump” outside the black hole, which we refer to as void formation processes. Such processes are initially exponentially suppressed, but dominate after a certain time scale, which can be used as a dynamical definition of the Page time. In the Hayden-Preskill protocol for young and old black holes, we show that void formation is also responsible for the transfer of information from the black hole to the radiation. We conjecture that void formation may provide a microscopic explanation for the recent semi-classical prescription of including islands in the calculation of the entanglement entropy of the radiation.

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The Page curve, the rise and fall of radiation entropy during unitary evaporation, is now measurable on quantum simulators: trapped-ion and superconducting platforms routinely prepare scrambled states of 10–50 qubits and estimate subsystem Rényi entropies by randomized measurements. Semiclassical gravity, including the quantum-extremal-surface and replica-wormhole machinery that derives the Page curve, is constructed in the G → 0 limit and makes no prediction at finite qubit number. We state a set of exact finite-N laws for a concrete evaporation protocol — the location and determinism of the fine-grained transition, its first-order (whole-island) character, a quantized entanglement barrier with a per-qubit mechanism identity, the barrier's monotone response to scrambling depth, and a one-particle "mode books" formula that predicts the transition time under partial (graybody) emission — together with the complete experimental design: register sizes, gate budgets (~84–348 two-qubit gates at 12–24 qubits), a one-line noise mitigation certified to 3× typical hardware error rates, a measurement plan in which no register larger than max(t, N_I − t + 1) qubits is ever measured, and a total shot budget of order 10⁵. Every prediction is stated with the condition that falsifies it. All numbers were fixed before any experiment; the reproduction package included with this deposit reproduces every claim, including the exact hardware-numbers table, in under a minute.
Quantum computers are emerging technologies expected to become important tools for exploring various aspects of fundamental physics in the future. Therefore, we pose the question of whether quantum computers can help us to study the Page curve and the black hole information dynamics, which has been a key focus in fundamental physics. In this regard, we rigorously examine the qubit transport model, a toy qubit model of black hole evaporation on IBM’s superconducting quantum computers, to shed light on this question. Specifically, we implement the quantum simulation of the scrambling dynamics in black holes using an efficient random unitary circuit. Furthermore, we employ the swap-based many-body interference protocol and the randomized measurement protocol to measure the entanglement entropy of Hawking radiation qubits in this model. Finally, by incorporating quantum error mitigation techniques into our challenging implementation of entanglement entropy measurement protocols on the IBM quantum hardware, we accurately determine the Rényi entropy in the qubit transport model, thus showcasing the utility of quantum computers for future investigations of complex quantum systems.
This tutorial develops the Page curve program from quantum entanglement and Hawking radiation to holographic entropy, replica wormholes, and random matrix theory. We derive the normalized random-state benchmark and its Marchenko–Pastur limit, explaining the roles of non-crossing partitions and spectral resolvents. We then distinguish averaging conditional entropies from calculating the entropy of a state with fluctuating energy or area. Gaussian broadening, finite reservoirs, and a quartic critical model illustrate how the preparation and the fluctuation law enter this distinction. Exact finite-dimensional examples separate averaged spectra from ensemble fluctuations and radiation data for one input from information about a quantum channel. Normalized annealed and quenched Rényi entropies have a common von Neumann limit, while their finite-order difference and connected replica moments measure additional ensemble information. Throughout, explicit calculations are distinguished from gravitational applications that require further assumptions. The resulting account provides a self-contained introduction to the established methods and to the remaining questions about microscopic information recovery.
The Page curve and the island formula describe a reorganization of quantum information in Hawking radiation across the Page time. We study how this reorganization appears in correlations among emissions collected at several retarded times. The late radiation is treated as a covariant multi-time process, or process tensor, and χisland=N∅−NRχ_{\mathrm{island}}=\mathcal{N}_{\varnothing}-\mathcal{N}_{R}χisland​=N∅​−NR​ measures the temporal memory removed when the early radiation register RRR is supplied. Operationally, this quantity is the corresponding change in the optimal distinguishability of the radiation process from a Markovian reference. We exhibit evaporation processes whose single-bin states and all process marginals below a finite order k∗≥3k_\ast\geq 3k∗​≥3 remain fixed and coincide with a CP-divisible reference, while χislandχ_{\mathrm{island}}χisland​ changes by order one at order k∗k_\astk∗​. The Page transition can therefore appear as a transition in the Markov order of the radiation process. When the interior mediates the late-time memory, entanglement-wedge reconstruction of the predictive interior from RRR Markovianizes the late process; conversely, when the late process has a minimal predictive memory and its future-affecting information is mediated by that memory, Markovianization by RRR supplies a simulator of its influence. A Hayden--Preskill evaporation comb and an evaporating-JT calculation realize this mechanism. The late radiation retains temporal memory across the transition, while the early radiation becomes sufficient for predicting its future multi-time statistics.
This undergraduate dissertation was completed as part of the BSc Computer Science degree at Newcastle University in May 2026. It investigates whether Page-curve-like behaviour can be reproduced in a simplified computational model of black hole evaporation using concepts from quantum information theory.The black hole information paradox arises from an apparent conflict between Hawking radiation and the unitary evolution required by quantum mechanics. Rather than attempting to simulate the full physical dynamics of an evaporating black hole, this project models the black hole and its emitted radiation as two subsystems of a finite-dimensional multi-qubit quantum system. The aim is to explore the information-theoretic behaviour associated with the Page curve through computational modelling of entanglement entropy.The model was implemented in Python using NumPy and Matplotlib. Random pure quantum states are generated and represented using density matrices, after which partial trace operations are used to obtain the reduced density matrix of the radiation subsystem. The von Neumann entropy of this subsystem is then calculated as the proportion of the overall quantum system assigned to radiation increases. Repeated simulations are averaged to examine the evolution of entanglement entropy over normalised evaporation time.The project evaluates the model using Page-curve shape, estimated Page time, variation of a mass-like evaporation parameter, scaling across different system sizes, statistical stability across repeated simulations, and validation tests using known quantum states. The baseline simulation reproduces the characteristic rise, peak, and fall associated with the Page curve, with maximum radiation entropy occurring close to the midpoint of the normalised evolution. Simulations across different qubit counts also produce broadly consistent Page-curve-like behaviour.The results should be interpreted as a computational demonstration of entanglement-entropy behaviour under a simplified information-theoretic model rather than as a complete physical simulation of black hole evaporation. The model does not include spacetime geometry, quantum field dynamics, back-reaction, or Hamiltonian time evolution. Its purpose is instead to provide a transparent and reproducible computational representation of the relationship between subsystem partitioning, quantum entanglement, entropy, and the Page curve.This record contains the final undergraduate dissertation.Degree: BSc Computer Science Institution: Newcastle University Year: 2026 Supervisor: Dr Jonte HanceThis work is an undergraduate dissertation and has not been peer-reviewed as a journal publication.
Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.
A bstract If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, without resorting to typicality. Our argument is based on what we refer to as the “operator gas” approach, which allows one to understand the evolution of the microstate of the black hole from generic features of the Heisenberg evolution of operators. One key feature which leads to the Page curve is the possibility of dynamical processes where operators in the “gas” can “jump” outside the black hole, which we refer to as void formation processes. Such processes are initially exponentially suppressed, but dominate after a certain time scale, which can be used as a dynamical definition of the Page time. In the Hayden-Preskill protocol for young and old black holes, we show that void formation is also responsible for the transfer of information from the black hole to the radiation. We conjecture that void formation may provide a microscopic explanation for the recent semi-classical prescription of including islands in the calculation of the entanglement entropy of the radiation.