Optimal Conversion from Classical to Quantum Randomness via Quantum Chaos (2410.05181v2)

Abstract: Quantum many-body systems provide a unique platform for exploring the rich interplay between chaos, randomness, and complexity. In a recently proposed paradigm known as deep thermalization, random quantum states of system A are generated by performing projective measurements on system B following chaotic Hamiltonian evolution acting jointly on AB. In this scheme, the randomness of the projected state ensemble arises from the intrinsic randomness of the outcomes when B is measured. Here we propose a modified scheme, in which classical randomness injected during the protocol is converted by quantum chaos into quantum randomness of the resulting state ensemble. We show that for generic chaotic systems this conversion is optimal in that each bit of injected classical entropy generates as much additional quantum randomness as adding an extra qubit to B. This significantly enhances the randomness of the projected ensemble without imposing additional demands on the quantum hardware. Our scheme can be easily implemented on typical analog quantum simulators, providing a more scalable route for generating quantum randomness valuable for many applications. In particular, we demonstrate that the accuracy of a shadow tomography protocol can be substantially improved.

• The paper demonstrates that injecting classical randomness optimally converts to quantum randomness via chaotic Hamiltonian dynamics.
• The study provides theoretical and numerical evidence showing exponential enhancement of quantum ensemble randomness characterized by k-designs.
• The approach enables quantum devices with limited qubits to generate highly complex random states for improved benchmarking and quantum communication.

Optimal Conversion from Classical to Quantum Randomness via Quantum Chaos

The paper "Optimal Conversion from Classical to Quantum Randomness via Quantum Chaos" proposes a novel approach to enhance quantum randomness in quantum many-body systems by utilizing a new scheme that optimally converts classical randomness into quantum randomness through chaotic dynamics. This work, rooted in quantum information science, targets optimizing the generation of quantum randomness, which is crucial for various applications such as randomized benchmarking, quantum communication, and shadow tomography.

Quantum Chaos and Random Net Generation

The authors introduce a modified scheme based on the recently developed deep thermalization paradigm. In the traditional setup, chaotic evolution coupled with measurements creates random quantum states. However, the randomness is primarily due to intrinsic uncertainties in measurement outcomes. The proposed scheme innovatively injects additional classical randomness into the system, which is then transformed into quantum randomness through chaotic Hamiltonian dynamics.

The paper demonstrates that this injection of classical randomness can be achieved optimally. Specifically, for chaotic systems, each bit of classical entropy introduced into the system can generate quantum randomness equivalent to adding an extra qubit to the measured bath in the absence of injected classical randomness. This strategy is shown to increase the randomness of the projected quantum state ensemble significantly without necessitating additional qubit resources.

Numerical and Theoretical Insights

The paper provides both theoretical and numerical evidence to support the optimality of the classical to quantum randomness conversion mechanism. Theoretically, the authors derive that ensemble randomness characterized by $k$-designs is exponentially improved by classical randomness. For a given chaotic system with a 1D mixed-field Ising Hamiltonian, the introduction of classical randomness is shown to increase the size of the projected ensemble by factors up to $2^{S_c}$, where $S_c$ is the classical entropy.

Numerically, the efficacy of the technique is exemplified through simulations involving chaotic Hamiltonians. The results, which align with derived theoretical bounds, confirm an exponential decrease in the Hilbert-Schmidt distance of the projected ensemble to the Haar ensemble, thus illustrating a stronger correlation to higher-order quantum designs.

Practical Implications and Applications

This research has substantial implications for both theoretical understanding and practical implementations of quantum randomness. Practically, the proposed method can be readily implemented on analog quantum simulators, which are prevalent in modern experimental setups. These simulations have the potential to generate highly complex random states pivotal in quantum computing, particularly in refining shadow tomography estimations.

For quantum devices that currently suffer from a limited number of physical qubits, this approach offers a way to leverage available classical resources to enhance the quality of the random quantum states generated. This is particularly beneficial in scenarios involving quantum verification, communication protocols, and quantum benchmarking tasks.

Conclusion and Future Directions

In conclusion, the paper successfully demonstrates a framework for optimal conversion of classical randomness into quantum randomness via chaotic dynamics. The findings hold potential for enhancing quantum randomness generation, thereby aiding various pivotal applications in quantum information processing. Future work may explore further optimizing this conversion process and exploring various Hamiltonian regimes to fully exploit the robustness and universality of this novel approach. Moreover, understanding the interplay between classical randomness and quantum resources can open new pathways in the development of more efficient quantum algorithms and protocols.