[Paper Review] Mixed Quantum-Semiclassical Simulation

Javier Gonzalez-Conde, Andrew Sornborger|arXiv (Cornell University)|Aug 30, 2023

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper proposes a hybrid quantum-semiclassical simulation framework using Koopman-von Neumann (KvN) formalism to encode semiclassical systems within a quantum simulation architecture. By leveraging the Hilbert space structure of KvN mechanics, the method enables efficient quantum simulation of mixed quantum-semiclassical (MQS) systems, demonstrating that field simulations can achieve resource advantages proportional to the ratio of quantum to semiclassical actions ($S_q/S_c$), with a notable potential for semiclassical gravity due to the large action ratio ($\sim 10^{-18}$).

ABSTRACT

We study the quantum simulation of mixed quantum-semiclassical (MQS) systems, of fundamental interest in many areas of physics, such as molecular scattering and gravitational backreaction. A basic question for these systems is whether quantum algorithms of MQS systems would be valuable at all, when one could instead study the full quantum-quantum system. We study MQS simulations in the context where a semiclassical system is encoded in a Koopman-von Neumann (KvN) Hamiltonian and a standard quantum Hamiltonian describes the quantum system. In this case, because KvN and quantum Hamiltonians are constructed with the same operators on a Hilbert space, standard theorems guaranteeing simulation efficiency apply. We show that, in this context, $ extit{many-body}$ MQS particle simulations give only nominal improvements in qubit resources over quantum-quantum simulations due to logarithmic scaling in the ratio, $S_q/S_c$, of actions between quantum and semiclassical systems. However, $ extit{field}$ simulations can give improvements proportional to the ratio of quantum to semiclassical actions, $S_q/S_c$. Of particular note, due to the ratio $S_q/S_c \sim 10^{-18}$ of particle and gravitational fields, this approach could be important for semiclassical gravity. We demonstrate our approach in a model of gravitational interaction, where a harmonic oscillator mediates the interaction between two spins. In particular, we demonstrate a lack of distillable entanglement generation between spins due to classical mediators, a distinct difference in dynamics relative to the fully quantum case.

Motivation & Objective

• To determine whether quantum algorithms for mixed quantum-semiclassical (MQS) systems offer a computational advantage over classical simulations.
• To investigate whether MQS simulations can outperform fully quantum-quantum simulations in terms of qubit resource usage.
• To develop a unified quantum simulation framework that combines standard quantum Hamiltonians with KvN-based semiclassical dynamics.
• To explore the role of action ratios ($S_q/S_c$) in determining resource efficiency for MQS simulations.
• To validate the approach through a proof-of-principle model of gravitational interaction mediated by a harmonic oscillator.

Proposed method

• Represent the semiclassical subsystem using the Koopman-von Neumann (KvN) formalism, which maps classical dynamics onto a Hilbert space with a Hermitian Hamiltonian.
• Encode both the quantum and semiclassical (KvN) systems within a common Hilbert space using standard quantum simulation techniques.
• Exploit the fact that KvN operators commute, enabling efficient time-evolution simulation via standard quantum algorithms.
• Apply Hamiltonian simulation algorithms (e.g., Trotter-Suzuki) to the combined KvN-quantum Hamiltonian, ensuring logarithmic scaling in system size.
• Use the ratio of actions ($S_q/S_c$) as a key parameter to assess resource efficiency relative to full quantum simulations.
• Implement a model of two spin-1/2 particles coupled via a harmonic oscillator, with the oscillator treated either quantum-mechanically or classically (KvN) to compare entanglement dynamics.

Figure 1: Examples of MQS systems of interest. (i) In molecular scattering, the Born-Oppenheimer approximation is used to describe the light electronic degrees of freedom quantum mechanically, but the heavy nuclear system semiclassically [ 1 , 2 , 3 , 4 ] . (ii) Backreaction in gravity is studied wi

Experimental results

Research questions

• RQ1 Can quantum algorithms for MQS systems provide a computational advantage over classical simulations, and if so, under what conditions?
• RQ2 Does the use of KvN formalism enable efficient quantum simulation of semiclassical systems within a unified quantum framework?
• RQ3 Under what circumstances do MQS simulations require fewer qubits than equivalent fully quantum-quantum simulations?
• RQ4 How does the ratio of quantum to semiclassical action ($S_q/S_c$) influence the resource efficiency of MQS simulations?
• RQ5 What differences in entanglement dynamics arise between quantum and classical mediators in a gravitational-like interaction model?

Key findings

• MQS simulations using the KvN formalism achieve logarithmic resource scaling with system size, enabling efficient simulation of classical dynamics within a quantum framework.
• For many-body particle systems, the resource advantage of MQS simulations over full quantum simulations is nominal, scaling logarithmically with the action ratio $S_q/S_c$.
• In field-theoretic settings, MQS simulations can achieve resource reductions proportional to the action ratio $S_q/S_c$, offering significant potential for systems with large $S_q/S_c$.
• The action ratio for particle and gravitational fields is estimated at $S_q/S_c \sim 10^{-18}$, suggesting that MQS simulations could be highly advantageous for semiclassical gravity.
• In the proof-of-principle gravitational model, classical mediators (KvN oscillator) fail to generate distillable entanglement between spins, in contrast to the quantum oscillator case.
• The simulation framework enables direct comparison of entanglement dynamics under quantum versus classical backreaction, highlighting fundamental differences in quantum information behavior.

Figure 2: Scheme followed to obtain a quantum representation of semiclassical-dynamics. First, we transform the state $\hat{\rho}$ , and Hamiltonian of our system $\hat{H}$ to phase space via the Wigner Transform. In phase space, we obtain a dynamical equation associated with a quasiprobability dist