Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuous Variable Hamiltonian Learning at Heisenberg Limit via Displacement-Random Unitary Transformation

Xi Huang, Lixing Zhang, Di Luo

TL;DR

The paper tackles the challenge of learning continuous-variable Hamiltonians in infinite-dimensional spaces by introducing Displacement-Random Unitary Transformation (D-RUT), which converts a general finite-order bosonic Hamiltonian into a number-conserving effective operator and enables Heisenberg-limit coefficient learning via Robust Phase Estimation. It provides a two-step learning pipeline: first recover single-mode terms, then extract multi-mode couplings, and demonstrates a hierarchical strategy that reduces estimator variance compared to simultaneous recovery. The approach is shown to be robust to SPAM errors and is extendable to first quantization through Bogoliubov transformations, mapping to a reference Bogoliubov basis and enabling retrieval of physical parameters with near-optimal scaling. Practical implementations include iterative and parallel D-RUT procedures for single- and two-mode first-quantized Hamiltonians, highlighting broad applicability to CV quantum sensing, simulation, and metrology. Overall, the work offers a versatile, experimentally feasible framework for precise CV Hamiltonian characterization at the Heisenberg limit.

Abstract

Characterizing the Hamiltonians of continuous-variable (CV) quantum systems is a fundamental challenge laden with difficulties arising from infinite-dimensional Hilbert spaces and unbounded operators. Existing protocols for achieving the Heisenberg limit precision are often restricted to specific Hamiltonian structures or demand experimentally challenging resources. In this work, we introduce an efficient and experimentally accessible protocol, the Displacement-Random Unitary Transformation (D-RUT), that learns the coefficients of general, arbitrary finite-order bosonic Hamiltonians with a total evolution time scaling as $O(1/ε)$ for a target precision $ε$ robust to SPAM error. For multi-mode systems, we develop a hierarchical coefficients recovering strategy with superior statistical efficiency. Furthermore, we extend our protocol to first quantization, enabling the learning of fundamental physical parameters from Hamiltonians expressed in position and momentum operators at the Heisenberg limit.

Continuous Variable Hamiltonian Learning at Heisenberg Limit via Displacement-Random Unitary Transformation

TL;DR

The paper tackles the challenge of learning continuous-variable Hamiltonians in infinite-dimensional spaces by introducing Displacement-Random Unitary Transformation (D-RUT), which converts a general finite-order bosonic Hamiltonian into a number-conserving effective operator and enables Heisenberg-limit coefficient learning via Robust Phase Estimation. It provides a two-step learning pipeline: first recover single-mode terms, then extract multi-mode couplings, and demonstrates a hierarchical strategy that reduces estimator variance compared to simultaneous recovery. The approach is shown to be robust to SPAM errors and is extendable to first quantization through Bogoliubov transformations, mapping to a reference Bogoliubov basis and enabling retrieval of physical parameters with near-optimal scaling. Practical implementations include iterative and parallel D-RUT procedures for single- and two-mode first-quantized Hamiltonians, highlighting broad applicability to CV quantum sensing, simulation, and metrology. Overall, the work offers a versatile, experimentally feasible framework for precise CV Hamiltonian characterization at the Heisenberg limit.

Abstract

Characterizing the Hamiltonians of continuous-variable (CV) quantum systems is a fundamental challenge laden with difficulties arising from infinite-dimensional Hilbert spaces and unbounded operators. Existing protocols for achieving the Heisenberg limit precision are often restricted to specific Hamiltonian structures or demand experimentally challenging resources. In this work, we introduce an efficient and experimentally accessible protocol, the Displacement-Random Unitary Transformation (D-RUT), that learns the coefficients of general, arbitrary finite-order bosonic Hamiltonians with a total evolution time scaling as for a target precision robust to SPAM error. For multi-mode systems, we develop a hierarchical coefficients recovering strategy with superior statistical efficiency. Furthermore, we extend our protocol to first quantization, enabling the learning of fundamental physical parameters from Hamiltonians expressed in position and momentum operators at the Heisenberg limit.

Paper Structure

This paper contains 26 sections, 2 theorems, 85 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Learning a Single Mode Hamiltonian via D-RUT
  4. The D-RUT Method and Measurement Protocol
  5. The D-RUT method
  6. Derivation of the Constant Term $C(\beta)$
  7. Measurement via Robust Phase Estimation
  8. Coefficient Recovery Strategy
  9. Error Propagation Analysis
  10. Error Propagation in Radial Interpolation
  11. Error Propagation in inverse Fourier Transform
  12. Robustness under SPAM Errors
  13. Bounding the Lipschitz Constant $L_C$
  14. Learning Multi-Mode Hamiltonians
  15. Hierarchical Recovering Strategy
  16. ...and 11 more sections

Key Result

Theorem 1

Given a unitary access to a generic multi-mode bosonic Hamiltonian in the form of Eq. eq:gen_H_b, there exists a protocol that learns all Hamiltonian coefficients up to a Root-Mean-Square Error (RMSE) $\epsilon$, such that:

Figures (1)

  • Figure 1: The D-RUT based learning algorithm for single mode coefficients.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2