[Paper Review] Heisenberg-limited Hamiltonian learning for interacting bosons

Haoya Li, Yu Tong+3 more|arXiv (Cornell University)|Jul 10, 2023

Quantum many-body systemsPhysics and Astronomy3 citations

TL;DR

This paper presents a Heisenberg-limited protocol for learning interacting bosonic Hamiltonians using coherent states, beam splitters, phase shifters, and homodyne measurements. By applying random unitaries during evolution to enforce symmetry and generate an effective Hamiltonian, the method achieves root mean squared error $\epsilon$ with $\mathcal{O}(1/\epsilon)$ total evolution time, independent of system size, and is robust to state-preparation and measurement errors.

ABSTRACT

We develop a protocol for learning a class of interacting bosonic Hamiltonians from dynamics with Heisenberg-limited scaling. For Hamiltonians with an underlying bounded-degree graph structure, we can learn all parameters with root mean squared error $ε$ using $\mathcal{O}(1/ε)$ total evolution time, which is independent of the system size, in a way that is robust against state-preparation and measurement error. In the protocol, we only use bosonic coherent states, beam splitters, phase shifters, and homodyne measurements, which are easy to implement on many experimental platforms. A key technique we develop is to apply random unitaries to enforce symmetry in the effective Hamiltonian, which may be of independent interest.

Motivation & Objective

• Address the challenge of achieving Heisenberg-limited precision in Hamiltonian learning for many-body bosonic systems.
• Overcome the limitations of standard quantum limit protocols that scale as $\mathcal{O}(\epsilon^{-2})$.
• Develop a scalable, experimentally feasible protocol using only coherent states, linear optics, and homodyne measurements.
• Ensure robustness against state-preparation and measurement (SPAM) errors in realistic experimental settings.
• Extend the use of random unitary control—previously applied to spin systems—to the bosonic regime with unbounded Hamiltonian terms.

Proposed method

• Apply random unitaries during time evolution to transform the dynamics into an effective Hamiltonian with enhanced symmetry.
• Use coherent states as initial states and perform homodyne measurements to extract observable data.
• Implement beam splitters and phase shifters as the primary control operations, avoiding the need for squeezing.
• Employ a randomized averaging procedure over random unitaries to suppress noise and stabilize the effective Hamiltonian.
• Derive bounds on the trace norm deviation between the true and effective dynamics using operator commutator techniques.
• Leverage the bounded-degree graph structure of the underlying bosonic mode network to ensure scalability and parameter identifiability.

Experimental results

Research questions

• RQ1 Can Heisenberg-limited scaling be achieved for learning interacting bosonic Hamiltonians under realistic experimental constraints?
• RQ2 How can random unitary control be adapted to bosonic systems with unbounded Hamiltonian terms?
• RQ3 What is the minimal resource cost (in evolution time) required to learn Hamiltonian parameters with precision $\epsilon$?
• RQ4 To what extent can the protocol tolerate state-preparation and measurement errors?
• RQ5 Can symmetry enforcement via random unitaries lead to effective Hamiltonians that are easier to learn while preserving the original parameters?

Key findings

• The protocol achieves Heisenberg-limited scaling: total evolution time scales as $\mathcal{O}(1/\epsilon)$ to reach root mean squared error $\epsilon$, independent of system size.
• For Hamiltonians with bounded-degree graph structure, the method learns all parameters with $\mathcal{O}(1/\epsilon)$ total evolution time.
• The protocol is robust against constant-amplitude SPAM errors, maintaining accuracy under realistic noise conditions.
• The effective Hamiltonian is derived via random unitary averaging, and the deviation from true dynamics is bounded by $\mathcal{O}(N^2 t^2 / r \cdot \max\{|\xi_{jklm}|, |\omega_i|, |h_{i,j}|\}^2)$.
• The technique of enforcing symmetry through random unitaries is generalizable and may benefit other quantum simulation tasks.
• The method avoids the need for squeezing or entangled probes, relying only on coherent states and linear optical operations.