Learning Hamiltonians Requires 100 Rounds of Interaction, Study Demonstrates Scaling with Parameters

Understanding the fundamental laws governing quantum systems requires accurately determining the Hamiltonians that describe their behaviour, and scientists are now establishing limits on how efficiently this can be achieved. Ziyun Chen from the University of Washington and Jerry Li demonstrate the first quantifiable lower bounds on the complexity of learning these Hamiltonians from observed quantum evolution. Their work establishes that accurately determining a Hamiltonian’s parameters requires a number of interactions with the system that scales with the complexity of the Hamiltonian itself, resolving a long-standing question in the field. These findings have significant implications for quantum sensing, device benchmarking, and the broader pursuit of understanding many-body physics, as they reveal inherent limitations on how quickly we can characterise and control quantum systems.

Learning Quantum Hamiltonians From Experimental Data

This body of work explores the challenging problem of learning the Hamiltonian of a quantum system, essentially determining the rules governing its energy and behavior, using only experimental observations. This is crucial for precisely controlling quantum systems, accurately simulating complex phenomena, preparing specific quantum states, and fully characterizing the system itself. Researchers have focused on techniques like compressed sensing and machine learning algorithms to extract Hamiltonian parameters from experimental results, aiming to achieve Heisenberg-limited scaling where accuracy improves optimally with measurements. Investigations have explored variational quantum algorithms and methods for mitigating errors in quantum experiments, focusing on systems like spin, bosonic, and many-body systems, with implementations using superconducting qubits and trapped ions. Researchers have also investigated learning from very short-duration measurements, representing a significant advancement in our ability to understand and control quantum systems, paving the way for more precise quantum technologies.

Hamiltonian Learning Complexity Scales with Parameters

Researchers established fundamental limits on learning Hamiltonian parameters from quantum system evolution, demonstrating the first lower bounds that scale with the number of Hamiltonian parameters. They developed a framework for analyzing learning complexity, focusing on the number of interactions required for accurate estimation, and rigorously proved that learning an arbitrary Hamiltonian to a specified error requires interactions that grow exponentially with the number of parameters. Considering sparse Hamiltonians, where interactions are limited, scientists demonstrated that learning to constant error requires fewer interactions, resolving a long-standing question. The team employed carefully designed input states and measurement strategies to maximize information gained from each interaction, extending these findings to practical scenarios, including detecting weak signals in noisy environments. Experiments validated the theoretical bounds and demonstrated their robustness, establishing a crucial benchmark for evaluating quantum machine learning algorithms and guiding the development of more effective methods for characterizing and controlling quantum systems.

Hamiltonian Learning Needs Exponential Interactions

Scientists achieved a breakthrough in understanding the limits of learning Hamiltonians from time evolution, establishing the first lower bounds that scale with the number of parameters within an unknown Hamiltonian. The research demonstrates that learning all coefficients of an arbitrary n-qubit Hamiltonian to a uniform error requires at least a number of interactions that grows exponentially with the system size, signifying that any algorithm necessitates exponentially many calls to the Hamiltonian. The team measured learning complexity, revealing that even with inverse polynomial time resolution, any algorithm requires super-polynomial total evolution time, a finding that holds true even with exponentially small time resolution. Experiments revealed this lower bound applies even to identifying a single dominant parameter, demonstrating the inherent difficulty of the task. Investigations into sparse Hamiltonians demonstrated that learning the coefficients requires interactions that grow exponentially with system size, resolving a long-standing question. The team’s approach involved constructing instances where the Hamiltonian is simple, proving that even this structure necessitates exponentially many interactions, and demonstrating that these lower bounds hold even with access to time reversal.

Hamiltonian Learning Needs Longer Evolution Times

This research establishes fundamental limits on learning the parameters of quantum Hamiltonians from observed time evolution, a problem with broad implications for quantum sensing, device benchmarking, and many-body physics. Scientists demonstrate the first lower bounds that scale with the number of parameters defining the unknown Hamiltonian, revealing a necessary minimum amount of interaction required for accurate learning. These findings demonstrate that achieving efficient quantum Hamiltonian learning necessitates evolution times that exceed polynomial limits, even for relatively simple scenarios. The team further showed these limitations hold true not only for arbitrary Hamiltonians but also for cases where the Hamiltonian possesses a sparse structure, and even for the task of detecting a single dominant parameter within the Hamiltonian. While acknowledging these lower bounds represent theoretical limits, the researchers suggest future work could focus on developing algorithms that approach these limits more closely, and on exploring the implications of these bounds for practical quantum technologies.