Ansatz-free Hamiltonian learning with Heisenberg-limited scaling

Abstract

Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term \emph{ansatz-free Hamiltonian learning}, remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system's real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.

Updated version with expanded explanations, added pseudocode, and new numerical demonstrations. 10 pages, 4 figures. HYH and MM contributed equally

References in corpus (25)

• Probing many-body dynamics on a 51-atom quantum simulator
• Logical quantum processor based on reconfigurable atom arrays
• Probing Topological Spin Liquids on a Programmable Quantum Simulator
• Quantum error correction below the surface code threshold
• Demonstration of fault-tolerant universal quantum gate operations
• Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors
• Symmetrised Characterisation of Noisy Quantum Processes
• Entanglement hamiltonians in two-dimensional conformal field theory
• Theoretical and Experimental Perspectives of Quantum Verification
• Learning many-body Hamiltonians with Heisenberg-limited scaling
• Entanglement Hamiltonians: from field theory, to lattice models and experiments
• Quantum advantages for Pauli channel estimation
• The learnability of Pauli noise
• Robust and Efficient Hamiltonian Learning
• The Negativity Hamiltonian: An operator characterization of mixed-state entanglement
• Pauli error estimation via Population Recovery
• Practical Black Box Hamiltonian Learning
• qSWIFT: High-order randomized compiler for Hamiltonian simulation
• Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix
• The advantage of quantum control in many-body Hamiltonian learning
• Robustly learning the Hamiltonian dynamics of a superconducting quantum processor
• Hamiltonian and Liouvillian learning in weakly-dissipative quantum many-body systems
• Structure learning of Hamiltonians from real-time evolution
• Learning interacting fermionic Hamiltonians at the Heisenberg limit
• Determining non-Hermitian parent Hamiltonian from a single eigenstate

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