Ansatz-free Hamiltonian learning with Heisenberg-limited scaling

TL;DR

This work resolves a long-standing open question by providing the first ansatz-free quantum algorithm for learning arbitrary sparse Hamiltonians with Heisenberg-limited scaling using only black-box evolution queries and simple quantum controls. The protocol alternates structure learning (identifying nonzero Pauli terms) with coefficient learning (estimating coefficients via Hamiltonian reshaping and robust frequency estimation), organized hierarchically to boost small terms while canceling large learned terms. It achieves Heisenberg-limited total time, with ancilla-assisted and ancilla-free variants, and remains robust to SPAM errors, as demonstrated on nonlocal spin models and Rydberg-based analog simulations. The work also proves fundamental trade-offs between total evolution time and quantum control, bridging Hamiltonian learning, quantum metrology, and verifiable quantum simulation, and laying groundwork for future open-system and noisy hardware applications.

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.

Figures (5)

• Figure 1: (a) Quantum circuit for the structure-learning subroutine $\mathcal{A}^I$. It prepares n pairs of 2-qubit Bell states between the original and ancillary systems via transversal gates. The original system then evolves coherently under the unknown Hamiltonian $H$ and the engineered Hamiltonian $H_*$, where $H_*$ consists of the large terms in $H$ learned in previous steps with an opposite sign. The combined system is then measured on the n-pair Bell basis. Nontrivial outcomes have probabilities proportional to $\mu_i^2$, enabling inference of the interaction structure. (b) Quantum circuit for the coefficient-learning subroutine $\mathcal{A}^{II}$. By inserting random Pauli gates from a designed set into the unknown Hamiltonian’s evolution, the time evolution of a specific term is approximated, allowing the interaction strength $\mu_s$ to be extracted via robust frequency estimation. (c) Combining these subroutines enables hierarchical coefficient estimation, achieving Heisenberg-limited scaling.
• Figure 2: Ansatz-free Hamiltonian learning with non-local and many-body interactions. (a) Learning a 1D disordered XY model with nearest-neighbor interactions ($X_iX_{i+1} + Y_iY_{i+1}$), supplemented by additional non-local cross-talk and an all-to-all coupling. The ansatz-free method (this work) is compared with the vanilla approach based on a nearest-neighbor ansatz HuangTongFangSu2023learning. Each run uses 2000 measurements for structure learning and 1000 for coefficient estimation. The inset shows total measurement time versus target precision; a fit yields $T_{\text{total}} \propto \text{precision}^{-0.96}$, consistent with Heisenberg-limited scaling. The red dashed box highlights failure of the vanilla method to recover non-local and many-body terms. False positives in structure learning (Pauli strings) are shown; coefficient learning confirms them as negligible. (b) Learning the Hamiltonian of a 1D neutral atom chain with $10\mu \text{m}$ spacing, uniform Rabi drive ($\Omega = 1.5/2\pi$ MHz), and global detuning ($\Delta = -4/2\pi$ MHz). Our method identifies the correct power-law decay from dipolar interactions after 5 rounds of structure and coefficient learning, using the same measurement counts as in (a). In contrast, the vanilla method captures only nearest-neighbor terms. All error bars indicate one standard deviation.
• Figure 3: Verifying effective Hamiltonians in engineered analog quantum simulations. (a) A 1D array of three Rydberg atoms spaced by $8.9~\mu\text{m}$ is driven by time-dependent Rabi frequency $\Omega(t)$ and global detuning $\Delta(t)$, engineered to approximate evolution under a symmetry-protected topological (SPT) Hamiltonian with three-body terms $Z_iX_{i+1}Z_{i+2}$, achieving a unitary fidelity of 90$\%$. Verifying the resulting effective Hamiltonian is essential for validating analog quantum simulations and pulse engineering. Applying our ansatz-free learning method, we identify ZXZ as the leading interaction. (b) Extending the same pulse to a five-atom array results in a different effective Hamiltonian. Our method successfully identifies all significant Pauli terms, revealing that the three-atom ansatz fails to generalize. True values are obtained via exact simulation; error bars represent one standard deviation.
• Figure 4: (a) Learning tree representation. Each node $u$ represents an experiment. Starting from the root experiment $r$, the number of child nodes depends on the possible POVM measurements. The transition probabilities are determined by Born's rule (also see \ref{['app_eq:transition_prob']}). After $N_\text{exp}$ experiments, one arrives at the leaves $l$ of the learning tree. (b) In each node u, the learning model prepares an arbitrary state $\rho_u$, applies discrete control channels $\mathcal{C}^u_{k}$, and queries real-time Hamiltonian evolutions with time $\tau_k^{u}$ multiple times. The protocols can also incorporate ancilla qubits (quantum memory).
• Figure S1: (a) Learning tree representation. Each node $u$ represents an experiment. Starting from the root experiment $r$, the number of child nodes depends on the possible POVM measurements. The transition probabilities are determined by \ref{['app_eq:transition_prob']}. After $N_\text{exp}$ experiments, one arrives at the leaves $l$ of the learning tree. (b) In each node u, the learning model prepares an arbitrary state $\rho_u$, applies discrete control channels $\mathcal{C}^u_{k}$, and queries real-time Hamiltonian evolutions with time $\tau_k^{u}$ multiple times. The protocols can also incorporate ancilla qubits (quantum memory).

Theorems & Definitions (25)

• Theorem 1: 2-copy Heisenberg-limited Hamiltonian learning algorithm
• Theorem 2: Single-copy Heisenberg-limited Hamiltonian learning algorithm
• Definition 1: Hierarchical learning subroutine
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof