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arXiv:2403.00069 · Physical Review A

Learning interacting fermionic Hamiltonians at the Heisenberg limit

Arjun Mirani, Patrick M. Hayden

Published Dec 23, 2024• 4 citations
arXivHTMLPDF

Abstract

Efficiently learning an unknown Hamiltonian given access to its dynamics is a problem of interest for quantum metrology, many-body physics, and machine learning. A fundamental question is whether learning can be performed at the Heisenberg limit, where the Hamiltonian evolution time scales inversely with the error $\ensuremath{\varepsilon}$ in the reconstructed parameters. The Heisenberg limit has previously been shown to be achievable for certain classes of qubit and bosonic Hamiltonians. Most recently, a Heisenberg-limited learning algorithm was proposed for a simplified class of fermionic Hubbard Hamiltonians restricted to real hopping amplitudes and zero chemical potential at all sites, along with on-site interactions. In this work, we provide an algorithm to learn a more general class of fermionic Hubbard Hamiltonians at the Heisenberg limit, allowing complex hopping amplitudes and nonzero chemical potentials in addition to the on-site interactions, thereby including several models of physical interest. The required evolution time across all experiments in our protocol is $O(1/\ensuremath{\varepsilon})$ and the number of experiments required to learn all the Hamiltonian parameters is $O[\text{polylog}(1/\ensuremath{\varepsilon})]$, independent of system size as long as each fermionic mode interacts with $O(1)$ other modes. Unlike prior algorithms for bosonic and fermionic Hamiltonians, to obey fermionic parity superselection constraints in our more general setting, our protocol utilizes $O(N)$ ancillary fermionic modes, where $N$ is the system size. Each experiment involves preparing fermionic Gaussian states, interleaving time evolution with fermionic linear optics unitaries, and performing local occupation number measurements on the fermionic modes. The protocol is robust to a constant amount of state preparation and measurement error.

DOI
10.1103/physreva.110.062421

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Efficiently learning an unknown Hamiltonian given access to its dynamics is a problem of interest for quantum metrology, many-body physics, and machine learning. A fundamental question is whether learning can be performed at the Heisenberg limit, where the Hamiltonian evolution time scales inversely with the error \ensuremathε\ensuremath{\varepsilon}\ensuremathε in the reconstructed parameters. The Heisenberg limit has previously been shown to be achievable for certain classes of qubit and bosonic Hamiltonians. Most recently, a Heisenberg-limited learning algorithm was proposed for a simplified class of fermionic Hubbard Hamiltonians restricted to real hopping amplitudes and zero chemical potential at all sites, along with on-site interactions. In this work, we provide an algorithm to learn a more general class of fermionic Hubbard Hamiltonians at the Heisenberg limit, allowing complex hopping amplitudes and nonzero chemical potentials in addition to the on-site interactions, thereby including several models of physical interest. The required evolution time across all experiments in our protocol is O(1/\ensuremathε)O(1/\ensuremath{\varepsilon})O(1/\ensuremathε) and the number of experiments required to learn all the Hamiltonian parameters is O[polylog(1/\ensuremathε)]O[\text{polylog}(1/\ensuremath{\varepsilon})]O[polylog(1/\ensuremathε)], independent of system size as long as each fermionic mode interacts with O(1)O(1)O(1) other modes. Unlike prior algorithms for bosonic and fermionic Hamiltonians, to obey fermionic parity superselection constraints in our more general setting, our protocol utilizes O(N)O(N)O(N) ancillary fermionic modes, where NNN is the system size. Each experiment involves preparing fermionic Gaussian states, interleaving time evolution with fermionic linear optics unitaries, and performing local occupation number measurements on the fermionic modes. The protocol is robust to a constant amount of state preparation and measurement error.
Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In this Letter, we propose the first algorithm to achieve the Heisenberg limit for learning an interacting N-qubit local Hamiltonian. After a total evolution time of O(ε^{-1}), the proposed algorithm can efficiently estimate any parameter in the N-qubit Hamiltonian to ε error with high probability. Our algorithm uses ideas from quantum simulation to decouple the unknown N-qubit Hamiltonian H into noninteracting patches and learns H using a quantum-enhanced divide-and-conquer approach. The proposed algorithm is robust against state preparation and measurement error, does not require eigenstates or thermal states, and only uses polylog(ε^{-1}) experiments. In contrast, the best existing algorithms require O(ε^{-2}) experiments and total evolution time. We prove a matching lower bound to establish the asymptotic optimality of our algorithm.
We revisit the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries. Given black-box query access to an unknown FLO, previous proposals required O~(n5/ε2)\widetilde{\mathcal{O}}(n^5 / \varepsilon^2)O(n5/ε2) queries, where nnn is the system size and ε\varepsilonε is the error in diamond distance. These algorithms also use unphysical operations (i.e., violating fermionic superselection rules) and/or nnn auxiliary modes to prepare Choi states of the FLO. In this work, we establish efficient and experimentally friendly protocols that obey superselection, use minimal ancilla (at most 111 extra mode), and exhibit improved dependence on both parameters nnn and ε\varepsilonε. For arbitrary (active) FLOs this algorithm makes at most O~(n4/ε)\widetilde{\mathcal{O}}(n^4 / \varepsilon)O(n4/ε) queries, while for number-conserving (passive) FLOs we show that O(n3/ε)\mathcal{O}(n^3 / \varepsilon)O(n3/ε) queries suffice. The complexity of the active case can be further reduced to O~(n3/ε)\widetilde{\mathcal{O}}(n^3 / \varepsilon)O(n3/ε) at the cost of using nnn ancilla. This marks the first FLO learning algorithm that attains Heisenberg scaling in precision. As a side result, we also demonstrate an improved copy complexity of O~(nη2/ε2)\widetilde{\mathcal{O}}(n η^2 / \varepsilon^2)O(nη2/ε2) for time-efficient state tomography of ηηη-particle Slater determinants in ε\varepsilonε trace distance, which may be of independent interest.
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 square error ϵ using O(1/ϵ){\mathcal{O}}(1/\epsilon )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.
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/ZX/ZX/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
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 , 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.
Hamiltonian learning provides a systematic framework for reconstructing unknown quantum dynamics. However, existing protocols typically assume direct measurement access to the entire system. With fast single-qubit control and a connected reference backbone, we show that a single measurable qubit suffices to learn all O(N)O(N)O(N) independent parameters of a bounded-degree two-body Hamiltonian on NNN qubits at the Heisenberg limit. Crucially, our protocol uses robust SWAP gates synthesized by quantum signal processing, enabling coherent transfer of states evolving under distant Hamiltonian parameters to the measurable qubit. This transfer requires no prior calibration of the Hamiltonian parameters of the intermediate links. A parallel learning architecture achieves total query time O~(N)\widetilde{O}(N)O(N) for an NNN-qubit chain, while retaining Heisenberg-limited precision scaling. On the chain, these scalings match the fundamental precision and information-propagation lower bounds up to logarithmic factors. The framework further extends to arbitrary bounded-degree interaction graphs. Our results establish a scalable route to learning an extensive number of many-body Hamiltonian parameters through only a local measurement interface.