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arXiv:2609.26596

Hamiltonian Learning and Certification via Eigenphase Engineering

Myeongjin Shin, Yu Tong

quant-phSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

We study Hamiltonian learning for $k$-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown $s$-sparse Hamiltonian with fixed locality $k$ and $\|H\|_{\mathrm{op}}\leqΛ$, our algorithm recovers its Pauli coefficients to $\ell_2$ error $\varepsilon$ in total evolution time $\widetilde{O}(s/\varepsilon)$, matching the best known scaling while allowing a near-maximal step size $\widetildeΘ(1/Λ)$. As an independent contribution, we strengthen the evolution-time lower bound to $Ω_k(s^{1-1/(2k)}/\varepsilon)$, placing our algorithm within a factor $\widetilde{O}(s^{1/(2k)})$ of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a $\sqrt n$ overhead in $\ell_2$ learning time for $n$ qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.

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We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to $e^{-itH}$ for an unknown $k$-local Hamiltonian $H$ and a fully specified target Hamiltonian $H_0$, the goal is to decide whether $H$ is exactly equal to $H_0$ or differs from $H_0$ by at least $\varepsilon$ in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general $n$-qubit, $k$-local, traceless Hamiltonians, our procedure uses $O(c^k/\varepsilon)$ total evolution time for a universal constant $c$, and succeeds with high probability. In particular, for $O(1)$-local Hamiltonians, the total evolution time becomes $Θ(1/\varepsilon)$, matching the known $Ω(1/\varepsilon)$ lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of $H$, required controlled access to $e^{-itH}$, or achieved near-optimal guarantees only in restricted settings such as the Ising case ($k=2$). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.

quant-pharXiv:2512.09778•Dec 11, 2025•0 citations

Heisenberg-limited Hamiltonian learning without short-time control

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Characterizing quantum systems by learning their underlying Hamiltonians is a central task in quantum information science. While recent algorithmic advances have achieved near-optimal efficiency in this task, they critically rely on accessing arbitrarily short-time dynamics. This reliance poses severe experimental challenges due to finite control bandwidth and transient pulse errors. In this work, we demonstrate that Heisenberg-limited Hamiltonian learning can be achieved without short-time control. We introduce a framework in which every query to the unknown dynamics has duration at least a prescribed minimum time $T$, and show that this restriction does not preclude Heisenberg-limited scaling. The key ingredient is a method for emulating the continuous quantum control required by iterative learning algorithms using only such lower-bounded evolution times. This reduces the learning task to sparse pure-state tomography. Notably, for logarithmically sparse Hamiltonians, our algorithm achieves the information-theoretically optimal $1/\varepsilon$ scaling in total evolution time for any arbitrary constant minimum evolution time $T$. For many-body (polynomially sparse) systems, we uncover a rigorous quantitative tradeoff, showing that the minimum required evolution time can be significantly relaxed from the standard limit at a polynomial cost in total evolution time. Our results affirmatively resolve a prominent open problem in the field and reveal that high-bandwidth, ultra-short pulses are not fundamentally necessary for optimal quantum learning.

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Learning about a Hamiltonian $H$ from its time evolution $e^{-iHt}$ is a fundamental task in quantum science. A flurry of recent work has developed powerful new algorithms with provable guarantees for this task, for a variety of natural settings. Despite this, relatively little is known about lower bounds for learning Hamiltonians. In particular, in the natural setting where we assume $H$ is a $k$-local Hamiltonian on $n$ qubits, all existing algorithms require total evolution time at least $n^{Ω(k)}$ to learn the parameters of $H$, and it remained open whether one could obtain even faster algorithms -- or at the very least, if one could obtain better runtimes for simpler tasks, such as estimating a single designated coefficient of the Hamiltonian. In this work we show the answer is essentially no, by obtaining strong lower bounds for these problems. We find that not only do $k$-local Hamiltonians require $n^{Ω(k)}$ time evolution or interactions to learn, but also that in several senses, learning anything about a Hamiltonian is just as hard as learning everything. In particular, we find the same $n^{Ω(k)}$ lower bound holds for learning a single coefficient of a $k$-local Hamiltonian $H$, even if the rest of $H$ is already known. We also show an $n^{Ω(k)}$ lower bound for the task of effective Hamiltonian learning, where one seeks only to learn a unitary that approximately implements time evolution of $H$. Several related lower bounds, such as for general sparse (but not necessarily local) $H$ are also given. On the technical side, we make a new connection between Hamiltonian learning lower bounds and the analysis of Boolean functions, where we introduce a novel extremal property that may be of independent interest.

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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.

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Optimal Ansatz-free Hamiltonian Learning In Situ

Taiqi Zhou, Weiyuan Gong

Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian $H$ with $||H||\leqΛ$ in total evolution time of $Θ(\fracΛ{ε^2}\log(\fracΛε))$. The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of $Ω(\fracΛ{ε^2}\log(\fracΛε))$. Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on $Λ$ instead of $\varepsilon$, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.

quant-pharXiv:2606.19486•Jun 19, 2026•0 citations
We study Hamiltonian learning for kkk-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown sss-sparse Hamiltonian with fixed locality kkk and ∥H∥op≤Λ\|H\|_{\mathrm{op}}\leqΛ∥H∥op​≤Λ, our algorithm recovers its Pauli coefficients to ℓ2\ell_2ℓ2​ error ε\varepsilonε in total evolution time O~(s/ε)\widetilde{O}(s/\varepsilon)O(s/ε), matching the best known scaling while allowing a near-maximal step size Θ~(1/Λ)\widetildeΘ(1/Λ)Θ(1/Λ). As an independent contribution, we strengthen the evolution-time lower bound to Ωk(s1−1/(2k)/ε)Ω_k(s^{1-1/(2k)}/\varepsilon)Ωk​(s1−1/(2k)/ε), placing our algorithm within a factor O~(s1/(2k))\widetilde{O}(s^{1/(2k)})O(s1/(2k)) of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a n\sqrt nn​ overhead in ℓ2\ell_2ℓ2​ learning time for nnn qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.
We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to e−itHe^{-itH}e−itH for an unknown kkk-local Hamiltonian HHH and a fully specified target Hamiltonian H0H_0H0​, the goal is to decide whether HHH is exactly equal to H0H_0H0​ or differs from H0H_0H0​ by at least ε\varepsilonε in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general nnn-qubit, kkk-local, traceless Hamiltonians, our procedure uses O(ck/ε)O(c^k/\varepsilon)O(ck/ε) total evolution time for a universal constant ccc, and succeeds with high probability. In particular, for O(1)O(1)O(1)-local Hamiltonians, the total evolution time becomes Θ(1/ε)Θ(1/\varepsilon)Θ(1/ε), matching the known Ω(1/ε)Ω(1/\varepsilon)Ω(1/ε) lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of HHH, required controlled access to e−itHe^{-itH}e−itH, or achieved near-optimal guarantees only in restricted settings such as the Ising case (k=2k=2k=2). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.
Characterizing quantum systems by learning their underlying Hamiltonians is a central task in quantum information science. While recent algorithmic advances have achieved near-optimal efficiency in this task, they critically rely on accessing arbitrarily short-time dynamics. This reliance poses severe experimental challenges due to finite control bandwidth and transient pulse errors. In this work, we demonstrate that Heisenberg-limited Hamiltonian learning can be achieved without short-time control. We introduce a framework in which every query to the unknown dynamics has duration at least a prescribed minimum time TTT, and show that this restriction does not preclude Heisenberg-limited scaling. The key ingredient is a method for emulating the continuous quantum control required by iterative learning algorithms using only such lower-bounded evolution times. This reduces the learning task to sparse pure-state tomography. Notably, for logarithmically sparse Hamiltonians, our algorithm achieves the information-theoretically optimal 1/ε1/\varepsilon1/ε scaling in total evolution time for any arbitrary constant minimum evolution time TTT. For many-body (polynomially sparse) systems, we uncover a rigorous quantitative tradeoff, showing that the minimum required evolution time can be significantly relaxed from the standard limit at a polynomial cost in total evolution time. Our results affirmatively resolve a prominent open problem in the field and reveal that high-bandwidth, ultra-short pulses are not fundamentally necessary for optimal quantum learning.
Learning about a Hamiltonian HHH from its time evolution e−iHte^{-iHt}e−iHt is a fundamental task in quantum science. A flurry of recent work has developed powerful new algorithms with provable guarantees for this task, for a variety of natural settings. Despite this, relatively little is known about lower bounds for learning Hamiltonians. In particular, in the natural setting where we assume HHH is a kkk-local Hamiltonian on nnn qubits, all existing algorithms require total evolution time at least nΩ(k)n^{Ω(k)}nΩ(k) to learn the parameters of HHH, and it remained open whether one could obtain even faster algorithms -- or at the very least, if one could obtain better runtimes for simpler tasks, such as estimating a single designated coefficient of the Hamiltonian. In this work we show the answer is essentially no, by obtaining strong lower bounds for these problems. We find that not only do kkk-local Hamiltonians require nΩ(k)n^{Ω(k)}nΩ(k) time evolution or interactions to learn, but also that in several senses, learning anything about a Hamiltonian is just as hard as learning everything. In particular, we find the same nΩ(k)n^{Ω(k)}nΩ(k) lower bound holds for learning a single coefficient of a kkk-local Hamiltonian HHH, even if the rest of HHH is already known. We also show an nΩ(k)n^{Ω(k)}nΩ(k) lower bound for the task of effective Hamiltonian learning, where one seeks only to learn a unitary that approximately implements time evolution of HHH. Several related lower bounds, such as for general sparse (but not necessarily local) HHH are also given. On the technical side, we make a new connection between Hamiltonian learning lower bounds and the analysis of Boolean functions, where we introduce a novel extremal property that may be of independent interest.
We study the problem of learning local Lindbladians from black-box access to the physical evolution, where the goal is to estimate all Hamiltonian and dissipative coefficients. For local Lindbladians with low-intersection dissipators and local dynamical strength at most ΛΛΛ, we design an algorithm that learns the coefficients to target accuracy ε\varepsilonε, whose number of channel uses and total evolution time scale as O~(Λ2/ε2)\widetilde{O}(Λ^2/\varepsilon^2)O(Λ2/ε2) and O~(Λ/ε2)\widetilde{O}(Λ/\varepsilon^2)O(Λ/ε2), respectively, with only logarithmic dependence on the number of qubits. The algorithm is built directly from finite-time channel probes. It runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions. The algorithm is non-adaptive, uses no ancillas, and requires only random product states as inputs followed by random Pauli measurements. The method does not require knowing the structure of the Lindbladian in advance. We prove matching lower bounds that establish the near-optimality of the algorithm in both resources. By constructing a family of single-qubit dephasing Lindbladians, we show that any algorithm, even an adaptive one with arbitrary ancillas and measurements, requires Ω(Λ2/ε2)Ω(Λ^2/\varepsilon^2)Ω(Λ2/ε2) channel uses and Ω(Λ/ε2)Ω(Λ/\varepsilon^2)Ω(Λ/ε2) total evolution time. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.
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.
Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian HHH with ∣∣H∣∣≤Λ||H||\leqΛ∣∣H∣∣≤Λ in total evolution time of Θ(Λε2log⁡(Λε))Θ(\fracΛ{ε^2}\log(\fracΛε))Θ(ε2Λ​log(εΛ​)). The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of Ω(Λε2log⁡(Λε))Ω(\fracΛ{ε^2}\log(\fracΛε))Ω(ε2Λ​log(εΛ​)). Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on ΛΛΛ instead of ε\varepsilonε, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.