Paper: Adaptive Learning of Quantum Hamiltonians (ICLR-2024)

Reviewer comment (Theory):
Oracle assumed is too strong and impractical. The oracle solves a very hard computational problem even in the classical setting, especially in the low temperature regime. The setting in this work is the quantum analogue of learning classical graphical models given an oracle that can always return the sufficient statistics given the energy function of a model. The main technical challenge that needs to be addressed in this field is the fact that there are no quantum versions of the pseudo-likelihood type methods that are used to learn classical Gibbs states. The existence of these methods in the quantum regime would bypass the need for these types of oracles. A very recent work (https://arxiv.org/abs/2310.02243) has shown that the computational complexity of solving this problem is polynomial in the size of the system, without assuming any strong oracles. This new method has much worse sample complexity, so a practical algorithm is still out of reach. In general oracles are useful to establish relative complexity of two tasks, inference and learning in this case. But for learning Gibbs states, from classical results, we already know that inference is much harder than learning. So the bounds established in this work are only marginally interesting.

Author response:
We thank the reviewer for a very informative report.

The reviewer's primary concern is the use of the adaptive oracle access model. Especially, the reviewer thought that the oracles assumed in this paper (referred to as adaptive oracles in the reply) were stronger than those in the Hamiltonian learning from Gibbs states (referred to as Gibbs oracles).
While adding adaptivity to the oracle does strengthen the requirements, we do not believe that the adaptive oracle is significantly different from the Gibbs oracle for the following reason. Both setups necessitate access to copies of Gibbs states or their local density matrices associated with a specific Hamiltonian. In the Hamiltonian learning from Gibbs states problem, the oracle is implicit in the problem, assuming that nature or a specific algorithm can prepare and provide such states for the true Hamiltonian. In the adaptive oracle setup, the Hamiltonian is adaptively chosen, but it shares the same locality as the true Hamiltonian. Thus, assuming a natural process or algorithm exists for preparing the true Hamiltonian, the same process could be employed for the adaptively chosen Hamiltonian. In this sense, it would be inaccurate to claim that Hamiltonian learning algorithms do not require strong oracles and that the adaptive oracle is significantly stronger than Gibbs oracles. We do agree with the reviewer that it is an important issue that needs to be addressed, and exploring under what conditions we can approximate the local density matrices of the Gibbs state is a fascinating question. This may, indeed, limit the algorithm to certain special cases where such an approximation is feasible.

The reviewer also said, "... from classical results, we know that inference is much harder than learning". This probably refers to the result of Montanari 2015. The claim is correct for "inference from sufficient statistics" and may not apply to the more general setting. The all-temperature learning algorithm (arXiv:2310.02243) the reviewer mentioned can be considered an *inference* algorithm where the data used is 2^\beta-local density matrices of the Gibbs state. In other words, it is possible to consider an inference algorithm where the local regions are larger than the locality of the Hamiltonian.

The reviewer asks for a comparison of the complexity of our algorithms with high-temperature learning results of arXiv:2108.04842. We will add a comparison in future revisions. However, we would like to emphasize that the focus of the paper is not to improve theoretical lower bounds at the expense of algorithm simplicity. Our paper aims to consider simpler algorithms grounded in classical theory and demonstrate how optimization heuristics can further enhance their practicality.

Reviewer comment (Novelty):
It is mostly using established techniques in optimization to solve a standard problem of matching sufficient statistics (essentially Maximum Likelihood). Novelty of this work mainly comes from the techniques used to establish the bounds on Hessian. The methods used to solve the problem itself are not new.

Author response:
We appreciate the reviewer's feedback.

Regarding the phase transition argument, we are convinced that adaptive oracles may be stronger in general. Thanks! However, in our algorithm, we initialize the Hamiltonian parameters with all zeros, corresponding to the infinite temperature case. The Hamiltonian then undergoes iterative updates towards the true Hamiltonian. Based on this process, we expect that the path of the Hamiltonian will remain in the same phase for most cases.

We do not see why the reviewer claimed that the problem we solve is strictly easier. Please see the discussion in the paper about these two problems for reasons that we consider them to be incomparable. Probably the reviewer was referring to the adaptive oracle again. We believe the study under this model is valuable for the following reasons.

First, the classical analogs of the algorithms we have used are state-of-the-art for the corresponding classical counterpart problems. The criticism regarding the adaptive oracle also applies to the classical case. In classical models, we also observe phase transition phenomena when the temperature changes, yet these classical algorithms have been proven to be valuable in practice. We expect a similar situation in the quantum case.

Second, our algorithm is much simpler compared to other existing quantum Hamiltonian learning algorithms. The simplicity of our algorithm also ensures that the quasi-Newton accelerations discussed in the paper perform well. This work is probably the first to consider quasi-Newton speedups in the Hamiltonian learning literature. We believe this is a must if any quantum Hamiltonian learning algorithm is ever used in practice because of its dramatic savings in computation time.

Third, quantum generalizations are significantly harder to analyze, and it is not clear that classical algorithms can be naturally generalized to the quantum case when the Hamiltonians are non-commutative. Our work developed techniques and methods to resolve the difficulties partially.

Anyway, we thank the reviewer for the helpful discussions. We will clarify the points in future revisions of the paper.

Reviewer comment (Theory):
Overall the method is not very efficient. Theory gives an exponential worst case run-time (which is absorbed into the oracle) and the field of quantum computing is too nascent to provide any high quality data to establish any practical claims regarding the algorithm. Overall the reviewer does not believe this work has made any good breakthroughs in solving this problem and considers it not a good fit for ICLR. The new Quantum BP method is interesting and might be a good research direction.

Author response:
We appreciate the reviewer's feedback.

Regarding the phase transition argument, we are convinced that adaptive oracles may be stronger in general. Thanks! However, in our algorithm, we initialize the Hamiltonian parameters with all zeros, corresponding to the infinite temperature case. The Hamiltonian then undergoes iterative updates towards the true Hamiltonian. Based on this process, we expect that the path of the Hamiltonian will remain in the same phase for most cases.

We do not see why the reviewer claimed that the problem we solve is strictly easier. Please see the discussion in the paper about these two problems for reasons that we consider them to be incomparable. Probably the reviewer was referring to the adaptive oracle again. We believe the study under this model is valuable for the following reasons.

First, the classical analogs of the algorithms we have used are state-of-the-art for the corresponding classical counterpart problems. The criticism regarding the adaptive oracle also applies to the classical case. In classical models, we also observe phase transition phenomena when the temperature changes, yet these classical algorithms have been proven to be valuable in practice. We expect a similar situation in the quantum case.

Second, our algorithm is much simpler compared to other existing quantum Hamiltonian learning algorithms. The simplicity of our algorithm also ensures that the quasi-Newton accelerations discussed in the paper perform well. This work is probably the first to consider quasi-Newton speedups in the Hamiltonian learning literature. We believe this is a must if any quantum Hamiltonian learning algorithm is ever used in practice because of its dramatic savings in computation time.

Third, quantum generalizations are significantly harder to analyze, and it is not clear that classical algorithms can be naturally generalized to the quantum case when the Hamiltonians are non-commutative. Our work developed techniques and methods to resolve the difficulties partially.

Anyway, we thank the reviewer for the helpful discussions. We will clarify the points in future revisions of the paper.

Reviewer comment (Novelty):
Problem setting is not clearly motivated. While Hamiltonian learning is a central problem in quantum computing, it is not clear to me why one would prefer to look at the Hamiltonian inference problem where one has to prepare adaptively a Gibbs state and measure observables in it.

Author response:
In terms of motivation, our goal is to explore the applicability of state-of-the-art classical algorithms to analogous problems in the quantum domain. We believe these classically inspired algorithms may offer practical advantages over existing quantum algorithms.

Reviewer comment (Theory):
Preparing adaptively Gibbs states and measuring observables in it is computationally hard — at least beyond some critical temperature, as discussed by the authors in the conclusions. See also the very recent work [https://arxiv.org/abs/2310.02243] for an efficient algorithm at all temperatures. It is then not clear whether this approach scales and is practical.

Author response:
We acknowledge the concern raised by the reviewer regarding the use of adaptive oracles, which was also questioned by another reviewer. While we agree that this is an area for improvement, we believe that the adaptive oracle assumption is only a slight strengthening of assuming the availability of Gibbs states for the true Hamiltonian. We argue that if there exists a natural process or algorithm capable of providing the Gibbs state for the true Hamiltonian (as required by the Hamiltonian learning problems), the same process should work for the adaptively chosen Hamiltonians in our algorithm, as these chosen Hamiltonians share the same locality as the true Hamiltonian.

Reviewer comment (Reproducibility):
I could not find details of the systems studied in the experiments.

Author response:
We are unsure what it refers to in your question about "systems studied" in the experiments. If the question pertains to the Hamiltonian systems considered in our numerical simulations, they are random 2-local Hamiltonians. If the reviewer was inquiring about the computing systems used for the experiments, we used a workstation with 32 CPUs (Intel(R) Xeon(R) Platinum 8369HB CPU @ 3.30GHz) and 64GB of memory.

Reviewer comment (Novelty):
Novelty is limited as it is an extension of the framework of [Anshu et al] to bound the spectrum of the Hessian.

Author response:
Regarding the novelty of our work, we indeed employ the work of Anshu et al. for the lower bound, while the upper bound of the spectrum serves as the main technical contribution of our paper.

Reviewer comment (Theory):
Page 1: I think that there is no $\times 1$ in the definition of $\alpha_j$ since $H_j$ is already acting on the whole Hilbert space.

Author response:
1. We will clarify on page 1 that H_j refers to a local matrix. In local Hamiltonian literature, omitting the tensor I part for each H_j is customary when writing H = \sum_j H_j. We will ensure proper clarification in the next update.

Reviewer comment (Novelty):
It seems that the paper's novelty is limited to some extent.

Author response:
We would like to express our gratitude to the reviewer for their comments and questions. The generalization of classical results to the quantum case encounters a significant challenge due to the non-commutativity of matrices within the matrix exponentials present in the log-partition function. This non-commutativity prevents the derivative of the matrix exponential function from having a simple closed-form formula. The technical innovation of our paper lies in the application of a newly proposed quantum belief propagation method (Lemma C.3) to provide bounds on the eigenvalues of the Hessian of the log-partition function, even in cases where we do not possess a formula for the Hessian. In classical scenarios, the corresponding part of the proof would be relatively straightforward.

Reviewer comment (Novelty):
Major results rely heavily on existing works such as Liang et al. (2004), with the main theorems in Section 4 being extensions of the analysis in the classical setting.

Author response:
We would like to express our gratitude to the reviewer for their comments and questions. The generalization of classical results to the quantum case encounters a significant challenge due to the non-commutativity of matrices within the matrix exponentials present in the log-partition function. This non-commutativity prevents the derivative of the matrix exponential function from having a simple closed-form formula. The technical innovation of our paper lies in the application of a newly proposed quantum belief propagation method (Lemma C.3) to provide bounds on the eigenvalues of the Hessian of the log-partition function, even in cases where we do not possess a formula for the Hessian. In classical scenarios, the corresponding part of the proof would be relatively straightforward.

Reviewer comment (Novelty):
Can you further explain the main novelty of the analysis compared to that of the classical settings?

Author response:
We would like to express our gratitude to the reviewer for their comments and questions. The generalization of classical results to the quantum case encounters a significant challenge due to the non-commutativity of matrices within the matrix exponentials present in the log-partition function. This non-commutativity prevents the derivative of the matrix exponential function from having a simple closed-form formula. The technical innovation of our paper lies in the application of a newly proposed quantum belief propagation method (Lemma C.3) to provide bounds on the eigenvalues of the Hessian of the log-partition function, even in cases where we do not possess a formula for the Hessian. In classical scenarios, the corresponding part of the proof would be relatively straightforward.