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- Montealegre-Mora, F., Boettiger, C., Walters, C.J., Cahill, C.L. (2024) Using machine learning to inform harvest control rules in complex fishery scenarios. arXiv:2412.12400.
Harvest management of size-structured stochastic populations is a long-standing and difficult problem for fishery science. This is particularly so for fish stocks with highly variable—also referred to as “spasmodic”—recruitment, i.e., stocks in which the quantity of young fish has a high variance, with some highly productive years having cohort sizes that are 10-50 times larger than the mean cohort size. Because of this variance in cohort sizes, the size structure of the population can play a very important role in informing harvest quotas. For example, if a formerly fishery population exhibits a sudden increase in size one year, the question of what to do with this information arises. Should harvest quotas be increased immediately? Should they be gradually increased, and if so, over which time-scale should they be increased? Should the fishery remain closed until the large incoming cohort reaches a spawning age? These sorts of questions were faced, e.g., by recreational Walleye fisheries in Alberta, Canada over the past decade.
In this paper, we use optimization techniques from Bayesian optimization and reinforcement learning (RL) to address these questions in the context of a realistic size-structured model for Walleye stocks. We compare how numerically optimized policies perform compared to a standard precautionary policy recommended by the DFO using several common utility functions. We find that numerically optimized policies can perform significantly better than standard precautionary policies in a variety of scenarios.
Additionally: Harvest control rules have traditionally only used the stock size to quantitatively determine harvest quotas. In this paper we go beyond this state-of-the-art and use RL to ask whether additional observations, such as mean fish size, can be used to inform harvest quotas. We surprisingly find that, in many scenarios, this additional information is not useful for this end. However, we find that in some cases, such as cases in which the relative sizes of harvested fish matter, additional size structure observations can allow for better harvest control.
- Montealegre-Mora, F., Lapeyrolerie, M., Chapman, M., Keller, A.G., Boettiger, C. (2023) Pretty darn good control: When approximate solutions are better than approximate models are better than approximate solutions. Bulletin of Mathematical Biology 85, 95. Available on arXiv. Open source code.
Complex control problems come up very frequently in environmental science. This can show up as questions such as “with how much intensity should we fish to balance profit with sustainability?”, or “what is a good culling strategy to manage an invasive species?”, or even, “how are my resources best spent in order to achieve a certain conservation goal?”
A classical approach is to leverage the results from optimal control theory here. This has a drawback: these results are usually only known for very simple and stylized mathematical models.
For fishery management, for instance, while very complex models are used to estimate stock sizes from a variety of variables, the model actually used for decision making—the dynamical model which is “fed” into the optimal control algorithm—is often very simple, summarizing complex environmental interactions into a few parameters of a single-species model. A model that considers, say, 3 or more species interacting with each other in non-linear ways is likely too complex for this approach.
In this paper we consider a different approach. Rather than performing optimal control on a stylized single-species model, we use tools from reinforcement learning (RL) to find non-optimal but highly competitive solutions to more complex models. That is, our RL algorithm tackles a more complex and expressive control problem, but at the price of not being guaranteed to find an optimal solution.
Which of these two paths produces a better solution? Does an approximate solution to a more accurate model perform better than an optimal solution to a simple and more inaccurate problem?
We investigate this question for the fishery management problem using moderately complex models of ecosystem dynamics—including up to 3 interacting species, and possibly time-varying parameters to model climate change. We find that for sufficiently complex models, RL-based solutions can outperform classical approaches both from an economical and ecological point of view.
- Montealegre-Mora, F., & Gross, D. (2022). Duality theory for Clifford tensor powers. arXiv:2208.01688.
The Clifford group – a subset of the quantum operations available in a quantum computer – has become a staple tool within the quantum computing community. These operations are typically the first to be implemented to high-fidelity in many quantum architectures. This has good reason to be: their accurate implementation is crucial if one is to achieve fault-tolerant quantum computing. There is a “folk result” stating that understanding the mathematical structure of the Clifford group leads to practical applications. For example, this knowledge has been useful towards finding fast classical simulators of quantum computers, for characterizing the noise processes affecting the quantum computer, and for devising new quantum cryptographic protocols.
In the past few years, knowledge of the structure of “tensor power representations” of the Clifford group has come in particularly handy in many applications. Up to now, the second, third and fourth tensor powers have been analyzed on a case-by-case basis. It is expected that knowledge about higher tensor powers will similarly lead to applications in quantum computing.
In this paper, we use an approach using a theory of duality to help clarify the structure of these representations for arbitrary tensor powers. We show how the mathematical literature on Howe duality can be generalized to cover a duality between the Clifford group and a certain discrete orthogonal group. This approach gives rather explicit statements about the structures of these representations. To showcase our results, we provide a full decomposition of the fifth tensor power representation of the Clifford group.
- Heimendahl, A., Montealegre-Mora, F., Vallentin, F., & Gross, D. (2021). Stabilizer extent is not multiplicative. Quantum, 5, 400. Open access.
Classically simulating a quantum computer is generally hard. However, some classes of quantum algorithms can be efficiently simulated. A recent simulation method focuses on the regime where the quantum circuit is dominated by Clifford gates, and only limited non-Clifford resources are available. The runtime of the algorithm scales exponentially in the amount of non-Clifford resources, this is quantified via a quantity called the stabilizer extent.
The stabilizer extent is, however, itself the outcome of an optimization problem with a super-exponential number of parameters. This limits the computation of the stabilizer extent, in general, to few-qubit states. On tensor-product states, the extent can be upper-bounded using the property that it is submultiplicative with respect to tensor-products (i.e. the extent of the product is not larger than the product of the extents).
The question of whether this upper bound is in fact an equality arises —whether the extent is multiplicative. Here we answer this question negatively. In fact, we answer it negatively for a general class of functions which includes the stabilizer extent and many other measures of non-Cliffordness commonly used.
- Montealegre-Mora, F., & Gross, D. (2021). Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes. Representation Theory of the American Mathematical Society, 25(8), 193–223. Open access.
Duality plays an important role in the mathematical field of representation theory. An important instance of this duality is the Theta correspondence —it relates the representation theory of two ubiquitous objects in the field, the symplectic group and the orthogonal group. This correspondence was previously only understood on certain ‘maximal rank’ representations. Here we use techniques from quantum information to extend this understanding to ‘rank-deficient’ representations as well.
- Montealegre-Mora, F., Rosset, D., Bancal, J.-D., & Gross, D. (2021). Certifying numerical decompositions of compact group representations. arXiv:2101.12244.
When numerically block-diagonalizing representations one is faced with the following question: to which extent does one trust that this numerical decomposition is close to the true one? While there exist block-diagonalization methods with such a guarantee, these are typically very expensive computationally. Under natural assumptions about the representation, the runtime of a commonly used method by Babai and Friedl scales as O(n^5), where n is the dimension of the representation.
Here we suggest an alternative approach: 1. use a heuristic to quickly find a decomposition — an example of such a heuristic is RepLAB, which inspired this work –, 2. certify that this decomposition is close to exact.
We provide an efficient and practical method for the second task. It works in the context of both finite and continuous compact groups. The runtime scales as O(n^3 + D d^2 log d), where d is the dimension of the largest irreducible block and D is the complexity of multiplying group elements together. An implementation of the algorithms interfacing with RepLAB is provided.
- Haferkamp, J., Montealegre-Mora, F., Heinrich, M., Eisert, J., Gross, D., & Roth, I. (2020). Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates. arXiv:2002.09524. Published in Communications in Mathematical Physics.
Unitary t-designs have found applications in several branches of theoretical physics: from quantum device characterization and quantum cryptography, to quantum thermodynamics and high energy physics. Unitary t-designs are probability distributions on the group of unitary matrices with a special property: they emulate up to the t-th moments of the flat distribution.
There exist efficient and experimentally-feasible constructions of low-order designs, with t=2 or 3. Producing higher-order designs is notoriously difficult and because of this, one resorts to approximate t-designs instead. The state of the art in this case is an algorithm due to Brandao, Harrow and Horodecki which produces t-designs on n qubits using O(n^2 t^9) random two-qubit unitaries.
Here we provide a new model for obtaining approximate unitary t-designs. This model is particularly well-suited for quantum computing contexts since the quantum circuits that generate it have an overwhelming majority of easy-to-implement Clifford gates. Only a very small amount of non-Clifford gates, O(t^4 log t), are needed. This number is surprisingly independent of the system size n. The total two-qubit gate count is O(n^2 t^4 log t).
Felipe Montealegre-Mora, PhD 