- 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.
- Rosset, D., Montealegre-Mora, F., & Bancal, J.-D. (2021). Replab: A computational/numerical approach to representation theory. In P. M.B., M. R., T. Z., W. P., & W.-K. W. (Eds.), Quantum Theory and Symmetries: Proceedings of the 11th International Symposium, Montreal, Canada (pp. 643–653). Springer. Available at arXiv:1911.09154.
Convex optimization is a widely used tool in science and engineering. These optimization problems often happen over high-dimensional spaces, which renders their runtime long in practice. However, considerable reductions can be obtained if the problem has symmetries. For this, the representation matrices which encode these symmetries must be simultaneously block-diagonalized.
Here we present RepLAB, a Matlab/Octave toolbox for numerically block-diagonalizing representations. Given as an input 1. a description of a semi-definite program and, 2. a description of the symmetry, it produces the symmetrized semi-definite program as an output. RepLAB can handle both finite and continuous representations.
- 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.
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).
- Vargas, W. E., Azofeifa, D. E., Clark, N., Solis, H., Montealegre-Mora, F., & Cambronero, M. (2014). Parametric formulation of the dielectric function of palladium and palladium hydride thin films. Applied optics, 53(24), 5294–5306.
The optical properties of thin metallic films — with a thickness of around 10nm — vary in the presence of Hydrogen in their sorroundings. In this experiment, we expose Palladium thin films to varying Hydrogen concentrations and determine how the shape of their dielectric function varies as Hydrogen pressure increases.