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Recently, many experiments have been conducted with the goal of demonstrating a quantum advantage over classical computation. One popular framework for these experiments is Gaussian boson sampling, where quadratic photonic input states are interfered via a linear optical unitary and subsequently measured in the Fock basis. In this paper, we study the modal entanglement of the output states in this framework just before the measurement stage. Specifically, we compute Page curves as measured by various Rényi- $\alpha$entropies, where the Page curve describes the entanglement between two partitioned groups of output modes averaged over all linear optical unitaries. We derive these formulas for $\alpha =1$(i.e., the von Neumann entropy) and, more generally, for all positive integer $\alpha$, in the asymptotic limit of infinite number of modes and for input states that are composed of single-mode-squeezed-vacuum state with equal squeezing strength. We then analyze the limiting behaviors when the squeezing is small and large. Having determined the averages, we then explicitly calculate the Rényi- $\alpha$variance for integers $\alpha >1$and are able to show that these entropies are weakly typical. Published by the American Physical Society2025 

Toric $t$-designs, or equivalently $t$-designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree $t$monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of $t$-designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense $B_{t}modm$sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state $2$-designs in dimension $d$of size exactly $d(d+1)$that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension $6$(and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information. 

In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to ${\mathbb{Z}}_d$qudits, including instances where $d$is a nonprime number. This capability allows the identification of topological orders that differ from the ${\mathbb{Z}}_d$toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for ${\mathbb{Z}}_p$qudits (with $p$being a prime) are equivalent to finite copies of ${\mathbb{Z}}_p$toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society2024 

As with classical computers, quantum computers require error-correction schemes to reliably perform useful large-scale calculations. The nature and frequency of errors depends on the quantum computing platform, and although there is a large literature on qubit-based coding, these are often not directly applicable to devices that store information in bosonic systems such as photonic resonators. Here, we introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, and we obtain multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that, at the same time, form averaging sets known as spherical designs. We also recast concatenations of Calderbank–Shor–Steane codes with cat codes as quantum spherical codes, which establishes a method to autonomously protect against dephasing noise. 

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t-designs from [Blume-Kohout et al., Commun.Math. Phys. 326, 755 (2014)], no state designs exist for t ≥ 2. Similarly, we prove that no CV unitary t-designs exist for t ≥ 2. We propose an alternative definition for CV state designs, which we call rigged t-designs, and provide explicit constructions for t ¼ 2. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. 

Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.