arXiv subjects

Joseph T. Iosue

Publications and source records attributed to Joseph T. Iosue.

At least 19 recordsLinked to original sources

arXiv2026

Cycle-Structure Generating Functions for Special Breakpoint Graphs

Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders. The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations. As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.

math.CO↗

arXiv2026

Higher moment theory and learnability of bosonic states

We present a sample- and time-efficient algorithm to learn any bosonic Fock state acted upon by an arbitrary Gaussian unitary. As a special case, this algorithm efficiently learns states produced in Fock state BosonSampling, thus resolving an open question put forth by Aaronson and Grewal (Aaronson, Grewal 2023). We further study a hierarchy of classes of states beyond Gaussian states that are specified by a finite number of their higher moments. Using the higher moments, we find a full spectrum of invariants under Gaussian unitaries, thereby providing necessary conditions for two states to be related by an arbitrary (including active, e.g. beyond linear optics) Gaussian unitary.

quant-ph↗

arXiv2026

Anticoncentration and entanglement in Gaussian boson sampling

Anticoncentration and entanglement are two properties used to study classical hardness or easiness of Gaussian boson sampling in varying regimes. In this paper, we consider three directions concerning anticoncentration and entanglement in Gaussian boson sampling: (1) We derive closed-form expressions for the second moment of hafnians of symmetric Gaussian products, and use this to precisely locate the anticoncentration transition as a function of the number of squeezed input modes. (2) We derive closed-form expressions for the Rényi-$α$ Page curves for Gaussian boson sampling. (3) We study unequal input squeezing parameters $(s_i)_i$, and prove estimates as well as monotonicity of the average-case Rényi-2 entropy Page curve in terms of the magnitudes $|s_i|$, which allow for extending equal squeezing results to unequal squeezing.

quant-ph↗

arXiv2026

Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes

Quantum low-density parity-check codes, such as the Kitaev toric code and bivariate bicycle codes, are often defined with periodic boundary conditions, which are difficult to realize in physical systems. In this paper, we present an algorithm for constructing all gapped boundaries and defects of two-dimensional Pauli stabilizer codes. Using the operator algebra formalism, we establish a one-to-one correspondence between the topological data, such as anyon fusion rules and topological spins, of two-dimensional bulk stabilizer codes and one-dimensional boundary anomalous subsystem codes. To make the operator algebra computationally accessible, we adapt Laurent polynomials and convert the tasks into matrix operations, e.g., the Hermite normal form for obtaining boundary anyons and the Smith normal form for determining fusion rules. This approach enables computers to automatically generate all possible gapped boundaries and defects for topological Pauli stabilizer codes through boundary anyon condensation and topological order completion. This streamlines the analysis of surface codes and associated logical operations for fault-tolerant quantum computation. Our algorithm applies to $\mathbb{Z}_d$ qudits for both prime and nonprime $d$, enabling exploration of topological phases beyond the Kitaev toric code. We have applied the algorithm and explicitly demonstrated the lattice constructions of 2 boundaries and 6 defects in the $\mathbb{Z}_2$ toric code, 3 boundaries and 22 defects in the $\mathbb{Z}_4$ toric code, 1 boundary and 2 defects in the double semion code, 1 boundary and 22 defects in the six-semion code, 6 boundaries and 270 defects in the color code, and 6 defects in the anomalous three-fermion code. Finally, we study the boundaries of bivariate bicycle codes, showing that they exhibit large logical dimensions and anyons with long translation periods.

quant-ph↗

arXiv2026

Entanglement and circuit complexity in finite-depth random linear optical networks

We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all $n$ modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the Rényi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in $L^2$ Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary $\tilde{\mathcal U}$ for the approximate implementation retains high output fidelity $|\langleψ|\mathcal U^\dagger \tilde{\mathcal U}|ψ\rangle|^2$ for pure states $|ψ\rangle$ with constrained expected photon-number.

quant-ph↗

arXiv2026

Superresolution in Quantum Noise Spectroscopy via Filter Design

Resolving signals with closely spaced frequencies is central to applications in communications, spectroscopy and sensing. Recent results have shown that quantum sensing protocols can exhibit superresolution, the ability to discriminate between spectral lines with arbitrarily small frequency separation. Here, we revisit this problem from the perspective of quantum control theory, utilizing the filter function formalism to derive general, analytic conditions on quantum control protocols for achieving superresolution. Building on these conditions, we develop an optimal control framework, the utility of which is demonstrated through numerical identification of superresolution control protocols in the presence of realistic, experimentally-relevant constraints. We further extend our results to entangled initial states and assess their potential advantage. Our approach is broadly applicable to a wide variety of quantum sensing platforms, and it provides a systematic path to discover novel protocols that surpass conventional resolution limits in these systems.

quant-ph↗

arXiv2025

Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking

Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the decohered cluster state -- a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry -- we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.

quant-ph↗

arXiv2025

Quantum-inspired Benchmark for Estimating Intrinsic Dimension

Machine learning models can generalize well on real-world datasets. According to the manifold hypothesis, this is possible because datasets lie on a latent manifold with small intrinsic dimension (ID). There exist many methods for ID estimation (IDE), but their estimates vary substantially. This warrants benchmarking IDE methods on manifolds that are more complex than those in existing benchmarks. We propose a Quantum-Inspired Intrinsic-dimension Estimation (QuIIEst) benchmark consisting of infinite families of topologically non-trivial manifolds with known ID. Our benchmark stems from a quantum-optical method of embedding arbitrary homogeneous spaces while allowing for curvature modification and additive noise. The IDE methods tested were generally less accurate on QuIIEst manifolds than on existing benchmarks under identical resource allocation. We also observe minimal performance degradation with increasingly non-uniform curvature, underscoring the benchmark's inherent difficulty. As a result of independent interest, we perform IDE on the fractal Hofstadter's butterfly and identify which methods are capable of extracting the effective dimension of a space that is not a manifold.

cs.LG↗

arXiv2025

Quantum-Centric Algorithm for Sample-Based Krylov Diagonalization

Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for pre- and early-fault-tolerant quantum computers. Here, we introduce a quantum diagonalization algorithm which combines two key ideas on quantum subspaces: a classical diagonalization based on quantum samples, and subspaces constructed with quantum Krylov states. We prove that our algorithm converges in polynomial time under the working assumptions of Krylov quantum diagonalization and sparseness of the ground state. We then demonstrate the scalability of our approach by performing the largest ground-state quantum simulation of impurity models using a Heron quantum processors and the Frontier supercomputer. We consider both the single-impurity Anderson model with 41 bath sites, and a system with 4 impurities and 7 bath sites per impurity. Our results are in excellent agreement with Density Matrix Renormalization Group calculations.

quant-ph↗

arXiv2025

Continuous-variable designs and design-based shadow tomography from random lattices

We investigate state designs for continuous-variable quantum systems using the aid of lattice-like quantum states. These are code states of Gottesman-Kitaev-Preskill (GKP) codes. We show that for an n-mode system, the set of all GKP states forms a rigged continuous-variable state 2-design. We use these lattice state designs to construct a continuous variable shadow tomography protocol, derive sample complexity bounds for both global- and local GKP shadows under reasonable physical assumptions, and provide the physical gadgets needed to implement this protocol.

quant-ph↗

arXiv2025

Extracting topological orders of generalized Pauli stabilizer codes in two dimensions

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 2d honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order.

quant-ph↗

arXiv2025

Average Rényi Entanglement Entropy in Gaussian Boson Sampling

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 work, 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-$α$ 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 $α= 1$ (i.e. the von Neumann entropy), and, more generally, for all positive integer $α$, 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-$α$ variance for integers $α> 1$, and we are able to show that these entropies are weakly typical.

quant-ph↗

arXiv2024

Projective toric designs, quantum state designs, and mutually unbiased bases

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 $B_t \bmod m$ 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.

quant-ph↗

arXiv2024

Continuous-variable quantum state designs: theory and applications

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 Comm. Math. Phys. 326, 755 (2014), no state designs exist for $t\geq2$. Similarly, we prove that no CV unitary $t$-designs exist for $t\geq 2$. We propose an alternative definition for CV state designs, which we call rigged $t$-designs, and provide explicit constructions for $t=2$. 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.

quant-ph↗

arXiv2024

The Second Moment of Hafnians in Gaussian Boson Sampling

Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector $2n = 80$. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.

quant-ph↗

arXiv2024

Transition of Anticoncentration in Gaussian Boson Sampling

Gaussian Boson Sampling is a promising method for experimental demonstrations of quantum advantage because it is easier to implement than other comparable schemes. While most of the properties of Gaussian Boson Sampling are understood to the same degree as for these other schemes, we understand relatively little about the statistical properties of its output distribution. The most relevant statistical property, from the perspective of demonstrating quantum advantage, is the anticoncentration of the output distribution as measured by its second moment. The degree of anticoncentration features in arguments for the complexity-theoretic hardness of Gaussian Boson Sampling, and it is also important to know when using cross-entropy benchmarking to verify experimental performance. In this work, we develop a graph-theoretic framework for analyzing the moments of the Gaussian Boson Sampling distribution. Using this framework, we show that Gaussian Boson Sampling undergoes a transition in anticoncentration as a function of the number of modes that are initially squeezed compared to the number of photons measured at the end of the circuit. When the number of initially squeezed modes scales sufficiently slowly with the number of photons, there is a lack of anticoncentration. However, if the number of initially squeezed modes scales quickly enough, the output probabilities anticoncentrate weakly.

quant-ph↗

arXiv2023

Quantum spherical codes

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, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring 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 CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise.

quant-ph↗

arXiv2023

Page curves and typical entanglement in linear optics

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.

quant-ph↗