arXiv subjects

Laura Shou

Publications and source records attributed to Laura Shou.

At least 19 recordsLinked to original sources

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 R\'enyi-$\alpha$ 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 R\'enyi-2 entropy Page curve in terms of the magnitudes $|s_i|$, which allow for extending equal squeezing results to unequal squeezing.

quant-ph↗

arXiv2026

Fast GHZ encoding with $1-o(1)$ fidelity

We prove that $(1-o(1))$-fidelity $N$-qubit Greenberger--Horne--Zeilinger (GHZ) encoding can be performed in time $O(\log N/N)$ using all-to-all 2-local Hamiltonians with bounded 2-qubit interaction terms. This saturates the theoretical lower bound $\Omega(\log N/N)$, and by rescaling, also saturates the lower bound on signaling time for Hamiltonians with power-law interaction strengths $1/r^\gamma$, $\gamma<d$ in dimension $d$. The protocol uses spin-squeezing dynamics combined with quantum signal processing.

quant-ph↗

arXiv2026

Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modes

Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.

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 R\'enyi-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 $|\langle\psi|\mathcal U^\dagger \tilde{\mathcal U}|\psi\rangle|^2$ for pure states $|\psi\rangle$ with constrained expected photon-number.

quant-ph↗

arXiv2026

Edge localization and Lifshitz tails for graphs with Ahlfors regular volume growth

In this work, we study the Anderson model on graphs with Ahlfors $\alpha$-regular volume growth. We show that, under mild regularity assumptions of the random distribution, Lifshitz-tail type estimates near the bottom of the spectrum lead to exponential decay of fractional moments of the Green's function and thus spectral and dynamical localization at low energies. This generalizes the result of [4] from the lattice $\mathbb{Z}^d$ to Ahlfors $\alpha$-regular graphs. In addition, we establish Lifshitz tail estimates for the integrated density of states, with the Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the underlying graphs, under certain assumptions on low lying eigenvalues of the Dirichlet and Neumann Laplacian on the graph. As an application, we verify all conditions on the Sierpinski gasket graph and obtain that, under mild regularity assumptions of the random distribution, for any fixed disorder, the Anderson model on the Sierpinski gasket graph has pure point spectrum and exhibits strong dynamical localization near the bottom of the spectrum.

math-ph↗

arXiv2026

Exactly Solvable Topological Phase Transition in a Quantum Dimer Model

We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a $2\times1$ periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled $\alpha$. We analytically show that the model exhibits a continuous quantum phase transition at $\alpha=3$, changing from a topological $\mathbb{Z}_2$ quantum spin liquid ($\alpha<3$) to a columnar ordered state ($\alpha>3$). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length $\xi\propto1/|\alpha-3|$ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological R\'enyi entropy of order $\infty$ (topological min-entropy) changes from $\log2$ for the quantum spin liquid phase $\alpha<3$, to $0$ for the ordered phase $\alpha>3$, thereby analytically confirming the topological nature of the phase transition.

cond-mat.str-el↗

arXiv2025

Quantum variance and fluctuations for Walsh-quantized baker's maps

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors $D\ge2$ except for $D=4$, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the empirical distribution of the scaled matrix element fluctuations $\sqrt{N}\{\langle \varphi^{(j)}|\operatorname{Op}_{k,\ell}(a)|\varphi^{(j)}\rangle-\int_{\mathbb{T}^2}a\}_{j=1}^{N}$ for a random eigenbasis $\{\varphi^{(j)}\}_{j=1}^{N}$ is asymptotically Gaussian in the semiclassical limit $N\to\infty$, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. We obtain a version of the Eigenstate Thermalization Hypothesis (ETH) for these eigenstates, including a limiting complex Gaussian distribution for the off-diagonal matrix elements, with variances also given in terms of classical correlations. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value $D=4$, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.

math-ph↗

arXiv2025

Proof of Hiding Conjecture in Gaussian Boson Sampling

Gaussian boson sampling (GBS) is a promising protocol for demonstrating quantum computational advantage. One of the key steps for proving classical hardness of GBS is the so-called ``hiding conjecture'', which asserts that one can ``hide'' a complex Gaussian matrix as a submatrix of the outer product of Haar unitary submatrices in total variation distance. In this paper, we prove the hiding conjecture for input states with the maximal number of squeezed states, which is a setup that has recently been realized experimentally [Madsen et al., Nature 606, 75 (2022)]. In this setting, the hiding conjecture states that a $o(\sqrt{M})\times o(\sqrt{M})$ submatrix of an $M\times M$ circular orthogonal ensemble (COE) random matrix can be well-approximated by a complex Gaussian matrix in total variation distance as $M\to\infty$. This is the first rigorous proof of the hiding property for GBS in the experimentally relevant regime, and puts the argument for hardness of classically simulating GBS with a maximal number of squeezed states on a comparable level to that of the conventional boson sampling of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)].

quant-ph↗

arXiv2025

Breakdown of the thermodynamic limit in quantum spin and dimer models

The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system's boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.

cond-mat.stat-mech↗

arXiv2025

Geometric Delocalization in Two Dimensions

We demonstrate the existence of transient two-dimensional surfaces where a random-walking particle escapes to infinity in contrast to localization in standard flat 2D space. We first prove that any rotationally symmetric 2D membrane embedded in flat 3D space cannot be transient. Then we formulate a criterion for the transience of a general asymmetric 2D membrane. We use it to explicitly construct a class of transient 2D manifolds with a non-trivial metric and height function but ``zero average curvature,'' which we dub tablecloth manifolds. The absence of the logarithmic infrared divergence of the Laplace-Beltrami operator in turn implies the absence of weak localization, non-existence of bound states in shallow potentials, and breakdown of the Mermin-Wagner theorem and Kosterlitz-Thouless transition on the tablecloth manifolds, which may be realizable in both quantum simulators and corrugated two-dimensional materials.

cond-mat.dis-nn↗

arXiv2024

Spectrum and Lifshitz tails for the Anderson model on the Sierpinski gasket graph

In this work, we study the Anderson model on the Sierpinski gasket graph. We first identify the almost sure spectrum of the Anderson model when the support of the random potential has no gaps. We then prove the existence of the integrated density states of the Anderson model and show that it has Lifshitz tails with Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the Sierpinski gasket graph.

math-ph↗

arXiv2024

Landscape estimates of the integrated density of states for Jacobi operators on graphs

We show the integrated density of states for a variety of Jacobi operators on graphs, such as the Anderson model and random hopping models on graphs with Gaussian heat kernel bounds, can be estimated from above and below in terms of the localization landscape counting function. Specific examples of these graphs include stacked and decorated lattices, graphs corresponding to band matrices, and aperiodic tiling graphs. The upper bound part of the landscape law also applies to the fractal Sierpinski gasket graph. As a consequence of the landscape law, we obtain landscape-based proofs of the Lifshitz tails in several models including random band matrix models, certain bond percolation Hamiltonians on $\mathbb{Z}^d$, and Jacobi operators on certain stacks of graphs. We also present intriguing numerical simulations exploring the behavior of the landscape counting function across various models.

math-ph↗

arXiv2024

Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems

We prove that the time required for sustained information scrambling in any Hamiltonian quantum system is universally at least logarithmic in the entanglement entropy of scrambled states. This addresses two foundational problems in nonequilibrium quantum dynamics. (1) It sets the earliest possible time for the applicability of equilibrium statistical mechanics in a quantum system coupled to a bath at a finite temperature. (2) It proves a version of the fast scrambling conjecture, originally motivated in models associated with black holes, as a fundamental property of quantum mechanics itself. Our result builds on a refinement of the energy-time uncertainty principle in terms of the infinite temperature spectral form factor in quantum chaos. We generalize this formulation to arbitrary initial states of the bath, including finite temperature states, by mapping Hamiltonian dynamics with any initial state to nonunitary dynamics at infinite temperature. A regularized spectral form factor emerges naturally from this procedure, whose decay is universally constrained by analyticity in complex time. This establishes an exact speed limit on information scrambling by the most general quantum mechanical Hamiltonian, without any restrictions on locality or the nature of interactions.

quant-ph↗

arXiv2023

Spectral anomalies and broken symmetries in maximally chaotic quantum maps

Spectral statistics such as the level spacing statistics and spectral form factor (SFF) are widely expected to accurately identify ``ergodicity'', including the presence of underlying macroscopic symmetries, in generic quantum systems ranging from quantized chaotic maps to interacting many-body systems. By studying various quantizations of maximally chaotic maps that break a discrete classical symmetry upon quantization, we demonstrate that this approach can be misleading and fail to detect macroscopic symmetries. Notably, the same classical map can exhibit signatures of different random matrix symmetry classes in short-range spectral statistics depending on the quantization. While the long-range spectral statistics encoded in the early time ramp of the SFF are more robust and correctly identify macroscopic symmetries in several common quantizations, we also demonstrate analytically and numerically that the presence of Berry-like phases in the quantization leads to spectral anomalies, which break this correspondence. Finally, we provide numerical evidence that long-range spectral rigidity remains directly correlated with ergodicity in the quantum dynamical sense of visiting a complete orthonormal basis.

quant-ph↗

arXiv2023

A localization-delocalization transition for nonhomogeneous random matrices

We consider $N\times N$ self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with $d$ nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization transition near the edge of the spectrum: when $d\gg\log N$ the random matrix possesses a delocalized approximate top eigenvector, while when $d\ll\log N$ any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix.

math.PR↗

arXiv2023

Chaotic Roots of the Modular Multiplication Dynamical System in Shor's Algorithm

Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the "maximally random" Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an "integrable" quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized A-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other "chaotic" quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity and chaos in and via quantum algorithms.

quant-ph↗

arXiv2023

Eigenstates and spectral projection for quantized baker's map

We extend the approach from [arXiv:2110.15301] to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker's map on the torus. The spectral window is allowed to shrink in the semiclassical (large dimension) limit. As a consequence, we obtain a strengthening of the quantum ergodic theorem from [arXiv:math-ph/0412058] to hold in shrinking spectral windows, a Weyl law on uniform spreading of eigenvalues, and statistics of random quasimodes. Using similar techniques, we also investigate random eigenbases of a different (non-Weyl) quantization, the Walsh-quantized baker's map, which has high degeneracies in its spectrum. For such random eigenbases, we prove that Gaussian eigenstate statistics and QUE hold with high probability in the semiclassical limit.

math-ph↗

arXiv2022

Landscape approximation of the ground state eigenvalue for graphs and random hopping models

We consider the localization landscape function $u$ and ground state eigenvalue $\lambda$ for operators on graphs. We first show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue if the operator satisfies certain semigroup kernel upper bounds. This implies general upper and lower bounds on the landscape product $\lambda\|u\|_\infty$ for several models, including the Anderson model and random hopping (bond-disordered) models, on graphs that are roughly isometric to $\mathbb{Z}^d$, as well as on some fractal-like graphs such as the Sierpinski gasket graph. Next, we specialize to a random hopping model on $\mathbb{Z}$, and show that as the size of the chain grows, the landscape product $\lambda\|u\|_\infty$ approaches $\pi^2/8$ for Bernoulli off-diagonal disorder, and has the same upper bound of $\pi^2/8$ for Uniform([0,1]) off-diagonal disorder. We also numerically study the random hopping model when the band width (hopping distance) is greater than one, and provide strong numerical evidence that a similar approximation holds for low-lying energies in the spectrum.

math-ph↗