Circular Orthogonal Ensemble Research Articles

Signatures of a sampling quantum advantage in driven quantum many-body systems

A crucial milestone in the field of quantum simulation and computation is to demonstrate that a quantum device can perform a computation task that is classically intractable. A key question is to identify setups that can achieve such goal within current technologies. In this work, we provide formal evidence that sampling bit-strings from a periodic evolution of a unitary drawn from the circular orthogonal ensemble (COE) cannot be efficiently simulated with classical computers. As the statistical properties of COE coincide with a large class of driven analog quantum systems thanks to the Floquet eigenstate thermalization hypothesis, our results indicate the possibility that those driven systems could constitute practical candidates for a sampling quantum advantage. To further support this, we give numerical examples of driven disordered Ising chains and 1D driven Bose–Hubbard model.

Three-fold way of entanglement dynamics in monitored quantum circuits

We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson’s three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan’s KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE.

Polarisation statistics of vector scattering matrices from the circular orthogonal ensemble

We study the polarisation properties of random N×N scattering matrices distributed according to the circular orthogonal ensemble. We interpret 2 × 2 sub-blocks of the scattering matrix as Jones matrices and study their statistical properties. Using the polar decomposition, we derive probability density functions for retardance and diattenuation from scattering matrices of arbitrary size and in the limit N→∞.

Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

The statistical properties of spectra of quantum systems within the framework of random matrix theory is widely used in many areas of physics. These properties are affected, if two or more sets of spectra are superposed, resulting from the discrete symmetries present in the system. Superposition of spectra of $m$ such circular orthogonal, unitary and symplectic ensembles are studied numerically using higher-order spacing ratios. For given $m$ and the Dyson index $\beta$, the modified index $\beta'$ is tabulated whose nearest neighbor spacing distribution is identical to that of $k$-th order spacing ratio. For the case of $m=2$ ($m=3$) in COE (CUE) a scaling relation between $\beta'$ and $k$ is given. For COE, it is conjectured that for $k=m+1$ ($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution the $\beta'$ is $m+2$ and $m-4$ respectively. Whereas in the case of CSE, for $k=m+1$ ($m\geq2$) and $k=m-1$-th ($m\geq3$) the $\beta'$ is $2m+3$ and $2(m-2)$ respectively. We also conjecture that for given $m$ ($k$) and $\beta$, the sequence of $\beta'$ as a function of $k$ ($m$) is unique. Strong numerical evidence in support of these results is presented. These results are tested on complex systems like the measured nuclear resonances, quantum chaotic kicked top and spin chains.

Immanants of blocks from random matrices in some unitary ensembles

The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong–Ou–Mandel effect and the boson sampling problem. In that context, it would be useful to know the distribution of the permanent and other immanants for random matrices, but that seems a difficult problem. We advance this program by calculating the average of the squared modulus of a generic immanant for blocks from random matrices in the unitary group, in the orthogonal group and in the circular orthogonal ensemble. In the case of the permanent in the unitary group, we also compute the variance. Our approach is based on Weingarten functions and factorizations of permutations. In the course of our calculations we are led to two curious conjectures relating dimensions of irreducible representations of the orthogonal and symplectic groups to the value of zonal and symplectic zonal polynomials at the identity.

Convergence of moments of twisted COE matrices

We investigate eigenvalue moments of matrices from the circular orthogonal ensemble multiplicatively perturbed by a permutation matrix. More precisely, we investigate the variance of the sum of the eigenvalues raised to power k for arbitrary but fixed k and in the limit of a large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the circular unitary ensemble with a particularly fast rate: the error is of order 1/N3 and the terms of orders 1/N and 1/N2 disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.

Asymptotics for Averages over Classical Orthogonal Ensembles

AbstractWe study the averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar-distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher–Hartwig singularities in cases where some of the singularities merge together and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the circular orthogonal and symplectic ensembles and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.

Statistics of the spectral form factor in the self-dual kicked Ising model

We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated. We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory if one identifies the appropriate ensemble of random matrices. We find that this ensemble is not the circular orthogonal one - composed of symmetric random unitary matrices and associated with time-reversal-invariant evolution operators - but is an ensemble of random matrices on a more restricted symmetric space (depending on the parity of the number of sites this space is either ${Sp(N)/U(N)}$ or ${O(2N)/{O(N)\!\times\!O(N)}}$). Even if the latter ensembles yield the same averaged spectral form factor as the circular orthogonal ensemble they show substantially enhanced fluctuations. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.

Loop-Erased Walks and Random Matrices

It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys Rev E 83:041127, 2011). We present further examples which give rise to various Cauchy-type ensembles. We also extend Fomin’s identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble.

Universal conductance fluctuations in Sierpinski carpets

We theoretically investigate the conductance fluctuation of two-terminal device in Sierpinski carpets. We find that, for the circular orthogonal ensemble (COE), the conductance fluctuation does not display a universal feature; but for circular unitary ensemble (CUE) without time-reversal symmetry or circular symplectic ensemble (CSE) without spin-rotational symmetry, the conductance fluctuation can reach an identical universal value of 0:74 ± 0:01(e2/h). We further find that the conductance distributions around the critical disorder strength for both CUE and CSE systems share the similar distribution forms. Our findings provide a better understanding of the electronic transport properties of the regular fractal structure.

Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos

The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin chains. Specifically, we consider the so called "self-dual" kicked Ising chains initialised in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the models considered are maximally chaotic: their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so called "minimal cut" bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel "duality-based" numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is non-integrable.

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability , we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.

Corner transfer matrices for 2D strongly coupled many-body Floquet systems

We develop, based on Baxter’s corner transfer matrices, a renormalizable numerically exact method for computation of the level density of the quasienergy spectra of two-dimensional (2D) locally interacting many-body Floquet systems. We demonstrate its functionality exemplified by the kicked 2D quantum Ising model. Using the method, we are able to treat systems of arbitrarily large finite size (for example lattices of the order of 108 spins). We clearly demonstrate that the density of the Floquet quasienergy spectrum tends to a flat function in the thermodynamic limit for generic values of model parameters. However, contrary to the prediction of random matrices of the circular orthogonal ensemble, the decay rates of the Fourier coefficients of the Floquet level density exhibit rich and non-trivial dependence on the system’s parameters. Remarkably, we find that the method is renormalizable and gives thermodynamically convergent results only in certain regions of the parameter space where the corner transfer matrices have effectively a finite rank for any system size. In the complementary regions, the corner transfer matrices effectively become of full rank and the method becomes non-renormalizable. This may indicate an interesting phase transition from an area- to volume-law of entanglement in the thermodynamic state of a Floquet system.

Singular values and evenness symmetry in random matrix theory

Abstract Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

Moments of a Single Entry of Circular Orthogonal Ensembles and Weingarten Calculus

Consider a symmetric unitary random matrix V = (v ij )1 ≤ i, j ≤ N from a circular orthogonal ensemble. In this paper, we study moments of a single entry v ij . For a diagonal entry v ii , we give the explicit values of the moments, and for an off-diagonal entry v ij , we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size N. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.

GENERAL MOMENTS OF MATRIX ELEMENTS FROM CIRCULAR ORTHOGONAL ENSEMBLES

The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.

Parameter-dependent spectral statistics of chaotic quantum graphs: Neumann versus circular orthogonal ensemble boundary conditions

The parameter-dependent spectral statistics of totally connected quantum graphs with n = 4-30 vertices, such as the parametric velocities correlation functions and the distribution of curvatures, are studied. The inverse participation ratio (IPR), an important measure of localization effects, was also numerically investigated. In the calculations, we successfully used two different theoretical approaches. The first approach was based on the graphs' eigenenergies and wave functions calculations, while the second one used the eigenphases and the eigenvectors of the bond scattering matrix S(k). We considered graphs with Neumann and circular orthogonal ensemble (COE) boundary conditions. We show that in contrast to large Neumann graphs, for which the departure of many parameter-dependent spectral statistics from the random matrix theory (RMT) predictions is observed, for large COE graphs, the spectral statistics and IPR are in good agreement with the RMT predictions.

Universal conductance fluctuation of mesoscopic systems in the metal-insulator crossover regime

We report a theoretical investigation on conductance fluctuation of mesoscopic systems. Extensive numerical simulations on quasi-one-dimensional, two-dimensional, and quantum dot systems with different symmetries [circular orthogonal ensemble, circular unitary ensemble (CUE), and circular symplectic ensemble (CSE)] indicate that the conductance fluctuation can reach a universal value in the crossover regime for systems with CUE and CSE symmetries. The conductance distribution is found to be a universal function from diffusive to localized regimes that depends only on the average conductance, dimensionality, and symmetry of the system. The numerical solution of DMPK equation agrees with our result in quasi-one dimension. Our numerical results in two dimensions suggest that this universal conductance fluctuation is related to the metal-insulator transition. In the localized regime with average conductance $⟨G⟩<0.3$, the conductance distribution seems to be superuniversal independent of dimensionality and symmetry.

Integrals over the circular ensembles relating to classical domains

Using the theory of classical domains and Jack polynomials, we generalize several integrals over the unitary group, due to Fyodorov and Khoruzhenko, to integrals over the circular orthogonal and circular symplectic ensembles. While in the unitary case a fundamental role is played by complex matrices, it is anti-symmetric and symmetric matrices which play the same role in the latter two cases. In our workings, a generalization of the evaluation of the integral of a Jack polynomial times the Selberg weight sought by Fyodorov and Khoruzhenko is identified.

The entries of circular orthogonal ensembles

Let V=(vij)n×n be a circular orthogonal ensemble. In this paper, for 1≤m≤o(n/log n), we give a bound for the tail probability of max1≤i,j≤m|vij−(1/n)yi′yj|, where Y=(y1,…,yn) is a certain n×n matrix whose entries are independent and identically distributed random variables with the standard complex normal distribution CN(0,1). In particular, this implies that, for a sequence of such matrices {Vn=(vij(n))n×n,n≥1}, as n→∞, nvij(n) converges in distribution to CN(0,1) for any i≥1,j≥1 with i≠j and nvii(n) converges in distribution to 2⋅CN(0,1) for any i≥1.