Designs via Free Probability

Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $ k $-ETH ($ k=1,2,\dots $), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, where the lowest order ETH (1-ETH) is the conventional ETH. As a non-trivial contribution of the higher-order ETH, we show that the $ k $-ETH with $ k\geq 2 $ implies a universal behavior of the $ k $th Renyi entanglement entropy of individual energy eigenstates. In particular, the Page correction of the entanglement entropy originates from the higher-order ETH, while as is well known, the volume law can be accounted for by the 1-ETH. We numerically verify that the 2-ETH approximately holds for a nonintegrable system, but does not hold in the integrable case. To further investigate the information-theoretic feature behind the $ k $-ETH, we introduce a concept named a partial unitary $ k $-design (PU $ k $-design), which is an approximation of the Haar random unitary up to the $ k $th moment, where partial means that only a limited number of observables are accessible. The $ k $-ETH is a special case of a PU $ k $-design for the ensemble of Hamiltonian dynamics with random-time sampling. In addition, we discuss the relationship between the higher-order ETH and information scrambling quantified by out-of-time-ordered correlators. Our framework provides a unified view on thermalization, entanglement entropy, and unitary $ k $-designs, leading to deeper characterization of higher-order quantum complexity.

We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two nonequivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. An analysis of finite size effects reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large.

Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry's conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity-generated by the Fibonacci word and its generalizations-for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.

We test the eigenstate thermalization hypothesis (ETH) in 1 + 1 -dimensional SU(2) lattice gauge theory (LGT) with one flavor of dynamical fermions. Using the loop-string-hadron framework of the LGT with a bosonic cutoff, we exactly diagonalize the Hamiltonian for finite size systems and calculate matrix elements (MEs) in the eigenbasis for both local and nonlocal operators. We analyze different indicators to identify the parameter space for quantum chaos at finite lattice sizes and investigate how the ETH behavior emerges in both the diagonal and off-diagonal MEs. Our investigations allow us to study various timescales of thermalization and the emergence of random matrix behavior, and highlight the interplays of the several diagnostics with each other. Furthermore, from the off-diagonal MEs, we extract a smooth function that is closely related to the spectral function for both local and nonlocal operators. We find numerical evidence of the spectral gap and the memory peak in the nonlocal operator case. Finally, we investigate aspects of subsystem ETH in the lattice gauge theory and identify certain features in the subsystem reduced density matrix that are unique to gauge theories.

Classical and quantum dynamics in the classically chaotic stadium billiard are compared through studies of the evolving phase-space distributions and expectation values. Good agreement between the two mechanics is found for all distributions other than vertically directed wave packets which contain significant contributions from pseudoregular quantum eigenstates. Despite formal arguments as to the absence ``quantum chaos'' the quantum dynamics clearly shows a time-averaged relaxation to equilibrium, over the timescale studied, which is consistent with the classical K-system behavior.

Tasks such as classification of data and determining the ground state of a Hamiltonian cannot be carried out through purely unitary quantum evolution. Instead, the inherent nonunitarity of the measurement process must be harnessed. Post-selection and its extensions provide a way to do this. However, they make inefficient use of time resources - a typical computation might require O(2m) measurements over m qubits to reach a desired accuracy and cannot be done intermittently on current (superconducting-based) NISQ devices. We propose a method inspired by thermalization that harnesses insensitivity to the details of the bath. We find a greater robustness to gate noise by coupling to this bath, with a similar cost in time and more qubits compared to alternate methods for inducing nonlinearity such as fixed-point quantum search for oblivious amplitude amplification. Post-selection on m ancillae qubits is replaced with tracing out O[logϵ/log(1-p)] (where p is the probability of a successful measurement) to attain the same accuracy as the post-selection circuit. We demonstrate this scheme on the quantum perceptron, quantum gearbox, and phase estimation algorithm. This method is particularly advantageous on current quantum computers involving superconducting circuits.

Motivated by the existence of exact many-body quantum scars in the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain, in this work we explore the connection between matrix product state (MPS) wave functions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wave function. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong eigenstate thermalization hypothesis. Using our approach, we recover the AKLT chain starting from the MPS of its ground state, and we derive the most general nearest-neighbor Hamiltonian that shares the AKLT quasiparticle tower of exact eigenstates. We further apply this formalism to other simple MPS wave functions, and derive families of Hamiltonians that exhibit AKLT-like quantum scars. As a consequence, we also construct a scar-preserving deformation that connects the AKLT chain to the integrable spin-1 pure biquadratic model. Finally, we also derive other families of Hamiltonians that exhibit types of exact quantum scars, including a $\\textit{U}$(1)-invariant perturbed Potts model.

Quantum kicked rotor and its variants: Chaos, localization and beyond

Free probability theory, which has became a main branch of random matrix theory, is a valuable tool for describing the asymptotic behavior of multiple systems, especially for large random matrices. In this paper, using free probability theory, a new cooperative scheme for spectrum sensing is proposed, which shows how asymptotic free behavior of random matrices and the property of Wishart distribution can be used to assist spectrum sensing for cognitive radios. Simulations over Rayleigh fading and AWGN channels demonstrate the proposed scheme has better detection performance compared with the energy detection techniques even for the case of a small sample of observations.

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper, we use another version of the techniques to extend the results of [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] to the case where the entries of the data matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, justifies the mean field approximation in the infinite width limit for the deep untrained neural networks and the property of the macroscopic universality of random matrix theory in this case.

Applications of realizations (aka linearizations) to free probability

This paper addresses extensions of the complex Ornstein–Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , ak are *-free \({\fancyscript{R}}\) -diagonal operators in a II1 factor, then \({D_t(a_{i_1}\cdots a_{i_n}) = e^{-nt} a_{i_1}\cdots a_{i_n}}\) defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1, . . . , ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1, . . . , ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: \({\|D_t\colon L^2\to L^\infty\| \sim t^{-1}}\) for small t > 0.

The topic of this thesis work is free probability theory. The main goal is to understand asymptotic eigenvalue distributions of large classes of random matrices. For the models discussed in [SpVa12], we obtain a quite general algorithm to plot their distributions. We also apply the tools from [BSTV14] to give a general numerical algorithm to compute the asymptotic distribution of some other types of matrix models, such as the block-linearly modified models which have been considered in [Au12,BaNe12,BaNe12b].\n\nClassical, free and non-commutative probability can be jointly understood through the combinatorics of multiplicative functions with respect to different lattices of set partitions. The second goal of this thesis is to survey on the basic topics on the combinatorics of free probability. Our basic reference is [NiSp06]. We present new results which allow to compute cumulants of products of free and Boolean independent random variables in terms of the posets of k-divisible set partitions [ArVa12]. We also find formulas relating the different types of cumulants in non-commutative probability [AHLV14].\n\nIn connection to random matrix theory, we make particular use of the combinatorial approach to operator-valued free probability ([NSS02]) to compute Cauchy-Stieljes transforms of the asymptotic eigenvalue distributions of the matrix ensembles introduced in [SpVa12]. We do this to show that our definition of a free deterministic equivalent as a concrete operator, introduced in [SpVa12], agrees with the more widespread notion of deterministic equivalents which are being used, for example, to describe recent matrix models for wireless communications [CoDe11].\n\nVoiculescu introduced free probability in 1985 in the context of operator algebras. In 1991, he found realizations of his free circular, semi-circular and Haar-unitary operators through limits of eigenvalue distributions of quite remarkable random matrix models, such as independent (self-adjoint and non-self-adjoint) Wigner and Haar-unitary random matrices. This allowed to understand Wigner's semicircle law as a special, single-variable case of a very general phenomenon on joint non-commutative distributions of large random matrices.\n\nIn 1995 he introduced operator-valued free probability, where the limiting behaviors of much more general random matrix models can be realized. A rich class of random matrix models arises from considering a polynomial P(x_1,...,x_n) in non-commutative indeterminates x_1,...,x_n,x_1^*,...,x_n^* and evaluating it on random and deterministic matrices. In this work we are specially concerned about these kind of models. We refer to them as polynomial models''.\n\nIf the inputs are (self-adjoint or non-self-adjoint) Wigner matrices, Wishart matrices, and deterministic matrices, we may consider a deterministic operator Q by evaluating P on certain operators (y_1,...,y_n) in the context of Voiculescu's free probability theory. Provided that the size of the matrices is large (but not necessarily too large), the spectral measure of the simplified model Q becomes a good approximation of the averaged eigenvalue distribution of P. The dimensions of the matrices can also be different. The free deterministic equivalent Q of P was defined in [SpVa12], based on the generalizations of [Vo91] to rectangular spaces [BG09,BG09b], and using mostly combinatorial tools from [NSS02].\n\nThe method of deterministic equivalents (DE) was introduced by Girko at the level of Cauchy-Stieltjes transforms of the considered matrix models. In contrast to DE, the simplification from P to Q can be explained very easily and does not require the polynomial to have a specific form. It will turn out that our definitions from [SpVa12] can be very effectively combined with all the elements of method described in [BMS13] for the distributions of polynomials on self-adjoint, square, asymptotically free random matrices.\n\nThroughout this work, we comment on how the different assumptions on the distributions that we input to the random matrix models affect the quality and the type of convergence of the model to its FDE. In particular, we discuss this at the combinatorial level: The different assumptions on the model determine the classes of partitions (or cumulants) that show up on the matrix sums, and hence the moments and the nature of the fixed point equations that we will get for its FDE.\n\nTo be numerically efficient one needs to understand how freeness restricts and extends to different operator-valued levels. For this, the combinatorial methods from [NSS02] are quite important.

The asymptotic behaviour of the mean spectral density for the product of N × N random unitary matrices distributed according to the Haar measure and a fixed diagonal matrix is investigated. The large N limit of the exact expression derived by Wei and Fyodorov (2008 J. Phys. A: Math. Theor. 41 502001) is calculated by a modification of the saddle point method. It is shown that in the bulk the result coincides with the one obtained within the free probability theory by Haagerup and Larsen (2000 J. Funct. Anal. 176 331). Close to the edge points of the bulk asymptotics, the mean density for large N is described by a universal function depending on a certain scaling variable. The large deviation formulae valid outside the bulk are also derived. Obtained formulae agree well with the results of direct numerical calculations.

We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these measures are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that generally the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.