Unitary Map Research Articles

Topological expansion of unitary integrals and maps

We study integrals on the unitary group with respect to the Haar measure. We give a combinatorial interpretation in terms of maps of the asymptotic topological expansion, established previously by Guionnet and Novak. The maps we introduce—the maps of unitary type—satisfy Tutte-like equations. It allows us to show that in the perturbative regime, they describe the different orders of the asymptotic topological expansion. Furthermore, they generalize the monotone Hurwitz numbers.

Quantization of locally compact groups associated with essentially bijective 1-cocycles

Given an extension [Formula: see text] of locally compact groups, with [Formula: see text] abelian, and a compatible essentially bijective [Formula: see text]-cocycle [Formula: see text], we define a dual unitary [Formula: see text]-cocycle on [Formula: see text] and show that the associated deformation of [Formula: see text] is a cocycle bicrossed product defined by a matched pair of subgroups of [Formula: see text]. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of [Formula: see text] on [Formula: see text] and a unitary quantization map [Formula: see text] of Kohn–Nirenberg type.

Large-N limit of the Segal–Bargmann transforms on the spheres

We study the large-N limit of the Segal–Bargmann transform on SN−1(N), the (N−1)-dimensional sphere of radius N, as a unitary map from the space of square-integrable functions with respect to the normalized spherical measure onto the space of holomorphic square-integrable functions with respect to a certain measure on the quadric. In particular, we give an explicit formulation and describe the geometric models for the limit of the domain, the limit of the range, and the limit of the transform when N tends to infinity. We show that the limiting transform is still a unitary map from the limiting domain onto the limiting range.

Quantum walk mixing is faster than classical on periodic lattices

The quantum mixing time is a critical factor affecting the efficiency of quantum sampling and algorithm performance. It refers to the minimum time required for a quantum walk to approach its limiting distribution closely and has implications across the areas of quantum computation. This work focuses on the continuous time quantum walk mixing on a regular graph, evolving according to the unitary map U=eiĀt, where the Hamiltonian Ā is the normalized adjacency matrix of the graph. In [Physical Review A 76, 042306 (2007).], Richter previously showed that this walk mixes in time O(ndlog(d)log(1/ϵ)) with O(log(d)log(1/ϵ)) intermediate measurements when the graph is the d−dimensional periodic lattice Zn×Zn×⋯×Zn. We extend this analysis to the periodic lattice L=Zn1×Zn2×⋯×Znd, relaxing the assumption that ni are identical. We provide two quantum walks on periodic lattices that achieve faster mixing compared to classical random walks: 1. A coordinate-wise quantum walk that mixes in O((∑i=1dni)log(d/ϵ)) time with O(dlog(d/ϵ)) measurements. 2. A continuous-time quantum walk with O(log(1/ϵ)) measurements that conjecturally mixes in O(∑i=1dni(log(n1))2log(1/ϵ)) time. Our results demonstrate a quadratic speedup over the classical mixing time O(dn12log(d/ϵ)) on the generalized periodic lattice L. We have provided analytical evidence and numerical simulations to support the conjectured faster mixing time of the continuous-time quantum walk algorithm. Making progress towards proving the general conjecture that quantum walks on regular graphs mix in O(δ−1/2log(N)log(1/ϵ)) time, where δ is the spectral gap and N is the number of vertices.

Quantum process tomography of unitary maps from time-delayed measurements

Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics. The overarching mathematical framework and feasibility guarantee of our method is provided by the Takens embedding theorem. We explain in detail how the reconstruction of a single-qubit Hamiltonian works in this setting and provide numerical methods and experiments for general few-qubit and lattice systems with local interactions. In particular, the method allows to find the Hamiltonian of a two qubit system by observing only one of the qubits.

Unitary maps on Hamiltonians of an electron moving in a plane and coherent state construction

In this work, we consider a model of an electron moving in a plane under uniform external magnetic and electric fields. We investigate the action of unitary maps on the associated quantum Hamiltonians and construct coherent states of Gazeau–Klauder type.

Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Białynicki-Birula (BB) approach, based on the Riemann–Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approaches are completely equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the BB quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell’s equations.

Lyapunov Spectrum Scaling for Classical Many-Body Dynamics Close to Integrability.

We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.

Bounding the quantum capacity with flagged extensions

In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum Q and private P capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than 1/2. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.

Quantum optimal control of ten-level nuclear spin qudits in Sr87

We study the ability to implement unitary maps on states of the $I=9/2$ nuclear spin in $^{87}\mathrm{Sr}$, a $d=10$ dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin resonance and a tensor ac Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline-earth-metal atoms, such as $^{87}\mathrm{Sr}$, have a very favorable figure of merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf Rabi frequency of ${\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{\ensuremath{\psi}}\ensuremath{\rangle}=0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.

Two-color optically addressed spatial light modulator as a generic spatiotemporal system.

Nonlinear spatiotemporal systems are the basis for countless physical phenomena in such diverse fields as ecology, optics, electronics, and neuroscience. The canonical approach to unify models originating from different fields is the normal form description, which determines the generic dynamical aspects and different bifurcation scenarios. Realizing different types of dynamical systems via one experimental platform that enables continuous transition between normal forms through tuning accessible system parameters is, therefore, highly relevant. Here, we show that a transmissive, optically addressed spatial light modulator under coherent optical illumination and optical feedback coupling allows tuning between pitchfork, transcritical, and saddle-node bifurcations of steady states. We demonstrate this by analytically deriving the system's dynamical equations in correspondence to the normal forms of the associated bifurcations and confirm these results via extensive numerical simulations. Our model describes a nematic liquid crystal device using nano-dimensional dichalcogenide (a-As 2S 3) glassy thin films as photo sensors and alignment layers, and we use device parameters obtained from experimental characterization. Optical coupling, for example, using diffraction, holography, or integrated unitary maps allows implementing a variety of system topologies of technological relevance for neural networks and potentially Ising or XY-Hamiltonian models with ultralow energy consumption.

Maps preserving the Douglas solution of operator equations

We consider bijective maps ϕ on the full operator algebra B(H) of an infinite dimensional Hilbert space with the property that, for every A,B,X∈B(H), X is the Douglas solution of the equation A=BX if and only if Y=ϕ(X) is the Douglas solution of the equation ϕ(A)=ϕ(B)Y. We prove that those maps are implemented by a unitary or anti-unitary map U, i.e., ϕ(A)=UAU⁎.

Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory

We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even though the measurement record is not informationally complete. We then compare this with the information generated and fidelities obtained by application of a different Haar random unitary at each time step. We give an upper bound on the maximal information that can be obtained in tomography and show that a covariance matrix taken from the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution saturates this bound. We find that physically, this corresponds to an application of a different Haar random unitary at each time step. We show that repeated application of random diagonal unitaries gives a covariance matrix in tomographic estimation that corresponds to a new ensemble of random matrices. We analytically and numerically estimate eigenvalues of this ensemble and show the information gain to be bounded from below by the Porter-Thomas distribution.

Orthogonal projections on hyperplanes intertwined with unitaries

Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes–no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.

Inferences on a multidimensional social hierarchy use a grid-like code.

Generalizing experiences to guide decision-making in novel situations is a hallmark of flexible behavior. Cognitive maps of an environment or task can theoretically afford such flexibility, but direct evidence has proven elusive. In this study, we found that discretely sampled abstract relationships between entities in an unseen two-dimensional social hierarchy are reconstructed into a unitary two-dimensional cognitive map in the hippocampus and entorhinal cortex. We further show that humans use a grid-like code in entorhinal cortex and medial prefrontal cortex for inferred direct trajectories between entities in the reconstructed abstract space during discrete decisions. These grid-like representations in the entorhinal cortex are associated with decision value computations in the medial prefrontal cortex and temporoparietal junction. Collectively, these findings show that grid-like representations are used by the human brain to infer novel solutions, even in abstract and discrete problems, and suggest a general mechanism underpinning flexible decision-making and generalization.

Schrödinger Equation in Moving Domains

We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation}where $\Omega(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $\Omega_0$ and a specific family of unitary maps $h^\sharp(t): L^2(\Omega(t),\mathbb{C})\longrightarrow L^2(\Omega_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }\Omega_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $\Omega(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $\Omega(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_\nu u(t)+i\langle\nu| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $\Omega(t)$.

Phase constants in the Fock–Goncharov quantum cluster varieties

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized to unitary maps between the Hilbert spaces intertwining the representations. These unitary intertwiners are described using the quantum dilogarithm function $$\Phi ^\hbar $$ . Algebraic relations among classical mutations are satisfied by the intertwiners up to complex constants. The present paper shows that these constants are 1. So the mapping class group representations resulting from the Chekhov–Fock–Goncharov quantum Teichmüller theory are genuine, not projective. During the course, the hexagon and the octagon operator identities for $$\Phi ^\hbar $$ are derived.

Quantum Cellular Automata, Tensor Networks, and Area Laws.

Quantum cellular automata are unitary maps that preserve locality and respect causality. We identify them, in any dimension, with simple tensor networks (projected entangled pair unitary) whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. We define other classes of nonunitary maps, the so-called quantum channels, that either respect causality or preserve locality. We show that, whereas the latter obey an area law for the number of quantum correlations they can create, as measured by the quantum mutual information, the former may violate it. We also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.

Bulk detection of time-dependent topological transitions in quenched chiral models

The topology of one-dimensional chiral systems is captured by the winding number of the Hamiltonian eigenstates. Here we show that this invariant can be read-out by measuring the mean chiral displacement of a single-particle wavefunction that is connected to a fully localized one via a unitary and translational-invariant map. Remarkably, this implies that the mean chiral displacement can detect the winding number even when the underlying Hamiltonian is quenched between different topological phases. We confirm experimentally these results in a quantum walk of structured light.

Floquet Anderson localization of two interacting discrete time quantum walks

We study the interplay of two interacting discrete time quantum walks in the presence of disorder. Each walk is described by a Floquet unitary map defined on a chain of two-level systems. Strong disorder induces a novel Anderson localization phase with a gapless Floquet spectrum and one unique localization length $\xi_1$ for all eigenstates for noninteracting walks. We add a local contact interaction which is parametrized by a phase shift $\gamma$. A wave packet is spreading subdiffusively beyond the bounds set by $\xi_1$ and saturates at a new length scale $\xi_2 \gg \xi_1$. In particular we find $\xi_2 \sim \xi_1^{1.2}$ for $\gamma=\pi$. We observe a nontrivial dependence of $\xi_2$ on $\gamma$, with a maximum value observed for $\gamma$-values which are shifted away from the expected strongest interaction case $\gamma=\pi$. The novel Anderson localization regime violates single parameter scaling for both interacting and noninteracting walks.