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Oscar Perdomo

Publications and source records attributed to Oscar Perdomo.

36 records · Page 2Linked to original sources
arXiv2020

Four qubits generated by Clifford gates

The Clifford group is the set of gates generated by the controlled not gates, the Hadamard gate and the P={{1,0},{0,i}} gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. In this paper we study the set of all 4-qubit Clifford states. We prove that there are 293760 states and their entanglement entropy must be either 0, 2/3, 1, 4/3 and 5/3. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We achieve this by splitting the 293760 states into 18 groups where each pair of states in a group can be connected with a local Clifford gate. We then study how the different CNOT gates act on the 18 groups. We also study the Clifford states with real entries under the action of the subgroup C_R of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in C_R. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.

quant-ph↗
arXiv2019

Entanglement types for two-qubit states with real amplitudes

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states --those locally equivalent to the Bell States --form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled, or so-called product states, are $π/4$ units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval $[0,π/4]$ with the point $0$ corresponding to the maximally entangled states. The point $π/4$ corresponds to the unentangled states which geometrically are described by a torus. Finally, for every $0< d < π/4$ the point $d$ corresponds to a disjoint pair of torus. We also show that if a state is $d$ units away from the maximally entangled states, then its entanglement entropy is $S(d) = 1- \log_2 \sqrt{\frac{(1+\sin 2 d)^{1+\sin 2 d}}{(1-\sin 2 d)^{-1+\sin 2 d}}}$. Finally, we also show how this geometrical interpretation allows us to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.

quant-ph↗
arXiv2019

A generative modeling approach for benchmarking and training shallow quantum circuits

Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterization of quantum devices and to train shallow circuits for generative tasks. The procedure leverages quantum hardware capabilities to its fullest extent by using native gates and their qubit connectivity. We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as "cat states". We further demonstrate that our approach can efficiently prepare approximate representations of coherent thermal states, wave functions that encode Boltzmann probabilities in their amplitudes. Finally, complementing proposals to characterize the power or usefulness of near-term quantum devices, such as IBM's quantum volume, we provide a new hardware-independent metric called the qBAS score. It is based on the performance yield in a specific sampling task on one of the canonical machine learning data sets known as Bars and Stripes. We show how entanglement is a key ingredient in encoding the patterns of this data set; an ideal benchmark for testing hardware starting at four qubits and up. We provide experimental results and evaluation of this metric to probe the trade off between several architectural circuit designs and circuit depths on an ion-trap quantum computer.

quant-ph↗
arXiv2019

Partition of 3-qubits using local gates

It is well known that local gates have smaller error than non-local gates. For this reason it is natural to make two states equivalent if they differ by a local gate. Since two states that differ by a local gate have the same entanglement entropy, then the entanglement entropy defines a function in the quotient space. In this paper we study this equivalence relation on (i) the set $\mathbb{R}\hbox{Q}(3)$ of 3-qubit states with real amplitudes, (ii) the set $Q\mathcal{C}$ of 3-qubit states that can be prepared with gates on the Clifford group, and (iii) the set $Q\mathbb{R}\mathcal{C}$ of 3-qubit states in $Q\mathcal{C}$ with real amplitudes. We show that the set $Q\mathcal{C}$ has 8460 states and the quotient space has 5 elements. We have $\frac{Q\mathcal{C}}{\sim}=\{S_{0},S_{2/3,1},S_{2/3,2},S_{2/3,3},S_{1}\}$. As usual, we will call the elements in the quotient space, orbits. We have that the orbit $S_0$ contains all the states that differ by a local gate with the state $|000\rangle$. There are 1728 states in $S_0$ and as expected, they have zero entanglement entropy. All the states in the orbits $S_{2/3,1},S_{2/3,2},S_{2/3,3}$ have entanglement entropy $2/3$ and each one of these orbits has 1152 states. Finally, the orbit $S_{1}$ has 3456 elements and all its states have maximum entanglement entropy equal to one. We also study how the controlled not gates $CNOT(1,2)$ and $CNOT(2,3)$ act on these orbits. For example, we show that when we apply a $CNOT(1,2)$ to all the states in $S_0$, then 960 states go back to the same orbit $S_0$ and 768 states go to the orbit $S_{2/3,1}$. Similar results are obtained for $\mathbb{R}Q\mathcal{C}$. We also show that the entanglement entropy function reaches its maximum value 1 in more than one point when acting on $\frac{\hbox{$\mathbb{R}$Q(3)} }{\sim}$.

quant-ph↗
arXiv2019

Spectrum of the Laplacian and the Jacobi operator on rotational cmc hypersurfaces of spheres

Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and we also prove that there are at least 2 positive eigenvalues of the Laplacian of $M$ smaller than $n$. When $H$ is not zero, we have that every non-flat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. Let $m$ be the number of points where the maximal distance from this profile curve to the origin is achieved (we assume that the coordinates of the plane containing the profile curve has been set up so that the axis of rotation goes through the origin). Let $l$ be be the wrapping number of the profile curve. We show that the number of negative eigenvalues of the operator $J$ counted with multiplicity is at least $(2l-1)n+(2m-1)$. This result was proven for the case $n=2$ by Rossman and Sultana. They called $m$ the number of bulges or the number of necks. We will slightly change the definition of $l$ to include immersed examples that contain the axis of rotation.

math.DG↗
arXiv2019

Robust Implementation of Generative Modeling with Parametrized Quantum Circuits

Although the performance of hybrid quantum-classical algorithms is highly dependent on the selection of the classical optimizer and the circuit ansatz, a robust and thorough assessment on-hardware of such features has been missing to date. From the optimizer perspective, the primary challenge lies in the solver's stochastic nature, and their significant variance over the random initialization. Therefore, a robust comparison requires that one perform several training curves for each solver before one can reach conclusions about their typical performance. Since each of the training curves requires the execution of thousands of quantum circuits in the quantum computer, such a robust study remained a steep challenge for most hybrid platforms available today. Here, we leverage on Rigetti's Quantum Cloud Services to overcome this implementation barrier, and we study the on-hardware performance of the data-driven quantum circuit learning (DDQCL) for three different state-of-the-art classical solvers, and on two-different circuit ansatze associated to different entangling connectivity graphs for the same task. Additionally, we assess the gains in performance from varying circuit depths. To evaluate the typical performance associated with each of these settings in this benchmark study, we use at least five independent runs of DDQCL towards the generation of quantum generative models capable of capturing the patterns of the canonical Bars and Stripes data set.

quant-ph↗
arXiv2015

A family of periodic solutions of the three body problem. Light version

In this paper we describe a 1-dimensional family of initial conditions Σthat provides reduced periodic solution of the three body problem. This family Σcontains a bifurcation point and extend the periodic solution described in (Perdomo, http://arxiv.org/pdf/1507.01100.pdf). This 1-dimensional family is the union of two embedded smooth curves. We will explain how the trajectories of the bodies in the solutions coming from one of the embedded curves have two symmetries while those coming from the other embedded curve only have one symmetry. The Round Taylor Method is a numerical method implemented by the author to keep track of the global error and the round-off error. A second version of this paper, same title with the "light version" part removed, will include analysis of the error of the solutions using the Round Taylor Method.

math.DS↗
arXiv2015

A small variation of the Taylor Method and periodic solutions of the 3-body problem

In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact solution. As an application we provide a rigorous proof of the construction/existence of a periodic solution of the three body problem. Some images of this periodic motion can be seen at https://www.youtube.com/watch?v=fSmQyeKcj5k

math.DS↗
arXiv2014

A family of Solutions of the N-Body Problem

In this paper we characterize all the solutions of the three body problem on which one body with mass $m_1$ remains in a fixed line and the other two bodies have the same mass $m_2$. We show that all the solutions with negative total energy (potential plus kinetic), never collide. "Explicit" solutions of the motion of all the three bodies are given in term of just one solution of an ordinary differential equation of order two of one real value function. We show that the solutions of a big subfamily of this family remains bounded for all time. The same results are shown for the n-body problem. When the body on the line remains constant, we have provide exact formulas for the solutions --for any n-- in the same way as we have for the two body problem. Once we have shown that there are infinitely many periodic solutions, we provide several examples of periodic examples for the 3-Body, 4-Body, 5-Body and 6-Body problem. This is a first version. Comments and suggestions for references are welcome. A YouTube video explaining one of this solutions has been posted on http://youtu.be/2Wpv6vpOxXk

math.DS↗
arXiv2014

Equilibrium Shapes of Cylindrical Rotating Liquid Drops

See http://www.youtube.com/watch?v=izbGXdjvK_I for a YouTube video showing part of the results in this paper.We will consider surfaces whose mean curvature at a point is a linear function of the square of the distance from that point to the vertical axis. We restrict ourselves here to surfaces which are cylinders over a curve in a horizontal plane. We describe the moduli space of the set of solutions which includes numerous properly immersed and embedded examples. We also analyze the stability of these cylindrical surfaces.

math.DG↗
arXiv2014

Rotating Drops with Helicoidal Symmetry

See http://youtu.be/Mf4IE8gWcJs for a YouTube video showing part of the results in this paper. We consider helicoidal immersions in the Euclidean space whose axis of symmetry is the z-axis that are solutions of the equation 2 H=Λ_0-a 1/2 R^2 where H is the mean curvature of the surface, R is the distance form the point in the surface to the z-axis and a is a real number. We refer to these surfaces as helicoidal rotating drops. We prove the existence of properly immersed solutions that contain the z-axis. We also show the existence of several families of embedded examples. We describe the set of possible solutions and we show that most of these solutions are not properly immerse and are dense in the region bounded by two concentric cylinders. We show that all properly immersed solutions, besides being invariant under a one parameter helicoidal group, they are invariant under a cyclic group of rotations of the variables x and y. The second variation of energy for the volume constrained problem with Dirichlet boundary conditions is also studied.

math.DG↗
arXiv2012

Rotational Surfaces in S^3 with constant mean curvature

Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} with mean curvature H. In 2010, Perdomo showed that for every H between cot(π/m) and (m^2-2)/(2(m^2-1)^1/2), there exists an embedded axially symmetric example with non constant principal curvatures that is invariant under the ciclic group Z_m. Andrews and Li, showed that these examples are the only non-isoparametric embedded examples in the family when H>0. In this paper we study those examples in the family with H<0. We prove that there are no embedded examples in the family when H<0 and we also prove that for every integer m>2 there is a properly immersed example in this family that contains a great circle and is invariant under the ciclic group Z_m. We will say that these examples contain the axis of symmetry. Finally we show that every non-isoparametric surface Σ_{H,C} is either properly immersed invariant under the ciclic group Z_m for some integer m>1 or it is dense in the region bounded by two isoparametric tori if the surface Σ_{H,C} does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface Σ_{H,C} contains the axis of symmetry.

math.DG↗
arXiv2009

CMC hypersurfaces on riemannian and semi-riemannian manifolds

In this paper we show explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-riemannian manifolds with constant sectional curvature. In particular, we prove that every h in [-1,-2 sqrt{n-1}/n) can be realized as the constant curvature of a complete immersion of S_1^{n-1} x R in the (n+1)-dimensional de Sitter space S_1^{n+1}. We provide 3 types of immersions with cmc in the Minkowski space, 5 types of immersion with cmc in the de Sitter space and 5 types of immersion with cmc in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.

math.DG↗
arXiv2009

Algebraic zero mean curvature varieties in semi-riemannian manifolds

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature in the de Sitter space.

math.DG↗
arXiv2009

Embedded constant mean curvature hypersurfaces on spheres

Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\fracπ{m}) and b_{m,n}=\frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}, then, there exists a non isoparametric, compact embedded hypersurface in S^{n+1} with constant mean curvature H that admits the group O(n)x Z_m in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Z_m are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m>2 and H is close to \cot(\fracπ{m}), the hypersurfaces look like a necklets made out of m spheres with (m+1) catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to b_{m,n} the hypersurface is close to an isoparametric hypersurface with the same mean curvature. As a consequence of the expression of these bounds for H, we have that every H different from 0,\pm\frac{1}{\sqrt{3}} can be realized as the mean curvature of a non isoparametric CMC surface in S^3. For hyperbolic spaces we prove that every non negative H can be realized as the mean curvature of an embedded CMC hypersurface in H^{n+1}, moreover we prove that when H>1 this hypersurface admits the group O(n)\times Z in its group of isometries. Here Z are the integer numbers. As a corollary of the properties proven for these hypersurfaces, for any n> 5, we construct non isoparametric compact minimal hypersurfaces in S^{n+1} which cone in R^{n+2} is stable. Also, we will prove that the stability index of every non isoparametric minimal hypersurface with two principal curvatures in S^{n+1} is greater than 2n+5.

math.DG↗
arXiv2008

A characterization of quadric constant mean curvature hypersurfaces of spheres

Let $ϕ:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=ϕ(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=ν(x),v$, where $ν:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $λ$, we have that $\ell_v=λf_v$, then, $ϕ(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.

math.DG↗
arXiv2007

Minimal tori with low nullity

The nullity of a minimal submanifold $M\subset S^{n}$ is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nullity $kn(M)$ of $M$. In the case of 2-dimensional tori $M$ in $S^{3}$, there is an additional naturally-defined 2-dimensional subspace; the dimension of the sum of the action of the rigid motions and this space is the natural nullity $nnt(M)$. In this paper we will study minimal tori in $S^{3}$ with natural nullity less than 8. We construct minimal immersions of the plane $R^{2}$ in $S^{3}$ that contain all possible examples of tori with $nnt(M)<8$. We prove that the examples of Lawson and Hsiang with $kn(M)=5$ also have $nnt(M)=5$, and we prove that if the $nnt(M)\le6$ then the group of isometries of $M$ is not trivial.

math.DG↗