Speaker: Rafael Freitas dos Santos (Center for Theoretical Physics PAS)
Abstract
In this talk, I’ll present a concept of nonclassicality exploring properties of contextuality and nonlocality. Quantum contextuality is one of the most intriguing properties of quantum theory and provides a notion of nonclassicality, while nonlocality is a special case of contextuality when the physical system is composed of spatially separated subsystems. Recently, certification schemes of quantum systems based on maximal violation of noncontextuality inequalities and Bell inequalities have been developed. I’ll give a broad overview of the basic mathematical concepts and then outline the main results about certification schemes based on the following articles:
https://doi.org/10.22331/q-2020-08-03-302,
https://doi.org/10.1103/PhysRevA.106.012431
https://doi.org/10.48550/arXiv.2212.07133.
Out of time ordered correlations, information scrambling and quantum entanglement
Date: Monday, March 20, 2023
Time: 14:15
Host: Quantum Chaos and Quantum Information (Jagiellonian University)
Abstract
We introduce a family of positive linear maps in the algebra of $3\times3$ complex matrices, which generalizes the seminal positive non-decomposable map originally proposed by Choi.
Necessary and sufficient conditions for decomposability are derived and demonstrated. The proposed maps offer a new method for the analysis of bound entangled states of two qutrits.
Pdf: https://arxiv.org/abs/2212.03807
Improved local models and new Bell inequalities
Date: Monday, March 6, 2023
Time: 14:15
Host: Quantum Chaos and Quantum Information (Jagiellonian University)
Abstract
In this work, we report a deterministic and exact algorithm to reverse any unknown qubit-encoding isometry operation. We present the semidefinite programming (SDP) to search the Choi matrix representing a quantum circuit reversing any unitary operation. We derive a quantum circuit transforming four calls of any qubit-unitary operation into its inverse operation by imposing the SU(2)×SU(2) symmetry on the Choi matrix. This algorithm only applies only for qubit-unitary operations, but we extend this algorithm to any qubit-encoding isometry operations. For that, we derive a subroutine to transform a unitary inversion algorithm to an isometry inversion algorithm by constructing a quantum circuit transforming finite sequential calls of any isometry operation into random unitary operations.
A random matrix model for random approximate t-designs
Speaker: Adam Sawicki (Centre for Theoretical Physics, Polish Academy of Sciences)
Abstract
For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mc{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mc{S}$ can be regarded as a $\delta(\nu_\mc{S},t)$-approximate $t$-design, $t\in\mathbb{Z}_+$. We propose a random matrix model that aims to describe the probability distribution of $\delta(\nu_\mathcal{S},t)$ for any $t$. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by $t$. We prove that, the operator norm of this matrix, $\delta({t})$, is the random variable to which $\sqrt{|\mathcal{S}|}\delta(\nu_\mc{S},t)$ converges in distribution when the number of elements in $\mc{S}$ grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$, for any $\epsilon>0$. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability $1$ there is $t\in\mathbb{Z}_+$ such that $\sup_{k\in\mathbb{Z}_{+}}\delta(k)=\delta(t)$. Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of $\mc{S}$. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities $\mathbb{P}(\sqrt{\mathcal{S}}\delta(\nu_\mathcal{S},t)>2+\epsilon)$ are bounded from above by the tail probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$ of our random matrix model. In particular our conjecture implies that a Haar random set $\mathcal{S}\subset U(d)$ satisfies the spectral gap conjecture with the probability $1$.
Speaker: Andris Ambainis (Center for Quantum Computer Science, Faculty of Computing, University of Latvia)
Abstract
Quantum algorithms are useful for a variety of problems in search and optimization. This line of work started with Grover’s quantum search algorithm which achieved a quadratic speedup over naive exhaustive search but has now developed far beyond it. In this talk, we describe three recent results in this area: (i) We show that, for any classical algorithm that uses a random walk to find an object with some property (by walking until the random walker reaches such an object), there is an almost quadratically faster quantum algorithm (arxiv:1903.07493). (ii) We show that the best-known exponential time algorithms for solving several NP-complete problems (such as Travelling Salesman Problem or TSP) can be improved quantumly (arxiv:1807.05209). For example, for the TSP, the best-known classical algorithm needs time O(2^n) but our quantum algorithm solves the problem in time O(1.728…^n). (iii) We show an almost quadratic quantum speedup for a number of geometric problems such as finding three points that are on the same line (arxiv:2004.08949).
About the speaker
Andris Ambainis is a professor of computer science at the University of Latvia. He has invented widely used methods for constructing quantum algorithms and has constructed record-breaking examples of quantum advantage for quantum computers. Andris Ambainis’ work has been recognized by an Advanced Grant from the European Research Council (2012) and the Grand Medal of the Latvian Academy of Sciences (2013).
Quantum correlations from the post-quantum perspective: When can we have security against a super-quantum adversary ?
Date: Monday, January 23, 2023
Time: 14:15
Host: Quantum Chaos and Quantum Information (Jagiellonian University)
Abstract
Approximate t-design are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitaries up to the order t. As such, they find numerous applications throughout quantum information, including randomized benchmarking , efficient estimation of properties of quantum states, decoupling, information transmission and quantum state discrimination. In this talk I will characterize how finite random gate-sets mimic the Haar measure.
The talk will be based on the joint word with Piotr Dulian.
arXiv:2210.07872
