arXiv subjects

Simone Severini

Publications and source records attributed to Simone Severini.

At least 37 records · Page 2Linked to original sources

arXiv2018

Weak Modular Product of Bipartite Graphs, Bicliques and Isomorphism

A 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Restricting to bipartite graphs and considering complete bipartite subgraphs (bicliques) therein, we study the combinatorics of the weak modular product. We identify cases where isomorphism is tractable using this approach, which we call Isomorphism via Biclique Enumeration (IvBE). We find that IvBE is polynomial for bipartite $2K_2$-free graphs and quasi-polynomial for families of bipartite graphs, where the largest induced matching and largest induced crown graph grows slowly in $n$, that is, $O(\mathrm{polylog }\, n)$. Furthermore, as expected a straightforward corollary of Kozen's theorem and Lovász's sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in $n$. However, we show that for balanced, bipartite graphs this is only true in a few trivial cases. In doing so we define a new graph product on bipartite graphs, the very weak modular product. The results pertaining to bicliques in bipartite graphs proved here may be of independent interest.

math.CO↗

arXiv2018

Some Ulam's reconstruction problems for quantum states

Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.

quant-ph↗

arXiv2018

Entanglement properties of quantum grid states

Grid states form a discrete set of mixed quantum states that can be described by graphs. We characterize the entanglement properties of these states and provide methods to evaluate entanglement criteria for grid states in a graphical way. With these ideas we find bound entangled grid states for two-particle systems of any dimension and multiparticle grid states that provide examples for the different aspects of genuine multiparticle entanglement. Our findings suggest that entanglement theory for grid states, although being a discrete set, has already a complexity similar to the one for general states.

quant-ph↗

arXiv2018

Quantum Walk Search on Kronecker Graphs

Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an "initiator" graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. In this paper, we explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's $O(\sqrt{N})$ time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search.

quant-ph↗

arXiv2018

Learning hard quantum distributions with variational autoencoders

Studying general quantum many-body systems is one of the major challenges in modern physics because it requires an amount of computational resources that scales exponentially with the size of the system.Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a new representation of states based on variational autoencoders. Variational autoencoders are a type of generative model in the form of a neural network. We probe the power of this representation by encoding probability distributions associated with states from different classes. Our simulations show that deep networks give a better representation for states that are hard to sample from, while providing no benefit for random states. This suggests that the probability distributions associated to hard quantum states might have a compositional structure that can be exploited by layered neural networks. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterising states of the size expected in first generation quantum hardware.

quant-ph↗

arXiv2018

Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian

Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits there is pretty good state transfer between the nodes at the j-th and (n-j+1)-th position if n is a power of 2. Moreover, this condition is also necessary for j=1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.

quant-ph↗

arXiv2018

Universal discriminative quantum neural networks

Quantum mechanics fundamentally forbids deterministic discrimination of quantum states and processes. However, the ability to optimally distinguish various classes of quantum data is an important primitive in quantum information science. In this work, we train near-term quantum circuits to classify data represented by non-orthogonal quantum probability distributions using the Adam stochastic optimization algorithm. This is achieved by iterative interactions of a classical device with a quantum processor to discover the parameters of an unknown non-unitary quantum circuit. This circuit learns to simulates the unknown structure of a generalized quantum measurement, or Positive-Operator-Value-Measure (POVM), that is required to optimally distinguish possible distributions of quantum inputs. Notably we use universal circuit topologies, with a theoretically motivated circuit design, which guarantees that our circuits can in principle learn to perform arbitrary input-output mappings. Our numerical simulations show that shallow quantum circuits could be trained to discriminate among various pure and mixed quantum states exhibiting a trade-off between minimizing erroneous and inconclusive outcomes with comparable performance to theoretically optimal POVMs. We train the circuit on different classes of quantum data and evaluate the generalization error on unseen mixed quantum states. This generalization power hence distinguishes our work from standard circuit optimization and provides an example of quantum machine learning for a task that has inherently no classical analogue.

quant-ph↗

arXiv2018

Constructing graphs with limited resources

We discuss the amount of physical resources required to construct a given graph, where vertices are added sequentially. We naturally identify information -- distinct into instructions and memory -- and randomness as resources. Not surprisingly, we show that, in this framework, threshold graphs are the simplest possible graphs, since the construction of threshold graphs requires a single bit of instructions for each vertex and no use of memory. Large instructions without memory do not bring any advantage. With one bit of instructions and one bit of memory for each vertex, we can construct a family of perfect graphs that strictly includes threshold graphs. We consider the case in which memory lasts for a single time step, and show that as well as the standard threshold graphs, linear forests are also producible. We show further that the number of random bits (with no memory or instructions) needed to construct any graph is asymptotically the same as required for the Erdős-Rényi random graph. We also briefly consider constructing trees in this scheme. The problem of defining a hierarchy of graphs in the proposed framework is fully open.

cs.DM↗

arXiv2018

Quantum linear systems algorithms: a primer

The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.

quant-ph↗

arXiv2018

Quantum machine learning: a classical perspective

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.

quant-ph↗

arXiv2017

Experimental learning of quantum states

The number of parameters describing a quantum state is well known to grow exponentially with the number of particles. This scaling clearly limits our ability to do tomography to systems with no more than a few qubits and has been used to argue against the universal validity of quantum mechanics itself. However, from a computational learning theory perspective, it can be shown that, in a probabilistic setting, quantum states can be approximately learned using only a linear number of measurements. Here we experimentally demonstrate this linear scaling in optical systems with up to 6 qubits. Our results highlight the power of computational learning theory to investigate quantum information, provide the first experimental demonstration that quantum states can be "probably approximately learned" with access to a number of copies of the state that scales linearly with the number of qubits, and pave the way to probing quantum states at new, larger scales.

quant-ph↗

arXiv2017

How to Suppress Dark States in Quantum Networks and Bio-Engineered Structures

Transport across quantum networks underlies many problems, from state transfer on a spin network to energy transport in photosynthetic complexes. However, networks can contain dark subspaces that block the transportation, and various methods used to enhance transfer on quantum networks can be viewed as equivalently avoiding, modifying, or destroying the dark subspace. Here, we exploit graph theoretical tools to identify the dark subspaces and show that asymptotically almost surely they do not exist for large networks, while for small ones they can be suppressed by properly perturbing the coupling rates between the network nodes. More specifically, we apply these results to describe the recently experimentally observed and robust transport behaviour of the electronic excitation travelling on a genetically-engineered light-harvesting cylinder (M13 virus) structure. We believe that these mainly topological tools may allow us to better infer which network structures and dynamics are more favourable to enhance transfer of energy and information towards novel quantum technologies.

quant-ph↗

arXiv2017

Quantum and non-signalling graph isomorphisms

We introduce a two-player nonlocal game, called the $(G,H)$-isomorphism game, where classical players can win with certainty if and only if the graphs $G$ and $H$ are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the $(G,H)$-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.

quant-ph↗

arXiv2017

Generalized Satisfiability Problems via Operator Assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations.

cs.LO↗

arXiv2016

Descriptive complexity of graph spectra

Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic $C^3$ are co-spectral, and this is not the case with $C^2$, nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.

cs.LO↗

arXiv2016

Note on von Neumann and Rényi entropies of a Graph

We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.

math.CO↗

arXiv2016

Permutation graphs and unique games

We study the value of unique games as a graph-theoretic parameter. This is obtained by labeling edges with permutations. We describe the classical value of a game as well as give a necessary and sufficient condition for the existence of an optimal assignment based on a generalisation of permutation graphs and graph bundles. In considering some special cases, we relate XOR games to EDGE BIPARTIZATION, and define an edge-labeling with permutations from Latin squares.

math.CO↗

arXiv2016

On moments of the integrated exponential Brownian motion

We present new exact expressions for a class of moments for the geometric Brownian motion, in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Ito's Wiener process. We then apply the obtained exact formulas to computing averages of the solution of the logistic stochastic differential equation via a series expansion, and compare the results to the solution obtained via Monte Carlo.

cond-mat.stat-mech↗