## Selected Papers

**Spectral Entropies as Information-Theoretic Tools for Complex Network Comparison**

Manlio De Domenico and Jacob Biamonte

Phys. Rev. X 6, 041062 (2016)

abstract, popular summary and open access PDF

Abstract

Any physical system can be viewed from the perspective that information is implicitly represented in its state. However, the quantification of this information when it comes to complex networks has remained largely elusive. In this work, we use techniques inspired by quantum statistical mechanics to define an entropy measure for complex networks and to develop a set of information-theoretic tools, based on network spectral properties, such as Rényi q entropy, generalized Kullback-Leibler and Jensen-Shannon divergences, the latter allowing us to define a natural distance measure between complex networks. First, we show that by minimizing the Kullback-Leibler divergence between an observed network and a parametric network model, inference of model parameter(s) by means of maximum-likelihood estimation can be achieved and model selection can be performed with appropriate information criteria. Second, we show that the information-theoretic metric quantifies the distance between pairs of networks and we can use it, for instance, to cluster the layers of a multilayer system. By applying this framework to networks corresponding to sites of the human microbiome, we perform hierarchical cluster analysis and recover with high accuracy existing community-based associations. Our results imply that spectral-based statistical inference in complex networks results in demonstrably superior performance as well as a conceptual backbone, filling a gap towards a network information theory.

Popular SummaryIn the realm of complex networks—for example, biological and quantum systems—a satisfactory definition of network entropy has thus far been hindered. The difficulty lies in realizing a tractable probability distribution that defines an interconnected system as a whole. A candidate starting point has been classical information theory, which is built centrally on the quantification of information through entropy. Despite its widespread applications, this approach is limited to applying entropy to a known network descriptor, resulting only in an alternative means to analyze a probability distribution. Here, we introduce a probability distribution representing a complex network that satisfies the same properties as a density matrix in quantum mechanics.

We build an information-theoretic framework that allows us to quantify the information content of complex networks. We define an entropy measure based on spectral properties of observed networks and their models, rather than relying on a subset of network descriptors such as mesoscale structure or degree distribution. Crucially, we introduce a metric distance to compare the units of interconnected systems such as multilayer networks. We apply our methodology to classifying human microbiome sites, and we explore our findings using numerical experiments. We show that our technique, which is built on ideas appearing in quantum statistical mechanics, is superior to previous efforts based on classical information theory. These previous efforts took into account only a subset of the possibilities (which our formulation inherently considers) and, furthermore, were subject to inconsistencies.

By showing that spectral methods are fundamental for analyzing and understanding complex networks, our framework introduces a set of spectral tools that we expect will have many statistical applications.

**Quantum Machine Learning**

Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe and Seth Lloyd

in review (2016) arXiv:1611.09347

**Chiral Quantum Walks**

Dawei Lu, Jacob D. Biamonte, Jun Li, Hang Li, Tomi H. Johnson, Ville Bergholm, Mauro Faccin, Zoltán Zimborás, Raymond Laflamme, Jonathan Baugh, Seth Lloyd

Physical Review A 93, 042302 (2016)

abstract and link

Given its importance to many other areas of physics, from condensed matter physics to thermodynamics, time-reversal symmetry has had relatively little influence on quantum information science. Here we develop a network-based picture of time-reversal theory, classifying Hamiltonians and quantum circuits as time-symmetric or not in terms of the elements and geometries of their underlying networks. Many of the typical circuits of quantum information science are found to exhibit time-asymmetry. Moreover, we show that time-asymmetry in circuits can be controlled using local gates only, and can simulate time-asymmetry in Hamiltonian evolution. We experimentally implement a fundamental example in which controlled time-reversal asymmetry in a palindromic quantum circuit leads to near-perfect transport. Our results pave the way for using time-symmetry breaking to control coherent transport, and imply that time-asymmetry represents an omnipresent yet poorly understood effect in quantum information science.

arXiv:1405.6209 [quant-ph]

**Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register**

Ya Wang et al.

ACS Nano 9, 7769 (2015)

abstract and link

Ab initiocomputation of molecular properties is one of the most promising applications of quantum computing. While this problem is widely believed to be intractable for classical computers, efficient quantum algorithms exist which have the potential to vastly accelerate research throughput in fields ranging from material science to drug discovery. Using a solid-state quantum register realized in a nitrogen-vacancy (NV) defect in diamond, we compute the bond dissociation curve of the minimal basis helium hydride cation, HeH^{+}. Moreover, we report an energy uncertainty (given our model basis) of the order of 10^{–14}hartree, which is 10 orders of magnitude below the desired chemical precision. As NV centers in diamond provide a robust and straightforward platform for quantum information processing, our work provides an important step toward a fully scalable solid-state implementation of a quantum chemistry simulator.arXiv:1405.2696 [quant-ph]

**Tensor Network Contractions for #SAT**

Jacob Biamonte, Jacob Turner and Jason Morton

Journal of Statistical Physics 160, 1389 (2015)

abstract and link

The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of

#SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is inP), determining the number of solutions is#P-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language forn-variable#SATinstances with complexity $$O((g+cd)^{O(1)} 2^c)$$ wherecis the number of COPY-tensors,gis the number of gates,

anddis the maximal degree of any COPY-tensor. Thus, counting problems can be solved efficiently when their tensor network expression has at most $$O(\log c)$$ COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SATinstance of the Dubois-Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.arXiv:1405.7375 [quant-ph]

**Tensor Networks and Graphical Calculus for Open Quantum Systems**

Chris Wood, Jacob Biamonte and David Cory

Quantum Information and Computation 15, 0579 (2015)

abstract and link

We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.

[PDF]

**Hamiltonian Gadgets with Reduced Resource Requirements**

Yudong Cao, Ryan Babbush, Sabre Kais and Jacob Biamonte

Physical Review A 91, 012315 (2015)

**Community Detection in Quantum Complex Networks**

Mauro Faccin, Piotr Migdał, Tomi Johnson, Ville Bergholm, Jacob Biamonte

Physical Review X 4, 041012 (2014)

abstract, popular summary and open access PDF

Abstract

Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, static or in interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems—a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.

Popular Summary

Real-life networks such as groups of animals and biochemical assemblies exhibit complex relationships that can benefit from systematic study. The macroscopic properties of a network cannot be easily deduced from knowledge of its microscopic properties. Such a deduction is aided by the identification of strongly connected subnetworks, called communities. For traditional networked systems, the problem of community detection has, accordingly, received a significant amount of attention, and a multitude of techniques are employed for this task, often based on dynamical processes within the network. No methods are currently known for community detection in quantum networks, despite a growing interest in large networks in quantum biology, transport, and communication. We extend the concept of community detection from classical to quantum systems, providing a crucial missing tool for analyzing quantum systems with a network structure.We argue that breaking down a quantum system into strongly correlated parts, i.e., a form of community partitioning, is an essential precursor for any simulation that aims to use this partitioning to reduce computational costs. We adapt traditional community detection methods that, as their starting point, use a measure of “closeness” of any two basic network components, denoted “nodes.” The computational costs of simulations scale exponentially with the number of nodes. We investigate quantum systems that are generally smaller than the classical systems typically studied, and we naturally ensure that the closeness measure captures relevant quantum effects, which can therefore lead to partitionings that are significantly different than those expected based on classical analyses. We partition nodes into communities using a quantum-walk process, which is akin to partitioning Hilbert space into orthogonal subspaces, illustrating our analyses on a light-harvesting complex.

We anticipate that our results will be useful for conducting numerical analyses of these systems.

**High fidelity spin entanglement using optimal control**

Florian Dolde, Ville Bergholm, Ya Wang, Ingmar Jakobi, Sebastien Pezzagna, Jan Meijer, Philipp Neumann, T. Schulte-Herbrüggen, Jacob Biamonte, Jörg Wrachtrup

Nature Communications 5, 3371 (2014)

**Degree Distribution in Quantum Walks on Complex Networks**

Mauro Faccin, Jacob Biamonte,Tomi Johnson, Sabre Kais, Piotr Migdał

Physical Review X 3, 041007 (2013)

abstract, popular summary and open access PDF

AbstractIn this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long-time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks, including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy, and at higher energies, the difference, the so-called

quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Rényi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.

Popular Summary

Imagine a web surfer mindlessly wandering from one page to another by clicking randomly on one of the many hyperlinks on each page they encounter. Where would they end up? This question and the answer to it actually are essential to how Google’s web-search engine decides the relative importance of the world’s webpages. Algorithmically, the world’s webpages are represented by a huge network of nodes (pages) and links (hyperlinks) and the mindless internet surfer by a “random walker.” Now, what happens if the random walker is quantum mechanical instead? This may sound like a question for science fiction, but it is actually part of a recent fundamental drive toward merging the science of complex networks—relevant to many scientific disciplines, including statistical physics, biology, computer science, and social science—with quantum mechanics. In this paper, we make one of the first steps in that drive: to uncover and delineate some of the fundamental connections and differences between classical and quantum networks, by developing and investigating a revealing toy model of quantum random walks on complex networks.It is well known that for a classical random walker on a complex network, the probability of finding the walker on a node after a long time is proportional to the probability of that node’s degree (or the number of links to other nodes), reflecting only the network’s connective topology. A quantum walker, however, brings conceptually nontrivial subtleties to the problem, including hallmark quantum effects such as quantum interference and the ability of a walker to be in a coherent superposition of states. In addition, the long-time state of a quantum walker depends on its initial state and most often does not converge to a steady state.

Here, we have constructed a model of a quantum walker on a network. The walker’s state is a multicomponent one, with the squared amplitude of the ith component representing the probability of finding the walker at node i of the network. This multicomponent state evolves in time according to a Schrödinger equation that has a correspondence with the classical walker. By investigating this model, we have succeeded in uncovering the following properties: (1) When the walker starts from its zero-energy ground state, the long-time average of probability of finding it at a node follows the classical result; (2) at higher energies, the walker’s long-time behavior deviates from the classical case, reflecting its quantumness, and this quantumness is quantitatively bounded by the initial energy of the walker and equal to Rényi entropy—a property associated with the network’s degree distribution.

Our paper thus provides the first analytical connection between classical and quantum walks on complex networks, as well as highlighting their differences. We see this work as the beginning of an exciting development that will involve quantum physics, graph and network theory, and the physics of stochastic processes.

[PDF]

**Solving search problems by strongly simulating quantum circuits**

T. H. Johnson, J. D. Biamonte, S. R. Clark, D. Jaksch

Scientific Reports 3, 1235 (2013)

**Tensor network methods for invariant theory**

Jacob Biamonte, Ville Bergholm, and Marco Lanzagorta,

J. Phys. A: Math. Theor. 46, 475301 (2013)

abstract and link

Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state factorization) when studying polynomial invariants. As our main example, we consider invariants of matrix product states. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Renyi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as matrix product states, contain excellent tools that can be applied in the study of invariants.

**Quantum Transport Enhancement by Time-Reversal Symmetry Breaking**

Zoltan Zimboras, Mauro Faccin, Zoltan Kadar, James Whitfield, Ben Lanyon, Jacob Biamonte

Scientific Reports 3, 2361 (2013)

abstract and open access PDF

Models of continuous time quantum walks, which implicitly use time-reversal symmetric Hamiltonians, have been intensely used to investigate the effectiveness of transport. Here we show how breaking time-reversal symmetry of the unitary dynamics in this model can enable directional control, enhancement, and suppression of quantum transport. Examples ranging from exciton transport to complex networks are presented. This opens new prospects for more efficient methods to transport energy and information.

**Undecidability in Tensor Network States**

Jason Morton and Jacob Biamonte

Physical Review A Rapid Communications 86, 030301(R) (2012)

**Ground State Spin Logic**

J. D. Whitfield, M. Faccin and J. D. Biamonte

EPL 99, 57004 (2012)

**Algebraically contractible topological tensor network states**

S. J. Denny, J. D. Biamonte, D. Jaksch and S. R. Clark

J. Phys. A: Math. Theor. 45, 015309 (2012)

**Categorical Tensor Network States**

Jacob Biamonte, Stephen Clark and Dieter Jaksch

AIP Advances 1(4), 042172 (2011)

abstract and link

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state S, we present a new and general method to factor S into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.

DOI: 10.1063/1.3672009

[PDF]

**Categorical Quantum Circuits**

Jacob Biamonte and Ville Bergholm

Journal of Physics A: Mathematical and Theoretical 44, 245304 (2011)

**Simulation of Electronic Structure Hamiltonians using Quantum Computers **

James Whitfield, Jacob Biamonte and Alan Aspuru-Guzik

Molecular Physics 109, 735 (2011)

**Adiabatic Quantum Simulators**

Jacob Biamonte, Ville Bergholm, James Whitfield, Joe Fitzsimons and Alan Aspuru-Guzik

AIP Advances 1, 022126 (2011)

**Non-perturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins**

Jacob Biamonte

Physical Review A 77, 052331 (2008)

**Realizable Hamiltonians for Universal Adiabatic Quantum Computers**

Jacob Biamonte and Peter Love

Physical Review A 78, 012352 (2008)

**Towards Quantum Chemistry on a Quantum Computer **

Benjamin P. Lanyon, James D. Whitfield, Geoff G. Gillet, Michael E. Goggin, Marcelo P. Almeida, Jacob D. Biamonte, Ben J. Powell, Marco Barbieri, Alán Aspuru-Guzik and Andrew G. White

Nature Chemistry 2, 106 (2009)

abstract and link

Exact first-principles calculations of molecular properties are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. A solution is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technology to calculate properties of the smallest molecular system: the hydrogen molecule in a minimal basis. We calculate the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chemical applications.

## Books and Chapters

**Tensor Networks for Entanglement Evolution**

Sebastian Meznaric, Jacob Biamonte

Advances in Chemical Physics, Volume 154, Chapter 17, Pages 561-574 (2013)

**Quantum Techniques for Stochastic Mechanics**

John C. Baez, Jacob Biamonte

Book preprint arXiv:1209.3632 (2016) [PDF]