“

—Robert FrostThe woods are lovely, dark and deep,But I have promises to keep,And miles to go before I sleep,And miles to go before I sleep“

From the first day I learned about quantum computing as an undergraduate student over two decades ago, I have not stopped studying.

I am deeply fascinated by the intersection of quantum physics and theoretical computer science and the associated mathematical methods used to describe and reason about quantum information processing systems.

Throughout my work, I have focused on the intersection of theory and physical hardware constraints. While my expertise lies on the theoretical side, I abstract the physical embodiment of the algorithm, the model of quantum computation, and consider hardware constraints as essential components shaping my work.

My research in quantum information science has taken me around the world, giving hundreds of invited talks and authoring several influential papers in the field. I would like to give you a glimpse into my story, not necessarily in chronological order, but I will mention some of the 100+ researchers I collaborated with and the locations where I conducted parts of my work.

For those who might be interested, here is a chronological list of all my publications with links.

As always, please feel free to reach out if you would like to discuss my research program.

We will add links to all heading (sections):

- Quantum machine learning & variational quantum algorithms
- Proving universality of quantum models of computation
- Quantum information and complex networks
- Quantum tensor networks
- Quantum algorithms simulating quantum chemistry
- Adiabatic and ground state-based quantum computing
- Contributions to the experimental development of the field
- References

## (1) Quantum machine learning & variational quantum algorithms

Arthur Samuel

“We must be careful not to confuse data with the abstractions we use to analyze them.”

Modern quantum processors have opened up the possibility of executing short quantum circuits, which consist of a fixed circuit structure with tunable gates (Kandala et al. 2017). This development led myself, along with other researchers, to explore the potential of quantum circuits as a new type of machine learning model based on quantum mechanics (Biamonte et al. 2017; Kardashin, Uvarov, and Biamonte 2021; Uvarov, Kardashin, and Biamonte 2020; Morales, Tlyachev, and Biamonte 2018). Although training quantum circuits shares similarities with training classical neural networks, there are differences.

One of the challenges encountered is analyzing the quantum to classical outer-loop optimization required for training variational algorithms and machine learning models. Many previous works in the field focused on proof-of-principle applications development (Kardashin et al. 2021). To address this, I introduced the concept of operator cardinality (Biamonte 2021; Biamonte 2022), which revealed exponential overheads in certain existing applications proposed for variational quantum algorithms. For example, the Hamiltonian

(1)

has terms when expanded in the local Pauli basis. Yet,

(2)

shares the same lowest eigenstate () as (1) yet has operator cardinality . Moreover in the work (PRA 103:L030401, 2021), it was shown that cardinality is invariant under conjugation by Clifford circuits. Of course, algebraic and spatial locality is not invariant under Clifford conjugation. Through my theory (PRA 103:L030401, 2021), I demonstrated that the variational model can emulate general quantum circuits, making it computationally universal as a feed-forward model. This finding contrasts with the work I coauthored on universality, which established that QAOA ansatze quantum circuits used in variational quantum computation can emulate any other quantum circuit (Morales, Biamonte, and Zimborás 2020).

Before my research, the literature on quantum approximate optimization was biased toward a statistically non-representative class of problem instances. Together with Vishwanathan and other collaborators, I discovered the concept of reachability deficits, which arise from increasing the constraint-to-variable ratio of random k-satisfiability instances and result in under-parameterization and failure of quantum approximate optimization (Akshay et al. 2020). We further investigated this effect in experimental data from Google (Akshay et al. 2021a) and found that as density increases, circuit depths beyond current experimental capacities become necessary for optimal performance (Harrigan et al. 2021).

Our subsequent discoveries shed light on abrupt training transitions and introduced penalty functions that cannot be learned below a certain number of circuit layers or depth, but can be learned perfectly by appending a single layer (Campos, Nasrallah, and Biamonte 2021). Although this effect was present in certain numerical data, it had not been explained in the literature. Our team provided the first analytical quantification to predict this effect (Campos et al. 2021).

Continuing our exploration with Vishwanathan, Campos, and Rabinovich we established that parameters concentrate in quantum approximate optimization (Campos et al. 2021). This means that training on a problem instance for -qubits produces a good approximation for training on a problem instance with qubits, where is . We developed a mathematical framework and derived analytical results to predict this concentration effect (Campos et al. 2021).

Building on the concept of layerwise training, we made an intriguing discovery. We found that single-layer training stops at critical values but can be recovered under certain types of noise (Akshay et al. 2021b). This finding suggests the potential of noise-assisted training. With Uvarov we also studied barren plateaus in variational quantum algorithms, providing some new bounds for their existence (Uvarov and Biamonte 2021).

Selected thematic papers (for a more comprehensive list, please refer to the references section or consult this list of all my publications):

**On Barren Plateaus and Cost Function Locality in Variational Quantum Algorithms**

A Uvarov and J Biamonte

Journal of Physics A: Mathematical and Theoretical 54, 245 (2021) DOI: 10.1088/1751-8121/abfac7

**Parameter Concentrations in Quantum Approximate Optimization**

V Akshay, D Rabinovich, E Campos, and J Biamonte

(Letter) Physical Review A 104:L010401 (2021) DOI: 10.1103/PhysRevA.104.L010401

**Training Saturation in Layerwise Quantum Approximate Optimisation**

E Campos, D Rabinovich, V Akshay, and J Biamonte

(Letter) Physical Review A 104:L030401 (2021) DOI: 10.1103/PhysRevA.104.L030401

**Reachability Deficits in Quantum Approximate Optimization**

V Akshay, H Philathong, M Morales, and J Biamonte

Physical Review Letters 124, 090504 (2020) DOI: 10.1103/PhysRevLett.124.090504

## (2) Proving universality of physical models of computation

Freeman Dyson

“For a physicist mathematics is not just a tool by means of which phenomena can be calculated,it is the main source of concepts and principles by means of whichnew theories can be created.”

In my research journey, I became an expert in proving the computational universality of experimentally relevant models of quantum computation and developing the necessary mathematical tools for these proofs. A significant focus of my work has been on the use of ground states as a resource for quantum computation, with applications in both adiabatic quantum computation and variational quantum algorithms.

Modern quantum algorithms rely on the availability of fixed and tunable quantum circuit structures (Kandala et al. 2017). To eventually understand the ultimate capacity of these algorithms and establish a theory of training quantum circuits, I proved that variational quantum computation is universal already without feedback (Biamonte 2021; Biamonte 2022). This achievement was recognized by the editors, stating that “this result brings the resources required for universal quantum computation closer to contemporary quantum processors [Letter, Editors’ Selection, PRA 103:3 L030401, (2021)].”

Along with other researchers, we established that certain short quantum circuits used in variational quantum computation can emulate any other quantum circuit. This refinement and extension of Lloyd’s result on quantum approximate optimization being universal have significantly contributed to the field (Morales, Biamonte, and Zimborás 2020).

On the other hand, I derived combinatorial bounds on the maximum possible entanglement across any bipartition of qubits acted upon by a quantum circuit composed of local unitaries and controlled NOT gates. These bounds, known as the combinatorial quantum circuit area law, provide some no-go results on what is not possible with variational quantum circuits (Biamonte 2021; Biamonte, Morales, and Koh 2020).

My research journey into adiabatic quantum computing can be traced back to a phone call I made to Dr. Geordie Rose, the founder of D-Wave Systems Inc., while completing my undergraduate thesis on quantum algorithms. I asked Dr. Rose to provide me with the most difficult theoretical problem his company was facing, which eventually led to my employment as one of the world’s first professional quantum applications scientists at D-Wave Systems Inc. During this time, I solved an open problem in Hamiltonian complexity, proving that the ground state energy problem of a tunable Heisenberg model is QMA-hard and enables universal adiabatic quantum computation (Biamonte and Love 2008). This work was patented by D-Wave Systems Inc. (International Patent WO/2008/122128, US Patent 60/910:445, 2007).

My collaboration with Peter Love resulted in the publication of this work (Biamonte and Love 2008). Building upon the contributions of renowned researchers such as Alexei Kitaev, Dorit Aharonov, and others, allowed me to influence both experimental and theoretical developments in ground state quantum computation. Government funding programs from IARPA and NRL in the United States have supported the experimental development of universal ground state models, leading to the realization of exotic couplers that serve as building blocks for a universal ground state quantum computer proposed in (Biamonte and Love 2008).

The golden thread connecting this research is the use of a ground state as a universal resource for quantum computation. The techniques used in both adiabatic and variational quantum computation share some similarities and build on some of the seminal theoretical findings in the field.

Selected thematic papers (for a more comprehensive list, please refer to the references section or consult this list of all my publications):

**Universal Variational Quantum Computation**

J Biamonte

(Letter) Physical Review A 103:L030401 (2021) DOI: 10.1103/PhysRevA.103.L030401

**On the Universality of the Quantum Approximate Optimization Algorithm**

M Morales, J Biamonte, and Z Zimborás

Quantum Information Processing 19, 291 (2020) DOI: 10.1007/s11128-020-02748-9

**Realizable Hamiltonians for Universal Adiabatic Quantum Computers**

J Biamonte and P Love

Physical Review A 78, 012352 (2008) DOI: 10.1103/PhysRevA.78.012352

## (3) Quantum information and complex networks

Stephen Hawking

I think the next [21st] century will be the century of complexity.

Despite outward simplicity, certain connectivity patterns of a graph—or network—often mirror physical properties that are meaningful, weaving a common thread across a range of phenomena. These phenomena span from the degree distribution of the internet (Albert and Barabási 2002), social dynamics, to systems guided by mass-action kinetics (Baez and Biamonte 2017). Indeed, the structure of a network can predict a broad range of noticeable behaviors (Newman 2010). Complex network theory evolved with graph theoretic models that are relevant to what we call complex systems (Albert and Barabási 2002; Newman 2010). While complex network theory characterizes a multitude of systems exceedingly well, these same tools often seem to fail for the majority of scenarios related to quantum information (Biamonte, Faccin, and De Domenico 2019). This failure led to the challenge of understanding why network theory falls short, and the quest for a new theory of quantum networks to supplement and supplant the traditional complex network theory (Bianconi and Barabási 2001; Perseguers et al. 2010; Biamonte, Faccin, and De Domenico 2019; De Domenico and Biamonte 2016; Zimboras et al. 2013; Faccin et al. 2013; Faccin et al. 2014; De Domenico and Biamonte 2016).

I have made substantial contributions to the application of complex networks and algebraic graph theory in quantum information science (Biamonte, Faccin, and De Domenico 2019). This emerging field is sometimes termed quantum complex networks, although this phrase is often used to denote quantum communication networks exclusively. My research has played a crucial role in sparking the subject, engendering several sub-fields that explore facets of the theory of quantum transport on graphs (De Domenico and Biamonte 2016; Zimboras et al. 2013; Faccin et al. 2013; Faccin et al. 2014; De Domenico and Biamonte 2016), and related topics that compare quantum and stochastic processes (Baez and Biamonte 2017).

Together with my collaborators, including Faccin and Bergholm, we merged ideas from complex network theory, creating several new analytical solutions to quantum dynamics problems (Faccin et al. 2013; Faccin et al. 2014). Specifically, Faccin et al. (2013) nurtured the low-energy theory of walks on graphs, contrasting the scale-free behavior of quantum versus stochastic walks on graphs. In Faccin et al. (2014), we introduced the quantum analogue of community detection. These results were published in a series of papers in Physical Review X (Faccin et al. 2013; Faccin et al. 2014; De Domenico and Biamonte 2016). In recognition of these works and others, I was invited to become a lifelong member of the Foundational Questions Institute, a nomination requested by peers with similar interests.

Working alongside John Baez from the University of California, Irvine, we proposed a mathematical framework for comparing stochastic and quantum processes (Bergholm and Biamonte 2011). This work was dedicated to utilizing mathematical techniques found in quantum theory and making them applicable to topics in mathematical biology and general chemical reaction networks. The paper I co-authored with De Domenico stimulated the sub-field currently examining information theoretic measures applied to complex network theory (De Domenico and Biamonte 2016), leading to numerous dedicated workshops and special issues in leading journals.

Selected thematic papers (for a more comprehensive list, please refer to the references section or consult this list of all my publications):

**Community Detection in Quantum Complex Networks**

M Faccin, P Migdał, T Johnson, V Bergholm, and J Biamonte

Physical Review X 4, 041012 (2014) DOI: 10.1103/PhysRevX.4.041012

**Quantum Techniques in Stochastic Mechanics**

J Baez and J Biamonte

World Scientific Publishing Co Pte Ltd, 276 pp. (2017) DOI: 10.1142/10623

**Complex Networks from Classical to Quantum**

J Biamonte, M Faccin, and M De Domenico

Communications Physics 2 (2019) DOI: 10.1038/s42005-019-0152-6

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

M De Domenico and J Biamonte

Physical Review X 6, 041062 (2016) DOI: 10.1103/PhysRevX.6.041062

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

Z Zimboras, M Faccin, Z Kadar, J Whitfield, B Lanyon, and J Biamonte

Scientific Reports 3, 2361 (2013) DOI: 10.1038/srep02361

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

M Faccin, T Johnson, S Kais, P Migdał, and J Biamonte

Physical Review X 3, 041007 (2013) DOI: 10.1103/PhysRevX.3.041007

## (4) Quantum tensor networks

Rodger Penrose

“What is particularly curious about quantum theory is that there can be actual physical effects arising from what philosophers refer to as counterfactuals-that is, things that might have happened, although they did not in fact happen.”

I have made substantial contributions to the theory of quantum tensor networks, which have primarily emerged from several European schools over the past decade and a half (Cirac et al. 2021). Tensor networks have a deep modern history in mathematical physics (Penrose 1971), in category theory (Selinger 2009), in computer science, algebraic logic and related disciplines (Baez and Stay 2010). These tensor network methods sprang from modern quantum theory as a tool to approximate quantum states, as detailed in the review by Cirac et al. (2021).

My early work with Bergholm introduced a model of quantum circuits that connected with the tensor networks employed in categorical quantum mechanics (Bergholm and Biamonte 2011). Later work with Clark and Jaksch led me to propose a theory of string bond states using related ideas, intersecting the tools of tensor networks used in categorical quantum mechanics with those found in condensed matter physics (Biamonte, Clark, and Jaksch 2011). This theory built upon earlier work including (Abramsky and Coecke 2004), Lafont’s algebraic theory of logic gates (Lafont 2003), and other facets of graphical tensor systems applied to quantum theory (Coecke and Duncan 2011; Coecke and Kissinger 2017; Biamonte 2017).

This framework of mine gave rise to a series of new findings and conclusions. In particular, while working with coauthors, I adapted these tools and found efficient tensor network descriptions of finite Abelian lattice gauge theories (Denny et al. 2012), thereby identifying a broad class of efficiently contractable tensor networks (Biamonte, Morton, and Turner 2015). In recognition of this work (Bergholm and Biamonte 2011; Biamonte, Clark, and Jaksch 2011), I received an EPSRC Doctoral Thesis Prize, administered by the Chancellor of the University of Oxford’s Mathematical and Physical Sciences Division in 2010.

In collaboration with Wood and Corey at the Institute for Quantum Computing at the University of Waterloo, we explored a graphical approach to open quantum systems (Wood, Biamonte, and Cory 2015), providing an alternative to Selinger’s graphical theory of completely positive maps (Selinger 2007). Additionally, I demonstrated that the ZX-calculus rewrite system (Coecke and Duncan 2011; Coecke and Kissinger 2017) admits a poly-time terminating rewrite sequence, thereby establishing the Gottesman–Knill theorem (Biamonte 2017; Biamonte 2022).

Later, Turner, Morton, and myself proved that #P-hard counting problems (and thus 2, 3-SAT decision problems) can be solved efficiently when their tensor network expression has at most 𝒪(ln 𝑐) COPY-tensors and polynomial bounded fan-out (Biamonte, Morton, and Turner 2015). Also, together with Bergholm and Lanzagorta, I proposed a diagrammatic theory of algebraic invariants applied to entanglement classification (Biamonte, Bergholm, and Lanzagorta 2013). I presented this work as a Shapiro Lecture, in the Department of Mathematics at Pennsylvania State University.

More recently, I have extended techniques found in projected entangled pair states (PEPS) to bound the bipartite entanglement possible in certain low-depth circuits generated by specific quantum processor topologies (where the topology is defined by the two-qubit interactions) (Biamonte, Morales, and Koh 2020; Biamonte 2021). With Kardashin and Uvarov, we also developed a quantum machine learning approach to contract tensors on a quantum computer (Kardashin, Uvarov, and Biamonte 2021).

My work, as well as the short courses and lecture notes I’ve given on tensor networks, have contributed to the growth of this vibrant multidisciplinary field. Today, tensor networks link condensed matter with quantum computation, as well as various other computer science approaches.

**Tensor Network Contractions for #SAT**

J Biamonte, J Morton, and J Turner

Journal of Statistical Physics 160, 1389 (2015) DOI: 10.1007/s10955-015-1276-z

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

C Wood, J Biamonte, and D Cory

Quantum Information & Computation 15, 759 (2015) DOI: 10.26421/QIC15.9-10-3

**Tensor Network Methods for Invariant Theory**

J Biamonte, V Bergholm, and M Lanzagorta

Journal of Physics A: Mathematical and Theoretical 46, 475301 (2013) DOI: 10.1088/1751-8113/46/47/475301

**Categorical Quantum Circuits**

V Bergholm and J Biamonte

Journal of Physics A: Mathematical and Theoretical 44, 245304 (2011) DOI: 10.1088/1751-8113/44/24/245304

## (5) Quantum algorithms simulating chemistry and physics

Richard Feynman

“Nature isn’t classical,~~dammit~~, and if you want to make a simulation of nature,

you’d better make it quantum mechanical,

and by golly it’s a wonderful problem,

because it doesn’t look so easy.”

Richard Feynman had proposed that, unlike classical computers, which would presumably experience an exponential slowdown when simulating quantum phenomena, a quantum simulator would not (Feynman 1985). Such early ideas were extended by Seth Lloyd (Lloyd 1996) and others over several decades. Today, a global research effort is focused on quantum emulation algorithms, largely in response to technology giants like Google and IBM making their processors available online and significant national funding initiatives (Biamonte, Dorozhkin, and Zacharov 2019). My early work in this field has stood the test of time in the vast quantum simulation literature, and my recent work is once again shaping the direction of this increasingly broad subject.

I’ve held the title of Fellow at Harvard University two separate times. While at Harvard, I collaborated with Whitfield (now at Dartmouth College) in Aspuru-Guzik’s group (now at University of Toronto), contributing to the seminal complexity analysis results of quantum algorithms for electronic structure problem instances (Whitfield, Biamonte, and Aspuru-Guzik 2011). This paper, now among the most cited in the field, received the Longuet-Higgins Paper Prize.

Many physical systems tend to minimize their free energies, and many remain close to, or in, their lowest energy state. Much of the focus on quantum simulation has therefore been on computing ground state energies (Biamonte et al. 2011). This common thread ties together many of my works on quantum simulation algorithms. In adiabatic quantum computation, we view ground states as a resource to perform computation, while in quantum simulation, we use a quantum computer to determine ground states. Lastly, variational quantum algorithms utilize methods from ground state quantum computation to create penalty functions.

Exploiting these connections, I proposed a type of quantum simulator partially based on adiabatic evolution (Biamonte et al. 2011). More recently, my colleagues and I proposed a means to find ground states of soliton models using variational algorithms (Kardashin et al. 2021). In collaboration with Uvarov, I suggested the merger of quantum simulation with quantum machine learning, resulting in a classifier detecting phases of matter (Uvarov, Kardashin, and Biamonte 2020). The totality and breadth of my work in quantum algorithms for simulation—as reflected by the citation record—underpin the research effort of the vibrant quantum computing applications community that has emerged today.

**Machine Learning Phase Transitions with a Quantum Processor**

A Uvarov, A Kardashin, and J Biamonte

Physical Review A 102, 012415 (2020) DOI: 10.1103/PhysRevA.102.012415

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

J Whitfield, J Biamonte, and A Aspuru-Guzik

Molecular Physics 109, 735 (2011) DOI: 10.1080/00268976.2011.552

## (6) Adiabatic and ground state-based quantum computing

Charles Darwin

“Believing as I do that man in the distant future will be a far more perfectcreature than he now is, it is an intolerable thought that he and all othersentient beings are doomed to complete annihilationafter such long-continued slow progress.”

I was one of the early researchers of the adiabatic approach to quantum computation, with a particular interest in universal adiabatic quantum computation. Many of the results from my research have remained relevant since their inception. My contributions include proving that a flux and capacitive coupling (tunable and ) can provide a universal resource for ground state quantum computation (Biamonte and Love 2008).

To put it simply, the two-body model Hamiltonian

(3)

is (i) computationally universal for adiabatic quantum computation and (ii) has a QMA-complete ground state energy decision problem. I also developed ground state spin-logic to program Hamiltonian ground states (Biamonte 2008; Whitfield, Faccin, and Biamonte 2012) and proposed methods to emulate physical systems using Hamiltonian ground states (Biamonte et al. 2011). This established a milestone result in Hamiltonian complexity theory which still remains the contemporary experimental target.

Adiabatic quantum computers—and other forms of ground state quantum computation—see the work of Farhi et al. (Farhi et al. 2001) and Kirkpatrick et al. (Kirkpatrick, Gelatt Jr, and Vecchi 1983)—utilise the low-energy configuration of a Hamiltonian system. They build on the principle that physical systems will minimise their free energy (Farhi et al. 2001; Van Dam, Mosca, and Vazirani 2001; Roland and Cerf 2002; Aharonov et al. 2008). My task was to develop methods to bootstrap this naturally occurring phenomena. The main questions that I had to answer were:

- How might one program Hamiltonian ground states?
- How might one best utilise physical processes to minimise a system’s Hamiltonian for computation?

In electronic structure calculations, for example, the lowest energy configuration of the 2nd quantised electronic Hamiltonian is of central interest. It has an expansion wherein the locality of Hamiltonian terms (the highest product terms) are bounded as some logarithmic function in the number of orbitals, using the Bravyi-Kitaev transform (Bravyi and Kitaev 2002).

I considered a tunable Ising model which assigns real values to binary edge assignments of a graph, written as

(4)

where corresponds to a real valued edge weight connecting a graph’s vertex and is an edge-dependent variable taking values and is the product of two edge assignments. A most elementary question that I faced early on with D-Wave’s annealer and elsewhere was: how might one define the couplings so that the lowest energy state of equation (4) is spanned by solutions to a problem of interest? This is often called, embedding.

Some problems have evident embedding(s), others are more complicated. My approach was to develop deductive, algebraic methods to create frameworks to solve embedding problems (and prove no-go theorems about possible embeddings (Biamonte 2008; Whitfield, Faccin, and Biamonte 2012)). I revived a theory of pseudo Boolean functions and developed a framework to embed logic gate operations into the lowest-energy sector of Ising spins (Biamonte 2008; Whitfield, Faccin, and Biamonte 2012). I used these and other methods to prove that three (and higher) body interaction Ising terms can be mimicked through the minimisation of 2- and 1-body Ising terms on slack spins (Biamonte 2008; Whitfield, Faccin, and Biamonte 2012). Working with Whitfield and Faccin (Whitfield, Faccin, and Biamonte 2012), I studied the representations of the symmetry groups of pseudo Boolean penalty functions (Biamonte 2008).

To program a universal adiabatic quantum computer, one faces a non-commuting embedding problem which requires the development of what is called Hamiltonian gadgets. Employing methods from infinite series, my collaborators, Cao and Kais, and I reduced the energy gap—a key physical resource—establishing the best known gadgets to date (Cao et al. 2015). Our work also implied several complexity theoretic results e.g. that the addition of the couplers again results in a universal model for adiabatic quantum computing. Furthermore, we established that one can simulate quantum chemistry with adiabatic quantum computers that have limited interaction types (Cao et al. 2015), thus enhancing my proposal to use adiabatic quantum computers as quantum simulators (Bergholm and Biamonte 2011).

**Hamiltonian Gadgets with Reduced Resource Requirements**

Y Cao, R Babbush, J Biamonte, and S Kais

Physical Review A 91, 012315 (2015) DOI: 10.1103/PhysRevA.91.012315

**Ground-State Spin Logic**

J Whitfield, M Faccin, and J Biamonte

Europhysics Letters 99, 57004 (2012) DOI: 10.1209/0295-5075/99/57004

**Nonperturbative k-Body to Two-Body Commuting Conversion Hamiltonians and Embedding Problem Instances into Ising Spins**

J Biamonte

Physical Review A 77, 052331 (2008) DOI: 10.1103/PhysRevA.77.052331

## (7) Contributions to the experimental development of the field

Yogi Berra

“In theory there is no difference between theory and practice. In practice there is.”

Apart from contributing to the theoretical foundations of several theoretical proposals that underpin such wide interest today, I also contributed to the experimental development of the field: I helped design and support several milestone quantum information processing demonstrations (Harris et al. 2007; Lanyon et al. 2010; Wang et al. 2015; Dolde et al. 2014; Lu et al. 2016; Palmieri et al. 2020; Borzenkova et al. 2021), beginning with numerical simulations of the first in situ tunable coupler for flux qubits while still at D-Wave Systems Inc. (Harris et al. 2007).

I led the theory side of the first experiment which enacted tomography using neural networks in a quantum photonics experiment (Palmieri et al. 2020). The paper developed feed forward neural networks to characterise measurement errors, resulting in several quantified improvements compared to existing state-of-the-art methods. This was done in partnership with the Kulik group at the Center for Quantum Technologies, at Moscow State University. Already being cited over 100 times, this paper became an instant staple of the literature on works focused on applying machine learning to solve problems in quantum theory. I also led the theory effort to demonstrate a variational algorithm simulating the Swinger model using a pair of polarization qubits (Borzenkova et al. 2021), again with the Kulik group of Moscow State University.

Collaborating with Bergholm, I developed the theory towards the early experimental implementation of optimal control. Collaborating with the Wrachtrup group of Stuttgart University, the software package was used to control quantum experiments based on Nitrogen vacancy centers in diamond (Dolde et al. 2014). The first result was precision control of a quantum memory which was demonstrated by creating, storing and then retrieving an entangled Bell pair. Next, I proposed the demonstration of a quantum algorithm for electronic structure calculations. Making use of a qubit-qutrit pair, the Wrachtrup group developed a quantum simulation of helium hydride cation using nitrogen vacancy centers in diamond based on my proposal (Wang et al. 2015).

I also proposed an experimental demonstration based on the theory of time-symmetry breaking to control quantum transport (Zimboras et al. 2013). I led the collaboration with the groups of Laflamme and Baugh of the Institute of Quantum Computing at the University of Waterloo, demonstrating the effect using nuclear magnetic resonance with three spin qubits (Lu et al. 2016).

Returning to on the theoretical gate counts in the work (Whitfield, Biamonte, and Aspuru-Guzik 2011) which received the Longuet-Higgins Prize: in collaboration with the White group of the University of Queensland, Australia, I supported the first experimental demonstration of quantum algorithms for electronic structure problems (Lanyon et al. 2010). The theory lead by Alan Aspuru-Guzik and the paper appears In Web of Science top 0.1% Highly Cited Paper Index for the field of Physics. Myself and James Whitfield were responsible for the theoretical appendix of the paper and for proposing the circuits that the experiments were based on. The calculation determined the complete energy spectrum to 20 bits of precision and discussed how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. The work remains a classic today.

My work also the experimental development of neighboring fields. My theory of algebraic embedding reductions (Biamonte 2008; Whitfield, Faccin, and Biamonte 201is responsible for essentially bringing the field of pseudo Boolean quadratic reductions into contact with quantum physics. Yet these same techniques found applications (Borders et al. 2019) in the new field creating -bits using high frequency instabilities in solid state electronics (called stochastic magnetic tunnel junctions)—see the citations therein (Borders et al. 2019). This early work of mine employed techniques from digital circuit theory to reduce penalty functions into quadratic Ising models such as Equation (4).

**Variational Simulation of Schwinger’s Hamiltonian with Polarization Qubits**

O Borzenkova, G Struchalin, A Kardashin, V Krasnikov, N Skryabin, S Straupe, S P Kulik, and J Biamonte

Applied Physics Letters 118:144002 (2021) DOI: 10.1063/5.0043322

**Experimental Neural Network Enhanced Quantum Tomography**

A Palmieri, E Kovlakov, F Bianchi, D Yudin, S Straupe, J Biamonte, and S Kulik

npj Quantum Information 6, 20 (2020) DOI: 10.1038/s41534-020-0248-6

**Chiral Quantum Walks**

D Lu, J Biamonte, J Li, H Li, T H Johnson, V Bergholm, M Faccin, Z Zimborás, R Laflamme, J Baugh, and S Lloyd

Physical Review A 93, 042302 (2016) DOI: 10.1103/PhysRevA.93.042302

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

Y Wang, F Dolde, J Biamonte, R Babbush, V Bergholm, S Yang, I Jakobi, P Neumann, A Aspuru-Guzik, J Whitfield, and J Wrachtrup

ACS Nano 9, 7769 (2015) DOI: 10.1021/acsnano.5b01651

**High-Fidelity Spin Entanglement Using Optimal Control**

F Dolde, V Bergholm, Y Wang, I Jakobi, B Naydenov, S Pezzagna, J Meijer, F Jelezko, P Neumann, T Schulte-Herbrüggen, J Biamonte, and J Wrachtrup

Nature Communications 5, 3371 (2014) DOI: 10.1038/ncomms4371

**Towards Quantum Chemistry on a Quantum Computer**

B Lanyon, J Whitfield, G Gillett, M Goggin, M Almeida, I Kassal, J Biamonte, M Mohseni, B Powell, M Barbieri, A Aspuru-Guzik, and A White

Nature Chemistry 2, 106 (2010) DOI: 10.1038/nchem.483

**Sign-and Magnitude-Tunable Coupler for Superconducting Flux Qubits**

R Harris, A Berkley, M Johnson, P Bunyk, S Govorkov, M Thom, S Uchaikin, A Wilson, J Chung, E Holtham, J Biamonte, A Smirnov, M Amin, and A Maassen van den Brink

Physical Review Letters 98, 177001 (2007) DOI: 10.1103/PhysRevLett.98.177001

## References

**On the Mathematical Structure of Quantum Models of Computation Based on Hamiltonian Minimisation**

J Biamonte Doctor of Physical and Mathematical Sciences (Dissertation), Moscow Institute of Physics and Technology, Department of Higher Mathematics, 242 pp. (2022) arXiv:2009.10088

**Reachability Deficits Implicit in Quantum Approximate Optimization of Graph Problems**

V Akshay, H Philathong, I Zacharov, and J Biamonte

Quantum 5:532 (2021) DOI: 10.22331/q-2021-08-30-532

**Parameter Concentrations in Quantum Approximate Optimization**

V Akshay, D Rabinovich, E Campos, and J Biamonte

Physical Review A 104:L010401 (2021) DOI: 10.1103/PhysRevA.104.L010401

**Universal Variational Quantum Computation**

J Biamonte

Physical Review A 103:L030401 (2021) DOI: 10.1103/PhysRevA.103.L030401

**Variational Simulation of Schwinger’s Hamiltonian with Polarization Qubits**

O Borzenkova, G Struchalin, A Kardashin, V Krasnikov, N Skryabin, S Straupe, S P Kulik, and J Biamonte

Applied Physics Letters 118:144002 (2021) DOI: 10.1063/5.0043322

**Abrupt Transitions in Variational Quantum Circuit Training**

E Campos, A Nasrallah, and J Biamonte

Physical Review A 103:032607 (2021) DOI: 10.1103/physreva.103.032607

**Training Saturation in Layerwise Quantum Approximate Optimisation**

E Campos, D Rabinovich, V Akshay, and J Biamonte

Physical Review A 104:L030401 (2021) DOI: 10.1103/PhysRevA.104.L030401

**Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, Theorems**

J Cirac, D Perez-Garcia, N Schuch, and F Verstraete

Reviews of Modern Physics 93:045003 (2021)

**Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor**

M Harrigan, K Sung, M Neeley, K Satzinger, F Arute, K Arya, J Atalaya, J Bardin, R Barends, S Boixo, et al.

Nature Physics 17:332 (2021)

**Numerical Hardware-Efficient Variational Quantum Simulation of a Soliton Solution**

A Kardashin, A Pervishko, J Biamonte, and D Yudin

Physical Review A 104:L020402 (2021) DOI: 10.1103/PhysRev

**On Barren Plateaus and Cost Function Locality in Variational Quantum Algorithms**

A Uvarov and J Biamonte

Journal of Physics A: Mathematical and Theoretical 54, 245 (2021) DOI: 10.1088/1751-8121/abfac7

**Reachability Deficits in Quantum Approximate Optimization**

V Akshay, H Philathong, M Morales, and J Biamonte

Physical Review Letters 124, 090504 (2020) DOI: 10.1103/PhysRevLett.124.090504

**Entanglement Scaling in Quantum Advantage Benchmarks**

J Biamonte, M Morales, and D Koh

Physical Review A 101, 012349 (2020) DOI: 10.1103/PhysRevA.101.012349

**On the Universality of the Quantum Approximate Optimization Algorithm**

M Morales, J Biamonte, and Z Zimborás

Quantum Information Processing 19, 291 (2020) DOI: 10.1007/s11128-020-02748-9

**Experimental Neural Network Enhanced Quantum Tomography**

A Palmieri, E Kovlakov, F Bianchi, D Yudin, S Straupe, J Biamonte, and S Kulik

npj Quantum Information 6, 20 (2020) DOI: 10.1038/s41534-020-0248-6

**Machine Learning Phase Transitions with a Quantum Processor**

A Uvarov, A Kardashin, and J Biamonte

Physical Review A 102, 012415 (2020) DOI: 10.1103/PhysRevA.102.012415

**Complex Networks from Classical to Quantum**

J Biamonte, M Faccin, and M De Domenico

Communications Physics 2 (2019) DOI: 10.1038/s42005-019-0152-6

**Integer Factorization Using Stochastic Magnetic Tunnel Junctions**

W Borders, A Pervaiz, S Fukami, K Camsari, H Ohno, and S Datta

Nature 573, 390 (2019)

**Variational Learning of Grover’s Quantum Search Algorithm**

M Morales, T Tlyachev, and J Biamonte

Physical Review A 98, 062333 (2018) DOI: 10.1103/PhysRevA.98.062333

**Quantum Techniques in Stochastic Mechanics**

J Baez and J Biamonte

World Scientific Publishing Co Pte Ltd, 276 pp. (2017) DOI: 10.1142/10623

**Charged String Tensor Networks**

J Biamonte

Proceedings of the National Academy of Sciences 114, 2447 (2017) DOI: 10.1073/pnas.1700736114

**Quantum Machine Learning**

J Biamonte, P Wittek, N Pancotti, P Rebentrost, N Wiebe, and S Lloyd

Nature 549, 195 (2017) DOI: 10.1038/nature23474

**Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning**

B Coecke and A Kissinger

Cambridge University Press (2017) DOI: 10.1017/9781316219317

**Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets**

A Kandala, A Mezzacapo, K Temme, M Takita, M Brink, J Chow, and J Gambetta

Nature 549, 242 (2017)

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

M De Domenico and J Biamonte

Physical Review X 6, 041062 (2016) DOI: 10.1103/PhysRevX.6.041062

**Chiral Quantum Walks**

D Lu, J Biamonte, J Li, H Li, T H Johnson, V Bergholm, M Faccin, Z Zimborás, R Laflamme, J Baugh, and S Lloyd

Physical Review A 93, 042302 (2016) DOI: 10.1103/PhysRevA.93.042302

**Tensor Network Contractions for #SAT**

J Biamonte, J Morton, and J Turner

Journal of Statistical Physics 160, 1389 (2015) DOI: 10.1007/s10955-015-1276-z

**Hamiltonian Gadgets with Reduced Resource Requirements**

Y Cao, R Babbush, J Biamonte, and S Kais

Physical Review A 91, 012315 (2015) DOI: 10.1103/PhysRevA.91.012315

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

Y Wang, F Dolde, J Biamonte, R Babbush, V Bergholm, S Yang, I Jakobi, P Neumann, A Aspuru-Guzik, J Whitfield, and J Wrachtrup

ACS Nano 9, 7769 (2015) DOI: 10.1021/acsnano.5b01651

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

C Wood, J Biamonte, and D Cory

Quantum Information & Computation 15, 759 (2015) DOI: 10.26421/QIC15.9-10-3

**High-Fidelity Spin Entanglement Using Optimal Control**

F Dolde, V Bergholm, Y Wang, I Jakobi, B Naydenov, S Pezzagna, J Meijer, F Jelezko, P Neumann, T Schulte-Herbrüggen, J Biamonte, and J Wrachtrup

Nature Communications 5, 3371 (2014) DOI: 10.1038/ncomms4371

**Community Detection in Quantum Complex Networks**

M Faccin, P Migdał, T Johnson, V Bergholm, and J Biamonte

Physical Review X 4, 041012 (2014) DOI: 10.1103/PhysRevX.4.041012

**Tensor Network Methods for Invariant Theory**

J Biamonte, V Bergholm, and M Lanzagorta

Journal of Physics A: Mathematical and Theoretical 46, 475301 (2013) DOI: 10.1088/1751-8113/46/47/475301

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

M Faccin, T Johnson, S Kais, P Migdał, and J Biamonte

Physical Review X 3, 041007 (2013) DOI: 10.1103/PhysRevX.3.041007

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

Z Zimboras, M Faccin, Z Kadar, J Whitfield, B Lanyon, and J Biamonte

Scientific Reports 3, 2361 (2013) DOI: 10.1038/srep02361

**Algebraically Contractible Topological Tensor Network States**

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

Journal of Physics A: Mathematical and Theoretical 45, 015309 (2012) DOI: 10.1088/1751-8113/45/1/015309

**Ground-State Spin Logic**

J Whitfield, M Faccin, and J Biamonte

Europhysics Letters 99, 57004 (2012) DOI: 10.1209/0295-5075/99/57004

**Categorical Quantum Circuits**

V Bergholm and J Biamonte

Journal of Physics A: Mathematical and Theoretical 44, 245304 (2011) DOI: 10.1088/1751-8113/44/24/245304

**Adiabatic Quantum Simulators**

J Biamonte, V Bergholm, J Whitfield, J Fitzsimons, and A Aspuru-Guzik

AIP Advances 1, 022126 (2011) DOI: 10.1063/1.3598408

**Categorical Tensor Network States**

J Biamonte, S Clark, and D Jaksch

AIP Advances 1, 042172 (2011) DOI: 10.1063/1.3672009

**Interacting Quantum Observables: Categorical Algebra and Diagrammatics**

B Coecke and R Duncan

New Journal of Physics 13, 043016 (2011)

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

J Whitfield, J Biamonte, and A Aspuru-Guzik

Molecular Physics 109, 735 (2011) DOI: 10.1080/00268976.2011.552

**Physics, Topology, Logic and Computation: A Rosetta Stone**

J Baez and M Stay Springer,

New Structures for Physics, 95–172 (2010)

**Towards Quantum Chemistry on a Quantum Computer**

B Lanyon, J Whitfield, G Gillett, M Goggin, M Almeida, I Kassal, J Biamonte, M Mohseni, B Powell, M Barbieri, A Aspuru-Guzik, and A White

Nature Chemistry 2, 106 (2010) DOI: 10.1038/nchem.483

**Networks: An Introduction**

M Newman

Oxford University Press 18, 772 (2010) DOI: 10.1162/ARTL_r_00062

**Quantum Random Networks**

S Perseguers, M Lewenstein, A Acı́n, and J Cirac

Nature Physics 6, 539 (2010)

**A Survey of Graphical Languages for Monoidal Categories**

P Selinger

Lecture Notes in Physics 813, 289 (2009) DOI: 10.1007/978-3-642-12821-9_4

**Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation**

D Aharonov, W Van Dam, J Kempe, Z Landau, S Lloyd, and O Regev

SIAM Review 50, 755 (2008)

**Nonperturbative k-Body to Two-Body Commuting Conversion Hamiltonians and Embedding Problem Instances into Ising Spins**

J Biamonte

Physical Review A 77, 052331 (2008) DOI: 10.1103/PhysRevA.77.052331

**Realizable Hamiltonians for Universal Adiabatic Quantum Computers**

J Biamonte and P Love

Physical Review A 78, 012352 (2008) DOI: 10.1103/PhysRevA.78.012352

**Sign-and Magnitude-Tunable Coupler for Superconducting Flux Qubits**

R Harris, A Berkley, M Johnson, P Bunyk, S Govorkov, M Thom, S Uchaikin, A Wilson, J Chung, E Holtham, J Biamonte, A Smirnov, M Amin, and A Maassen van den Brink

Physical Review Letters 98, 177001 (2007) DOI: 10.1103/PhysRevLett.98.177001

**Dagger Compact Closed Categories and Completely Positive Maps**

P Selinger

Electronic Notes in Theoretical Computer Science 170, 139 (2007)

**The Complexity of the Local Hamiltonian Problem**

J Kempe, A Kitaev, and O Regev

SIAM Journal on Computing 35, 1070 (2006)

**The Complexity of Quantum Spin Systems on a Two-Dimensional Square Lattice**

R Oliveira and B Terhal

Quantum Information & Computation 8, (2005) DOI: 10.26421/QIC8.10-2

**A Categorical Semantics of Quantum Protocols**

S Abramsky and B Coecke

Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 415–425 (2004)

**Towards an Algebraic Theory of Boolean Circuits**

Y Lafont

Journal of Pure and Applied Algebra 184, 257 (2003)

**Statistical Mechanics of Complex Networks**

R Albert and A Barabási

Reviews of Modern Physics 74, 47 (2002)

**Fermionic Quantum Computation**

S Bravyi and A Kitaev

Annals of Physics 298, 210 (2002)

**Classical and Quantum Computation**

A Kitaev, A Shen, M Vyalyi, and M Vyalyi

American Mathematical Society, (2002)

**Quantum Search by Local Adiabatic Evolution**

J Roland and N Cerf

Physical Review A 65, 042308 (2002)

**Bose-Einstein Condensation in Complex Networks**

G Bianconi and A Barabási

Physical Review Letters 86, 5632 (2001)

**A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-complete Problem**

E Farhi, J Goldstone, S Gutmann, J Lapan, A Lundgren, and D Preda

Science 292, 472 (2001)

**How Powerful is Adiabatic Quantum Computation?**

W Van Dam, M Mosca, and U Vazirani

Proceedings 42nd IEEE Symposium on Foundations of Computer Science, 279 (2001)

**Universal Quantum Simulators**

S Lloyd

Science 273, 1073 (1996)

**Quantum Mechanical Computers**

R Feynman

Optics News 11, 11 (1985)

**Optimization by Simulated Annealing**

S Kirkpatrick, C Gelatt Jr, and M Vecchi

Science 220, 671 (1983)

**Applications of Negative Dimensional Tensors**

R Penrose

Combinatorial Mathematics and Its Applications 1, 221 (1971)