Papers

Bias-Field Digitized Counterdiabatic Quantum Algorithm for Higher-Order Binary Optimization

Sebastián V. Romero, Anne-Maria Visuri, Alejandro Gomez Cadavid, Enrique Solano, and Narendra N. Hegade

We present an enhanced bias-field digitized counterdiabatic quantum optimization (BF-DCQO) algorithm to address higher-order unconstrained binary optimization (HUBO) problems. Combinatorial optimization plays a crucial role in many industrial applications, yet classical computing often struggles with complex instances. By encoding these problems as Ising spin glasses and leveraging the advancements in quantum computing technologies, quantum optimization methods emerge as a promising alternative. We apply BF-DCQO with an enhanced bias term to a HUBO problem featuring three-local terms in the Ising spin-glass model. Our protocol is experimentally validated using 156 qubits on an IBM quantum processor with a heavy-hex architecture. In the studied instances, the results outperform standard methods, including the quantum approximate optimization algorithm (QAOA), quantum annealing, simulated annealing, and Tabu search. Furthermore, we perform an MPS simulation and provide numerical evidence of the feasibility of a similar HUBO problem on a 433-qubit Osprey-like quantum processor. Both studied cases, the experiment on 156 qubits and the simulation on 433 qubits, can be considered as the start of the commercial quantum advantage era, Kipu dixit, and even more when extended soon to denser industry-level HUBO problems.

Bias-field digitized counterdiabatic quantum optimization

Alejandro Gomez Cadavid, Archismita Dalal, Anton Simen, Enrique Solano, and Narendra N. Hegade

We introduce a method for solving combinatorial optimization problems on digital quantum computers, where we incorporate auxiliary counterdiabatic (CD) terms into the adiabatic Hamiltonian, while integrating bias terms derived from an iterative digitized counterdiabatic quantum algorithm. We call this protocol bias-field digitized counterdiabatic quantum optimization (BF-DCQO). Designed to effectively tackle large-scale combinatorial optimization problems, BF-DCQO demonstrates resilience against the limitations posed by the restricted coherence times of current quantum processors and shows clear enhancement even in the presence of noise. Additionally, our purely quantum approach eliminates the dependency on classical optimization required in hybrid classical-quantum schemes, thereby circumventing the trainability issues often associated with variational quantum algorithms. Through the analysis of an all-to-all connected general Ising spin-glass problem, we exhibit a polynomial scaling enhancement in ground state success probability compared to traditional DCQO and finitetime adiabatic quantum optimization methods. Furthermore, it achieves scaling improvements in ground state success probabilities, increasing by up to two orders of magnitude, and offers an average 1.3x better approximation ratio than the quantum approximate optimization algorithm for the problem sizes studied. We validate these findings through experimental implementations on both trapped-ion quantum computers and superconducting processors, tackling a maximum weighted independent set problem with 36 qubits and a spin-glass on a heavyhex lattice with 100 qubits, respectively. These results mark a significant advancement in gate-based quantum computing, employing a fully quantum algorithmic approach.

Digital-analog counterdiabatic quantum optimization with trapped ions

Shubham Kumar, Narendra N Hegade, Murilo Henrique de Oliveira, Enrique Solano, Alejandro Gomez Cadavid and F Albarrán-Arriagada

We introduce a hardware-specific, problem-dependent digital-analog quantum algorithm of a counterdiabatic quantum dynamics tailored for optimization problems. Specifically, we focus on trapped-ion architectures, taking advantage from global Mølmer–Sørensen gates as the analog interactions complemented by digital gates, both of which are available in the state-of-the-art technologies. We show an optimal configuration of analog blocks and digital steps leading to a substantial reduction in circuit depth compared to the purely digital approach. This implies that, using the proposed encoding, we can address larger optimization problem instances, requiring more qubits, while preserving the coherence time of current devices. Furthermore, we study the minimum gate fidelity required by the analog blocks to outperform the purely digital simulation, finding that it is below the best fidelity reported in the literature. To validate the performance of the digital-analog encoding, we tackle the maximum independent set problem, showing that it requires fewer resources compared to the digital case. This hybrid co-design approach paves the way towards quantum advantage for efficient solutions of quantum optimization problems.

Efficient digitized counterdiabatic quantum optimization algorithm within the impulse regime for portfolio optimization

Alejandro Gomez Cadavid, Iraitz Montalban, Archismita Dalal, Enrique Solano, and Narendra N. Hegade

We propose a faster digital quantum algorithm for portfolio optimization using the digitized counterdiabatic quantum optimization paradigm in the impulse regime, that is, where the counterdiabatic terms are dominant. Our approach notably reduces the circuit depth requirement of the algorithm and enhances the solution accuracy, making it suitable for current quantum processors. We apply this protocol to a real-case scenario of portfolio optimization with 20 assets, using purely quantum and hybrid classical-quantum paradigms. We demonstrate the advantages of our protocol using up to 20 qubits on an IonQ trapped-ion quantum computer. By benchmarking our method against the standard quantum approximate optimization algorithm and finite-time digitized adiabatic algorithms, we obtain a significant reduction in the circuit depth by factors of 2.5 to 40, while minimizing the dependence on the classical optimization subroutine. Besides portfolio optimization, the proposed method is applicable to a large class of combinatorial optimization problems.

Digitized Counterdiabatic Quantum Algorithms for Logistics Scheduling

Archismita Dalal, Iraitz Montalban, Narendra N. Hegade, Alejandro Gomez Cadavid, Enrique Solano, Abhishek Awasthi, Davide Vodola, Caitlin Jones, Horst Weiss and Gernot Füchsel

We study a job shop scheduling problem for an automatized robot in a high-throughput laboratory and a travelling salesperson problem with recently proposed digitized counterdiabatic quantum optimization (DCQO) algorithms. In DCQO, we find the solution of an optimization problem via an adiabatic quantum dynamics, which is accelerated with counterdiabatic protocols. Thereafter, we digitize the global unitary to encode it in a digital quantum computer. For the job-shop scheduling problem, we aim at finding the optimal schedule for a robot executing a number of tasks under specific constraints, such that the total execution time of the process is minimized. For the traveling salesperson problem, the goal is to find the path that covers all cities and is associated with the shortest traveling distance. We consider both hybrid and pure versions of DCQO algorithms and benchmark the performance against digitized quantum annealing and the quantum approximate optimization algorithm (QAOA). In comparison to QAOA, the DCQO solution is improved by several orders of magnitude in success probability using the same number of two-qubit gates. Moreover, we experimentally implement our algorithms on superconducting and trapped-ion quantum processors. Our results demonstrate that circuit compression using counterdiabatic protocols is amenable to current NISQ hardware and can solve logistics scheduling problems, where other digital quantum algorithms show insufficient performance.

Analog Counterdiabatic Quantum Computing

Qi Zhang, Narendra N. Hegade, Alejandro Gomez Cadavid, Lucas Lassabli`ere, Jan Trautmann, S´ebastien Perseguers, Enrique Solano, Lo¨ıc Henriet, Eric Michon

We propose analog counterdiabatic quantum computing (ACQC) to tackle combinatorial optimization problems on neutral-atom quantum processors. While these devices allow for the use of hundreds of qubits, adiabatic quan- tum computing struggles with non-adiabatic errors, which are inevitable due to the hardware’s restricted coherence time. We design counterdiabatic proto- cols to circumvent those limitations via ACQC on analog quantum devices with ground-Rydberg qubits. To demonstrate the effectiveness of our paradigm, we experimentally apply it to the maximum independent set (MIS) problem with up to 100 qubits and show an enhancement in the approximation ratio with a short evolution time. We believe ACQC establishes a path toward quantum advantage for a variety of industry use cases.

Physics-informed neural networks for an optimal counterdiabatic quantum computation

Antonio Ferrer-Sánchez, Carlos Flores-Garrigos, Carlos Hernani-Morales, José J Orquín-Marqués, Narendra N Hegade, Alejandro Gomez Cadavid, Iraitz Montalban, Enrique Solano, Yolanda Vives-Gilabert and José D Martín-Guerrero

A novel methodology that leverages physics-informed neural networks to optimize quantum circuits in systems with NQ qubits by addressing the counterdiabatic (CD) protocol is introduced. The primary purpose is to employ physics-inspired deep learning techniques for accurately modeling the time evolution of various physical observables within quantum systems. To achieve this, we integrate essential physical information into an underlying neural network to effectively tackle the problem. Specifically, the imposition of the solution to meet the principle of least action, along with the hermiticity condition on all physical observables, among others, ensuring the acquisition of appropriate CD terms based on underlying physics. This approach provides a reliable alternative to previous methodologies relying on classical numerical approximations, eliminating their inherent constraints. The proposed method offers a versatile framework for optimizing physical observables relevant to the problem, such as the scheduling function, gauge potential, temporal evolution of energy levels, among others. This methodology has been successfully applied to 2-qubit representing H2 molecule using the STO-3G basis, demonstrating the derivation of a desirable decomposition for non-adiabatic terms through a linear combination of Pauli operators. This attribute confers significant advantages for practical implementation within quantum computing algorithms.

Digital-analog quantum convolutional neural networks for image classification

Anton Simen, Carlos Flores-Garrigos, Narendra N. Hegade, Iraitz Montalban, Yolanda Vives-Gilabert, Eric Michon, Qi Zhang Enrique Solano and José D. Martín-Guerrero

We propose digital-analog quantum kernels for enhancing the detection of complex features in the classification of images. We consider multipartite-entangled analog blocks, stemming from native Ising interactions in neutral-atom quantum processors, and individual operations as digital steps to implement the protocol. To further improve the detection of complex features, we apply multiple quantum kernels by varying the qubit connectivity according to the hardware constraints. An architecture that combines nontrainable quantum kernels and standard convolutional neural networks is used to classify realistic medical images, from breast cancer and pneumonia diseases, with a significantly reduced number of parameters. Despite this fact, the model exhibits better performance than its classical counterparts and achieves comparable metrics according to public benchmarks. These findings highlight the potential of digital-analog quantum convolutions in extracting complex and meaningful features from images, positioning them as a candidate model for addressing challenging classification problems.

Single-layer digitized-counterdiabatic quantum optimization for p-spin models

Huijie Guan, Fei Zhou, Francisco Albarrán-Arriagada, Xi Chen, Enrique Solano, Narendra N Hegade and He-Liang Huang

Quantum computing holds the potential for quantum advantage in optimization problems, which requires advances in quantum algorithms and hardware specifications. Adiabatic quantum optimization is conceptually a valid solution that suffers from limited hardware coherence times. In this sense, counterdiabatic quantum protocols provide a shortcut to this process, steering the system along its ground state with fast-changing Hamiltonian. In this work, we take full advantage of a digitized-counterdiabatic quantum optimization algorithm to find an optimal solution of the p-spin model up to four-local interactions. We choose a suitable scheduling function and initial Hamiltonian such that a single-layer quantum circuit suffices to produce a good ground-state overlap. By further optimizing parameters using variational methods, we solve with unit accuracy two-spin, three-spin, and four-spin problems for 100%, 93%, and 83% of instances, respectively. As a particular case of the latter, we also solve factorization problems involving 5, 9, and 12 qubits. Due to the low computational overhead, our compact approach may become a valuable tool towards quantum advantage in the NISQ era.

Codesigned counterdiabatic quantum optimization on a photonic quantum processor

Xiao-Wen Shang, Xuan Chen, Narendra N. Hegade, Ze-Feng Lan, Xuan-Kun Li, Hao Tang, Yu-Quan Peng, Enrique Solano and Xian-Min Jin

Codesign, an integral part of computer architecture referring to the information interaction in hardware-software stack, is able to boost the algorithm mapping and execution in the computer hardware. This well applies to the noisy intermediate-scale quantum era, where quantum algorithms and quantum processors both need to be shaped to allow for advantages in experimental implementations. The state-of-the-art quantum adiabatic optimization algorithm faces challenges for scaling up, where the deteriorating optimization performance is not necessarily alleviated by increasing the circuit depth given the noise in the hardware. The counterdiabatic term can be introduced to accelerate the convergence, but decomposing the unitary operator corresponding to the counterdiabatic terms into one and two-qubit gates may add additional burden to the digital circuit depth. In this work, we focus on the counterdiabatic protocol with a codesigned approach to implement this algorithm on a photonic quantum processor. The tunable Mach-Zehnder interferometer mesh provides rich programmable parameters for local and global manipulation, making it able to perform arbitrary unitary evolutions. Accordingly, we directly implement the unitary operation associated to the counterdiabatic quantum optimization on our processor without prior digitization. Furthermore, we develop and implement an optimized counterdiabatic method by tackling the higherorder many-body interaction terms. Moreover, we benchmark the performance in the case of factorization, by comparing the final success probability and the convergence speed. In conclusion, we experimentally demonstrate the advantages of a codesigned mapping of counterdiabatic quantum dynamics for quantum computing on photonic platforms.

Microwave quantum memcapacitor effect

Xinyu Qiu, Shubham Kumar, Francisco A. Cárdenas-López, Gabriel Alvarado Barrios, Enrique Solano & Francisco Albarrán-Arriagada

Developing the field of neuromorphic quantum computing necessitates designing scalable quantum memory devices. Here, we propose a superconducting quantum memory device in the microwave regime, termed a microwave quantum memcapacitor. It comprises two linked resonators, the primary one is coupled to a Superconducting Quantum Interference Device, which allows for the modulation of the resonator properties through external magnetic flux. The auxiliary resonator, operated through weak measurements, provides feedback to the primary resonator, ensuring stable memory behavior. This device operates with a classical input in one cavity while reading the response in the other, serving as a fundamental building block toward arrays of microwave quantum memcapacitors. We observe that a bipartite setup can retain its memory behavior and gains entanglement and quantum correlations. Our findings pave the way for the experimental implementation of memcapacitive superconducting quantum devices and memory device arrays for neuromorphic quantum computing.

Efficient DCQO Algorithm within the Impulse Regime for Portfolio Optimization

Alejandro Gomez Cadavid, Iraitz Montalban, Archismita Dalal, Enrique Solano, Narendra N. Hegade

We propose a faster digital quantum algorithm for portfolio optimization using the digitized-counterdiabatic quantum optimization (DCQO) paradigm in the impulse regime, that is, where the counterdiabatic terms are dominant. Our approach notably reduces the circuit depth requirement of the algorithm and enhances the solution accuracy, making it suitable for current quantum processors. We apply this protocol to a real-case scenario of portfolio optimization with 20 assets, using purely quantum and hybrid classical-quantum paradigms. We experimentally demonstrate the advantages of our protocol using up to 20 qubits on an IonQ trapped-ion quantum computer. By benchmarking our method against the standard quantum approximate optimization algorithm and finite-time digitized-adiabatic algorithms, we obtain a significant reduction in the circuit depth by factors of 2.5 to 40, while minimizing the dependence on the classical optimization subroutine. Besides portfolio optimization, the proposed method is applicable to a large class of combinatorial optimization problems.

Digital-analog quantum computing of fermion-boson models in superconducting circuits

Shubham Kumar, Narendra N. Hegade, Enrique Solano, Francisco Albarrán-Arriagada, Gabriel Alvarado Barrios

We proposed the digital-analog encoding for fermion-boson models using superconducting circuits outperforming digital methods in terms of circuit depth which scales quadratically with the smaller dimension of the lattice, (e.g. for a 5X5 or 5X10000 lattice, the scaling remains 25) and, for 1D chains we have a constant depth of 9, paving the way for advancements in material science, energy, and pharmaceuticals.

Digitized-counterdiabatic quantum factorization

Narendra N. Hegade, Enrique Solano.

This work shows how a simple adaptation of the work carried out by by B. Yan et al., arXiv:2212.12372 (2022) can be outperformed with a non-hybrid digitized-counterdiabatic by a factor of 6, factorizing 48-bit integer using trapped-ion hardware.

Digitized-Counterdiabatic Quantum Algorithm for Protein Folding

Pranav Chandarana, Narendra N. Hegade, Iraitz Montalban, Enrique Solano, Xi Chen.

We apply our method of Hybrid Digitized Counterdiabatic Quantum Computing (hybrid DCQC) to proteins with up to 9 amino acids, using up to 17 qubits on quantum hardware. Specifically, we benchmark our quantum algorithm with Quantinuum’s trapped ions, Google’s and IBM’s superconducting circuits, obtaining high success probabilities with low-depth circuits as required in the NISQ era.

Digitized-Counterdiabatic Quantum Optimization

Narendra N. Hegade, Xi Chen, Enrique Solano

Kipu’s battlehorse for combinatorial optimization outperforming state-of-the-art techniques to solve some of the most challenging industry problems.

Digitized-counterdiabatic quantum approximate optimization algorithm

Pranav Chandarana, Narendra N. Hegade, K. Paul, Francisco Albarrán-Arriaga, Enrique Solano, Adolfo del Campo, Xi Chen.

QAOA being one of the most use variational techniques for combinatorial optimization, addition of counterdiabatic protocols shows enhancement from solving combinatorial optimization problems to finding the ground state of many-body quantum systems.

Approximating the quantum aproximate optimization algorithm with digital-analog interactions

David Headley, Thorge Müller, Ana Martin, Enrique Solano, Mikel Sanz, Frank K. Wilhelm.

By embracing the analog capacity for multi-qubit interactions, we exploit the recently proposed digital-analog quantum computation paradigm, in which the versatility of programmable universal quantum computers and the error resilience of quantum simulators are combined to improve platforms for quantum computation, specially suited to the variational quantum approximate optimisation algorithm.

Pioneering Quantum Algorithm for solving Black-Scholes Equation

J. Gonzáles-Conde, A. Rodríguez-Rozas, E. Solano, and M. Sanz

“Quantum Algorithm for Pricing Financial Derivatives”

General theory of digital-analog quantum computing (DAQC)

Adrian Parra-Rodriguez, Pavel Lougovski, Lucas Lamata, Enrique Solano, and Mikel Sanz, “Digital-Analog Quantum Computation”, Phys. Rev. A 101, 022305 (2020).

This paper is the kick-off of the general theory on Digital-Analog Quantum Computing. Essentially, we proved there that we can be universal with DAQC, although we will use this mostly for bespoke Co-Design Quantum Computers and application to key use cases. I think this paper will be a reference for decades in quantum computing, until error correction may become meaningful, perhaps next century.

Implementing quantum Fourier transform via digital-analog methods (DAQC)

A. Martin, L. Lamata, E. Solano, and M. Sanz, “Digital-analog quantum algorithm for the quantum Fourier transform”, Phys. Rev. Research 2, 013012 (2020).

A key example of the power of DAQC applied to the quantum Fourier transform algorithm, which is the basis of quantum phase estimation article, which is at the basis of Shor algorithm, most quantum chemistry algorithms, and material design algorithms. A pillar for what comes soon from Co-Design QC as quantum products fo key use cases.

Enhancing connectivity through digital-analog approach (DAQC)

A. Galicia, B. Ramon, E. Solano, and M. Sanz, “Enhanced connectivity of quantum hardware with digital-analog control”, arXiv:1912.09331, accepted in Phys. Rev. Research (2020).

Another key paper on the unpredictable flexibility and power of DAQC methods.

Digital-analog quantum simulation of quantum approximate optimization algorithm (DAQS)

D. Headley, T. Müller, A. Martin, E. Solano, M. Sanz, and F. K. Wilhelm, “Approximating the Quantum Approximate Optimisation Algorithm”, arXiv:2002.12215 (2020).

A masterpiece of Co-Design Quantum Computers developed with Mercedes Benz researchers, Saarbrücken researchers that coordinate the Quantum Computing European consortium, and our QUTIS Center in Bilbao, Spain, where most of these ideas were developed in last 10 years. We proved that all what other quantum software/hardware companies are proposing for QAOA is misusing the available quantum hardware and quantum software.

Pioneering Connection between Active Learning and Quantum Information

Y.-C. Ding, J.-D. Martín-Guerrero, M. Sanz, R. Magdalena-Benedicto, X. Chen, and E. Solano, “Retrieving Quantum Information with Active Learning”, Phys. Rev. Lett. 124, 140504 (2020).

Pioneering Quantum Computing Realization of Models of Financial Crashes

Y.-C. Ding, L. Lamata, J.-D. Martín-Guerrero, E. Lisazo, S. Mugel, R. Orús, E. Solano, and M. Sanz, “Towards Prediction of Financial Crashes with a D-Wave Quantum Computer”, arXiv:1904.05808 (2020).

Pioneering Quantum Computing Implementation of Pricing Financial Derivatives

A. Martin, B. Candelas, A. Rodríguez-Rozas, J.-D. Martín-Guerrero, X. Chen, L. Lamata, R. Orús, E. Solano, and M. Sanz, “Towards Pricing Financial Derivatives with an IBM Quantum Computer”, arXiv:1904.0583 (2020).

Reaching quantum supremacy via co-design approach (CDQC)

F. Hu, L. Lamata, C. Wang, X. Chen, E. Solano, and M. Sanz, “Quantum Supremacy in Cryptography with a Low-Connectivity Quantum Annealer”, arXiv:1906.08140 (2019).

A prove that we can reach quantum supremacy and quantum advantage with variants, some of them even simpler, of D-Wave architectures. A key result for cryptography.

Digital-analog quantum computation of scattering in quantum electodynamics in trapped ions (CDQS)

X. Zhang, K. Zhang, Y. Shen, J. Zhang, M.-H. Yung, J. Casanova, J. S. Pedernales, L. Lamata, E. Solano, and K. Kim, “Fermion-antifermion scattering via boson exchange in a trapped ion”, Nat. Comm. 9, 195 (2018).

An impressive implementation in the lab of our proposals on CDQS

Review article on digital-analog quantum simulations (DAQS)

Lucas Lamata, Adrián Parra-Rodriguez, Mikel Sanz, and Enrique Solano, “Digital-Analog Quantum Simulations with Superconducting Circuits”, Advances in Physics X: 3, 1457981 (2018).

A review article on what we had achieved up to 2018 on DAQS proposals.

Pioneering proposal for a nonlinear non-Markovian quantum element (DAQS)

P. Pfeiffer, I. L. Egusquiza, M. Di Ventra, M. Sanz, and E. Solano, “Quantum Memristors”, Sci. Rep. 6, 29507 (2016).

Here, we invented the Quantum Memristor, as a new fundamental quantum device in superconducting circuits for opening the field of Neuromorphic Quantum Computing.

Digital-analog quantum computing for mapping quantum chemistry and biomolecules on a physical architecture (DAQS)

L. García-Álvarez, U. Las Heras, A. Mezzacapo, M. Sanz, E. Solano, and L. Lamata, “Quantum chemistry and charge transport in biomolecules with superconducting circuits”, Sci. Rep. 6, 27836 (2016).

Digital-analog mapping of spin models on a trapped-ion architecture (DAQS)

I. Arrazola, J. S. Pedernales, L. Lamata, and E. Solano, “Digital-Analog Quantum Simulation of Spin Models in Trapped Ions”, Sci. Rep. 6, 30534 (2016).

These original ideas for trapped ions inspired us to go ahead with further models in superconducting circuits and other quantum platforms.

Digital-analog quantum simulation of interacting fermions via exchange of bosons in quantum field theories (DAQS)

L. García-Álvarez, J. Casanova, A. Mezzacapo, I. L. Egusquiza, L. Lamata, G. Romero, and E. Solano, “Fermion-Fermion Scattering in Quantum Field Theory with superconducting circuits”, Phys. Rev. Lett. 114, 070502 (2015).

In this work, we were able to describe a modular architecture for quantum computation of scattering processes between fermions, like electrons, via exchange of a continuum of bosonic modes, like photons in open air. Our proposal shows that you may reach quantum advantage with rather few quantum elements (qubits, cavities, open transmission lines), while the proposal of John Preskill and others, published in Science 2011 requires millions of qubits.

Mapping unphysical operations on a physical architecture in ion traps (CDQS)

X. Zhang, Y. Shen, J. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time Reversal and Charge Conjugation in an Embedding Quantum Simulator”, Nat. Commun. 6, 7917 (2015).

Key experiment proving our prediction on how modular co-design concepts work nicely in the lab, in this case trapped ions.

Digital-analog quantum simulator of fluid dynamics (DAQS)

A. Mezzacapo, M. Sanz, L. Lamata, I. L. Egusquiza, S. Succi, and E. Solano, “Quantum Simulator for Transport Phenomena in Fluid Flows”, Sci. Rep. 5, 13153 (2015).

The pioneeing article dealing with the problem on how to approach quantum simulation/computation of fluid dynamics model. The highly nonlinear case may only be develop in Co-Design Quantum Computers.

Digital-analog quantum simulation of quantum chemistry (DAQS)

M.-H. Yung, J. Casanova, A. Mezzacapo, J. McClean, L. Lamata, A. Aspuru-Guzik, and E. Solano, “From transistor to trapped-ion computers for quantum chemistry”, Sci. Rep. 4, 3589 (2014).

This is the pioneering paper where celebrated VQE algorithm was applied to quantum chemistry models in trapped ions. These ideas were taken by top experimental groups in trapped ions (IQOQI, Innsbruck, Austria) and superconducting circuits (Google, Santa Barbara, US), and implemented in the lab.

Transforming an analog block into another one by single-qubit pulses (DAQS)

J. S. Pedernales, R. Di Candia, D. Ballester, E. Solano, “Quantum Simulations of Relativistic Quantum Physics in Circuit QED”, New J. Phys. 15, 055008 (2013).

Here, we applied a known co-design method and found a mysterious coincidence in mathematical structure between light-matter interactions and relativistic Dirac equations, similar to the ones appearing in monolayer and bilayer graphene.

Embedding quantum simulators to measure entanglement (EQS)

R. Di Candia, B. Mejia, H. Castillo, J. S. Pedernales, J. Casanova, and E. Solano, “Embedding Quantum Simulators for Quantum Computation of Entanglement”, Phys. Rev. Lett 111, 240502 (2013).

In this manuscript, we develop the full theory of embedding quantum simulators/computers to reproduce antilinear operations, which are unphysical but present in useful applications. In particular, it is present in the definition of measures of entanglement, so here we manage to prove that measuring entanglement in dynamical systems works better in quantum computers with Co-Design embedding concepts.

Transforming an analog block into another one by single-qubit pulses (DAQS)

D. Ballester, G. Romero, J. J. García-Ripoll, F. Deppe, and E. Solano, “Quantum simulation of the ultrastrong coupling dynamics in circuit QED”, Phys. Rev. X 2, 021007 (2012).

Here, we developed first tools for creating Co-Design counter-rotating terms, associated with a pseudo-violation of energy conservation, onto a conventional light-matter interaction model. In that way we could move towards collective Dicke model and towards superradiance phenomena.

Quantum simulation of materials with interacting fermions and bosons (DAQS)

A. Mezzacapo, J. Casanova, L. Lamata, and E. Solano, “Digital Quantum Simulation of the Holstein Model in Trapped Ions”, Phys. Rev. Lett. 109, 200501 (2012).

In this work, we developed the pioneering Co-Design quantum simulation/computation of materials involving fermions coupled to bosons, where bosons are represented by bosons, as it should be for efficiency reasons. Along these lines, many important models in material design for quantum computers can reach quantum advantage with a few dozens of quantum elements, be qubits, qutrits, cavity modes, or open transmission lines.

Mapping unphysical operations on a physical architecture (EQS)

J. Casanova, C. Sabín, J. León, I. L. Egusquiza, R. Gerritsma, C. Roos, J. J. García-Ripoll, and E. Solano, “Quantum Simulation of the Majorana Equation and Unphysical Operations”, Phys. Rev. X 1, 021018 (2011).

Here, we enhanced the architecture possibilities of quantum simulations/computations involving mathematical models that do not have a direct mapping on a physical system. This is the case, for example, of Black-Scholes equations in financial models. It is this paper the original proposal of what later we named Embedding Quantum Simulators or, equivalently, Embedding Quantum Computers. This requires ancillary qubits with a suitable mapping of model onto architecture that was inexistent previously.

Quantum simulation of quantum field theories for trapped ions (DAQS)

J. Casanova, L. Lamata, I. L. Egusquiza, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, and E. Solano, “Quantum simulation of quantum field theories in trapped ions”, Phys. Rev. Lett. 107, 260501 (2011).

In this work, we develop for the first time the possibility of combining qubits and motional modes in trapped ions for a quantum computation of quantum field theories in the co-design paradigm. Harmonic oscillator excitations are not mapped onto qubits, in-built harmonic oscillators in the architecture represent harmonic oscillators in the model. This proposal was tested in the lab and published later.