Density-matrix renormalization group: a pedagogical introduction

The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership amongst students and researchers who are taking their first steps in numerical simulations via DMRG.Comment: 30 pages, 17 figure

From Heisenberg to Hubbard: An initial state for the shallow quantum simulation of correlated electrons

The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-$\frac{1}{2}$ Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-$\frac{1}{2}$ wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is bound to be most relevant.Comment: Main text: 4 pages, 3 figures. Supp. Mat.: 10 pages, 9 figure

One-to-one correspondence between thermal structure factors and coupling constants of general bilinear Hamiltonians

A theorem that establishes a one-to-one relation between zero-temperature static spin-spin correlators and coupling constants for a general class of quantum spin Hamiltonians bilinear in the spin operators has been recently established by Quintanilla, using an argument in the spirit of the Hohenberg-Kohn theorem in density functional theory. Quintanilla's theorem gives a firm theoretical foundation to quantum spin Hamiltonian learning using spin structure factors as input data. Here we extend the validity of the theorem in two directions. First, following the same approach as Mermin, the proof is extended to the case of finite-temperature spin structure factors, thus ensuring that the application of this theorem to experimental data is sound. Second, we note that this theorem applies to all types of Hamiltonians expressed as sums of bilinear operators, so that it can also relate the density-density correlators to the Coulomb matrix elements for interacting electrons in the lowest Landau level.J.F.R. acknowledges financial support from FCT (Grant No. PTDC/FIS-MAC/2045/2021), FEDER / Junta de Andalucía — Consejería de Transformación Económica, Industria, Conocimiento y Universidades, (Grant No. P18-FR-4834), and Generalitat Valenciana funding Prometeo20XXX, MICIIN-Spain (Grant No. PID2019-109539GB-C41). B.M. acknowledges support from the FCT PhD Scholarship No. SFRH/BD/08444/2020

From Heisenberg to Hubbard: An initial state for the shallow quantum simulation of correlated electrons

The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin- 1/2 Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin- 1/2 wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doubloon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin- 1/2 state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.B.M. acknowledges financial support from Fundação para a Ciência e a Tecnologia (FCT)–Portugal through Ph.D. Scholarship No. SFRH/BD/08444/2020. J.F.R. acknowledges financial support from FCT (Grant No. PTDC/FISMAC/2045/2021), the Generalitat Valenciana funding No. Prometeo2021/017 and No. MFA/2022/045, and funding from MICIIN-Spain (Grants No. PID2019-109539GB-C41 and No. PID2022-141712NB-C22)

Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging

The controlled-SWAP and controlled-controlled-NOT gates are at the heart of the original proposal of reversible classical computation by Fredkin and Toffoli. Their widespread use in quantum computation, both in the implementation of classical logic subroutines of quantum algorithms and in quantum schemes with no direct classical counterparts, have made it imperative early on to pursue their efficient decomposition in terms of the lower-level gate sets native to different physical platforms. Here, we add to this body of literature by providing several logically equivalent CNOT-count-optimal circuits for the Toffoli and Fredkin gates under all-to-all and linear qubit connectivity, the latter with two different routings for control and target qubits. We then demonstrate how these decompositions can be employed on near-term quantum computers to mitigate coherent errors via equivalent circuit averaging. We also consider the case where the three qubits on which the Toffoli or Fredkin gates act nontrivially are not adjacent, proposing a novel scheme to reorder them that saves one CNOT for every SWAP. This scheme also finds use in the shallow implementation of long-range CNOTs. Our results highlight the importance of considering different entanglement structures and connectivity constraints when designing efficient quantum circuits.Comment: Main text: 10 pages, 8 figures. Appendix: 4 sections, 5 figures. QASM files will be made available in open-source online platform upon next update of preprin

Berry phase estimation in gate-based adiabatic quantum simulation

Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here, we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is extensible to an interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.B.M. and J.F.R. acknowledge the FCT Functionalized Graphene for Quantum Technologies project (PTDC/FIS-NAN/4662/2014). G.C. acknowledges support from Fundação para a Ciência e Tecnologia (FCT) Ph.D. Scholarship No. SFRH/BD/138806/2018

Produção de bioetanol a partir de polpa de alfarroba: destoxificação de polifenóis por fotocatálise

Tese de mestrado em Engenharia da Energia e do Ambiente, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2013O aumento das necessidades energéticas a nível mundial e o elevado consumo de combustíveis fósseis tem levado à procura de fontes de energia renováveis, surgindo assim os biocombustíveis (e.g. bioetanol) como uma possível alternativa aos combustíveis fosseis. Nesta perspetiva, a polpa de alfarroba surge como um resíduo endógeno adequado à produção de bioetanol por conter uma elevada concentração de açúcares solúveis (40-50% peso seco (p.s.)), além de uma fração lenhocelulósica (aproximadamente 20% p.s.) que poderá ainda ser fermentada após hidrólise. No entanto, trabalhos recentemente realizados no LNEG mostraram que os baixos rendimentos obtidos na hidrólise enzimática da polpa de alfarroba podiam resultar do alto conteúdo em polifenóis na polpa. Assim, a finalidade do presente trabalho foi aplicar um método de destoxificação, designado de fotocatálise, à polpa de alfarroba para remover os polifenóis e avaliar o impacto deste processo no posterior rendimento da hidrólise enzimática. A polpa de alfarroba foi primeiramente caracterizada e processada de modo a se obterem três subprodutos diferentes: polpa de alfarroba inteira (PAI), xarope de alfarroba e polpa de alfarroba extratada (PAE). A PAE e os respetivos xaropes foram obtidos após extração dos açúcares solúveis da PAI por contacto direto com água, usando uma razão sólido:líquido de 1:3 e duas temperaturas de extração, 25 e 50ºC. Em seguida, aplicou-se o método fotocatalítico a cada um dos subprodutos mencionados. Os resultados mostraram que a fotocatálise removeu totalmente os polifenóis presentes nos xaropes (236,4 e 488,4 mg EAG/l). Contrariamente, na PAE a aplicação da fotocatálise levou a um aumento significativo dos polifenóis solúveis que foi de 6,30 mg EAG/g (p.s. PAE) para 20,64 mg EAG/g (p.s. PAE) na polpa extraída a 25ºC. No entanto, quer na PAE quer nos xaropes de alfarroba a cor castanho-alaranjada característica dos polifenóis corados existentes na polpa de alfarroba desapareceu após aplicação da fotocatálise. Por fim, aplicou-se a hidrólise enzimática à PAE tratada e não tratada por fotocatálise, tendo os resultados mostrado um aumento no rendimento da hidrólise de 22% para 38%.The worldwide increasing demand for energy has been leading to the exhaustion of fossil fuels reserves. This situation is pushing the search for alternative renewable energies, such as biofuels that can be directly used in the transportation sector. Carob pulp, which is an endogenous biomass waste that contains high levels of readily fermentable sugars (40-50% w/dw) and also a lignocellulosic fraction (ca 20% w/dw) that can be further fermented after being hydrolyzed, arises as a very suitable feedstock for bioethanol production. However, in a recent work carried out at LNEG, results suggested that the low yields obtained in the enzymatic hydrolysis of the lignocellulosic fraction of carob pulp were due to its high polyphenolic content. Thus, the aim of the present work was to apply a photocatalytic method to carob pulp to degrade its polyphenols and to evaluate the impact of this treatment on the yield of the enzymatic hydrolysis process. Carob pulp was first characterized for its composition and then the material was processed into three products: whole carob pulp (WCP), carob syrup and extracted carob pulp (ECP). ECP and carob syrups were obtained by extraction of the WCP soluble sugars by direct contact with water, using a solids: liquid ratio of 1:3 and temperatures of 25 and 50 ºC, respectively. Results showed that carob syrups extracted at 25ºC and at 50ºC contained 236.4 and 488.4 mg GAE/l, respectively, while the ECP at 25ºC and 50ºC contained 6.30 and 7.37 mg GAE/g (d.w. of ECP), respectively. Detoxification of carob syrups by photocatalysis showed that the soluble polyphenols were successfully removed, while in the ECP at 25ºC the polyphenolic content increased from 6.30 to 20.64 mg GAE/g (d.w. of ECP). Nevertheless, both substrates (ECP and carob syrup) lost their intense orange-brown color, which is believed to correspond to colored polyphenols, after photocatalysis. Finally, enzymatic hydrolysis was applied to ECP treated and not-treated by photocatalysis and results showed a significant increase on the yield of the process (from 22% to 38%)

Quantum circuits to measure scalar spin chirality

The scalar spin chirality is a three-body physical observable that plays an outstanding role both in classical magnetism, characterizing non-co-planar spin textures, and in quantum magnetism, as an order parameter for chiral spin liquids. In quantum information, the scalar spin chirality is a witness of genuine tripartite entanglement. Here we propose an indirect measurement scheme, based on the Hadamard test, to estimate the scalar spin chirality for general quantum states. We apply our method to study chirality in two types of quantum states: generic one-magnon states of a ferromagnet, and the ground state of a model with competing symmetric and antisymmetric exchange. We show a single-shot determination of the scalar chirality is possible for chirality eigenstates, via quantum phase estimation with a single auxiliary qutrit. Our approach provides a unified theory of chirality in classical and quantum magnetism.L.I.R. thanks the New Talents program of Fundação Calouste Gulbenkian for financial support. B.M. acknowledges financial support from Fundação para a Ciência e a Tecnologia (FCT)—Portugal through the Ph.D. scholarship No. SFRH/BD/08444/2020. E.F.G. acknowledges support from FCT via project CEECINST/00062/2018, and from the Digital Horizon Europe project FoQaCiA, GA No. 101070558. J.F.R. acknowledges financial support from FCT (Grant No. PTDC/FIS-MAC/2045/2021), Science National Foundation (Switzerland) Sinergia (Grant Pimag), Generalitat Valenciana funding Prometeo2021/017 and MFA/2022/045, and funding from MICIIN-Spain (Grant No. PID2019-109539GB-C41)

Quantum circuits to measure scalar spin chirality

The scalar spin chirality is a three-body physical observable that plays an outstanding role both in classical magnetism, characterizing non-coplanar spin textures, and in quantum magnetism, as an order parameter for chiral spin liquids. In quantum information, the scalar spin chirality is a witness of genuine tripartite entanglement. Here we propose an indirect measurement scheme, based on the Hadamard test, to estimate the scalar spin chirality for general quantum states. We apply our method to study chirality in two types of quantum states: generic one-magnon states of a ferromagnet, and the ground state of a model with competing symmetric and antisymmetric exchange. We show a single-shot determination of the scalar chirality is possible for chirality eigenstates, via quantum phase estimation with a single auxiliary qutrit. Our approach provides a unified theory of chirality in classical and quantum magnetism.Comment: 5 pages, 3 figure