Stijn De baerdemacker

• PhD Physics
• Chair at University of New Brunswick

Finding the ground state of spin glasses is a challenging problem with broad implications. Many hard optimization problems, including NP-complete problems, can be mapped, for instance, to the Ising spin glass model. We present a graph-based approach that allows for accurate state initialization of a frustrated triangular spin-lattice with up to 20...

We show that the behavior of stochastic gradient descent is related to Bayesian statistics by showing that SGD is effectively diffusion on a fractal landscape, where the fractal dimension can be accounted for in a purely Bayesian way. By doing this we show that SGD can be regarded as a modified Bayesian sampler which accounts for accessibility cons...

We present a new application of the Generator Coordinate Method (GCM) as an electronic structure method for strong electron correlation in molecular systems. We identify spin fluctuations as an important generator coordinate responsible for strong static electron correlation that is associated with bond-breaking processes. Spin-constrained Unrestri...

In deep learning methods, especially in the context of chemistry, there is an increasing urgency to uncover the hidden learning mechanisms often dubbed as “black box." In this work, we...

We introduce an electronic structure approach for spin symmetry breaking and restoration from the mean-field level. The spin-projected constrained-unrestricted Hartree-Fock (SPcUHF) method restores the broken spin symmetry inherent in spin-constrained-UHF determinants by employing a non-orthogonal Configuration Interaction (NOCI) projection method....

Wavefunction forms based on products of electron pairs are usually constructed as closed-shell singlets, which is insufficient when the molecular state has a nonzero spin or when the chemistry is determined by d- or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...

Understanding the electronic structure of amino acids is crucial for understanding protein structure and function. We interfaced quantum information indices and ab initio quantum chemistry methods to quantify the locality correlation between neighboring amino acids. Quantum chemical information from the 1-, 2- and multi-orbital Reduced Density Matr...

Wavefunction forms based on products of electron pairs are usually constructed as closed-shell singlets, which is insufficient when the molecular state has a nonzero spin or when the chemistry is determined by $d$- or $f-$electrons. A set of two-electron forms are considered as explicit couplings of second-quantized operators to open-shell singlets...

We present an overview of the mathematical structure of geminal theory within the seniority formalism and bi-variational principle. Named after the constellation, geminal wavefunctions provide the mean-field like representation of paired-electron wavefunctions in quantum chemistry, tying in with the Lewis picture of chemical bonding via electron pa...

In deep learning methods, especially in the context of chemistry, there is an increasing urgency to uncover the hidden learning mechanisms often dubbed as ``black box." In this work, we show that graph models built on computational chemical data behave similar to natural language processing (NLP) models built on text data. Crucially, we show that a...

We explore transfer learning models from a pre-trained graph convolutional neural network representation of molecules, obtained from SchNet, to predict ¹³C-NMR, pKa, and log S solubility. SchNet learns a graph representation of a molecule by associating each atom with an “embedding vector” and interacts the atom-embeddings with each other by levera...

Graph neural nets, such as SchNet, [Schütt et al., J. Chem. Phys., 2018, 148, 241722], and AIMNet, [Zubatyuk et al., Sci. Adv., 2019, 5, 8] provide accurate predictions of chemical quantities without invoking any direct physical or chemical principles. These methods learn a hidden statistical representation of molecular systems in an end-to-end fas...

Graph convolutional neural nets, such as SchNet, [Schütt et al, Journal of Chemical Physics, 2018, 148, 241722], provide accurate predictions of chemical quantities without invoking any direct physical or chemical principles. These methods learn a hidden statistical representation of molecular systems in an end-to-end fashion; from xyz coordinates...

We explore transfer learning models from a pre-trained graph convoluntional neural network representation of molecules, obtained from SchNet, 1 to predict 13 C-NMR, pKa, and logS sol- ubility. SchNet learns a graph representation of a molecule by associating each atom with an “embedding vector” and interacts the atom-embeddings with each other by l...

The variational quantum eigensolver (VQE) algorithm recently became a popular method to compute quantum chemical properties of molecules on noisy intermediate scale quantum (NISQ) devices. The VQE is a hybrid quantum-classical algorithm in which the variational wavefunction is implemented as a quantum circuit on the quantum device, whereas the opti...

We present a mathematical analysis of the spin-constrained Hartree-Fock solutions (CHF) of the 2-site Hubbard model. The analysis sheds light on the spin symmetry breaking process around the Coulson-Fischer point. CHF states are useful as input states for the Generator Coordinate Method (GCM) in which CHF states can be used as a basis for multiconf...

In spin frustrated H3-rings, variationally minimal single Slater determinant descriptions break S^2 and S^z symmetries in an effort to simultaneously minimise the interactions between all electrons. Given the underlying spin dynamics, it remains unclear how one can move beyond these symmetry-broken mean-field states and efficiently introduce electr...

For any unitary matrix there exists a ZXZ decomposition, according to a theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists a block-ZXZ decomposition, according to a theorem by Führ and Rzeszotnik. We conjecture that these two decompositions are merely special cases of a set of decompositions, one for every divisor of the...

The failure of many approximate electronic structure methods can be traced to their erroneous description of fractional charge and spin redistributions in the asymptotic limit towards infinity, where violations of the flat-plane conditions lead to delocalization and static correlation errors. Although the energetic consequences of the flat-planes a...

We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the sam...

We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the sam...

Clar's aromatic -sextet rule is a widely used qualitative method for assessing the electronic structure of polycyclic benzenoid hydrocarbons. Unfortunately, many of the quantum chemical concordances for this rule have a limited range of applicability. Here, we show that the fundamental probabilities associated with a distribution of electrons over...

For any unitary matrix there exists a ZXZ decomposition, according to a theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists a block-ZXZ decomposition, according to a theorem by F\"uhr and Rzeszotnik. We conjecture that these two decompositions are merely special cases of a set of decompositions, one for every divisor of t...

We employ tensor network methods for the study of the seniority quantum number – defined as the number of unpaired electrons in a many-body wave function – in molecular systems. Seniority-zero methods recently emerged as promising candidates to treat strong static correlations in molecular systems, but are prone to deficiencies related to dynamical...

Ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian are employed as a wavefunction Ansatz to model strong electron correlation in quantum chemistry. This wavefunction is a product of weakly interacting pairs of electrons. While other geminal wavefunctions may only be employed in a projected Schrödinger equation, the prese...

We employ tensor network methods for the study of the seniority quantum number - defined as the number of unpaired electrons in a many-body wave function - in molecular systems. Seniority-zero methods recently emerged as promising candidates to treat strong static correlations in molecular systems, but are prone to deficiencies related to dynamical...

Ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian are employed as a wavefunction ansatz to model strong electron correlation in quantum chemistry. This wavefunction is a product of weakly-interacting pairs of electrons. While other geminal wavefunctions may only be employed in a projected Schr\"{o}dinger equation, the p...

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of 2 (say 2w) deserve special attention, as they represent quant...

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent qu...

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent qu...

As approximations to the wave functions governing quantum chemical systems become more and more complex, it is becoming increasingly important to devise descriptors that help understand the practical results of those approximations by condensing information in insightful ways. Quantum chemical descriptors that are able to capture the statistical si...

Birkhoff’s theorem tells how any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. Similar theorems on unitary matrices reveal a connection between quantum circuits and linear classical reversible circuits. It triggers the question whether a quantum computer can be regarded as a superposition of classical reversi...

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension n of the unitary matrix equals a power of a prime p, i....

We establish the most general class of spin- ¹2 integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- ¹2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Ri...

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension~$n$ of the unitary matrix equals a power of a prime $p$...

We establish the most general class of spin-1/2 integrable Richardson-Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin-1/2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Ri...

The design of a quantum computer and the design of a classical computer can be based on quite similar circuit designs. The former is based on the subgroup structure of the infinite group of unitary matrices, whereas the latter is based on the subgroup structure of the finite group of permutation matrices. Because these two groups display similariti...

At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation. At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation. Whereas an arbitrary quantum circuit, acting on w qubits, is d...

This work proposes the variational determination of two-electron reduced density matrices corresponding to the ground state of N-electron systems within the doubly occupied-configuration-interaction methodology. The P, Q, and G two-index N-representability conditions have been extended to the T1 and T2 (T2′) three-index ones and the resulting optim...

At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation. Whereas an arbitrary quantum circuit, acting on w qubits, is described by an n X n unitary matrix with n=2^w, a reversible classical circuit, acting on w bits, is described by a 2^w X 2^w permutation matrix...

We summarize the previous chapters: In Chapter 2, we presented three synthesis methods for classical reversible circuits, based on three decompositions of an arbitrary permutation matrix, with increasing efficiency: a primal matrix decomposition, a dual matrix decomposition, and a refined matrix decomposition. They lead to three circuits, each comp...

We show how adiabatically varying the driving frequency of a periodically-driven many-body quantum system can induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian. It is argued that these can be understood as adiabatic transitions in the Floquet spectrum at quasi-degeneracies. Using the central spin model as a...

Adiabatically varying the driving frequency of a periodically-driven many-body quantum system can induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian, corresponding to adiabatic transitions in the Floquet spectrum and presenting a general tool in quantum many-body control. Using the central spin model as an a...

Background: The nuclear many-body system is a strongly correlated quantum system, posing serious challenges for perturbative approaches starting from uncorrelated reference states. The last decade has witnessed considerable progress in the accurate treatment of pairing correlations, one of the major components in medium-sized nuclei, reaching accur...

We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is show...

We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is show...

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schr\"odinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not $N$-representable. That is, the response 2...

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schr\"odinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not $N$-representable. That is, the response 2...

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining th...

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining th...

Density matrix embedding theory (DMET) is a relatively new technique for the calculation of strongly correlated systems. Recently, block product DMET (BPDMET) was introduced for the study of spin systems such as the antiferromagnetic J1−J2 model on the square lattice. In this paper, we extend the variational Ansatz of BPDMET using spin-state optimi...

We discuss some strategies for extending recent geminal-based methods to open-shells by replacing the geminal-creation operators with more general composite boson creation operators, and even creation operators that mix fermionic and bosonic components. We also discuss the utility of symmetry-breaking and restoration, but using a projective (not a...

Density matrix embedding theory (DMET) is a relatively new technique for the calculation of strongly correlated systems. Recently cluster DMET (CDMET) was introduced for the study of spin systems such as the anti-ferromagnetic $J_1-J_2$ model on the square lattice. In this paper, we study the Kitaev-Heisenberg model on the honeycomb lattice using t...

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.

By systematically inflating the group of \(n \times n\) permutation matrices to the group of \(n \times n\) unitary matrices, we can see how classical computing is embedded in quantum computing. In this process, an important role is played by two subgroups of the unitary group U(n), i.e. XU(n) and ZU(n). Here, XU(n) consists of all \(n \times n\) u...

Given an arbitrary 2w×2w unitary matrix U, a powerful matrix decomposition can be applied, leading to four different syntheses of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bi...

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition,...

We study the weak-pairing phase in a finite-size two-dimensional $p_x+ip_y$ superfluid interacting with an environment. This interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian. We present the exact wave function and solve the resulting Bethe ansatz equations, from which it is shown how resonances a...

Any matrix of the unitary group U(n) can be decomposed into matrices from two subgroups, denoted XU(n) and ZU(n). This leads to decompositions of an arbitrary quantum circuit into NEGATOR circuits and PHASOR circuits. The NEGATOR circuits are closely related to classical reversible computation.

Given an arbitrary $2^\omega \times 2^w$ unitary matrix, a powerful matrix decomposition can be applied, leading to the synthesis of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by F\"uhr and Rzeszotnik, generalizing the scaling of single-bit unitary gates $(\omega=1)$ to gates with...

The theory of Maximum Probability Domains (MPDs) is formulated for the Hubbard model in terms of projection operators and generating functions for both exact eigenstates as well as Slater determinants. A fast MPD analysis procedure is proposed, which is subsequently used to analyse numerical results for the Hubbard model. It is shown that the essen...

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime dimensions.

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime dimensions.

Starting from integrable $su(2)$ (quasi-)spin Richardson-Gaudin XXZ models we derive several properties of integrable spin models coupled to a bosonic mode. We focus on the Dicke-Jaynes-Cummings-Gaudin models and the two-channel $(p+ip)$-wave pairing Hamiltonian. The pseudo-deformation of the underlying $su(2)$ algebra is here introduced as a way t...

A class of polynomial scaling methods that approximate Doubly Occupied Configuration Interaction (DOCI) wave functions and improve the description of dynamic correlation is introduced. The accuracy of the resulting wave functions is analysed by comparing energies and studying the overlap between the newly developed methods and full configuration in...

We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability -, -, and -conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied...

As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit Φ() and the 1-qubit NEGATOR circuit N(). Both are roots of the IDENTITY circuit. Indeed: both Φ(0) and N(0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of con...

CheMPS2, our spin-adapted implementation of the density matrix renormalization group (DMRG) for ab initio quantum chemistry (Wouters et al., 2014), has several new features. A speed-up of the augmented Hessian Newton–Raphson DMRG self-consistent field (DMRG-SCF) routine is achieved with the direct inversion of the iterative subspace (DIIS). For ext...

Classical reversible circuits, acting on $w$~bits, are represented by permutation matrices of size $2^w \times 2^w$. Those matrices form the group P($2^w$), isomorphic to the symmetric group {\bf S}$_{2^w}$. The permutation group P($n$), isomorphic to {\bf S}$_n$, contains cycles with length~$p$, ranging from~1 to $L(n)$, where $L(n)$ is the so-cal...

We propose an extension of the numerical approach for integrable Richardson-Gaudin models based on a new set of eigenvalue-based variables. Starting solely from the Gaudin algebra, the approach is generalized towards the full class of XXZ Richardson-Gaudin models. This allows for a fast and robust numerical determination of the spectral properties...

The Dicke model is derived in the contraction limit of a pseudo-deformation of the quasispin algebra in the su(2)-based Richardson-Gaudin models. Likewise, the integrability of the Dicke model is established by constructing the full set of conserved charges, the form of the Bethe Ansatz state, and the associated Richardson-Gaudin equations. Thanks...

Quantum computation on w qubits is represented by the infinite unitary group U(2w); classical reversible computation on w bits is represented by the finite symmetric group S2w. In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU(n) and ZU(n) of the unitary group U(n)...

The geometric and electronic structure of the MIL-47(V) metal-organic framework (MOF) is investigated by using ab initio density functional theory (DFT) calculations. Special focus is placed on the relation between the spin configuration and the properties of the MOF. The ground state is found to be antiferromagnetic, with an equilibrium volume of...

We introduce new non-variational orbital optimization schemes for the antisymmetric product of one-reference orbital geminal (AP1roG) wave function (also known as pair-coupled cluster doubles) that are extensions to our recently proposed projected seniority-two (PS2-AP1roG) orbital optimization method [J. Chem. Phys. 214114, 140 2014)]. These appro...

We present a new, non-variational orbital-optimization scheme for the antisymmetric product of one-reference orbital geminal wave function. Our approach is motivated by the observation that an orbital-optimized seniority-zero configuration interaction (CI) expansion yields similar results to an orbital-optimized seniority-zero-plus-two CI expansion...

We present an efficient approach to the electron correlation problem that is well suited for strongly interacting many-body systems, but requires only mean-field-like computational cost. The performance of our approach is illustrated for one-dimensional Hubbard rings with different numbers of sites, and for the nonrelativistic quantum-chemical Hami...

Any matrix of the unitary group U(n) can (up to a global phase) be decomposed into 2n-1 matrices from two subgroups, denoted XU(n) and ZU(n). This leads to the decomposition of an arbitrary quantum circuit into NEGATOR circuits and PHASOR circuits. The NEGATOR circuits are closely related to classical reversible computation.

The orbital dependence of closed-shell wavefunction energies is investigated by performing doubly-occupied configuration interaction (DOCI) calculations, representing the most general class of these wavefunctions. Different local minima are examined for planar hydrogen clusters containing two, four, and six electrons applying (spin) symmetry-broken...

Recently, interest has increased in the hyperbolic family of integrable Richardson-Gaudin (RG) models. It was pointed out that a particular linear combination of the integrals of motion of the hyperbolic RG model leads to a Hamiltonian that describes p-wave pairing in a two-dimensional system. Such an interaction is found to be present in fermionic...

A new multireference perturbation approach has been developed for the recently proposed AP1roG scheme, a computationally facile parametrization of an antisymmetric product of nonorthogonal geminals. This perturbation theory of second-order closely follows the biorthogonal treatment from multiconfiguration perturbation theory as introduced by Surján...

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary un...

Pairing correlations in the even-even A=102-130 Sn isotopes are discussed, based on the Richardson-Gaudin variables in an exact Woods-Saxon plus reduced BCS pairing framework. The integrability of the model sheds light on the pairing correlations, in particular on the previously reported sub-shell structure.

The reduced density matrix is variationally optimized for the two-dimensional Hubbard model. Exploiting all symmetries present in the system, we have been able to study $6\times6$ lattices at various fillings and different values for the on-site repulsion, using the highly accurate but computationally expensive three-index conditions. To reduce the...

We propose an approach to the electronic structure problem based on noninteracting electron pairs that has similar computational cost to conventional methods based on noninteracting electrons. In stark contrast to other approaches, the wave function is an antisymmetric product of nonorthogonal geminals, but the geminals are structured so the projec...

Between (classical) reversible computation and quantum computation there exists an intermediate computational world, represented by unitary matrices that have all line sums equal to 1. All of these quantum circuits can be synthesized with the help of merely two building blocks: the NEGATOR and the singly controlled square root of NOT.

Inspired by the wavefunction forms of exactly solvable algebraic Hamiltonians, we present several wavefunction ansatze. These wavefunction forms are exact for two-electron systems; they are size consistent; they include the (generalized) antisymmetrized geminal power, the antisymmetrized product of strongly orthogonal geminals, and a Slater determi...

The pair condensation energy of a finite-size superconducting particle is studied as a function of two control parameters. The first control parameter is the shape of the particle, and the second parameter is a position-dependent impurity introduced in the particle. Whereas the former parameter is known to induce strong fluctuations in the condensa...

The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand,...

The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in...

The quantum gates called 'k th root of NOT' and 'controlled k th root of NOT' can be applied to synthesize circuits, both classical reversible circuits and quantum circuits. Such circuits, acting on w qubits, fill a (2w-1)2-dimensional subspace of the (2w)2 -dimensional space U(2w) of the 2w × 2w unitary matrices and thus describe computers situate...