arXiv subjects

Gustavo E. Scuseria

Publications and source records attributed to Gustavo E. Scuseria.

At least 37 records · Page 2Linked to original sources

arXiv2023

Restoring permutational invariance in the Jordan-Wigner transformation

The Jordan-Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labeling of the spins. One consequence of this dependence is that approximate solutions of a Jordan-Wigner--transformed Hamiltonian may depend on the (physically inconsequential) labeling of the spins. In this work, we turn to an extended Jordan-Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree-Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan-Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. We show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and $J_1-J_2$ Heisenberg Hamiltonians.

cond-mat.str-el↗

arXiv2023

Exploring Spin AGP Ansatze for Strongly Correlated Spin Systems

The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree--Fock--Bogoliubov (HFB), and number-projected Bardeen--Cooper--Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin $su(2)$ algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and $\mathrm{J_1-J_2}$ Heisenberg models, both in 1 and 2 dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.

cond-mat.str-el↗

arXiv2023

Symmetry-projected cluster mean-field theory applied to spin systems

We introduce $S_z$ spin-projection based on cluster mean-field theory and apply it to the ground state of strongly-correlated spin systems. In cluster mean-field, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is $S_z$ symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all $S_z$ sectors, breaking $S_z$ symmetry. In addition, a projection scheme is used to restore global $S_z$, which gives rise to $S_z$ spin-projected generalized cluster mean-field (S$_z$GcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D $J_1-J_2$ and $XXZ$ Heisenberg models. Our results indicate that the new methods (GcMF and S$_z$GcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.

cond-mat.str-el↗

arXiv2023

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this paper, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

physics.chem-ph↗

arXiv2023

State preparation of AGP on a quantum computer without number projection

The antisymmetrized geminal power (AGP) is equivalent to the number projected Bardeen-Cooper-Schrieffer (PBCS) wavefunction. It is also an elementary symmetric polynomial (ESP) state. We generalize previous research on deterministically implementing the Dicke state to a state preparation algorithm for an ESP state, or equivalently AGP, on a quantum computer. Our method is deterministic and has polynomial cost, and it does not rely on number symmetry breaking and restoration. We also show that our circuit is equivalent to a disentangled unitary paired coupled cluster operator and a layer of unitary Jastrow operator acting on a single Slater determinant. The method presented herein highlights the ability of disentangled unitary coupled cluster to capture non-trivial entanglement properties that are hardly accessible with traditional Hartree-Fock based electronic structure methods.

quant-ph↗

arXiv2023

Thermofield theory for finite-temperature electronic structure

Wave-function methods have offered a robust, systematically improvable means to study ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogs of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space, which is a formidable task. Besides, excited-state theories are generally not as well studied as ground-state ones. In this mini-review, we present an overview of a finite-temperature wave function formalism based on thermofield dynamics to overcome these difficulties. Thermofield dynamics allows us to map the equilibrium thermal density matrix to a pure state, i.e., a single wave function, albeit in an expanded Hilbert space. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we have developed a procedure to generalize ground-state wave function theories to finite temperatures. As explicit examples, we highlight formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of fermions in the grand-canonical ensemble. To assess the quality of these approximations, we also show benchmark studies for the one-dimensional Hubbard model, while comparing against exact results. We will see that the thermal methods perform similarly to their ground-state counterparts, while merely adding a pre-factor to the asymptotic computational cost. They also inherit all the properties, good or bad, from the ground-state methods, signifying the robustness of our formalism and the scope for future development.

cond-mat.str-el↗

arXiv2022

Strong-Weak Duality via Jordan-Wigner Transformation: Using Fermionic Methods for Strongly Correlated $su(2)$ Spin Systems

The Jordan-Wigner transformation establishes a duality between $su(2)$ and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree-Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J$_1$-J$_2$ Heisenberg (in 1D and 2D) indicate that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams, and provides a better starting point for addressing challenging spin problems.

cond-mat.str-el↗

arXiv2022

AGP-based unitary coupled cluster theory for quantum computers

Electronic structure methods typically benefit from symmetry breaking and restoration, specially in the strong correlation regime. The same goes for Ansätze on a quantum computer. We develop a unitary coupled cluster method on the antisymmetrized geminal power (AGP) -- a state formally equivalent to the number-projected Bardeen--Cooper--Schrieffer wavefunction. We demonstrate our method for the single-band Fermi--Hubbard Hamiltonian in one and two dimensions. We also explore post-selection as a state preparation step to obtain correlated AGP and prove that it scales no worse than $\mathcal{O}(\sqrt{M})$ in the number of measurements, thereby making it a less expensive alternative to gauge integration to restore particle number symmetry.

quant-ph↗

arXiv2022

Coupled cluster and perturbation theories based on a cluster mean-field reference applied to strongly correlated spin systems

We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly-correlated spin systems. In cluster mean-field, the ground state wavefunction is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allows for a straightforward improvement which uses perturbation theory and coupled-cluster to account for inter-cluster correlations. We present benchmark calculations on the 1D chain and 2D square $J_1-J_2$ Heisenberg model, using cluster mean-field, perturbation theory and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4 and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly-correlated systems. Some ways to improve upon the methods presented in this work are discussed.

cond-mat.str-el↗

arXiv2022

A Power Series Approximation in Symmetry Projected Coupled Cluster Theory

Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly-correlated systems. On the other hand, single-reference methods such as configuration interaction or coupled cluster theory can handle weakly-correlated problems but cannot properly account for strong correlations. Ideally, we would like to combine these approaches in a symmetry-projected coupled cluster approach, but this is far from straightforward. In this work, we provide an alternative formulation to identify the so-called disentangled cluster operators which arise when we combine these two methodological strands. Our formulation shows promising results for model systems and small molecules.

cond-mat.str-el↗

arXiv2022

Thermal coupled cluster theory for SU(2) systems

Coupled cluster (CC) has established itself as a powerful theory to study correlated quantum many-body systems. Finite-temperature generalizations of CC theory have attracted considerable interest and have been shown to work as nicely as the ground-state theory. However, most of these recent developments address only fermionic or bosonic systems. The distinct structure of the $su(2)$ algebra requires the development of a similar thermal CC theory for spin degrees of freedom. In this paper, we provide a formulation of our thermofield-inspired thermal CC for SU(2) systems. We apply the thermal CC to the Lipkin-Meshkov-Glick system as well as the one-dimensional transverse field Ising model as benchmark applications to highlight the accuracy of thermal CC in the study of finite-temperature phase diagrams in SU(2) systems.

cond-mat.str-el↗

arXiv2021

Construction of linearly independent non-orthogonal AGP states

We show how to construct a linearly independent set of antisymmetrized geminal power (AGP) states, which allows us to rewrite our recently introduced geminal replacement models as linear combinations of non-orthogonal AGPs. This greatly simplifies the evaluation of matrix elements and permits us to introduce an AGP-based selective configuration interaction method, which can reach arbitrary excitation levels relative to a reference AGP, balancing accuracy and cost as we see fit.

physics.chem-ph↗

arXiv2021

Exploring non-linear correlators on AGP

Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential ansatze in this paper. We investigate two approaches in particular. Similar to Phys. Rev. B 91, 041114(R) (2015), we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting Schrodinger's equation. The second approach is based on approximating the unitary pair-hopper ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.

cond-mat.str-el↗

arXiv2020

Correlating AGP on a quantum computer

For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose and implement a unitary correlator on AGP and benchmark it on the ground state of the pairing Hamiltonian. The results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.

quant-ph↗

arXiv2020

Geminal replacement models based on AGP

The antisymmetrized geminal power (AGP) wavefunction has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J-CI, provides excellent accuracy in energies when applied to the pairing Hamiltonian.

physics.chem-ph↗

arXiv2020

Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems

We present a wave function representation for the canonical ensemble thermal density matrix by projecting the thermofield double state against the desired number of particles. The resulting canonical thermal state obeys an imaginary time-evolution equation. Starting with the mean-field approximation, where the canonical thermal state becomes an antisymmetrized geminal power wave function, we explore two different schemes to add correlation: by number-projecting a correlated grand-canonical thermal state, and by adding correlation to the number-projected mean-field state. As benchmark examples, we use number-projected configuration interaction and an AGP-based perturbation theory to study the Hydrogen molecule in a minimal basis and the six-site Hubbard model.

physics.chem-ph↗

arXiv2020

Correlating the Antisymmetrized Geminal Power Wave Function

Strong pairing correlations are responsible for superconductivity and off-diagonal long range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.

physics.chem-ph↗

arXiv2020

On a dual representation of the Goldstone manifold

An intrinsic wavefunction with a broken continuous symmetry can be rotated with no energy penalty leading to an infinite set of degenerate states known as a Goldstone manifold. In this work, we show that a dual representation of such manifold exists that is sampled by an infinite set of non-degenerate states. A proof that both representations are equivalent is provided. From the work of Peierls and Yoccoz (Proc. Phys. Soc. A {\bf 70}, 381 (1957)), it is known that collective states with good symmetries can be obtained from the Goldstone manifold using a generator coordinate trial wavefunction. We show that an analogous generator coordinate can be used in the dual representation; we provide numerical evidence using an intrinsic wavefunction with particle number symmetry-breaking for the electronic structure of the Be atom and one with $\hat{S}^z$ symmetry-breaking for a H$_5$ ring. We discuss how the dual representation can be used to evaluate expectation values of symmetry-projected states when the norm $|\langle Φ| \hat{P}^q | Φ\rangle|$ becomes very small.

physics.chem-ph↗