Fermion Operators Research Articles

Fermi operator expansion method for nuclei and inhomogeneous matter with a nuclear energy density functional

The nuclear energy density functional method at finite temperature is a useful tool for studies of nuclear structure at high excitation, and also for researches of nuclear matter involved in explosive stellar phenomena and neutron stars. However, its unrestricted calculation requires large computational costs for the three-dimensional coordinate-space solvers, especially for the Hamiltonian matrix diagonalization and (or) the Gram-Schmidt orthonormalization of the single-particle wave functions. We test numerical performance of the Fermi operator expansion method, that requires neither the diagonalization nor the Gram-Schmidt orthonormalization, for finite nuclei and inhomogeneous nuclear matter. The method is applied to isolated finite N=Z nuclei and to non-uniform symmetric nuclear matter at finite temperature, which turns out be very effective with the three-dimensional coordinate-space representation, especially at high temperature. The Fermi operator expansion method is a useful tool for studies of various nuclear phases at finite temperature with the energy density functional calculations. The method is suitable for massively parallel computing with distributed memory. Furthermore, when the space size is large, the calculation may benefit from its order-N scaling property.

Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver

Measuring the expectation value of the molecular electronic Hamiltonian is one of the challenging parts of the variational quantum eigensolver. A widely used strategy is to express the Hamiltonian as a sum of measurable fragments using fermionic operator algebra. Such fragments have an advantage of conserving molecular symmetries that can be used for error mitigation. The number of measurements required to obtain the Hamiltonian expectation value is proportional to a sum of fragment variances. Here, we introduce a new method for lowering the fragments' variances by exploiting flexibility in the fragments' form. Due to idempotency of the occupation number operators, some parts of two-electron fragments can be turned into one-electron fragments, which then can be partially collected in a purely one-electron fragment. This repartitioning does not affect the expectation value of the Hamiltonian but has non-vanishing contributions to the variance of each fragment. The proposed method finds the optimal repartitioning by employing variances estimated using a classically efficient proxy for the quantum wavefunction. Numerical tests on several molecules show that repartitioning of one-electron terms lowers the number of measurements by more than an order of magnitude.

Stationary states of open XX -spin chains

We study an open quantum spin chain of arbitrary length with nearest neighbor $XX$ interactions of strength $g$, immersed in an external constant magnetic field $\mathrm{\ensuremath{\Delta}}$ along the $z$ direction, whose end spins are weakly coupled to two heat baths at different temperatures. In the so-called global approach, namely, without neglecting interspin interactions, using standard weak-coupling limit techniques, we first derive the open chain master equation written in terms of fermionic mode operators. Then, we focus on the study of the dependence of the resulting open dynamics from the ratio $r\ensuremath{\equiv}g/\mathrm{\ensuremath{\Delta}}$. By increasing $r$, some of the chain Bohr transition frequencies become negative; when this occurs, both the generator of the dissipative time evolution and its stationary states behave discontinuously. As a consequence, the asymptotic spin and heat flows also exhibit discontinuities, but in a different way: while source terms in the spin flow continuity equation show jumps, the heat flow instead is continuous but with discontinuous first derivatives with respect to $r$. These two behaviors might be experimentally accessible; in particular, they could discriminate between the global and the local approaches to open quantum spin chains. Indeed, the latter one, which neglects interspin interactions in the derivation of the master equation, does not show any kind of discontinuous behavior.

Efficient quantum computation of molecular forces and other energy gradients

While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices we demonstrate how low rank factorizations and other tomography schemes can be optimized for energy derivative calculations. We perform numerics revealing that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling incorporating state-of-the-art techniques for block encoding fermionic operators. Our results suggest that the calculation of forces on a single nucleus may be of similar cost to estimating energies of chemical systems, but that further developments are needed for quantum computers to meaningfully assist with molecular dynamics simulations.

Operatorial Formulation of a Model of Spatially Distributed Competing Populations

This paper deals with the application of the mathematical apparatus of quantum mechanics for the formulation of an operatorial model of a couple of populations spatially distributed over a one-dimensional region. The two populations interact with a competitive mechanism and are able to diffuse over the region. A nonlocal competition effect is also included. In more detail, we consider a one-dimensional region divided in N cells where the actors, represented by annihilation, creation, and a number fermionic operators, interact. The dynamics is governed by a self-adjoint and time-independent Hamiltonian operator describing the various interactions. The results of some numerical simulations are presented and discussed. The recently introduced variant of the standard Heisenberg approach, named (H,ρ)-induced dynamics, is also used in order to take into account some changes in time of the attitudes of the two populations, and obtain more realistic dynamical outcomes.

Exact weak bosonic zero modes in a spin/fermion chain

We study an exactly solvable one-dimensional spin-$\frac{1}{2}$ model which can support weak zero modes in its ground state manifold. The spin chain has staggered XXZ-type and ZZ-type spin interaction on neighboring bonds and is thus dubbed the (XXZ,Z) chain. The model is equivalent to an interacting fermionic chain by Jordan-Wigner transformation. We study the phase diagram of the system and work out the conditions and properties of its weak zero modes. In the fermion chain representation, these weak zero modes are given by even-order polynomials of Majorana fermion operators and are thus bosonic. The fermionic chain Hamiltonian contains only fermion hopping and interaction terms and may have potential realization in experiments.

Quantum computing of the Li6 nucleus via ordered unitary coupled clusters

The variational quantum eigensolver (VQE) is an algorithm to compute ground and excited state energy of quantum many-body systems. A key component of the algorithm and an active research area is the construction of a parametrized trial wavefunction -- a so called variational ansatz. The wavefunction parametrization should be expressive enough, i.e. represent the true eigenstate of a quantum system for some choice of parameter values. On the other hand, it should be trainable, i.e. the number of parameters should not grow exponentially with the size of the system. Here, we apply VQE to the problem of finding ground and excited state energies of the odd-odd nucleus $^6$Li. We study the effects of ordering fermionic excitation operators in the unitary coupled clusters ansatz on the VQE algorithm convergence by using only operators preserving the $J_z$ quantum number. The accuracy is improved by two order of magnitude in the case of descending order. We first compute optimal ansatz parameter values using a classical state-vector simulator with arbitrary measurement accuracy and then use those values to evaluate energy eigenstates of $^6$Li on a superconducting quantum chip from IBM. We post-process the results by using error mitigation techniques and are able to reproduce the exact energy with an error of 3.8% and 0.1% for the ground state and for the first excited state of $^6$Li, respectively.

Higher-dimensional Jordan-Wigner transformation and auxiliary Majorana fermions

We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination numbers and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular, and cubic lattices for spin-$1/2$ fermions. We also discuss the relation to some quantum spin liquid models.

Invariant quadratic operators associated with linear canonical transformations and their eigenstates

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.

Lindbladian dynamics of the Sachdev-Ye-Kitaev model

We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are non-random while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large $N$, where $N$ is the number of Majorana fermion operators, and also in the limit of large $N$ and $M$, where $M$ is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite $N$, we also study the distribution of the eigenvalues of the SYK Lindbladians.

Representation of the fermionic boundary operator

The boundary operator is a linear operator that acts on a collection of high-dimensional binary points (simplices) and maps them to their boundaries. This boundary map is one of the key components in numerous applications, including differential equations, machine learning, computational geometry, machine vision and control systems. We consider the problem of representing the full boundary operator on a quantum computer. We first prove that the boundary operator has a special structure in the form of a complete sum of fermionic creation and annihilation operators. We then use the fact that these operators pairwise anticommute to produce an $O(n)$-depth circuit that exactly implements the boundary operator without any Trotterization or Taylor series approximation errors. Having fewer errors reduces the number of shots required to obtain desired accuracies.

Topological properties of minimally doubled fermions in two spacetime dimensions

The two-dimensional Schwinger model is used to explore how lattice fermion operators perceive the global topological charge $q \in \mathbb{Z}$ of a given background gauge field. We focus on Karsten-Wilczek and Borici-Creutz fermions, which are minimally doubled, and compare them to Wilson, Brillouin, naive, staggered and Adams fermions. For each operator the eigenvalue spectrum in a background with $q \neq 0$ is determined along with the chiralities of the eigenmodes, and the spectral flow of the pertinent hermitean operator is worked out. We find that Karsten-Wilczek and Borici-Creutz fermions perceive the global topological charge $q$ in the same way as staggered and naive fermions do.

Index Theorems, Generalized Hall Currents, and Topology for Gapless Defect Fermions.

We show how the index of the fermion operator from the Euclidean action can be used to uncover the existence of gapless modes living on defects (such as edges and vortices) in topological insulators and superconductors. The 1-loop Feynman diagram that computes the index reveals an analog of the quantum Hall current flowing on and off the defect-even in systems without conserved currents or chiral anomalies-and makes explicit the interplay between topology in momentum and coordinate space. We provide several explicit examples.

Spin functional renormalization group for dimerized quantum spin systems

We investigate dimerized quantum spin systems using the spin functional renormalization group approach proposed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] which directly focuses on the physical spin correlation functions and avoids the representation of the spins in terms of fermionic or bosonic auxiliary operators. Starting from decoupled dimers as initial condition for the renormalization group flow equations, we obtain the spectrum of the triplet excitations as well as the magnetization in the quantum paramagnetic, ferromagnetic, and thermally disordered phases at all temperatures. Moreover, we compute the full phase diagram of a weakly coupled dimerized spin system in three dimensions, including the correct mean field critical exponents at the two quantum critical points.

Second-Quantized Fermionic Operators with Polylogarithmic Qubit and Gate Complexity

A new qubit encoding method for second-quantized fermion interactions in systems of relevance for quantum chemistry is described, which can achieve an exponential improvement over prior second-quantized strategies in the limit of large basis sets.

Revisiting the fermionic dark matter absorption on electron target

We perform a systematic study of the fermionic DM absorption interactions on electron target in the context of effective field theory. The fermionic DM absorption is not just sensitive to sub-MeV DM with efficient energy release, but also gives a unique signature with clear peak in the electron recoil spectrum whose shape is largely determined by the atomic effects. Fitting with the Xenon1T and PandaX-II data prefers DM mass at mχ = 59 keV and 105 keV, respectively, while the cut-off scale is probed up to around 1 TeV. The DM overproduction in the early Universe, the invisible decay effect on the cosmological evolution, and the visible decay signal collected by the astrophysical X(gamma)-ray observations (Insight-HXMT, NuSTAR, HEAO-1, and INTEGRAL) are thoroughly explored to constrain the DM absorption interactions. With stringent bounds on the tensor and pseudo-scalar operators, the other fermionic DM operators are of particular interest at tonne-scale direct detection experiments such as PandaX-4T, XENONnT, and LZ.

Edge Z3 parafermions in fermionic lattices

Parafermions modes are non-Abelian anyons which were introduced as $\mathbb{Z}_N$ generalizations of $\mathbb{Z}_2$ Majorana states. In particular, $\mathbb{Z}_3$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of theoretical work on $\mathbb{Z}_3$ parafermions has relied on bosonization methods or parafermionic quasi-particles. In this work, we introduce a representation of $\mathbb{Z}_3$ parafermions in terms of purely fermionic models operators in the t-J regime. We establish the equivalency of a family of lattice fermionic models written in the $t-J$ model basis with a Kitaev-like chain supporting free $\mathbb{Z}_3$ parafermonic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing $\mathbb{Z}_3$ parafermions in strongly interacting electronic systems.

Fermionic quantum field theories as probabilistic cellular automata

A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.

Modular symmetry in the SMEFT

We study the modular symmetric standard model effective field theory. We employ the stringy Ansatz on coupling structure that 4-point couplings ${y}^{(4)}$ of matter fields are written by a product of 3-point couplings ${y}^{(3)}$ of matter fields, i.e., ${y}^{(4)}={y}^{(3)}{y}^{(3)}$. In this framework, we discuss the flavor structure of bilinear fermion operators and 4-fermion operators, where the holomorphic and antiholomorphic modular forms appear. From the Ansatz, the ${A}_{4}$ modular-invariant semileptonic four-fermion operator $[{\overline{E}}_{R}\mathrm{\ensuremath{\Gamma}}{E}_{R}][{\overline{D}}_{R}\mathrm{\ensuremath{\Gamma}}{D}_{R}]$ does not lead to the flavor changing (FC) processes since this operator would be constructed in terms of gauge couplings $g$ as ${y}^{(3)}\ensuremath{\sim}g$. The chirality flipped bilinear operator $[{\overline{D}}_{R}\mathrm{\ensuremath{\Gamma}}{D}_{L}]$ also does not lead FC if the mediated mode corresponds to the Higgs boson ${H}_{d}$. In this case, the flavor structure of this operator is the exactly same as the mass matrix. On the other hand, if the flavor structure of the operator is not the exactly same as the mass matrix, the situation would change drastically. Then, we obtain the nontrivial relations of FC transitions at nearby fixed points $\ensuremath{\tau}=i,\ensuremath{\omega},i\ensuremath{\infty}$, which are testable in the future. As an application, we discuss the relations of the lepton flavor violation processes $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma},\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\gamma}$ and $\ensuremath{\tau}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ at nearby ${\ensuremath{\tau}}_{e}=i$, where the successful lepton mass matrix was obtained. We also study the flavor changing 4-quark operators in the ${A}_{4}$ modular symmetry of quarks. As a result, the minimal flavor violation could be realized by taking relevant specific parameter sets of order one.

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.