Two-body Hamiltonian Research Articles

Universal quantum processors in spin systems via robust local pulse sequences

We propose a protocol to realize quantum simulation and computation in spin systems with long-range interactions. Our approach relies on the local addressing of single spins with external fields parametrized by Walsh functions. This enables a mapping from a class of target Hamiltonians, defined by the graph structure of their interactions, to pulse sequences. We then obtain a recipe to implement arbitrary two-body Hamiltonians and universal quantum circuits. Performance guarantees are provided in terms of bounds on Trotter errors and total number of pulses. Additionally, Walsh pulse sequences are shown to be robust against various types of pulse errors, in contrast to previous hybrid digital-analog schemes of quantum computation. We demonstrate and numerically benchmark our protocol with examples from the dynamics of spin models, quantum error correction and quantum optimization algorithms.

Accurate Simulation of Efimov Physics in Ultracold Atomic Gases with Realistic Three-Body Multichannel Interactions

We give a detailed and self-contained description of a recently developed theoretical and numerical method for the simulation of three identical bosonic alkali-metal atoms near a Feshbach resonance, where the Efimov effect is induced. The method is based on a direct construction of the off-shell two-body transition matrix from exact eigenfunctions of the embedded two-body Hamiltonians, obtained using realistic parameterizations of the interaction potentials which accurately reproduce the molecular energy levels. The transition matrix is then inserted into the appropriate three-body integral equations, which may be efficiently solved on a computer. We focus especially on the power of our method in including rigorously the effects of multichannel physics on the three-body problem, which are usually accounted for only by various approximations. We demonstrate the method for 7Li, where we recently showed that a correct inclusion of this multichannel physics resolves the long-standing disagreement between theory and experiment regarding the Efimovian three-body parameter. We analyze the Efimovian enhancement of the three-body recombination rate on both sides of the Feshbach resonance, revealing strong sensitivity to the spin structure of the model thus indicating the prevalence of three-body spin-exchange physics. Finally, we discuss an extension of our methodology to the calculation of three-body bound-state energies.

Hamiltonian quantum generative adversarial networks

We propose Hamiltonian quantum generative adversarial networks (HQuGANs) to learn to generate unknown input quantum states using two competing quantum optimal controls. The game-theoretic framework of the algorithm is inspired by the success of classical generative adversarial networks in learning high-dimensional distributions. The quantum optimal control approach not only makes the algorithm naturally adaptable to the experimental constraints of near-term hardware, but also offers a more natural characterization of overparameterization compared to the circuit model. We numerically demonstrate the capabilities of the proposed framework to learn various highly entangled many-body quantum states, using simple two-body Hamiltonians and under experimentally relevant constraints such as low-bandwidth controls. We analyze the computational cost of implementing HQuGANs on quantum computers and show how the framework can be extended to learn quantum dynamics. Furthermore, we introduce a cost function that circumvents the problem of mode collapse that prevents convergence of HQuGANs and demonstrate how to accelerate the convergence of them when generating a pure state. Published by the American Physical Society 2024

Digital-analog quantum computation with arbitrary two-body Hamiltonians

Digital-analog quantum computation with arbitrary two-body Hamiltonians

Binary Dynamics through the Fifth Power of Spin at O(G^{2}).

We use a previously developed scattering-amplitudes-based framework for determining two-body Hamiltonians for generic binary systems with arbitrary spin S. By construction this formalism bypasses difficulties with unphysical singularities or higher-time derivatives. This framework has been previously used to obtain the exact velocity dependence of the O(G^{2}) quadratic-in-spin two-body Hamiltonian. We first evaluate the S^{3} scattering angle and two-body Hamiltonian at this order in G, including not only all operators corresponding to the usual worldline operators, but also an additional set due to an interesting subtlety. We then evaluate S^{4} and S^{5} contributions at O(G^{2}) which we confirm by comparing against aligned-spin results. We conjecture that a certain shift symmetry together with a constraint on the high-energy growth of the scattering amplitude specify the Wilson coefficients for the Kerr black hole to all orders in the spin and confirm that they reproduce the previously obtained results through S^{4}.

Conservative Binary Dynamics with a Spinning Black Hole at O(G^{3}) from Scattering Amplitudes.

We compute the conservative two-body Hamiltonian of a compact binary system with a spinning black hole through O(G^{3}) to all orders in velocity, including linear and quadratic spin terms. To obtain our results we calculate the classical limit of the two-loop amplitude for the scattering of a massive scalar particle with a massive spin-1 particle minimally coupled to gravity. We employ modern scattering amplitude and loop integration techniques, in particular numerical unitarity, integration-by-parts identities, and the method of regions. The conservative potential in terms of rest-frame spin vectors is extracted by matching to a nonrelativistic effective field theory. We also apply the Kosower-Maybee-O'Connell (KMOC) formalism to calculate the impulse in the covariant spin formalism directly from the amplitude. We work systematically in conventional dimensional regularization and explicitly evaluate all divergent integrals that appear in full- and effective-theory amplitudes, as well as in the phase-space integrals that arise in the KMOC formalism.

Extending the VDPC+BCS formalism by including three-body forces* *Supported by the National Natural Science Foundation of China (11405109)

Recently, Jia proposed a formalism to apply the variational principle to a coherent-pair condensate for a two-body Hamiltonian. The present study extends this formalism by including three-body forces. The result is the same as the so-called variation after particle-number projection in the BCS case, but now, the particle number is always conserved, and the time-consuming projection is avoided. Specifically, analytical formulas of the average energy are derived along with its gradient for a three-body Hamiltonian in terms of the coherent-pair structure. Gradient vanishment is required to obtain analytical expressions for the pair structure at the energy minimum. The new algorithm iterates on these pair-structure expressions to minimize energy for a three-body Hamiltonian. The new code is numerically demonstrated when applied to realistic two-body forces and random three-body forces in large model spaces. The average energy can be minimized to practically any arbitrary precision.

Analytic exposition of the graviton modes in fractional quantum Hall effects and its physical implications

Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this work, we derived a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (nullspaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin $\nu = 1/5$ and the Gaffnian $\nu = 2/5$ phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a "phase transition" even when the ground state is invariant. We discuss the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the "hollow-core" modes, as well as their experimental consequences.

Chaos in the Bose–Hubbard model and random two-body Hamiltonians

We investigate the chaotic phase of the Bose–Hubbard model (Pausch et al 2021 Phys. Rev. Lett. 126 150601) in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose–Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.

Classical observables from coherent-spin amplitudes

The quantum field-theoretic approach to classical observables due to Kosower, Maybee and O’Connell provides a rigorous pathway from on-shell scattering amplitudes to classical perturbation theory. In this paper, we promote this formalism to describe general classical spinning objects by using coherent spin states. Our approach is fully covariant with respect to the massive little group SU(2) and is therefore completely synergistic with the massive spinor-helicity formalism. We apply this approach to classical two-body scattering due gravitational interaction. Starting from the coherent-spin elastic-scattering amplitude, we derive the classical impulse and spin kick observables to first post-Minkowskian order but to all orders in the angular momenta of the massive spinning objects. From the same amplitude, we also extract an effective two-body Hamiltonian, which can be used beyond the scattering setting. As a cross-check, we rederive the classical observables in the center-of-mass frame by integrating the Hamiltonian equations of motion to the leading order in Newton’s constant.

Leading nonlinear tidal effects and scattering amplitudes

We present the two-body Hamiltonian and associated eikonal phase, to leading post-Minkowskian order, for infinitely many tidal deformations described by operators with arbitrary powers of the curvature tensor. Scattering amplitudes in momentum and position space provide systematic complementary approaches. For the tidal operators quadratic in curvature, which describe the linear response to an external gravitational field, we work out the leading post-Minkowskian contributions using a basis of operators with arbitrary numbers of derivatives which are in one-to-one correspondence with the worldline multipole operators. Explicit examples are used to show that the same techniques apply to both bodies interacting tidally with a spinning particle, for which we find the leading contributions from quadratic in curvature tidal operators with an arbitrary number of derivatives, and to effective field theory extensions of general relativity. We also note that the leading post-Minkowskian order contributions from higher-dimension operators manifest double-copy relations. Finally, we comment on the structure of higher-order corrections.

Tidal effects for spinning particles

Expanding on the recent derivation of tidal actions for scalar particles, we present here the action for a tidally deformed spin-1/2 particle. Focusing on operators containing two powers of the Weyl tensor, we combine the Hilbert series with an on-shell amplitude basis to construct the tidal action. With the tidal action in hand, we compute the leading-post-Minkowskian tidal contributions to the spin-1/2–spin-1/2 amplitude, arising at mathcal{O} (G2). Our amplitudes provide evidence that the observed long range spin-universality for the scattering of two point particles extends to the scattering of tidally deformed objects. From the scattering amplitude we find the conservative two-body Hamiltonian, linear and angular impulses, eikonal phase, spin kick, and aligned-spin scattering angle. We present analogous results in the electromagnetic case along the way.

In-medium T matrix in case of pairing instability

The diagonalization technique for two-body propagator in nuclear matter is generalized by inclusion of the two-hole continuum. In the approach, the total effective two-body Hamiltonian is introduced which allows us to find the complete propagator as well as the reaction $T$ matrix at many energies and relative momenta simultaneously from a single diagonalization procedure for the Hamiltonian matrix. Explicit treatment of bound and continuum states allows us to determine forward- and backward-propagating parts of the $T$ matrix in a closed form which simplifies strongly the evaluation of single-particle and single-hole self-energies. Special attention is paid to the bound-state contribution to the reaction matrix in case of pairing instability when complex conjugated eigenvalues of the effective Hamiltonian appear. A generalization of the approach to calculation of pairing gaps in case of nonzero center-of-mass momentum of the pair is also presented.

Exact three-colored quantum scars from geometric frustration

Non-equilibrium properties of quantum materials present many intriguing properties, among them athermal behavior, which violates the eigenstate thermalization hypothesis. Such behavior has primarily been observed in disordered systems. More recently, experimental and theoretical evidence for athermal eigenstates, known as "quantum scars" has emerged in non-integrable disorder-free models in one dimension with constrained dynamics. In this work, we show the existence of quantum scar eigenstates and investigate their dynamical properties in many simple two-body Hamiltonians with "staggered" interactions, involving ferromagnetic and antiferromagnetic motifs, in arbitrary dimensions. These magnetic models include simple modifications of widely studied ones (e.g., the XXZ model) on a variety of frustrated and unfrustrated lattices. We demonstrate our ideas by focusing on the two dimensional frustrated spin-1/2 kagome antiferromagnet, which was previously shown to harbor a special exactly solvable point with "three-coloring" ground states in its phase diagram. For appropriately chosen initial product states -- for example, those which correspond to any state of valid three-colors -- we show the presence of robust quantum revivals, which survive the addition of anisotropic terms. We also suggest avenues for future experiments which may see this effect in real materials.

Local Two-Body Parent Hamiltonians for the Entire Jain Sequence.

Using an algebra of second-quantized operators, we develop local two-body parent Hamiltonians for all unprojected Jain states at filling factor n/(2np+1), with integer n and (half-)integer p. We rigorously establish that these states are uniquely stabilized and that zero mode counting reproduces mode counting in the associated edge conformal field theory. We further establish the organizing "entangled Pauli principle" behind the resulting zero mode paradigm and unveil an emergent SU(n) symmetry characteristic of the fixed point physics of the Jain quantum Hall fluid.

Explicit properties of the simplest inhomogeneous matrix-product-state including the Riemann metric of the MPS manifold

We consider the simplest inhomogeneous matrix-product-state for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site k are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix of the spin k alone. The two angles associated to the bond parametrize the entanglement properties of the Schmidt decomposition across the bond . Explicit results are given for the reduced density matrix of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix of two sites at distance r that is needed to evaluate the spin–spin correlations at distance r. The global structure of the MPS manifold as parametrized by these angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.

Consistent large-scale shell-model analysis of the two-neutrino ββ and single β branchings in 48Ca and 96Zr

Two-neutrino double-beta-decay matrix elements M2ν and single beta-decay branching ratios were calculated for 48Ca and 96Zr in the interacting nuclear shell model using large single-particle valence spaces with well-tested two-body Hamiltonians. For 48Ca the matrix element M2ν=0.0511 is obtained, which is 5.5% smaller than the previously reported value of 0.0539. For 96Zr this work reports the first large-scale shell-model calculation of the nuclear matrix element, yielding a value M2ν=0.0747 with extreme single-state dominance. These matrix elements, combined with the available ββ-decay half-life data, yield effective values of the weak axial coupling which in turn are used to produce in a consistent way the β-decay branching ratios of (7.5±2.8)% for 48Ca and (18.4±0.09)% for 96Zr. These are larger than obtained in previous studies, implying that the detection of the β-decay branches could be possible in dedicated experiments sometime in the (near) future.

On the virtual level of two-body interactions and applications to three-body systems in higher dimensions

We consider a system of three particles in dimension d = 4 interacting via short-range potentials, where the two-body Hamiltonians have a resonance at the bottom of the essential spectrum. We prove that the three-body Hamiltonian can only admit a finite number of eigenvalues. This is in contrast to the case of three-dimensional bosons and two-dimensional fermions, where the existence of resonances leads to the infiniteness of the discrete spectrum.

Black hole binary dynamics from the double copy and effective theory

We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.

Projected shell model analysis of structural evolution and chaoticity in fast-rotating nuclei

Rotationally-induced chaotic motion in fast-rotating nuclei is investigated using the projected shell model with an aggressively extended multi-quasiparticle configuration space. The two-body Hamiltonian in separable forms with quadrupole–quadrupole plus pairing forces is diagonalized in superimposed states with more than 2000 angular momentum projected configurations. By taking 164Yb as an example, it is shown that the experimental energies and moment of inertia (MoI) for the yrast band can be clearly described. The variations in MoI are explained in terms of band crossings among the 2-quasiparticle (qp), 4-qp, 6-qp, and 8-qp bands, corresponding to successive pair-breakings caused by rotation. Moreover, the chaotic motion in different spin and excitation regions is studied by quantitatively analyzing the branching number. It is found that the degree of chaoticity increases monotonically with spin and excitation energy. At the highest spins, bands built by different numbers of quasiparticles, with their own characteristic rotational behavior originally at low spins, tend to rotate uniformly with the same rotational frequency, indicating an emergent nuclear collectivity appearing from an extremely chaotic system. The important role played by nucleon pairing in increasing the chaoticity is emphasized.