Many-body States Research Articles

Exact eigenstates of multicomponent Hubbard models: SU( N ) magnetic η pairing, weak ergodicity breaking, and partial integrability

We construct exact eigenstates of multicomponent Hubbard models in arbitrary dimensions by generalizing the η-pairing mechanism. Our models include the SU(N) Hubbard model as a special case. Unlike the conventional two-component case, the generalized η-pairing mechanism permits the construction of eigenstates that feature off-diagonal long-range order and magnetic long-range order. These states form fragmented fermionic condensates due to a simultaneous condensation of multicomponent η pairs. While the η-pairing states in the SU(2) Hubbard model are based on the η-pairing symmetry, the exact eigenstates in the N-component system with N≥3 arise not from symmetry of the Hamiltonian but from a spectrum generating algebra defined in a restricted Hilbert space. We exploit this fact to show that the generalized η-pairing eigenstates do not satisfy the eigenstate thermalization hypothesis and serve as quantum many-body scar states. This result indicates a weak breakdown of ergodicity in the N-component Hubbard models for N≥3. Furthermore, we show that these exact eigenstates constitute integrable subsectors in which the Hubbard Hamiltonian effectively reduces to a noninteracting model. This partial integrability causes various multicomponent Hubbard models to weakly break ergodicity. We propose a method of harnessing dissipation to distill the integrable part of the dynamics and elucidate a mechanism of nonthermalization caused by dissipation. This work establishes the coexistence of off-diagonal long-range order and SU(N) magnetism in excited eigenstates of the multicomponent Hubbard models, which presents a possibility of novel out-of-equilibrium pairing states of multicomponent fermions. These models unveil a unique feature of quantum thermalization of multicomponent fermions, which can experimentally be tested with cold-atom quantum simulators. Published by the American Physical Society 2024

State preparation by shallow circuits using feed forward

Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, constant-depth, quantum circuits that perform 1 qubit gates and nearest−neighbor 2 qubit gates.We introduce a new computational model called Local Alternating Quantum Classical Computations(LAQCC). In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent LAQCC circuit, and that any LAQCC circuit can be simulated by a QNC1circuit. Next, we conjecture the non-simulatability of LAQCC by showing that LAQCC contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any LAQCC circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting LAQCC implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the W-state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of n-bit strings of Hamming weight k, for k=O(n), but fails for higher k due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger k requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.

High temperature condensation and many body localization of blended ferrons in antiferromagnetic clusters

Abstract This paper presents a multiferron system and analyzes the conditions under which it forms a localized many-body state. We determined the self-trapping potential of this system and showed that the ferrons pair up to form bound biferron states, which interact through their relative spin cloud and settle into two different configurations mimicking the Bose-Einstein Condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) states. The external B-field strengthens the magnetic coupling, which favors the pairing of the ferrons. It also creates a vortex, providing good confinement of the particles and favoring condensation. The system's state functions are well described by the nonlinear Schr"{o}dinger equation, whose solution is a bell-shaped soliton representing the density of particle distribution in the condensate. The system's critical temperature is obtained as a function of the biferron energy shift and thus strongly depends on the applied field.

Interaction Between Gravitational Waves and Trapped Bose–Einstein Condensates

Inspired by recent proposals for detecting gravitational waves by using Bose–Einstein condensates (BECs), we investigated the interplay between these two phenomena. A gravitational wave induces a phase shift in the fidelity amplitude of the many-body quantum state. We investigated the enhancement of the phase shift in the case of Bose condensates confined by an anisotropic harmonic potential, considering both ideal and interacting BECs.

Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers

Abstract Interpreting the representation and generalization powers has been a long-standing issue in the field of machine learning (ML) and artificial intelligence. This work contributes to uncovering the emergence of universal scaling laws in quantum-probabilistic ML. We take the generative tensor network (GTN) in the form of a matrix product state as an example and show that with an untrained GTN (such as a random TN state), the negative logarithmic likelihood (NLL) $L$ generally increases linearly with the number of features $M$, i.e., $L \simeq k M + const$. This is a consequence of the so-called “catastrophe of orthogonality,” which states that quantum many-body states tend to become exponentially orthogonal to each other as $M$ increases. We reveal that while gaining information through training, the linear scaling law is suppressed by a negative quadratic correction, leading to $L \simeq \beta M - \alpha M^2 + const$. The scaling coefficients exhibit logarithmic relationships with the number of training samples and the number of quantum channels $\chi$. The emergence of the quadratic correction term in NLL for the testing (training) set can be regarded as evidence of the generalization (representation) power of GTN. Over-parameterization can be identified by the deviation in the values of $\alpha$ between training and testing sets while increasing $\chi$. We further investigate how orthogonality in the quantum feature map relates to the satisfaction of quantum probabilistic interpretation, as well as to the representation and generalization powers of GTN. The unveiling of universal scaling laws in quantum-probabilistic ML would be a valuable step toward establishing a white-box ML scheme interpreted within the quantum probabilistic framework.

Enhancing some separability criteria via equiangular tight frames

Abstract Improving the detection ability of quantum entanglement is one of the es-
sential tasks in quantum computing and quantum information. Finite tight frames
play a fundamental role in a wide variety of areas, and generally, each application
requires a specific class of frames and is closely related to quantum measurement. It
is worth noting that a maximal set of complex equiangular vectors is closely related
to a symmetric informationally complete measurement. Hence, our goal in this work
is to propose a series of separability criteria assigned to a finite tight frame and some
well-known inequalities in different quantum systems, respectively. In addition, some
tighter criteria to detect entanglement for many-body quantum states are presented in
arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detec-
tion criteria is illustrated through some detailed examples.

Evidence of Zero-Field Wigner Solids in Ultrathin Films of Cadmium Arsenide

The quantum Wigner crystal is a many-body state where Coulombic repulsion quenches the kinetic energy of electrons, causing them to crystallize into a lattice. Experimental realization of a quantum Wigner crystal at zero magnetic field has been a long-sought goal. Here, we report on the experimental evidence of a Wigner solid in ultra-thin films of cadmium arsenide (Cd3As2) at zero magnetic field. We show that a finite bias depins the domains and produces an unusually sharp-threshold current-voltage behavior. Hysteresis and voltage fluctuations point to domain motion across the pinning potential and disappear at finite temperature as thermal fluctuations overcome the potential. The application of a small magnetic field destroys the Wigner solid, pointing to an unconventional origin. We use Landau-level spectroscopy to show that the formation of the Wigner solid is closely connected to a topological transition as the film thickness is reduced. Published by the American Physical Society 2024

Dual-Isometric Projected Entangled Pair States.

Efficient characterization of higher dimensional many-body physical states presents significant challenges. In this Letter, we propose a new class of projected entangled pair states (PEPS) that incorporates two isometric conditions. This new class facilitates the efficient calculation of general local observables and certain two-point correlation functions, which have been previously shown to be intractable for general PEPS, or PEPS with only a single isometric constraint. Despite incorporating two isometric conditions, our class preserves the rich physical structure while enhancing the analytical capabilities. It features a large set of tunable parameters, with only a subleading correction compared to that of general PEPS. Furthermore, we analytically demonstrate that this class can encode universal quantum computation and can represent a transition from topological to trivial order.

Quantum many-body scars in few-body dipole-dipole interactions

We simulate the dynamics of Rydberg atoms resonantly exchanging energy via two-, three-, and four-body dipole-dipole interactions in a one-dimensional array. Using simplified models of a realistic experimental system, we study the initial-state survival probability, mean level spacing, spread of entanglement, and properties of the energy eigenstates. By exploring a range of disorders and interaction strengths, we find regions in parameter space where the three- and four-body dynamics either fail to thermalize or do so slowly. The interplay between the stronger hopping and weaker field-tuned interactions gives rise to quantum many-body scar states, which play a critical role in slowing the dynamics of the three- and four-body interactions. Published by the American Physical Society 2024

Covariant Representation of Spin and Entanglement—A Review and Reformulation

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.

Atomistic first-principles modeling of single donor spin-qubit

Using an impurity atom in crystal silicon as a spin-1/2 qubit has been made experimentally possible recently where the impurity atom acts as a quantum dot (QD). Quantum transport in and out of such a donor QD occurs in the sequential tunneling regime where a physical quantity of importance is the charging (addition) energy, which measures the energy necessary for adding an electron into the donor QD. In this work, we present a first-principles method to quantitatively predict the addition energy of the donor QD. Using density functional theory (DFT), we determine the impurity states that serve as the basis set for subsequent exact diagonalization calculation of the many-body states and energies of the donor QD. Due to the large effective Bohr radius of the conduction electrons in Si, very large supercells containing more than 10 000 atoms must be used to obtain accurate results. For the donor QD of a phosphorus impurity in bulk Si, the combined DFT and exact diagonalization predicts the first addition energy to be 53 meV, in good agreement with the corresponding experimental value. For the donor QD of an arsenic impurity in Si, the first addition energy is predicted to be 44.2 meV. The calculated many-body wave functions provide a vivid electronic picture of the donor QD.

Feedback-based quantum algorithm inspired by counterdiabatic driving

In recent quantum algorithmic developments, a feedback-based approach has shown promise for preparing quantum many-body system ground states and solving combinatorial optimization problems. This method utilizes quantum Lyapunov control to iteratively construct quantum circuits. Here, we propose a substantial enhancement by implementing a protocol that uses ideas from quantum Lyapunov control and the counterdiabatic driving protocol, a key concept from quantum adiabaticity. Our approach introduces an additional control field inspired by counterdiabatic driving. We apply our algorithm to prepare ground states in one-dimensional quantum Ising spin chains. Comprehensive simulations demonstrate a remarkable acceleration in population transfer to low-energy states within a significantly reduced time frame compared to conventional feedback-based quantum algorithms. This acceleration translates to a reduced quantum circuit depth, a critical metric for potential quantum computer implementation. We validate our algorithm on the IBM cloud computer, highlighting its efficacy in expediting quantum computations for many-body systems and combinatorial optimization problems. Published by the American Physical Society 2024

Realizing multiband states with ultracold dipolar quantum simulators

The manipulation of dipolar interactions within ultracold molecular ensembles represents a pivotal advancement in experimental physics, aiming at the emulation of quantum phenomena unattainable through mere contact interactions. Our study uncovers regimes of multiband occupation which allow to probe more realistic, complex long-range interacting lattice models with ultracold dipolar simulators. By mapping out experimentally relevant ranges of potential depths, interaction strengths, particle fillings, and geometric configurations, we calculate the agreement between the state prepared in the quantum simulator and a target lattice state. We do so by separately calculating numerically exact many-body wave functions in the continuous space and single- or multiband lattice representations, and building their many-body state overlaps. Our findings reveal that for shallow lattices and stronger interactions above half filling, multiband population increases, resulting in fundamentally different ground states than the ones observed in simple lowest-band descriptions, e.g., striped vs checkerboard states. A wide range of probed parameter regimes in its turn provides a systematic and quantitative blueprint for realizing multiband states with two-dimensional quantum simulators employing ultracold dipolar molecules. Published by the American Physical Society 2024

Many-body localized hidden generative models

Quantum generative models hold the promise of accelerating or improving machine learning tasks by leveraging the probabilistic nature of quantum states, but the successful optimization of these models remains a difficult challenge. To tackle this challenge, we present a new architecture for quantum generative modeling that combines insights from classical machine learning and quantum phases of matter. In particular, our model utilizes both many-body localized (MBL) dynamics and hidden units to improve the optimization of the model. We demonstrate the applicability of our model on a diverse set of classical and quantum tasks, including a toy version of MNIST handwritten digits, quantum data obtained from quantum many-body states, and nonlocal parity data. Our architecture and algorithm provide novel strategies of utilizing quantum many-body systems as learning resources and reveal a powerful connection between disorder, interaction, and learning in quantum many-body systems. Published by the American Physical Society 2024

Proposal for simulating quantum spin models with the Dzyaloshinskii-Moriya interaction using Rydberg atoms and the construction of asymptotic quantum many-body scar states

Proposal for simulating quantum spin models with the Dzyaloshinskii-Moriya interaction using Rydberg atoms and the construction of asymptotic quantum many-body scar states

Learning quantum properties from short-range correlations using multi-task networks

Characterizing multipartite quantum systems is crucial for quantum computing and many-body physics. The problem, however, becomes challenging when the system size is large and the properties of interest involve correlations among a large number of particles. Here we introduce a neural network model that can predict various quantum properties of many-body quantum states with constant correlation length, using only measurement data from a small number of neighboring sites. The model is based on the technique of multi-task learning, which we show to offer several advantages over traditional single-task approaches. Through numerical experiments, we show that multi-task learning can be applied to sufficiently regular states to predict global properties, like string order parameters, from the observation of short-range correlations, and to distinguish between quantum phases that cannot be distinguished by single-task networks. Remarkably, our model appears to be able to transfer information learnt from lower dimensional quantum systems to higher dimensional ones, and to make accurate predictions for Hamiltonians that were not seen in the training.

Construction of Continuous Collective Energy Landscapes for Large Amplitude Nuclear Many-Body Problems.

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.

Phase transitions in quantum many-body scars

We propose a type of phase transition in quantum many-body systems that occurs in highly excited quantum many-body scar states while most of the spectrum is largely unaffected. Such scar-state phase transitions can be realized by embedding a matrix product state, known to undergo a phase transition, as a scar state into the thermal spectrum of a parent Hamiltonian. We find numerically that the mechanism for the scar-state phase transition involves the formation or presence of low-entropy states at energies similar to the scar state in the vicinity of the phase transition point. Published by the American Physical Society 2024

Specialising neural-network quantum states for the Bose Hubbard model

Abstract Projected variational wavefunctions such as the Gutzwiller, many-body correlator and Jastrow ansatzes have provided crucial insight into the nature of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two or more spatial dimensions. However, these ansatzes have no obvious tractable and systematic way of being improved. A promising alternative is to use Neural-network quantum states (NQS) based on Restricted Boltzmann Machines (RBMs). With binary visible and hidden units NQS have proven to be a highly effective at describing quantum states of interacting spin- 1 2 lattice systems. The application of NQS to bosonic systems has so far been based on one-hot encoding from machine learning where the multi-valued site occupation is distributed across several binary-valued visible units of an RBM. Compared to spin- 1 2 systems one-hot encoding greatly increases the number of variational parameters whilst also making their physical interpretation opaque. Here we revisit the construction of NQS for bosonic systems by reformulating a one-hot encoded RBM into a correlation operator applied to a reference state, analogous to the structure of the projected variational ansatzes. In this form we then propose a number of specialisations of the RBM motivated by the physics of the BHM and the ability to capture exactly the projected variational ansatzes. We analyse in detail the variational performance of these new RBM variants for a 10 × 10 BHM, using both a standard Bose condensate state and a pre-optimised Jastrow + many-body correlator state as the reference state of the calculation. Several of our new ansatzes give robust results as nearly good as one-hot encoding across the regimes of the BHM, but at a substantially reduced cost. Such specialised NQS are thus primed tackle bosonic lattice problems beyond the accuracy of classic variational wavefunctions.

Rapidly rotating quantum droplets confined in a harmonic potential

We consider a “symmetric” quantum droplet in two spatial dimensions, which rotates in a harmonic potential, focusing mostly on the limit of “rapid” rotation. We examine this problem using a purely numerical approach, as well as a semianalytic Wigner-Seitz approximation (first developed by Baym, Pethick, and their co-workers) for the description of the state with a vortex lattice. Within this approximation we assume that each vortex occupies a cylindrical cell, with the vortex-core size treated as a variational parameter. Working with a fixed angular momentum, as the angular momentum increases and depending on the atom number, the droplet accommodates none, few, or many vortices, before it turns to center-of-mass excitation. For the case of a “large” droplet, working with a fixed rotational frequency of the trap Ω, as Ω approaches the trap frequency ω, a vortex lattice forms, the number of vortices increases, the mean spacing between them decreases, while the “size” of each vortex increases as compared to the size of each cell. In contrast to the well-known problem of contact interactions, where we have melting of the vortex lattice and highly correlated many-body states, here no melting of the vortex lattice is present, even when Ω=ω. This difference is due to the fact that the droplet is self-bound. For Ω=ω, the “smoothed” density distribution becomes a flat top, very much like the static unconfined droplet. When Ω exceeds ω, the droplet maintains its shape and escapes to infinity, via center-of-mass motion. Published by the American Physical Society 2024