Quantum Hilbert Space Research Articles

Robust finite-temperature many-body scarring on a quantum computer

Mechanisms for suppressing thermalization in disorder-free many-body systems, such as Hilbert space fragmentation and quantum many-body scars, have recently attracted much interest in foundations of quantum statistical physics and potential quantum information processing applications. However, their sensitivity to realistic effects such as finite temperature remains largely unexplored. Here, we have utilized IBM's Kolkata quantum processor to demonstrate an unexpected robustness of quantum many-body scars at finite temperatures when the system is prepared in a thermal Gibbs ensemble. We identify such robustness in the PXP model, which describes quantum many-body scars in experimental systems of Rydberg atom arrays and ultracold atoms in tilted Bose-Hubbard optical lattices. By contrast, other theoretical models which host exact quantum many-body scars are found to lack such robustness and their scarring properties quickly decay with temperature. Our study sheds light on the important differences between scarred models in terms of their algebraic structures, which impacts their resilience to finite temperature. Published by the American Physical Society 2024

A quantum mechanical example for Hodge theory

On the basis of (i) the discrete and continuous symmetries (and corresponding conserved charges), (ii) the ensuing algebraic structures of the symmetry operators and conserved charges, and (iii) a few basic concepts behind the subject of differential geometry, we show that the celebrated Friedberg-Lee-Pang-Ren (FLPR) quantum mechanical model (describing the motion of a single non-relativistic particle of unit mass under the influence of a general spatial 2D rotationally invariant potential) provides a tractable physical example for the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the symmetry operators and conserved charges lead to the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. We concisely mention the Hodge decomposition theorem in the quantum Hilbert space of states and choose the harmonic states as the real physical states of our theory. We discuss the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST and (anti-)co-BRST charges and the physical consequences that ensue from them.

QPEN: Quantum Projection and Quantum Entanglement Enhanced Network for Cross-Lingual Aspect-Based Sentiment Analysis

Aspect-based sentiment analysis (ABSA) has attracted much attention due to its wide application scenarios. Most previous studies have focused solely on monolingual ABSA, posing a formidable challenge when extending ABSA applications to multilingual scenarios. In this paper, we study upgrading monolingual ABSA to cross-lingual ABSA. Existing methods usually exploit pre-trained cross-lingual language to model cross-lingual ABSA, and enhance the model with translation data. However, the low-resource languages might be under-represented during the pre-training phase, and the translation-enhanced methods heavily rely on the quality of the translation and label projection. Inspired by the observation that quantum entanglement can correlate multiple single systems, we map the monolingual expression to the quantum Hilbert space as a single quantum system, and then utilize quantum entanglement and quantum measurement to achieve cross-lingual ABSA. Specifically, we propose a novel quantum neural model named QPEN (short for quantum projection and quantum entanglement enhanced network). It is equipped with a proposed quantum projection module that projects aspects as quantum superposition on a complex-valued Hilbert space. Furthermore, a quantum entanglement module is proposed in QPEN to share language-specific features between different languages without transmission. We conducted simulation experiments on the classical computer, and experimental results on SemEval-2016 dataset demonstrate that our method achieves state-of-the-art performance in terms of F1-scores for five languages.

Null Hamiltonian Yang–Mills theory: Soft Symmetries and Memory as Superselection

AbstractSoft symmetries for Yang–Mills theory are shown to correspond to the residual Hamiltonian action of the gauge group on the Ashtekar–Streubel phase space, which is the result of a partial symplectic reduction. The associated momentum map is the electromagnetic memory in the Abelian theory, or a nonlinear, gauge-equivariant, generalisation thereof in the non-Abelian case. This result follows from an application of Hamiltonian reduction by stages, enabled by the existence of a natural normal subgroup of the gauge group on a null codimension-1 submanifold with boundaries. The first stage is coisotropic reduction of the Gauss constraint, and it yields a symplectic extension of the Ashtekar–Streubel phase space (up to a covering). Hamiltonian reduction of the residual gauge action leads to the fully reduced phase space of the theory. This is a Poisson manifold, whose symplectic leaves, called superselection sectors, are labelled by the (gauge classes of the generalised) electric flux across the boundary. In this framework, the Ashtekar–Streubel phase space arises as an intermediate reduction stage that enforces the superselection of the electric flux at only one of the two boundary components. These results provide a natural, purely Hamiltonian, explanation of the existence of soft symmetries as a byproduct of partial symplectic reduction, as well as a motivation for the expected decomposition of the quantum Hilbert space of states into irreducible representations labelled by the Casimirs of the Poisson structure on the reduced phase space.

Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models

Quantum kinetically constrained models have recently attracted significant attention due to their anomalous dynamics and thermalization. In this work, we introduce a hitherto unexplored family of kinetically constrained models featuring conserved particle number and strong inversion-symmetry breaking due to facilitated hopping. We demonstrate that these models provide a generic example of so-called quantum Hilbert space fragmentation, that is manifested in disconnected sectors in the Hilbert space that are not apparent in the computational basis. Quantum Hilbert space fragmentation leads to an exponential in system size number of eigenstates with exactly zero entanglement entropy across several bipartite cuts. These eigenstates can be probed dynamically using quenches from simple initial product states. In addition, we study the particle spreading under unitary dynamics launched from the domain wall state, and find faster than diffusive dynamics at high particle densities, that crosses over into logarithmically slow relaxation at smaller densities. Using a classically simulable cellular automaton, we reproduce the logarithmic dynamics observed in the quantum case. Our work suggests that particle conserving constrained models with inversion symmetry breaking realize so far unexplored dynamical behavior and invite their further theoretical and experimental studies.

A hybrid quantum–classical neural network for learning transferable visual representation

State-of-the-art quantum machine learning (QML) algorithms fail to offer practical advantages over their notoriously powerful classical counterparts, due to the limited learning capabilities of QML algorithms, the constrained computational resources available on today’s noisy intermediate-scale quantum (NISQ) devices, and the empirically designed circuit ansatz for QML models. In this work, we address these challenges by proposing a hybrid quantum–classical neural network (CaNN), which we call QCLIP, for Quantum Contrastive Language-Image Pre-Training. Rather than training a supervised QML model to predict human annotations, QCLIP focuses on more practical transferable visual representation learning, where the developed model can be generalized to work on unseen downstream datasets. QCLIP is implemented by using CaNNs to generate low-dimensional data feature embeddings followed by quantum neural networks to adapt and generalize the learned representation in the quantum Hilbert space. Experimental results show that the hybrid QCLIP model can be efficiently trained for representation learning. We evaluate the representation transfer capability of QCLIP against the classical Contrastive Language-Image Pre-Training model on various datasets. Simulation results and real-device results on NISQ IBM_Auckland quantum computer both show that the proposed QCLIP model outperforms the classical CLIP model in all test cases. As the field of QML on NISQ devices is continually evolving, we anticipate that this work will serve as a valuable foundation for future research and advancements in this promising area.

Origin of Hilbert-space quantum scars in unconstrained models

Origin of Hilbert-space quantum scars in unconstrained models

A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators

Abstract In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems.

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

RR-BS correspondence—a localization phenomenon of the index in geometric quantization

In the Kostant-Souriau formulation of geometric quantization, one needs a supplemental structure, so called a polarization, in order to construct a quantum Hilbert space. So, it is natural to ask whether the obtained quantum Hilbert space depends on the choice of polarizations. This is a survey article for recent progress in the study of this problem and its generalization.

Quantum interference on the non-commutative plane and the quantum-to-classical transition

We explore a possible link between the structure of space at short length scales and the emergence of classical phenomena at macroscopic scales. To this end we adopt the paradigm of non-commutative space at short length scales and explicitly compute the outcomes of a double slit experiment and a von Neumann measurement in the non-commutative plane. A very consistent picture of a continuous quantum-to-classical transition emerges. The mechanism driving this transition is standard decoherence, which arises quite naturally from the tensor product structure of the non-commutative quantum Hilbert space. The double slit calculation enables us to establish a lower bound on the non-commutative parameter for this transition to become effective at particle numbers of the order of Avogadro’s number. Similarly, the result of the von Neumann measurement establishes a criterion involving the non-commutative parameter, apparatus size and coupling between system and apparatus for classicality to emerge.

Dynamical emergence of SUq(2) from the regularization of (2+1)D gravity with a cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.

Local Quantum Theory with Fluids in Space-Time

In 1948, Schwinger developed a local Lorentz-covariant formulation of relativistic quantum electrodynamics in space-time which is fundamentally inconsistent with any delocalized interpretation of quantum mechanics. An interpretation compatible with Schwinger’s theory is presented, which reproduces all of the standard empirical predictions of conventional delocalized quantum theory in configuration space. This is an explicit, unambiguous, and Lorentz-covariant “local hidden variable theory” in space-time, whose existence proves definitively that such theories are possible. This does not conflict with Bell’s theorem because it is a local many-worlds theory. Each physical system is characterized by a wave-field, which is a set of indexed piece-wise single-particle wavefunctions in space-time, each with its own coefficient, along with a memory which contains the separate local Hilbert-space quantum state at each event in space-time. Each single-particle wavefunction of a fundamental system describes the motion of a portion of a conserved fluid in space-time, with the fluid decomposing into many classical point particles, each following a world-line and recording a local memory. Local interactions between two systems take the form of local boundary conditions between the differently indexed pieces of those systems’ wave-fields, with new indexes encoding each orthogonal outcome of the interaction. The general machinery is introduced, including the local mechanisms for entanglement and interference. The experience of collapse, Born rule probability, and environmental decoherence are discussed, and a number of illustrative examples are given.

Quantum machine learning beyond kernel methods

Machine learning algorithms based on parametrized quantum circuits are prime candidates for near-term applications on noisy quantum computers. In this direction, various types of quantum machine learning models have been introduced and studied extensively. Yet, our understanding of how these models compare, both mutually and to classical models, remains limited. In this work, we identify a constructive framework that captures all standard models based on parametrized quantum circuits: that of linear quantum models. In particular, we show using tools from quantum information theory how data re-uploading circuits, an apparent outlier of this framework, can be efficiently mapped into the simpler picture of linear models in quantum Hilbert spaces. Furthermore, we analyze the experimentally-relevant resource requirements of these models in terms of qubit number and amount of data needed to learn. Based on recent results from classical machine learning, we prove that linear quantum models must utilize exponentially more qubits than data re-uploading models in order to solve certain learning tasks, while kernel methods additionally require exponentially more data points. Our results provide a more comprehensive view of quantum machine learning models as well as insights on the compatibility of different models with NISQ constraints.

A quantum-enhanced support vector machine for galaxy classification

Abstract Galaxy morphology, a key tracer of the evolution of a galaxy’s physical structure, has motivated extensive research on machine learning techniques for efficient and accurate galaxy classification. The emergence of quantum computers has generated optimism about the potential for significantly improving the accuracy of such classifications by leveraging the large dimensionality of quantum Hilbert space. This paper presents a quantum-enhanced support vector machine (SVM) algorithm for classifying galaxies based on their morphology. The algorithm requires the computation of a kernel matrix, a task that is performed on a simulated quantum computer using a quantum circuit conjectured to be intractable on classical computers. The result shows similar performance between classical and quantum-enhanced SVM algorithms. For a training size of 40k, the receiver operating characteristic curve for differentiating ellipticals and spirals has an under-curve area of 0.946 ± 0.005 for both classical and quantum-enhanced algorithms. Additionally, we demonstrate for a small data set that the performance of a noise-mitigated quantum SVM algorithm on a quantum device is in agreement with simulation. Finally, a necessary condition for achieving a potential quantum advantage is presented. This investigation is among the very first applications of quantum machine learning in astronomy and highlights their potential for further application in this field.

Unleashing the Potentials of Quantum Probability Theory for Customer Experience Analytics

In information systems research, the advantages of Customer Experience (CX) and its contribution to organizations are largely recognized. The CX analytics evaluate how customers perceive products, ranging from their functional usage to their cognitive states regarding the product, such as emotions, sentiment, and satisfaction. The most recent research in psychology reveals that cognition analytics research based on Classical Probability Theory (CPT) and statistical learning, which is used to evaluate people’s cognitive states, is limited due to their reliance on rational decision-making. However, the cognitive attitudes of customers are characterized by uncertainty and entanglement, resulting in irrational decision-making bias. What is captured by traditional CPT-based data science in the context of cognition aspects of CX analytics is only a small portion of what should be captured. Current CX analytics efforts fall far short of their full potential. In this paper, we set a novel research direction for CX analytics by Quantum Probability Theory (QPT). QPT-based analytics have been introduced recently in psychology research and reveal better cognition assessment under uncertainty, with a high level of irrational behavior. Adopting recent advances in the psychology domain, this paper develops a vision and sets a research agenda for expanding the application of CX analytics by QPT to overcome CPT shortcomings, identifies research areas that contribute to the vision, and proposes elements of a future research agenda. To stimulate debate and research QPT-CX analytics, we attempt a preliminary characterization of the novel method by introducing a QPT-based rich mathematical framework for CX cognitive modeling based on quantum superposition, Bloch sphere, and Hilbert space. We demonstrate the implementation of the QPT-CX model by the use case of customers’ emotional motivator assessments while implementing quantum vector space with a set of mathematical axioms for CX analytics. Finally, we outline the key advantages of quantum CX over classical by supporting theoretical proof for each key.

On the Role of Inconsistency in Quantum Foundational Debate and Hilbert Space Formulation

This article is intended mainly to develop an expository outline of an inherently inconsistent reasoning in the development of quantum mechanics during 1920s, which set up the background of proposing different variants of quantum logic a bit later. We will discuss here two of the quantum logical variants with reference to Hilbert space formulation, based on the proposals of Bohr and Schrödinger as a result of addressing the same kernel of difficulties and will give a relative comparison. Our presentation is fairly informal, as our goal here is to simply sketch the central ideas leaving further details for other occasions.Quanta 2022; 11: 28–41.

Operad structures in geometric quantization of the moduli space of spatial polygons

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ in a special case.

Clustering using matrix product states

The matrix product state has been demonstrated to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions for one-dimensional interacting quantum many-body systems. Inspired by this success, here we propose a clustering algorithm based on the matrix product state, which first maps the classical data into quantum states represented as matrix product states, and then minimizes the loss function using a variational matrix product states algorithm in the enlarged space. We demonstrate this algorithm by applying it to several commonly used machine learning data sets, showing that this algorithm could reach higher learning precision and that it is less likely to be trapped in local minima compared to the standard K-means algorithm. We also show that this algorithm can achieve state-of-the-art learning precision on popular computer vision data sets when used in combination with better initialization schemes.

Hilbert Space Fragmentation and Commutant Algebras

We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using the language of commutant algebras, the algebra of all operators that commute with each term of the Hamiltonian or each gate of the circuit. We provide a precise definition of Hilbert space fragmentation in this formalism as the case where the dimension of the commutant algebra grows exponentially with the system size. Fragmentation can hence be distinguished from systems with conventional symmetries such as $U(1)$ or $SU(2)$, where the dimension of the commutant algebra grows polynomially with the system size. Further, the commutant algebra language also helps distinguish between "classical" and "quantum" Hilbert space fragmentation, where the former refers to fragmentation in the product state basis. We explicitly construct the commutant algebra in several systems exhibiting classical fragmentation, including the $t-J_z$ model and the spin-1 dipole-conserving model, and we illustrate the connection to previously-studied "Statistically Localized Integrals of Motion" (SLIOMs). We also revisit the Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely studied in the literature, and show that they exhibit quantum Hilbert space fragmentation. Finally, we study the contribution of the full commutant algebra to the Mazur bounds in various cases. In fragmented systems, we use expressions for the commutant to analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that agree with previous numerical results. In addition, we are able to rigorously show the localization of the on-site spin operator in the spin-1 dipole-conserving model.