Space Of Quantum States Research Articles

Musings on SVD and pseudo entanglement entropies

Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy that involve post-selection. In this work we analyze their properties as measures on the spaces of quantum states and argue that their excess provides useful characterization of a difference between two (i.e. pre-selected and post-selected) states, which shares certain features and in certain cases can be identified as a metric. In particular, when applied to link complement states that are associated to topological links via Chern-Simons theory, these generalized entropies and their excess provide a novel quantification of a difference between corresponding links. We discuss the dependence of such entropy measures on the level of Chern-Simons theory and determine their asymptotic values for certain link states. We find that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also consider properties of entropy measures for simpler quantum mechanical systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.

Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman's Path-Integral Measure for Quantum Fields in the Presence of Gravity

Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman's Path-Integral Measure for Quantum Fields in the Presence of Gravity

Quantum computing and tensor networks for laminate design: A novel approach to stacking sequence retrieval

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters, which is typically the second phase in a common bi-level optimization procedure for minimizing the weight of composite structures. To adapt stacking sequence retrieval for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed numerical state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. For the DMRG algorithm, we derived a matrix product operator representation of the loss function Hamiltonian and the penalty terms. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

Application of quantum machine learning in a Higgs physics study at the CEPC

Machine learning has blossomed in recent decades and has become essential in many fields. It significantly solved some problems in particle physics — particle reconstruction, event classification, etc. However, it is now time to break the limitation of conventional machine learning with quantum computing. A support-vector machine algorithm with a quantum kernel estimator (QSVM-Kernel) leverages high-dimensional quantum state space to identify a signal from backgrounds. In this study, we have pioneered employing this quantum machine learning algorithm to study the [Formula: see text] process at the Circular Electron–Positron Collider (CEPC), a proposed Higgs factory to study electroweak symmetry breaking of particle physics. Using 6 qubits on quantum computer simulators, we optimized the QSVM-Kernel algorithm and obtained a classification performance similar to the classical support-vector machine algorithm. Furthermore, we have validated the QSVM-Kernel algorithm using 6-qubits on quantum computer hardware from both IBM and Origin Quantum: the classification performances of both are approaching noiseless quantum computer simulators. In addition, the Origin Quantum hardware results are similar to the IBM Quantum hardware within the uncertainties in our study. Our study shows that state-of-the-art quantum computing technologies could be utilized by particle physics, a branch of fundamental science that relies on big experimental data.

Engineering Four-Qubit Fuel States for Protecting Quantum Thermalization Machine from Decoherence

Decoherence is a major issue in quantum information processing, degrading the performance of tasks or even precluding them. Quantum error-correcting codes, creating decoherence-free subspaces, and the quantum Zeno effect are among the major means for protecting quantum systems from decoherence. Increasing the number of qubits of a quantum system to be utilized in a quantum information task as a resource expands the quantum state space. This creates the opportunity to engineer the quantum state of the system in a way that improves the performance of the task and even to protect the system against decoherence. Here, we consider a quantum thermalization machine and four-qubit atomic states as its resource. Taking into account the realistic conditions such as cavity loss and atomic decoherence due to ambient temperature, we design a quantum state for the atomic resource as a classical mixture of Dicke and W states. We show that using the mixture probability as the control parameter, the negative effects of the inevitable decoherence on the machine performance almost vanish. Our work paves the way for optimizing resource systems consisting of a higher number of atoms.

Noncommutativity in Configuration Space Induced by a Conjugate Magnetic Field in Phase Space

An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.

Quantum Computing Algorithms for Nonlinear Optimization Problems

The increasing complexity of real-world optimization problems highlights the importance of this research since classical algorithms are unable to provide efficient answers in these cases. Innovative methods for fast and scalable resolution of nonlinear optimization problems are required because these problems are prevalent in many fields. The potential for quantum computing to speed up optimization processes and overcome classical limitations is great, owing to its superposition principles and intrinsic parallelism. The integration of quantum algorithms (I-QA) into real-world applications, however, will not always be smooth sailing. There are significant challenges associated with preserving quantum coherence, correcting errors, and working within hardware limits. To enable the simultaneous exploration of solution spaces through quantum parallelism, this research proposes the Hybrid Quantum Gradient -Classical Approach (HQG-CA), which makes use of parameterized quantum circuits to represent probable solutions. Additionally, improves convergence rates through applying quantum gradient information to direct optimization in the quantum state space. Optimization of portfolios in finance, adjustment of model parameters in machine learning, and optimization of routes in logistics are a few examples of the many industries that find use for HQG-CA. These applications are explored in this abstract, which highlights the revolutionary potential of HQG-CA to solve optimization problems in the real world. The effectiveness of HQG-CA is assessed through a thorough simulation experiment. Performance measures such algorithmic speedup, solution accuracy, and scalability are discussed, which is based on extensive testing and comparison with classical alternatives. The present research provides a comprehensive evaluation of HQG-CA's potential for tackling nonlinear optimization problems.

Quantum motion algebra

A notion of Quantum Motion Algebra (QMA) allows to construct quantum state spaces for various physical systems moving under a given group of motion. The main idea of QMA is a construction of a group algebra with involution generated by a group of motion G. After defining a linear and nonnegative functional on this algebra one can construct the appropriate quantum state space by means of the Gelfand-Naimark-Segal theorem. The QMA method can be also applied in the modeling of physical systems requiring additional degrees of freedom or additional constraints.The presented paper gives a brief description of the QMA method. As an example of the QMA application, we present a model of nuclear collective pairing where the nonnegative functional is generated by a temperature dependent quantum density operator.

Tristochastic operations and products of quantum states

The notion of convolution of two probability vectors, corresponding to a coincidence experiment can be extended to a family of binary operations determined by (tri)stochastic tensors, to describe Markov chains of a higher order. The problem of associativity, commutativity, and the existence of neutral elements and inverses for such operations acting on classical states is analyzed. For a more general setup of multi-stochastic tensors, we present the characterization of their probability eigenvectors. Similar results are obtained for the quantum case: we analyze tristochastic channels, which induce binary operations defined in the space of quantum states. Studying coherifications of tristochastic tensors we propose a quantum analogue of the convolution of probability vectors defined for two arbitrary density matrices of the same size. Possible applications of this notion to construct schemes of error mitigation or building blocks in quantum convolutional neural networks are discussed.

Chiral Loop Quantum Supergravity and Black Hole Entropy

Recent work has shown that local supersymmetry on a spacetime boundary in N-extended AdS supergravity in chiral variables implies coupling to a boundary OSp(N|2)C super Chern–Simons theory. Consequently there has been a proposal to define and calculate the entropy S for the boundary, in the supersymmetric version of loop quantum gravity, for the minimal case N=1, via this super Chern–Simons theory. We give an overview of how supergravity can be treated in loop quantum gravity. We review the calculation of the dimensions of the quantum state spaces of UOSp(1|2) super Chern–Simons theory with punctures, and its analytical continuation, for the fixed quantum super area of the surface, to OSp(1|2)C. The result is S=aH/4 for large (super) areas. Lower order corrections can also be determined. We begin also a discussion of the statistical mechanics of the surface degrees of freedom by calculating the grand canonical partition function at zero chemical potential. This is a new result.

A topic-aware classifier based on a hybrid quantum-classical model

In the era of Large Language Models, there is still potential for improvement in current Natural Language Processing (NLP) methods in terms of verifiability and consistency. NLP classical approaches are computationally expensive due to their high-power consumption, computing power, and storage requirements. Another computationally efficient approach to NLP is categorical quantum mechanics, which combines grammatical structure and individual word meaning to deduce the sentence meaning. As both quantum theory and natural language use vector space to describe states which are more efficient on quantum hardware, QNLP models can achieve up to quadratic speedup over classical direct calculation methods. In recent years, there is significant progress in utilizing quantum features such as superposition and entanglement to represent linguistic meaning on quantum hardware. Earlier research work has already demonstrated QNLP’s potential quantum advantage in terms of speeding up search, enhancing classification tasks’ accuracy and providing an exponentially large quantum state space in which complex linguistic structures can be efficiently embedded. In this work, a QNLP model is used to determine if two sentences are related to the same topic or not. By comparing our QNLP model to a classical tensor network-based one, our model improved training accuracy by up to 45% and validation accuracy by 35%, respectively. The QNLP model convergence is also studied when varying: first, the problem size, second, parametrized quantum circuits used for model’s training, and last, the backend quantum simulator noise model. The experimental results show that strongly entangled ansatz designs result in fastest model convergence.

Creating nonlocality using geometric phases between partially distinguishable photons

The geometric (Berry-Pancharatnam) phase originates from the intrinsic geometry of the space of quantum states and can be observed in different situations, such as a cyclic evolution of a quantum system. Here, we utilize the geometric phase to obtain a surprising insight: It is possible to create nonlocal correlations in a fixed interferometer with independent photon inputs by varying the photons' internal states. In particular, we consider a cyclic interferometer that is fixed, i.e., that has no variable internal phase shifts or subsequent measurement settings. Instead, the measurement choices of the different parties correspond to the internal states of the input photons which influence the observed correlations via a collective $N$-photon geometric phase, constituting a different approach for the generation of nonlocality with respect to the usual paradigm. We observe a trade-off between the geometric phases and the visibility of the many-photon interference, impeding the generation of nonlocality. However, by making use of the dynamical quantum Zeno effect, we show that nonlocality can be created in the fixed cyclic interferometer using 12 (or more) independent photons.

Curves in quantum state space, geometric phases, and the brachistophase

Given a curve in quantum spin state space, we inquire what is the relation between its geometry and the geometric phase accumulated along it. Motivated by Mukunda and Simon’s result that geodesics (in the standard Fubini-Study metric) do not accumulate geometric phase, we find a general expression for the derivatives (of various orders) of the geometric phase in terms of the covariant derivatives of the curve. As an application of our results, we put forward the brachistophase problem: given a quantum state, find the (appropriately normalized) Hamiltonian that maximizes the accumulated geometric phase after time τ—we find an analytical solution for all spin values, valid for small τ. For example, the optimal evolution of a spin coherent state consists of a single Majorana star separating from the rest and tracing out a circle on the Majorana sphere.

Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences.

We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.

Inherent color symmetry in quantum Yang-Mills theory

We present the basic non-perturbative structure of the space of classical dynamical solutions and corresponding one particle quantum states in SU(3) Yang-Mills theory. It has been demonstrated that the Weyl group of su(3) algebra plays an important role in constructing non-perturbative solutions and leads to profound changes in the structure of the classical and quantum Yang-Mills theory. We show that the Weyl group as a non-trivial color subgroup of SU(3) admits singlet irreducible representations on a space of classical dynamical solutions which lead to strict concepts of one particle quantum states for gluons and quarks. The Yang-Mills theory is a non-linear theory and, in general, it is not possible to construct a Hilbert space of classical solutions and quantum states as a linear vector space, so, usually, a perturbative approach is applied. We propose a non-perturbative approach based on Weyl symmetric solutions to full non-linear equations of motion and construct a full space of dynamical solutions representing an infinite but countable solution space classified by a finite set of integer numbers. It has been proved that the Weyl singlet structure of classical solutions provides the existence of a stable non-degenerate vacuum which serves as a main precondition of the color confinement phenomenon. Some physical implications in quantum chromodynamics are considered.

Quantum receiver enhanced by adaptive learning.

Quantum receivers aim to effectively navigate the vast quantum-state space to endow quantum information processing capabilities unmatched by classical receivers. To date, only a handful of quantum receivers have been constructed to tackle the problem of discriminating coherent states. Quantum receivers designed by analytical approaches, however, are incapable of effectively adapting to diverse environmental conditions, resulting in their quickly diminishing performance as the operational complexities increase. Here, we present a general architecture, dubbed the quantum receiver enhanced by adaptive learning, to adapt quantum receiver structures to diverse operational conditions. The adaptively learned quantum receiver is experimentally implemented in a hardware platform with record-high efficiency. Combining the architecture and the experimental advances, the error rate is reduced up to 40% over the standard quantum limit in two coherent-state encoding schemes.

Geometric quantum thermodynamics.

Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We focus on microcanonical and canonical ensembles, looking at the geometric counterpart of Gibbs ensembles for distributions on the space of quantum states. We show that one can define quantum heat and work in an intrinsic way, including single-trajectory work. We reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the first and second laws of thermodynamics and Jarzynki's fluctuation theorem. Overall, this results in a more transparent physics than conventionally available. The mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned. The experimental relevance is brought out via a stochastic model for chiral molecules (in the two-state approximation) and Josephson junctions. Numerically, we demonstrate this invariably leads to the emergence of the geometric canonical ensemble.

Three principles of quantum computing

The point of building a quantum computer is that it allows to model living things with predictive power and gives the opportunity to control life. Its scaling means not just the improvement of the instrument part, but also, mainly, mathematical and software tools, and our understanding of the QC problem. The first principle of quantum modeling is the reduction of reality to finite-dimensional models similar to QED in optical cavities. The second principle is a strict limitation of the so-called Feynman principle, the number of qubits in the standard formulation of the QC. This means treating decoherence exclusively as a limitation of the memory of a classical modeling computer, and introducing corresponding progressive restrictions on the working area of the Hilbert space of quantum states as the model expands. The third principle is similarity in processes of different nature. The quantum nature of reality is manifested in this principle; its nature is quantum nonlocality, which is the main property that ensures the prospects of quantum physical devices and their radical advantage over classical ones.

Variational quantum extreme learning machine

Extreme learning machine (ELM), with fast training speed and high generalization performance, has been widely used in many fields. However, it becomes inefficient or even impossible to process data with extremely large feature spaces, which is expected to be solved by quantum computing with an exponentially large quantum state space. Here, we propose a novel variational quantum extreme learning machine (VQELM). In detail, we design a special feature mapping method to achieve nonlinear transformation of the input data, replacing the hard-to-construct activation function on quantum devices. Considering that the Harrow-Hassidim-Lloyd algorithm is difficult to solve the ELM parameters on near-term quantum devices, we adopt a variational framework to facilitate implementation on the near-term noisy intermediate scale quantum computer. On both classification and regression tasks, our proposed method outperforms classical ELM in classical simulations. Moreover, the classification tasks achieved on IBM quantum simulator also show comparable classification accuracy. The final analysis shows that our proposed algorithm has an exponential improvement over classical ones for high-dimensional data processing, and is a powerful application of quantum machine learning on near-term quantum devices.