Operator Product Research Articles

Systematic approach to dynamic balancing using coloured petri nets for production lines in advanced manufacturing systems

The assembly line is widely used in the manufacturing of products that involve tasks with defined and repetitive sequences. Its efficiency greatly impacts the overall manufacturing process, necessitating resource optimization. Balancing activities among workstations is a design challenge, especially in mixed-product assembly lines, due to significant variations in task durations and short-term changes in product sequencing. This paper introduces a balancing system that considers product sequencing effects on resource utilization. The system identifies specific sets of sequenced activities for each product type and operator, enhancing workstation flexibility. It employs balancing strategies utilizing a precedence diagram and a heuristic algorithm based on positional weights. Modeling and simulation are performed using Coloured Petri Nets. A case study of a truck assembly line in Brazil illustrates the main features of this approach. Simulation results demonstrate that it is possible to reduce dependent occupancy and sequencing, meet cycle time requirements, and achieve a 12.5% reduction in resource utilization while maintaining acceptable average occupancy, ultimately establishing a system beneficial for assembly line managers.

A spin on the bulk locality of Slightly Broken Higher Spin theories

In this paper, we investigate if it is possible to express correlation functions in Large N Chern-Simons (CS) matter theories/ Slightly Broken Higher Spin (SBHS) theories purely in terms of single trace twist conformal blocks (TCBs). For this, we first develop the machinery for spinning TCBs. We do this both by explicitly solving the spinning TCB eigenvalue equation taking into account consistency with the operator product expansion (OPE) and crossing symmetry, and also by employing weight shifting and spin raising operators and acting with them on scalar seeds. Using these results we show that spinning correlators in theories with exact higher spin symmetry can be entirely expressed in terms of single trace TCBs. However, when the higher spin symmetry is slightly broken at large-N, even though the scalar four-point function is given by single-trace TCBs, the spinning correlators in general, are not. Our results suggest that it may be possible to identify a sub-sector of SBHS theory which has a local bulk dual.

Scaling Violation in Power Corrections to Energy Correlators from the Light-Ray Operator Product Expansion

In recent years, energy correlators have emerged as a powerful tool to explore the field theoretic structure of strong interactions at particle colliders. In this Letter we initiate a novel study of the nonperturbative power corrections to the projected N-point energy correlators in the limit where the angle between the detectors is small. Using the light-ray operator product expansion as a guiding principle, we derive the power corrections in terms of two nonperturbative quantities describing the fragmentation of quarks and gluons. In analogy with their perturbative leading-power counterpart, we show that power corrections obey a classical scaling behavior that is violated at the quantum level. This crucially results in a dependence on the hard scale Q of the problem that is calculable in perturbation theory. Our analytic predictions are successfully tested against Monte Carlo simulations for both lepton and hadron colliders, marking a significant step forward in the understanding of these observables. Published by the American Physical Society 2024

Tensor quantum programming

Abstract Running quantum algorithms often involves implementing complex quantum circuits with such a large number of multi-qubit gates that the challenge of tackling practical applications appears daunting. In this article, we propose a novel approach called Tensor Quantum Programming, which leverages tensor networks (TNs) for hybrid quantum computing. Our key insight is that the primary challenge of algorithms based on TNs lies in their high ranks (bond dimensions). Quantum computing offers a potential solution to this challenge, as an ideal quantum computer can represent tensors with arbitrarily high ranks in contrast to classical counterparts, which indicates the way towards quantum advantage. While tensor-based vector-encoding and state-readout are known procedures, the matrix-encoding required for performing matrix-vector multiplications directly on quantum devices is much less studied. We introduce an algorithm that encodes matrix product operators into quantum circuits with a depth that scales linearly with the number of qubits. We demonstrate the algorithm’s performance for matrices commonly encountered in differential equations, optimization problems, and quantum chemistry, for systems involving up to 50 qubits.

Bjorken polarised sum rule with IR-finite QCD coupling

We evaluate the Bjorken polarised sum rule (BSR) Γ‾1p−n(Q2) with truncated Operator Product Expansion (OPE) up to the dimension D=4 terms; the leading-twist (D=0) contribution is evaluated with a renormalon motivated resummation, where we use the IR-finite running coupling free of Landau singularities. The free residue parameters at the D=2 and D=4 terms are then extracted by fits to the experimental BSR data. In comparison to the case when perturbative QCD coupling is used in this approach (cf. Refs. [1,2]), many advantages become manifest in the present case: (a) no regularisation in the resummation integral is needed; (b) the Q2-interval in the fit can be significantly extended to lower Q2-values; (c) the extracted residue parameter values are more stable, this being reflected in the significantly reduced experimental uncertainties of these values and the suppressed (almost negligible) central extracted value of the D=4 residue parameter (μ6).

Symmetric group gauge theories and simple gauge/string dualities

Abstract We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are coded in one partial permutation algebra. We propose a generalization of the partial permutation algebra to the symmetric orbifold topological quantum field theory of any seed theory and show that the theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov–Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

Tensor-networks-based learning of probabilistic cellular automata dynamics

Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle issues in the classical domain. This work focuses on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes, and generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bitwise difference between the rules and whether one is much more likely than the other. Published by the American Physical Society 2024

Treelike process tensor contraction for automated compression of environments

The algorithm “automated compression of environments” (ACE) [M. Cygorek , ] provides a versatile way of simulating an extremely broad class of open quantum systems. This is achieved by encapsulating the influence of the environment, which is determined by the interaction Hamiltonian(s) and initial states, into compact process tensor matrix product operator (PT-MPO) representations. The generality of the ACE method comes at high numerical cost. Here, we demonstrate that orders-of-magnitude improvement of ACE is possible by changing the order of PT-MPO contraction from a sequential to a treelike scheme. The problem of combining two partial PT-MPOs with large inner bonds is solved by a preselection approach. The drawbacks of the preselection approach are that the MPO compression is suboptimal and that it is more prone to error accumulation than sequential combination and compression. We therefore also identify strategies to mitigate these disadvantages by fine-tuning compression parameters. This results in a scheme that is similar in compression efficiency and accuracy to the original ACE algorithm, yet is significantly faster. Our numerical experiments reach similar conclusions for bosonic and fermionic test cases, suggesting that our findings are characteristic of the combination of PT-MPOs more generally. Published by the American Physical Society 2024

On Compactness of Products of Toeplitz Operators

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols f and g, we show that TfTg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_fT_g$$\\end{document} is compact if and only if fg vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of fg on the whole polydisc might not imply the compactness of TfTg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_fT_g$$\\end{document}. On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

Efficient higher-order matrix product operators for time evolution

We introduce a systematic construction of higher-order matrix product operator (MPO) approximations of the time evolution operator for generic (short and long range) one-dimensional Hamiltonians. We demonstrate the utility of our construction, by showing an order of magnitude speedup in simulation cost compared to conventional first-order MPO time evolution schemes.

Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies

We discuss peculiarities of the Schwinger-DeWitt technique for quantum effective action, associated with the origin of dimensionally regularized double-pole divergences of the one-loop functional determinant for massive Proca model in a curved spacetime. These divergences have the form of the total-derivative term generated by integration by parts in the functional trace of the heat kernel for the Proca vector field operator. Because of the nonminimal structure of second-order derivatives in this operator, its vector field heat kernel has a nontrivial form, involving the convolution of the scalar d’Alembertian Green’s function with its heat kernel. Moreover, its asymptotic expansion is very different from the universal predictions of Gilkey-Seeley heat kernel theory because the Proca operator violates one of the basic assumptions of this theory—the nondegeneracy of the principal symbol of an elliptic operator. This modification of the asymptotic expansion explains the origin of double-pole total-derivative terms. Another hypostasis of such terms is in the problem of multiplicative determinant anomalies—lack of factorization of the functional determinant of a product of differential operators into the product of their individual determinants. We demonstrate that this anomaly should have the form of total-derivative terms and check this statement by calculating divergent parts of functional determinants for products of minimal and nonminimal second-order differential operators in curved spacetime. Published by the American Physical Society 2024

Fusion of irreducible modules in the periodic Temperley–Lieb algebra

We propose a new family \mathsf{Y}_{k,\ell,x,y,[z,w]}𝖸k,ℓ,x,y,[z,w] of modules over the enlarged periodic Temperley–Lieb algebra \mathsf{\mathcal EPTL}_N(\beta)ℰ𝖯𝖳𝖫N(β). These modules are built from link states with two marked points, similarly to the modules \mathsf{X}_{k,\ell,x,y,z}𝖷k,ℓ,x,y,z that we constructed in a previous paper. They however differ in the way that defects connect pairwise. We analyse the decomposition of \mathsf{Y}_{k,\ell,x,y,[z,w]}𝖸k,ℓ,x,y,[z,w] over the irreducible standard modules \mathsf{W}_{k,x}𝖶k,x for generic values of the parameters zz and ww, and use it to deduce the fusion rules for the fusion \mathsf{W} × \mathsf{W}𝖶×𝖶 of standard modules. These turn out to be more symmetric than those obtained previously using the modules \mathsf{X}_{k,\ell,x,y,z}𝖷k,ℓ,x,y,z. From the work of Graham and Lehrer, it is known that, for \beta = -q-q^{-1}β=−q−q−1 where qq is not a root of unity, there exists a set of non-generic values of the twist yy for which the standard module \mathsf{W}_{\ell,y}𝖶ℓ,y is indecomposable yet reducible with two composition factors: a radical submodule \mathsf{R}_{\ell,y}𝖱ℓ,y and a quotient module \mathsf{Q}_{\ell,y}𝖰ℓ,y. Here, we construct the fusion products \mathsf{W}×\mathsf{R}𝖶×𝖱, \mathsf{W}×\mathsf{Q}𝖶×𝖰 and \mathsf{Q} × \mathsf{Q}𝖰×𝖰, and analyse their decomposition over indecomposable modules. For the fusions involving the quotient modules \mathsf{Q}𝖰, we find very simple results reminiscent of \mathfrak{sl}(2)𝔰𝔩(2) fusion rules. This construction with modules \mathsf{Y}_{k,\ell,x,y,[z,w]}𝖸k,ℓ,x,y,[z,w] is a good lattice regularization of the operator product expansion in the underlying logarithmic bulk conformal field theory. Indeed, it fits with the correspondence between standard modules and connectivity operators, and is useful for the calculation of their correlation functions. Remarkably, we show that the fusion rules \mathsf{W}×\mathsf{Q}𝖶×𝖰 and \mathsf{Q}×\mathsf{Q}𝖰×𝖰 are consistent with the known fusion rules of degenerate primary fields.

Simulating the Landau-Zener sweep in deeply sub-Ohmic environments.

With the goal to study dissipative Landau-Zener (LZ) sweeps in realistic solid-state qubits, we utilize novel methods from non-Markovian open quantum system dynamics that enable reliable long-time simulations for sub-Ohmic environments. In particular, we combine a novel representation of the dynamical propagator, the uniform time evolving matrix product operator method, with a stochastic realization of finite temperature fluctuations. The latter greatly reduces the computational cost for the matrix product operator approach, enabling convergence in the experimentally relevant deeply sub-Ohmic regime. Our method allows the exact simulation of dynamical protocols with long operation times, such as the LZ sweep, in challenging parameter regimes that are realized in current experimental platforms.

Area-driven Boolean bi-decomposition by function approximation

Bi-decomposition rewrites logic functions as the composition of simpler components. It is related to Boolean division, where a given function is rewritten as the product of a divisor and a quotient, but bi-decomposition can be defined for any Boolean operation of two operands. The key questions are how to find a good divisor and then how to compute the quotient. In this article, we select the divisor by approximation of the original function and then characterize by an incompletely specified function the full flexibility of the quotient for each binary operator. We target area-driven exact bi-decomposition, and we apply it to the bi-decomposition of Sum-of-Products (SOP) forms. We report experiments that exhibit significant gains in literals of SOP forms when rewritten as bi-decompositions with respect to the product operator. This suggests the application of this framework to other logic forms and binary operations, both for exact and approximate implementations.

Locally purified density operators for noisy quantum circuits

Abstract Simulating open quantum systems is essential for exploring novel quantum phenomena and evaluating noisy quantum circuits. In this Letter, we address the problem of whether mixed states generated from noisy quantum circuits can be efficiently represented by locally purified density operators (LPDOs). We map an LPDO of N qubits to a pure state of size 2 × N defined on a ladder and introduce a unified method for managing virtual and Kraus bonds. We numerically simulate noisy random quantum circuits with depths up to d = 40 using fidelity and entanglement entropy as accuracy measures. LPDO representation proves to be effective in describing mixed states in both quantum and classical regions but encounters significant challenges at the quantum-classical critical point, limiting its applicability to the quantum region exclusively. In contrast, the matrix product operator (MPO) successfully characterizes the entanglement trend throughout the simulation, while truncation in MPOs breaks the positivity condition required for a physical density matrix. This work advances our understanding of efficient mixed-state representation in open quantum systems and provides insights into the entanglement structure of noisy quantum circuits.

Decomposition of nonlinear collision operator in quantum Lattice Boltzmann algorithm

We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.

On p-numerical radius inequalities for sums and products of operators

On p-numerical radius inequalities for sums and products of operators

Conformal bootstrap equations from the embedding space operator product expansion

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

Site selection for biomass-solar hybrid renewable energy facilities: Spatial modelling based on fuzzy logic-geographic information systems

The aim of this study is to propose a fuzzy logic-geographic information systems (GIS) based spatial model within the context of hybrid renewable energy, rural areas, and spatial planning approach in the site selection process of renewable energy facilities. In this context, potential suitability index and site selection suitability index for biomass-solar hybrid renewable energy facilities are determined within the spatial model, through a case study conducted in Türkiye. The study involves a two-stage process: the first stage focuses on establishing the national-level potential, and the second stage involves site selection in areas with high potential. Suitable locations are determined based on the optimal suitability index values, which are derived from fuzzy logic operators including AND, OR, SUM, PRODUCT, and GAMMA, applied in the GIS. The study results indicate that by assigning different γ values to the fuzzy GAMMA operator, results for the fuzzy AND (γ = 0.45–0.60), fuzzy OR (γ = 1.00), fuzzy SUM (γ = 1.00), and fuzzy PRODUCT (γ = 0.00–0.15) operators can be obtained. Furthermore, as the γ value increases, the suitability index also rises, and the optimal site selection occurs at γ = 0.90. The performance of the fuzzy GAMMA operator in big data sets and two-stage processes demonstrates its applicability in site selection.

Operator product states on tensor powers of $$C^*$$-algebras

Operator product states on tensor powers of $$C^*$$-algebras