Maximum Cutting Research Articles

The soaring kite: a tale of two punctured tori

We consider the 5-mass kite family of self-energy Feynman integrals and present a systematic approach for constructing an ε-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the locations of relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be extracted from integrals over maximal cuts. A boundary value is provided such that the differential equation is systematically solved in terms of iterated integrals over g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.

On-shell functions on the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM

We study on-shell functions in the kinematic space for the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM. We construct BCFW bridges that help us build bigger on-shell functions. As a consequence, we provide on-shell diagram formulations for BCFW shifts that correspond to various mass configurations. We will use this to calculate the quadruple cut for the one-loop amplitude on the Coulomb branch and maximal cuts for higher-loops. We make preliminary comments on finding the inequivalent set of on-shell functions for the Coulomb branch.

Two-loop non-planar four-point topology with massive internal loop

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an ϵ-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen’s iterated integrals.

Noisecut: a python package for noise-tolerant classification of binary data using prior knowledge integration and max-cut solutions

BackgroundClassification of binary data arises naturally in many clinical applications, such as patient risk stratification through ICD codes. One of the key practical challenges in data classification using machine learning is to avoid overfitting. Overfitting in supervised learning primarily occurs when a model learns random variations from noisy labels in training data rather than the underlying patterns. While traditional methods such as regularization and early stopping have demonstrated effectiveness in interpolation tasks, addressing overfitting in the classification of binary data, in which predictions always amount to extrapolation, demands extrapolation-enhanced strategies. One such approach is hybrid mechanistic/data-driven modeling, which integrates prior knowledge on input features into the learning process, enhancing the model’s ability to extrapolate.ResultsWe present NoiseCut, a Python package for noise-tolerant classification of binary data by employing a hybrid modeling approach that leverages solutions of defined max-cut problems. In a comparative analysis conducted on synthetically generated binary datasets, NoiseCut exhibits better overfitting prevention compared to the early stopping technique employed by different supervised machine learning algorithms. The noise tolerance of NoiseCut stems from a dropout strategy that leverages prior knowledge of input features and is further enhanced by the integration of max-cut problems into the learning process.ConclusionsNoiseCut is a Python package for the implementation of hybrid modeling for the classification of binary data. It facilitates the integration of mechanistic knowledge on the input features into learning from data in a structured manner and proves to be a valuable classification tool when the available training data is noisy and/or limited in size. This advantage is especially prominent in medical and biomedical applications where data scarcity and noise are common challenges. The codebase, illustrations, and documentation for NoiseCut are accessible for download at https://pypi.org/project/noisecut/. The implementation detailed in this paper corresponds to the version 0.2.1 release of the software.

A CMOS-compatible oscillation-based VO2 Ising machine solver

Phase-encoded oscillating neural networks offer compelling advantages over metal-oxide-semiconductor-based technology for tackling complex optimization problems, with promising potential for ultralow power consumption and exceptionally rapid computational performance. In this work, we investigate the ability of these networks to solve optimization problems belonging to the nondeterministic polynomial time complexity class using nanoscale vanadium-dioxide-based oscillators integrated onto a Silicon platform. Specifically, we demonstrate how the dynamic behavior of coupled vanadium dioxide devices can effectively solve combinatorial optimization problems, including Graph Coloring, Max-cut, and Max-3SAT problems. The electrical mappings of these problems are derived from the equivalent Ising Hamiltonian formulation to design circuits with up to nine crossbar vanadium dioxide oscillators. Using sub-harmonic injection locking techniques, we binarize the solution space provided by the oscillators and demonstrate that graphs with high connection density (η > 0.4) converge more easily towards the optimal solution due to the small spectral radius of the problem’s equivalent adjacency matrix. Our findings indicate that these systems achieve stability within 25 oscillation cycles and exhibit power efficiency and potential for scaling that surpasses available commercial options and other technologies under study. These results pave the way for accelerated parallel computing enabled by large-scale networks of interconnected oscillators.

Ultralow Power Electronic Analog of a Biological Fitzhugh-Nagumo Neuron.

Here, we introduce an electronic circuit that mimics the functionality of a biological spiking neuron following the Fitzhugh-Nagumo (FN) model. The circuit consists of a tunnel diode that exhibits negative differential resistance (NDR) and an active inductive element implemented by a single MOSFET. The FN neuron converts a DC voltage excitation into voltage spikes analogous to biological action potentials. We predict an energy cost of 2 aJ/cycle through detailed simulation and modeling for these FN neurons. Such an FN neuron is CMOS compatible and enables ultralow power oscillatory and spiking neural network hardware. We demonstrate that FN neurons can be used for oscillator-based computing in a coupled oscillator network to form an oscillator Ising machine (OIM) that can solve computationally hard NP-complete max-cut problems while showing robustness toward process variations.

Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes

Variational quantum algorithms (VQAs) represent a promising approach to utilizing current quantum computing infrastructures. VQAs are based on a parameterized quantum circuit optimized in a closed loop via a classical algorithm. This hybrid approach reduces the quantum processing unit load but comes at the cost of a classical optimization that can feature a flat energy landscape. Existing optimization techniques, including either imaginary time-propagation, natural gradient, or momentum-based approaches, are promising candidates but place either a significant burden on the quantum device or suffer frequently from slow convergence. In this work, we propose the quantum Broyden adaptive natural gradient (qBang) approach, a novel optimizer that aims to distill the best aspects of existing approaches. By employing the Broyden approach to approximate updates in the Fisher information matrix and combining it with a momentum-based algorithm, qBang reduces quantum-resource requirements while performing better than more resource-demanding alternatives. Benchmarks for the barren plateau, quantum chemistry, and the max-cut problem demonstrate an overall stable performance with a clear improvement over existing techniques in the case of flat (but not exponentially flat) optimization landscapes. qBang introduces a new development strategy for gradient-based VQAs with a plethora of possible improvements.

General spatial photonic Ising machine based on the interaction matrix eigendecomposition method

The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.

Parametric Frequency Divider Based Ising Machines.

We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.

Multilevel Leapfrogging Initialization Strategy for Quantum Approximate Optimization Algorithm

AbstractRecently, Zhou et al. have proposed an Interpolation‐based (INTERP) strategy to generate the initial parameters for Quantum Approximate Optimization Algorithm (QAOA). INTERP guesses the initial parameters at level by applying interpolation to the optimized parameters at level , achieving better performance than random initialization (RI). Nevertheless, INTERP consumes extensive costs for deep QAOA because it necessitates optimization at each level depth. To address it, a Multilevel Leapfrogging Interpolation (MLI) strategy is proposed. MLI produces initial parameters from level to () at level , omitting the optimization rounds from level to . MLI executes optimization at few levels rather than each level, and this operation is called Multilevel Leapfrogging optimization (M‐Leap). The performance of MLI is investigated on the Maxcut problem. The simulation results demonstrate MLI achieves the same quasi‐optima as INTERP while consuming 1/2 of costs required by INTERP. Besides, for MLI, where there is no RI except for level 1, the greedy‐MLI strategy is presented. The simulation results suggest greedy‐MLI has better stability than INTERP and MLI beyond obtaining the quasi‐optima. According to the efficiency of finding the quasi‐optima, the idea of M‐Leap might be extended to other training tasks, especially those requiring numerous optimizations.

Highly Connected Graphs Have Highly Connected Spanning Bipartite Subgraphs

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper bounded by some function of $k$. The best upper bound for $f(k,n)$ is by Delcourt and Ferber who proved that $f(k,n) \le 10^{10}k^3 \log n$. Here it is proved that $f(k,n) \le 22k^2 \log n$. For larger $k$, stronger bounds hold. In the linear regime, it is proved that for any $0 < c < \frac{1}{2}$ and all sufficiently large $n$, $f(\lfloor cn \rfloor, n) \le 30\sqrt{c}n$. In the polynomial regime, it is proved that for any $\frac{1}{3} \le \alpha < 1$ and all sufficiently large $n$, $f(\lfloor n^\alpha \rfloor ,n) \le 9n^{(1+\alpha)/2}$.

Ferroelectric compute-in-memory annealer for combinatorial optimization problems

Computationally hard combinatorial optimization problems (COPs) are ubiquitous in many applications. Various digital annealers, dynamical Ising machines, and quantum/photonic systems have been developed for solving COPs, but they still suffer from the memory access issue, scalability, restricted applicability to certain types of COPs, and VLSI-incompatibility, respectively. Here we report a ferroelectric field effect transistor (FeFET) based compute-in-memory (CiM) annealer for solving larger-scale COPs efficiently. Our CiM annealer converts COPs into quadratic unconstrained binary optimization (QUBO) formulations, and uniquely accelerates in-situ the core vector-matrix-vector (VMV) multiplication operations of QUBO formulations in a single step. Specifically, the three-terminal FeFET structure allows for lossless compression of the stored QUBO matrix, achieving a remarkably 75% chip size saving when solving Max-Cut problems. A multi-epoch simulated annealing (MESA) algorithm is proposed for efficient annealing, achieving up to 27% better solution and ~ 2X speedup than conventional simulated annealing. Experimental validation is performed using the first integrated FeFET chip on 28nm HKMG CMOS technology, indicating great promise of FeFET CiM array in solving general COPs.

A hybrid modeling framework for generalizable and interpretable predictions of ICU mortality across multiple hospitals

The development of reliable mortality risk stratification models is an active research area in computational healthcare. Mortality risk stratification provides a standard to assist physicians in evaluating a patient’s condition or prognosis objectively. Particular interest lies in methods that are transparent to clinical interpretation and that retain predictive power once validated across diverse datasets they were not trained on. This study addresses the challenge of consolidating numerous ICD codes for predictive modeling of ICU mortality, employing a hybrid modeling approach that integrates mechanistic, clinical knowledge with mathematical and machine learning models . A tree-structured network connecting independent modules that carry clinical meaning is implemented for interpretability. Our training strategy utilizes graph-theoretic methods for data analysis, aiming to identify the functions of individual black-box modules within the tree-structured network by harnessing solutions from specific max-cut problems. The trained model is then validated on external datasets from different hospitals, demonstrating successful generalization capabilities, particularly in binary-feature datasets where label assessment involves extrapolation.

Prescriptive unitarity from positive geometries

In this paper, we define the momentum amplituhedron in the four-dimensional split-signature space of dual momenta. It encodes scattering amplitudes at tree level and loop integrands for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills in the planar sector. In this description, every point in the tree-level geometry is specified by a null polygon. Using the null structure of this kinematic space, we find a geometry whose canonical differential form produces loop-amplitude integrands. Remarkably, at one loop it is a curvy version of a simple polytope, whose vertices are specified by maximal cuts of the amplitude. This construction allows us to find novel formulae for the one-loop integrands for amplitudes with any multiplicity and helicity. The formulae obtained in this way agree with the ones derived via prescriptive unitarity. It makes prescriptive unitarity naturally emerge from this geometric description.

An expressive ansatz for low-depth quantum approximate optimisation

The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.

Identifying similar-bicliques in bipartite graphs

Bipartite graphs have been widely used to model the relationship between entities of different types, where vertices are partitioned into two disjoint sets/sides. Finding dense subgraphs in a bipartite graph is of great significance and encompasses many applications. However, none of the existing dense bipartite subgraph models consider similarity between vertices from the same side, and as a result, the identified results may include vertices that are not similar to each other. In this work, we formulate the notion of similar-biclique which is a special kind of biclique where all vertices from a designated side are similar to each other and aim to enumerate all similar-bicliques. The naive approach of first enumerating all maximal bicliques and then extracting all maximal similar-bicliques from them is inefficient, as enumerating maximal bicliques is already time consuming. We propose a backtracking algorithm MSBE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{MSBE}$$\\end{document} to directly enumerate maximal similar-bicliques and power it by vertex reduction and optimization techniques. In addition, we design a novel index structure to speed up a time-critical operation of MSBE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{MSBE}$$\\end{document}, as well as to speed up vertex reduction. Efficient index construction algorithms are developed. To handle dynamic graph updates, we also propose algorithms and optimization techniques for maintaining our index. Finally, we parallelize our index construction algorithms to exploit multiple CPU cores. Extensive experiments on 17 bipartite graphs as well as case studies are conducted to demonstrate the effectiveness and efficiency of our model and algorithms.

Parameter Setting in Quantum Approximate Optimization of Weighted Problems

Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth p=1 applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for p≥1 we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for p=1 the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only 1.1 percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.4x on average.

Systematic study on the dependence of the warm-start quantum approximate optimization algorithm on approximate solutions

Quantum approximate optimization algorithm (QAOA) is a promising hybrid quantum-classical algorithm to solve combinatorial optimization problems in the era of noisy intermediate-scale quantum computers. Recently it has been revealed that warm-start approaches can improve the performance of QAOA, where approximate solutions are obtained by classical algorithms in advance and incorporated into the initial state and/or unitary ansatz. In this work, we study in detail how the accuracy of approximate solutions affects the performance of the warm-start QAOA (WS-QAOA). We numerically find that in typical MAX-CUT problems, WS-QAOA achieves higher fidelity (probability that exact solutions are observed) and approximation ratio than QAOA as the Hamming distance of approximate solutions to the exact ones becomes smaller. We reveal that this could be quantitatively attributed to the initial state of the ansatz. We also solve MAX-CUT problems by WS-QAOA with approximate solutions obtained via QAOA, having higher fidelity and approximation ratio than QAOA especially when the circuit is relatively shallow. We believe that our study may deepen understanding of the performance of WS-QAOA and also provide a guide as to the necessary quality of approximate solutions.

A critical probability for biclique partition of Gn,p

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G) with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take G=Gn,p, where p is constant and less than a certain threshold value p0≈0.312. This verifies a conjecture of Chung and Peng for these values of p. We also show that if p0<p<1/2 then bp(Gn,p)=n−(1+Θ(1))α(Gn,p) with high probability.

Convergence of digitized-counterdiabatic QAOA: circuit depth versus free parameters

Recently, digitized-counterdiabatic (CD) quantum approximate optimization algorithm (QAOA) has been proposed to make QAOA converge to the solution of an optimization problem in fewer steps, inspired by Trotterized CD driving in continuous-time quantum annealing. In this paper, we critically revisit this approach by focusing on the paradigmatic weighted and unweighted one-dimensional MaxCut problem. We study two variants of QAOA with first and second-order CD corrections. Our results show that, indeed, higher order CD corrections allow for a quicker convergence to the exact solution of the problem at hand by increasing the complexity of the variational cost function. Remarkably, however, the total number of free parameters needed to achieve this result is independent of the particular QAOA variant analyzed for the problems considered.