### Conveners

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Zohreh Davoudi (University of Maryland)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Arata Yamamoto

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Xiao-Yong Jin (Argonne National Laboratory)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Yannick Meurice (University of Iowa)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Masafumi Fukuma (Kyoto University)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Lukas Kades (Universität Heidelberg)
- Chris Culver (University of Liverpool)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Taku Izubuchi (Brookhaven National Laboratory)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Fu-Jiun Jiang (National Taiwan Normal University)

#### Algorithms (including Machine Learning, Quantum Computing, Tensor Networks)

- Karl Jansen
- Gert Aarts (Swansea University)

We discuss recent progress in Tensor Lattice Field Theory and economical, symmetry preserving, truncations suitable for quantum computations/simulations. We focus on spin and gauge models with continuous Abelian symmetries such as the Abelian Higgs model and emphasize noise-robust implementations of Gauss's law. We discuss recent progress concerning the comparison between field digitizations...

The paradigm of effective field theory is one of the most powerful tools available in physics. While most commonly employed in parametrizing renormalization group flow, it is also of great utility in describing dispersive systems such as $K_0 - \bar{K}_0$ states that both oscillate and decay. Of particular interest for the lattice community is the study of field theories off the real axis of...

Some aspects of quantum systems with non-unitary dynamics are well-described by non-Hermitian effective Hamiltonians. Such systems contain a wealth of interesting physics such as their phase structure, eg. QCD at finite Baryon density, which describes cores of neutron stars. Classical simulation of general non-Hermitian Hamiltonians is rendered difficult, and in some cases, impossible due to...

Open lattice field theories are useful in describing many physical systems. Yet their implementation in traditional quantum computing is hindered by the requirement of Hermiticity. One method used to overcome this is embedding the non-Hermitian system within a larger Hermitian system by introducing ancillary qubits. We implement the transverse Ising Model with an addition of an imaginary...

The possibility for near-term quantum simulations in lattice field theory depends upon efficiently using the limited resources available. In this talk, we will discuss how approximating lattice gauge theories like SU(3) with discrete subgroups can be theoretically analyzed as a lattice effective field theory. Further, methods for implementation upon quantum hardware will be covered. Numerical...

The Schwinger model is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we analyze low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for...

Computing conformal dimensions $D(j_L,j_R)$ of local fields that transform in an irreducible representation of $SU(2) \times SU(2)$ labeled with $(j_L,j_R)$ at the $O(4)$ Wilson-Fisher fixed point has become interesting recently, especially when $j_L$, $j_R$ become large. These calculations are challenging in the traditional lattice $O(4)$ model. We can overcome these difficulties by using a...

We provide strong evidence that the asymptotically free (1+1)-dimensional non-linear O(3) sigma model can be regularized using a quantum lattice Hamiltonian, referred to as the "Heisenberg-comb", that acts on a Hilbert space with only two qubits per spatial lattice site. The Heisenberg-comb consists of a spin-half anti-ferromagnetic Heisenberg-chain coupled anti-ferromagnetically to a second...

The gauge-invariant formulations of lattice field theories provide a way to study real-time dynamics using a smaller effective Hilbert space. This allows for more information to be encoded for the same quantum resources as a non-gauge invariant forumlation which will be important for simulations on Noisy Intermediate Scale Quantum (NISQ) computers. While qubit-based hardware is currently the...

Quantum technologies offer the prospect to efficiently simulate sign-problem afflicted phenomena, such as topological terms, chemical potentials, and out-of-equilibrium dynamics. In this work, we derive the 3+1D topological $\theta$-term for Abelian and non-Abelian lattice gauge theories in the Hamiltonian formulation, paving the way towards Hamiltonian-based simulations of such terms on...

In order to simulate quantum field theories using quantum computers, a regularization of the target space of the field theory must be obtained which admits a representation in terms of qubits. For the 1+1 dimensional nonlinear sigma model, there have been several proposals for how such a regularization may be achieved. The fuzzy sphere regularization proposes to represent the Hilbert space of...

Quantum simulation has the promise of enabling access to Minkowski-time dynamical observables in quantum field theories. Progress in devising and benchmarking quantum-simulation proposals, in form of analog protocols or digital algorithms, is ongoing, and increasingly complex theories are being targeted towards the goal of simulating QCD. In this talk, I will introduce a hybrid analog-digital...

We introduce a new method to calculate phase shifts on noisy intermediate scale quantum (NISQ) hardware platforms using a wave packet edge time delay. The method uses the early and intermediate stages of the collision because the standard method based on the asymptotic out-state behavior is unreachable using today’s NISQ platforms. The calculation was implemented on a 4-site transverse Ising...

Dimensionally reducing gauge theories like QED or Yang-Mills theory on small spatial tori often yields simple quantum mechanical models that retain some of the interesting structure of the parent gauge theory. 2D electrodynamics with massive charge-N matter, for example, leads to the quantum mechanics of a particle on a circle with a Z_N potential and a theta-term. This model, despite being...

Future quantum computers may serve as a tool to access non-perturbative real-time correlation functions. In this talk, we discuss the prospects of using these to study Compton scattering for arbitrary kinematics. In particular, the need to restrict the size of the spacetime in quantum computers prohibits a naive determination of such amplitudes. However, we present a practical solution to this...

We present a complete and scalable quantum algorithm for the simulation of SU(2) gauge bosons coupled to fermionic matter in one spatial dimension. To represent the gauge fields, we find it is more practical to start from their Schwinger boson formulation, rather than the more conventional Kogut-Susskind rigid rotor formulation. Within this framework, and taking Trotter-Suzuki decomposition...

We develop a gauge covariant neural network for four dimensional non-abelian gauge theory, which realizes a map between rank-2 tensor valued vector fields. We also find the conventional smearing procedures for gauge fields can be regarded as this neural network with fixed parameters. We developed a formula to train the network as an extension of the delta rule, which is used in machine...

Highly oscillatory path integrals are common in lattice field theory. They crop up as sign problems and as signal to noise problems and prevent Monte Carlo calculations of both lattice QCD at finite chemical potential and real-time dynamics. A general method for treating highly oscillatory path integrals has emerged in which the domain of integration of the path integral is deformed into a...

Complex contour deformations of the path integral have previously been shown to mitigate extensive sign problems associated with non-zero chemical potential and real-time evolution in lattice field theories. This talk details recent extensions of this method to observables affected by signal-to-noise problems in theories with real actions. Contour deformations are shown to result in...

In this work, we obtained the finite temperature Bottomonium interaction potential from the first principle lattice-NRQCD calculation of Bottomonium mass and width [Phys.Lett.B 800, 135119 (2020)]. We find that the HTL complex potential is disfavored by the lattice result, which motives us to employ a model-independent parameterization --- the Deep Neural Network (DNN) --- to represent the...

Statistical modeling plays a key role in lattice field theory calculations. Examples including extracting masses from correlation functions or taking the chiral-continuum limit of a matrix element. We discuss the method of model averaging, a way to account for uncertainty due to model variations, from the perspective of Bayesian statistics. Statistical formulas are derived for model-averaged...

The lattice formulation of finite-temperature field theory is readily extended,

via the Schwinger-Keldysh contour, to accomodate the definition of real-time

observables. Unfortunately, this extension also induces a maximally severe sign

problem, obstructing the computation of, for example, the shear viscosity. In

the large-N limit of certain field theories, including $O(N)$-symmetric...

Reduced staggered fermions afford a very economical lattice fermion formulation yielding just two Dirac fermions in the continuum limit. They have also been used to construct models capable of symmetric mass generation. However, generically they suffer from sign problems. We discuss an application of the tensor renormalization group, a sign problem free method, to such models. We make a...

We present results of tensor network simulations of the three-dimensional O(2) model at nonzero chemical potential and temperature, which were computed using the higher order tensor renormalization group method. This also includes some enhancements to the method which take care of anisotropic tensors. Some special care was also taken to reduce the systematic error on the computation of the observables.

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm used to find the spectrum of a Hamiltonian using the variational method. In particular, this procedure can be used to study LGT in the Hamiltonian formulation. Bayesian Optimization (BO) based on Gaussian Process Regression (GPR) is a powerful algorithm for finding the global minimum of the energy with a very low...

Quantum computing allows for the study of real-time dynamics of non-perturbative quantum field theories while avoiding the sign problem in conventional lattice approaches. Current and near-future quantum devices are severely limited by noise, making investigations of simple low-dimensional lattice systems ideal testbeds for algorithm development. Considering simple supersymmetric systems,...

In this talk, we will be looking at quantum circuits comprising parametric gates and analyze their expressivity in terms of the space of states that can be generated by a given circuit. In particular, we will be considering parametric quantum circuits (PQCs) for use in variational quantum simulations (VQS). In such a setting, the design of PQCs is subject to two competing drivers. On one hand,...

We reformulate the continuous space Schrödinger equation in terms of spin Hamiltonians. For the kinetic energy operator, the critical concept facilitating the reduction in model complexity is the idea of position encoding. Binary encoding of position produces a Heisenberg-like model and yields exponential improvement in space complexity when compared to classical computing. Encoding with a...

We provide a modification to the textbook's quantum phase estimation algorithm (QPEA) inspired on classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrate an improvement. Moreover, with similar...

A long standing problem associated with performing lattice gauge theory calculations on GPU hardware is latency for both global memory transfers and MPI data transfers. Mitigating these latencies with data compression techniques can vastly improve the performance of solvers and help to combat strong scaling. In this talk we discuss a new gauge field compression technique in which the SU(N)...

We show that using the multi-splitting algorithm as a preconditioner for the domain wall Dirac linear operator, arising in lattice QCD, effectively reduces the inter-node communication cost, at the expense of performing more on-node floating point and memory operations. Correctly including the boundary \textit{snake} terms, the preconditioner is implemented in the QUDA framework, where it is...

The era of exascale computing enables the generation of ever-finer gauge configurations, capturing gauge-fermion physics with unprecedented accuracy. This approach to the continuum comes with a super-linear increase in the cost of the iterative Krylov solve of the Dirac fermion operator, the phenomena of critical slowing down. Multi-grid methods are the optimal approach to addressing this...

Lattice QCD at nonzero baryon density is a big challenge in hadron physics. In this talk, I discuss the quantum computation of lattice gauge theory at nonzero density. I present some results of a benchmark test based on the quantum variational algorithm.

Finding ground states and low-lying excitations of Hamiltonians is one of the most important problems that can be solved with near-term quantum computers. It can be utilized in fields ranging from optimization over chemistry and material science to particle physics. In this work, we propose an efficient error mitigation scheme that is independent of the Hamiltonian and the concrete noise...

We will present how to calculate propagators on Chroma+QUDA with overlap, HISQ (highly improved staggered quark) and twisted-mass fermion actions. Those actions are not fully supported before and now most of the actions can be used in Chroma framework efficiently. The multi-grid speed-up for the HISQ and Twisted mass actions using QUDA will also be presented.

We apply the complex Langevin method (CLM) to overcome the sign problem in 4D SU(2) gauge theory with a theta term extending our previous work on the 2D U(1) case. The topology freezing problem can be solved by using open boundary conditions in all spatial directions, and the criterion for justifying the CLM is satisfied even for large $\theta$ as far as the lattice spacing is sufficiently...

Hadron spectral functions are important quantities as they carry all the information of hadrons. Unfortunately they cannot be computed directly on the lattice, and can only be extracted from Euclidean two-point temporal correlation functions which are directly computable in lattice QCD. The extraction of spectral functions from correlation functions is an inverse problem, and the most commonly...

A universal supervised neural network (NN) relevant to study phase transitions

is constructed. The validity of the built NN is examined by applying it to

calculate the criticalities of several three-dimensional (3D) and two-dimensional (2D) models including the 3D classical $O(3)$ model, the 3D 5-state ferromagnetic Potts model, a 3D dimerized quantum antiferromagnetic Heisenberg model as...

We present a novel method to evaluate the real-time path integral for 1+1 dimensional real scalar phi^4 theory. The method can be combined with tensor network coarse-graining schemes. As a demonstration we will show numerical results of two-point correlator in a small lattice.

This work seeks to enable more detailed numerical probes into nuclear structure using the standard model through lattice QCD. The computational cost required to compute nuclear correlations functions grows exponentially in the number of quarks, leaving the study of many large nuclear bound states inaccessible. However, these tensor expressions exhibit a high degree of permutation symmetry that...

The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the perspective of quantum field theory. In this talk we will demonstrate, through the Hammersley-Clifford theorem, that the $\phi^{4}$ scalar field...

With increasing interest in machine learning techniques for lattice gauge theory, it becomes important to develop suitable neural network architectures that are compatible with the fundamental symmetries that lie at the heart of lattice QCD. We propose a novel neural network architecture called lattice gauge equivariant convolutional neural networks (L-CNNs) [1] that can be applied to problems...

In recent years, the use of machine learning has become increasingly popular in the context of lattice field theories. An essential element of such theories is represented by symmetries, whose inclusion in the neural network properties can lead to high reward in terms of performance and generalizability. A fundamental symmetry that usually characterizes physical systems on a lattice with...

We interpret machine learning functions as physical observables, opening up the possibility to apply "standard" statistical-mechanics methods to outputs from neural networks. This includes histogram reweighting and finite-size scaling, to analyse phase transitions quantitatively, as well as the incorporation of predictive functions as conjugate variables coupled to an external field within the...

In this talk, I will discuss how thermodynamic observables of lattice field theories can be estimated using machine learning. Specifically¸ deep generative models are used to estimate the absolute value of the free energy. This is in contrast to MCMC-based methods which are limited to estimating differences of free energies. These methods come with the same asymptotic guarantees as the...

Machine learning is becoming an established area of research in lattice field theories, with prominent applications to phase classification, configuration generation, and noise reduction. When moving beyond the toy models, scalable methods to learn phases of matter are needed. In this talk, we compare two possible avenues to speed up the methods for classifying phase transitions in the 2D...

We present recent results [1] in which we apply unconditionally stable stochastic partial differential equations solvers [2] to complex Langevin in real-time [3]. This allows us to avoid runaway solutions in principle and enables simulations at relatively large Langevin step size. We show that implicit schemes act as a regulator of the underlying path integral and give a heuristic estimate of...

The numerical sign problem is one of the major obstacles to the first-principles calculations for important physical systems, such as finite-density QCD, strongly-correlated electron systems and frustrated spin systems, as well as for the real-time dynamics of quantum systems. The tempered Lefschetz thimble method (TLTM) [1] was proposed as a versatile algorithm towards solving the numerical...

We numerically study the Hamiltonian lattice formulation of the Schwinger model with two fermion flavors using matrix product states. Keeping the mass of the first flavor at a fixed positive value, we tune the mass of the second flavor through a range of negative values, thus exploring a regime where conventional Monte Carlo methods suffer from the sign problem. Our results show signatures of...

Partially twisted boundary conditions are widely used for improving the momentum resolution in lattice computations of hadronic correlation functions. The method is however expensive since every additional twist requires computing additional propagators. We propose a novel variance reduction technique that exploits statistical correlations between correlators at different twists to reduce the...

We demonstrate that classical bit-flip correction can be employed to mitigate measurement errors on quantum computers. Importantly, our method can be applied to any operator, any number of qubits, and any realistic bit-flip probability. Starting with the example of the longitudinal Ising model, we then generalize to arbitrary operators and test our method both numerically and experimentally on...

Readout errors are among the most dominant sources of error on current noisy intermediate-scale quantum devices. Recently an efficient, scaleable method for mitigating such errors has been developed. Here, we benchmark this correction protocol on IBM's and Rigetti's quantum devices. Measuring observables in the computational basis, we demonstrate how the mitigation procedure improves the...

In modern lattice simulations, conventional update algorithms do not allow for tunneling between topological sectors at fine lattice spacings. We compare the viability of multiple (less commonly used) algorithms with respect to proper sampling of all topological sectors in the Schwinger model. We briefly comment on the prospects of applying these methods to 4-dimensional SU(3) simulations.

Standard sampling algorithms for lattice QCD suffer from topology freezing (or critical slowing down) when approaching the continuum limit, thus leading to poor sampling of the distinct topological sectors. I will present a modified Hamiltonian Monte Carlo (HMC) algorithm that triggers topological sector jumps during the assembly of Markov chain of lattice configurations. We study its...

Flow models are emerging as a promising approach to sampling complicated probability distributions via machine learning in a way that can be made asymptotically exact. For applications to lattice field theory in particular, success has been demonstrated in proof-of-principle studies of scalar theories, gauge theories, and thermodynamic systems. This work develops approaches which enable...

We propose a continuous differentiable and invertible gauge field transformation parametrized with neural networks. We apply this technique to 2D U(1) pure gauge system, combine the transformation and HMC, and train the neural network field transformation for improved tunneling of topological sectors during the gauge generation. We present the properties of the trained transformation applied...

Flow-based models were previously successfully applied for the generation of configurations in lattice (gauge) field theories in two dimensions. In this work, we discuss further development of this approach for lattice gauge theories in four dimensions. We show several implementations and apply improvements to the approach. We study different masking patterns and choices of frozen loops, as...

Critical slowing down presents a critical obstacle to lattice QCD calculation at the smaller lattice spacings made possible by Exascale computers. Inspired by the concept of Fourier acceleration, we study a version of the Riemannian Manifold HMC (RMHMC) algorithm in which the canonical mass term of the HMC algorithm is replaced by a rational function of the gauge-covariant, QCD Laplace...

For an asymptotically free theory, a promising strategy for eliminating Critical Slowing Down (CSD) is naïve Fourier acceleration. This requires the introduction of gauge-fixing into the action, in order to isolate the asymptotically decoupled Fourier modes. In this talk, we present our approach and results from a gauge-fixed Fourier-accelerated hybrid Monte Carlo algorithm, using an action...

We introduce LeapfrogLayers, an invertible neural network architecture that can be trained to efficiently sample the topology of a 2D U(1) lattice gauge theory. We show an improvement in the integrated autocorrelation time of the topological charge when compared with traditional HMC, and propose methods for scaling our model to larger lattice volumes.

We have proposed a concrete experimental setup for quantum simulating the $(1+1)$-dimensional Abelian Higgs model in J. Zhang *et al.* (2018), where the Hamiltonian in the electric field representation can be implemented on a multi-leg ladder with a single atom in each rung. The finite-size scaling of the energy gap can be measured and its universal behavior can be extracted at large enough...

The tensor network framework is an alternative to traditional lattice-based methods where one can simulate the statics as well as dynamics of lattice gauge theories (LGTs) without encountering the sign problem.

So far, however, most tensor network studies of non-abelian LGTs have been restricted in scope to one spatial dimension with open boundary conditions.

This restriction is lifted in...

The classical O(2) model is the zero-gauge-coupling limit of compact scalar quantum electrodynamics in Euclidean spacetime. We obtain two dual representations of the O(2) model, where the field quantum numbers on the plaquettes determine the charge quantum numbers on the links according to Gauss's law. Taking the time continuum limit, we study the Hamiltonians in the two representations with a...

The $q$-state clock model is a classical spin model that corresponds to the Ising model (when $q= 2$) and the XY model (when $q\rightarrow\infty$). The integer-$q$ clock model has been studied extensively and has been shown to have a single phase transition when $q = 2,3,4$ and two phase transitions when $q > 4$.We define an extended-$q$ clock model that reduces to the ordinary $q$-state clock...

It has been recently shown how explicit low-storage Lie group integrators can be built from classical low-storage methods of Williamson's type. We discuss a one-parameter family of three-stage third-order methods and show how a coefficient scheme can be chosen specifically for optimal integration of the gradient flow. We also illustrate how two low-storage fourth-order integrators can be used...

We present regression and compression algorithms for lattice QCD data utilizing the efficient binary optimization ability of quantum annealers. In the regression algorithm, we encode the correlation between the independent and dependent variables into a dictionary optimized for sparse reconstruction. The trained correlation pattern is used to predict lattice QCD observables of unseen lattice...

Previous work has shown that renormalization group blocking of a

2+1 flavor DWF ensemble with 1/a = 2 GeV can produce an ensemble

with 1/a = 1 GeV, with physical quantities on the blocked 1 GeV

ensemble within a few percent of their values on an independently

generated 1 GeV ensemble. This has led us to investigate using the

blocked ensemble DWF operator as a coarse-grid operator for...

I review multigrid algorithms for domain wall fermions and discuss the development status of a domain decomposed hybrid monte carlo appropriate for chiral fermions and tailored for use on GPU accelerated computing nodes.

Graphene can be modeled by the Hubbard model on a honeycomb lattice. However, this system suffers strongly from the sign problem if a chemical potential is included.

Tensor network methods are not affected by this problem. We use the imaginary time evolution of a fermionic Projected Entangled Pair State, which allows to simulate both parity sectors independently. Incorporating the fermionic...

We applied tensor network methods to study strongly coupled U(N) in its dimer formulation. We investigated the chiral condensate as a function of the quark mass and the degree of the symmetry group, and find good agreement with Monte Carlo simulations.

We calculate the free energy of the CP(1) model with a topological term using the tensor renormalization group (TRG) method. TRG calculations allow to use large volumes and to determine the phase structure in the presence of the topological term. In this talk, we will focus on the systematic errors appearing in the calculations and compare our results to previous work. Our TRG calculations...

For many physical systems, the computation of observables amounts to solving an integral over a strongly oscillating complex-valued function. This so-called sign problem renders the numerical evaluation of these integrals a hard computational problem. Complex Langevin dynamics is one numerical method for tackling the sign problem. In this talk, I introduce a generalized framework for this...

Multilevel techniques in lattice were introduced twenty years ago by Lüscher and Weisz as a way to overcome exponential signal-to-noise decay in lattice gauge theory. It is known that the algorithm performs well when the correlation length of the system is small, and less favourably when it is large. In this project, the transition between these regimes is studied. The 2D-Ising Model is used...

We discuss the theoretical foundations of non-equilibrium Monte Carlo simulations based on Jarzynski's equality and present, as an example of application, the determination of the running coupling in the Schrödinger-functional scheme.

To facilitate future efficient calculation of the connected and disconnected parts of the vector-vector correlator and other observables in QCD+QED with C* boundary conditions, dilution and distillation noise reduction techniques were implemented into the OpenQxD program. To find the low-lying eigenmodes of the gauge-covariant Laplacian that form the distillation sub-space, the PRIMME...

OpenQ$^\ast$D code has been used by the RC$^\ast$ collaboration for the generation of fully dynamical QCD+QED gauge configurations with C$^\ast$ boundary conditions. In this talk, optimization of solvers provided with the openQ$^\ast$D package relevant for porting the code on GPU-accelerated supercomputing platforms is discussed. We present the analysis of the current implementations of the...

Lattice QCD simulations directly at physical masses of dynamical light, strange and charm quarks are highly desirable in order to remove systematic errors due to chiral extrapolations. However such simulations are still challenging. We discuss the adaption of efficient algorithms, like higher order integrators or multi-grid methods, within the molecular dynamics of the Hybrid Monte Carlo...

Algebraic multigrid methods have become state of the art at solving discretizations of the Dirac equation in Lattice QCD, in particular, when systems are ill-conditioned. Lattice QCD is bound by the use of supercomputers to speed up their simulations, where algebraic multigrid methods have brought a way of pushing the computational boundaries at large-scale and they open the possibility of...

At physical light quark masses, efficient linear solvers are crucial for carrying out the millions of inversions of the Dirac matrix required for obtaining high statistics in quark correlation functions. Adaptive algebraic multi-grid methods have proven to be very efficient in such cases, exhibiting mild critical slowing down towards very light quark masses and outperforming traditional solver...