### Conveners

#### Algorithms: Algorithms I

- Urs Wenger (University of Bern)

#### Algorithms: Algorithms II

- Simone Bacchio (The Cyprus Institute)

#### Algorithms: Algorithms III

- Michael Fromm

#### Algorithms: Algorithms IV

- Alexei Bazavov (Michigan State University)

#### Algorithms: Algorithms V

- Lena Funcke (MIT)

#### Algorithms: Algorithms VI

- Akio Tomiya (IPUT Osaka)

#### Algorithms: Algorithms VII

- Fernando Romero-Lopez (MIT)

#### Algorithms: Algorithms VIII

- Jacques Bloch (University of Regensburg)

Emerging sampling algorithms based on normalizing flows have the potential to solve ergodicity problems in lattice calculations. Furthermore, it has been noted that flows can be used to compute thermodynamic quantities which are difficult to access with traditional methods. This suggests that they are also applicable to the density-of-states approach to complex action problems. In particular,...

Normalizing flows (NFs) are a class of machine-learning algorithms that can be used to efficiently evaluate posterior approximations of statistical distributions. NFs work by constructing invertible and differentiable transformations that map sufficiently simple distributions to the target distribution, and provide a new, promising route to study quantum field theories regularized on a...

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to...

In this talk, we discuss gauge-equivariant architectures for flow-based sampling in fermionic lattice field theories with pseudofermions. We also discuss how flow-based sampling approaches can be improved by combination with standard techniques such as even/odd preconditioning and the Hasenbusch factorization. Numerical demonstrations in two-dimensional U(1) and SU(3) theories with $N_f=2$...

Automatic Differentiation (AD) techniques allows to determine the Taylor expansion of any deterministic function. The generalization of these techniques to stochastic problems is not trivial. In this work we explore two approaches to extend the ideas of AD to stochastic processes, one based on reweighting and another one based on the ideas of numerical stochastic perturbation theory using the...

The numerical sign problem has been a major obstacle to first-principles calculations of many important systems, including QCD at finite density. The worldvolume tempered Lefschetz thimble method is a HMC algorithm which solves both the sign problem and the ergodicity problems simultaneously. In this algorithm, configurations explore the extended configuration space (worldvolume) that includes...

I will describe recent progress in the development of custom machine learning architectures based on flow models for the efficient sampling of gauge field configurations. I will present updates on the status of this program and outline the challenges and potential of the approach.

We present our attempts to control the sign problem by the path optimization method with emphasis on efficiency of the neural network. We found a gauge invariant neural network is successful in the 2-dimensional U(1) gauge theory with a complex coupling. We also investigate possibility of the improvement in the learning process.

We present a novel strategy to strongly reduce the severity of the sign problem, using line integrals along paths of changing imaginary action. Highly oscillating regions along these paths cancel out, decreasing their contributions. As a result, sampling with standard Monte-Carlo techniques becomes possible in cases which otherwise requires methods taking advantage of complex analysis, such as...

At fine lattice spacings, lattice simulations are plagued by slow (topological) modes that give rise to large autocorrelation times. These in turn lead to statistical and systematic errors that are difficult to estimate. We study the problem and possible algorithmic solutions in 4-dimensional SU(3) gauge theory, with special focus on instanton updates and metadynamics.

A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which sampling is trivial. A series of seminal studies have demonstrated that approximations of trivializing maps can be 'machine-learned' by a class of invertible...

The recent introduction of machine learning tecniques, especially normalizing flows, for the sampling of lattice gauge theories has shed some hope on improving the sampling efficiency of the traditional HMC algorithm. However, naive usage of normalizing flows has been shown to lead to bad scaling with the volume. In this talk we propose using local normalizing flows at a scale given by the...

The study of real-time evolution of quantum field theories is known to be an extremely challenging problem for classical computers. Due to a fundamentally different computational strategy, quantum computers hold the promise of allowing for detailed studies of these dynamics from first principles. However, much like with classical computations, it is important that quantum algorithms do not...

We propose a variational quantum eigensolver suitable for exploring the phase structure of the multi-flavor Schwinger model in the presence of a chemical potential. The parametric ansatz we design incorporates the symmetries of the model and can be implemented on both measurement-based and circuit-based quantum hardware. We numerically demonstrate that our ansatz is able to capture the phase...

With the long term perspective of using quantum computers for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of $\mathrm{SU}(2)$, such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing...

Simulating SU$(N)$ gauge theories on a quantum computer requires some form of digitization of the gauge degrees of freedom. Recently, we have proposed discretisation schemes, which offer in contrast to finite subgroups the possibility to freely refine the discretisation. Here we present an approach to define the corresponding canonical momentum operators. We present results on the restoration...

Sign problems in Monte Carlo simulations have long hindered studies of phase diagrams of lattice gauge theories (LGTs) at finite densities. Quantum computation of LGTs does not encounter sign problems, but preparing thermal states needed for a complete phase-diagram analysis on quantum devices is a difficult and resource-intensive process. Thermal Pure Quantum (TPQ) states have been proposed...

Quantum computing is a promising new computational paradigm which may allow one to address exponentially hard problems inaccessible in Euclidean lattice QCD. Those include real-time dynamics, matter at non-zero baryon density, field theories with non-trivial CP-violating terms and can often be traced to the sign problem that makes stochastic sampling methods inapplicable. As a prototypical...

The Hamiltonian approach can be used successfully to study the real time evolution of a non-Abelian lattice gauge theory on the available noisy quantum computers. In this talk, results from the real time evolution of SU(2) pure gauge theory on IBM hardware are presented. The long real time evolution spanning dozens of Trotter steps with hundreds of CNOT gates and the observation of a traveling...

Studies of the Schwinger model in the Hamiltonian formulation have hitherto used the Kogut-Susskind staggered approach. However, Wilson fermions offer an alternative approach and are often used in Monte Carlo simulations. Tensor networks allow the exploration of the Schwinger model even with a topological θ-term, where Monte Carlo methods would suffer from the sign problem. Here, we study the...

Quantum simulations of QCD require digitization of the infinite-dimensional gluon field. Schemes for doing this with the minimum amount of qubits are desirable. A practical digitization for SU(3) gauge theories via its discrete subgroup S(1080) has been shown to allow classical simulations down to a=0.08 fm and reproduce thermal and glueball spectrum using modified and improved actions. ...

Open quantum systems are good models of many interesting physical systems. Non-Hermitian Hamiltonians are known to describe, or at least approximate some of these open quantum systems well. Recently, there has been an increase in interest in quantum algorithms for simulating such Hamiltonians, such as the Quantum Imaginary Time Evolution algorithm, and other ones based on trace preserving...

Using D-Wave's quantum annealer as a computing platform, we study lattice gauge theory with discrete gauge groups. As digitization of continuous gauge groups necessarily involves an approximation of the symmetry, we extend the formalism of previous studies on the annealer to finite, simply reducible gauge groups. As an example we use the dihedral group $D_n$ with $n=3,4$ on a two plaquette...

Future quantum computers will enable the study of real-time dynamics of non-perturbative quantum field theories without the introduction of the sign problem. We present ongoing progress on low-dimensional lattice systems which will serve as suitable testbeds for near-term quantum devices. The two systems studied to date are 0+1 dimensional supersymmetric quantum mechanics and the Wess-Zumino...

We present a tensor-network method for strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a...

We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group with the initial tensors and...

Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining...

Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a $q$-state clock model (infinite coupling). In the latter case, a...

Worldline representations were established as a powerful tool for studying bosonic lattice field theories at finite density. For fermions, however, the worldlines still may carry signs that originate from the Dirac algebra and from the Grassmann nature of the fermion fields. We show that a density of states approach can be set up to deal with this remaining sign problem, where finite density...

The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "Rényi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering...

We develop a method to improve on the statistical errors for higher moments using machine learning techniques. We present here results for the dual representation of the Ising model with an external field, derived via the high temperature expansion.

We compare two ways of measuring the same set of observables via machine learning: the first gives any higher moments but has larger statistical...

Supervised machine learning with a decoder-only CNN architecture is used to interpolate the chiral condensate in QCD simulations with five degenerate quark flavors in the HISQ action. From this a model for the probability distribution of the chiral condensate as function of lattice volume, light quark mass and gauge coupling is obtained. Using the model, first order and crossover regions can...

Deep generative models such as normalizing flows are suggested as alternatives to standard methods for generating lattice gauge field configurations. Previous studies on normalizing flows demonstrate proof of principle for simple models in two dimensions. However, further studies indicate that the training cost can be, in general, very high for large lattices. The poor scaling traits of...

Many fascinating systems suffer from a severe (complex action) sign problem preventing us from simulating them with Markov Chain Monte Carlo. One promising method to alleviate the sign problem is the transformation towards Lefschetz Thimbles. Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian. In this...

In Monte Carlo simulations of lattice quantum field theories, if the variance of an estimator of a particular quantity is formally infinite, or very large compared to the square of the mean, then expectation of the estimator can not be reliably obtained using the given sampling procedure. A particularly simple example is given by the Gross-Neveu model where Monte Carlo calculations involve the...

The study of autocorrelation times of various meson operators and the topological charge revealed the presence of hidden harmonic oscillations of the autocorrelations (for the HMC).

These modes can be extracted by smoothing the observables with respect to the Monte Carlo time. While this smoothing procedure removes the largest share of the operator's signal, it can not be excluded that...

When lattice QCD is formulated in sectors of fixed quark numbers, the canonical fermion determinants can be expressed explicitly in terms of transfer matrices. This in turn provides a complete factorization of the fermion determinants in temporal direction. Here we present this factorization for Wilson-type fermions and provide explicit constructions of the transfer matrices. Possible...

The trace of a function $f(A)$, in our case matrix inverse $A^{-1}$, can be estimated stochastically using samples $\tau^*A^{-1}\tau$ if the components of the random vectors $\tau$ obey an appropriate probability distribution, for example when $\tau$ is an i.i.d random vector with each component taking the value $\pm 1$ at equal probability $0.5$, this is known as Hutchinson estimator. This...

In lattice QCD, the trace of the inverse of the discretized Dirac operator appears in the disconnected fermion loop contribution to an observable. As simulation methods get more and more precise, these contributions become increasingly important. Hence, we consider here the problem of computing the trace $\mathrm{tr}( D^{-1} )$, with $D$ the Dirac operator.

The Hutchinson method, which is...

Most Monte Carlo algorithms generally applied to lattice gauge theories, among other fields, satisfy the detailed balance condition (DBC) or break it in a very controlled way. While DBC is not essential to correctly simulate a given probability distribution, it ensures the proper convergence after the system has equilibrated. While being powerful from this perspective, it puts strong...

In this talk I will outline a strategy to include the effects of the electromagnetic interactions of the sea quarks in QCD+QED. When computing leading order corrections in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected diagrams which are not easily computed using stochastic estimators. An analysis of their variance can help construct better estimators for...

We develop digital quantum algorithms for simulating a 1+1 dimensional SU(2) lattice gauge theory in the Schwinger boson and loop-string-hadron (LSH) formulations. These algorithms complement and improve on the algorithm by Kan & Nam (arXiv:2107.12769) based on the angular momentum basis, which generalized an earlier algorithm for a U(1) gauge theory (the Schwinger model) [Quantum 4, 306...

Recently we introduced a new gradient flow based beta-function which is

defined over infinite Euclidean space-time to calculate and integrate

infinitesimal scale changes in RG flows. It can be applied in high-

precision determination of the strong coupling at the Z-pole in QCD. In

this talk we will discuss the results and challenges of the method

applied to quenched QCD ( pure Yang-Mill...

We report results on the Schwinger model at finite temperature and density using a variational algorithm for near-term quantum devices. We adapt β-VQE, a classical-quantum hybrid algorithm with a neural network, to evaluate thermal and quantum expectation values and study the phase diagram for the massless Schwinger model along with the temperature and density. By comparing the exact...

In our work we study lattice QED in 2+1 dimensions,

which serves as a toy model for 3+1-dimensional QCD due to similarities in the behaviour of running coupling.

Moreover, the theory exhibits a rich and interesting phenomenology in itself and can be extended by including a topological term, non-zero matter density and time evolution.

Our main goal is to match physical quantities, such as...

The Ansatz for studying 2+1-dimensional QED on a quantum computer is described. This comprises the transposition of the system onto a quantum circuit, and the Jordan-Wigner transformation for the numerical implementation of fermionic degrees of freedom. In order to find the low-lying eigenvalues of a given Hamiltonian and hence the mass-gap, we discuss an extension of the Variational Quantum...