Quantum Simulations

Some selected publications

The path to perform quantum simulations in 2+1 and 3+1 dimensions
A resource efficient approach for quantum and classical simulations of gauge theories in particle physics
Jan F. Haase, Luca Dellantonio, Alessio Celi, Danny Paulson, Angus Kan, Karl Jansen, Christine A. Muschik
Published in Quantum 5 (2021) 393
arXiv:2006.14160

This work has formulated a new resource efficient truncation scheme

Figure: Discrete approximation of a continuous distribution of states in the magnetic representation. For a given l, L controls the resolution of the approximation, which is always centred around the vacuum. Black circles represent the U(1) group, the violet 2L+1 edged polygon the Z(2L+1) group. Blue lines (solid and dashed) mark the 2L+1 states of Z(2L+1), while only the 2l+1 states indicated with the solid lines are kept after truncating. Red and green markers are pictorial representations of states in U(1) while the light blue areas correspond to their binned approximation.


Abstract
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2+1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.

The starting point to combine Quantum Computing with classical MC simulations
Strategies for the Determination of the Running Coupling of (2+1)-dimensional QED with Quantum Computing physics
Giuseppe Clemente, Arianna Crippa, Karl Jansen
To be published
arXiv:2206.12454

This work has quantum computed the relevant energy gap to match eventually with classical Markov Chain Monte Carlo simulations

Figure: Results for spectral gap as a function of the coupling g (massless case) in the electric basis. VQD results (dots) and ED (lines); the shaded area corresponds to the region where we could not obtain results with enough precision to estimate the gap reliably.


Abstract
We propose to utilize NISQ-era quantum devices to compute short distance quantities in (2+1)-dimensional QED and to combine them with large volume Monte Carlo simulations and perturbation theory. On the quantum computing side, we perform a calculation of the mass gap in the small and intermediate regime, demonstrating, in the latter case, that it can be resolved reliably. The so obtained mass gap can be used to match corresponding results from Monte Carlo simulations, which can be used eventually to set the physical scale. In this paper we provide the setup for the quantum computation and show results for the mass gap and the plaquette expectation value. In addition, we discuss some ideas that can be applied to the computation of the running coupling. Since the theory is asymptotically free, it would serve as a training ground for future studies of QCD in (3+1)-dimensions on quantum computers.

A first go for oneway computing in high energy physics with VQE
A measurement-based variational quantum eigensolver
Ryan R. Ferguson, Luca Dellantonio, Karl Jansen, Abdulrahim Al Balushi, Wolfgang Dür, Christine A. Muschik
Phys.Rev.Lett. 126 (2021) 22, 220501
arXiv:2010.13940

A different kind of quantum computing the Schwinger model: one-way computing

Figure: (a) Variation of a problem-specific ansatz state by `edge decoration'. An ansatz graph state starts the MB-VQE in a suitable corner of Hilbert space (choice of green island). Next, a classical algorithm explores the neighbourhood (runner on black arrow). The variational optimization exploits a custom state that is obtained by decorating the edges of the ansatz state with auxiliary qubits (orange circles). Their measurement in rotated bases with variational parameters transforms the state vector into the output state. (b) Direct translation of a VQE circuit into a MB-VQE. Left: circuit consisting of Clifford gates (black) and single-qubit parametric gates (`knobs'). Right: corresponding MB-VQE, where the Clifford-part of the circuit has been performed beforehand. The custom state consists of output (white circles) and auxiliary (orange circles) qubits only; the latter are measured in rotated bases and are related to the `knobs' in the circuit.


Abstract
Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entagled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit-based to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times.

My first quantum computing project on the Rigetti machine
Zeta-regularized vacuum expectation values
Tobias Hartung, Karl Jansen
Published in J.Math.Phys. 60 (2019) 9, 093504
arXiv:1808.06784

The amazing work on zeta-regularization, receives much too less attention and recognition

Figure: This figure shows the relative truncation error of the Hamiltonian expectation value computed for a number of qubits ranging from 1 to 11 and compared to the expectation value with 12 qubits as. The fit to an exponential functions indicates exponential convergence. We have chosen m=q=1 for the mass of the electron m and the electric coupling constant q.


Abstract
It has recently been shown that vacuum expectation values and Feynman path integrals can be regularized using Fourier integral operator ζ-function, yet the physical meaning of these ζ-regularized objects was unknown. Here we show that ζ-regularized vacuum expectations appear as continuum limits using a certain discretization scheme. Furthermore, we study the rate of convergence for the discretization scheme using the example of a one-dimensional hydrogen atom in (-π,π) which we evaluate classically, using the Rigetti Quantum Virtual Machine, and on the Rigetti 8Q quantum chip Agave device. We also provide the free radiation field as an example for the computation of ζ-regularized vacuum expectation values in a gauge theory.

The review of quantum simulation for gauge theores
Simulating Lattice Gauge Theories within Quantum Technologies
M.C. Ba&tilden;uls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller
Published in Eur.Phys.J.D 74 (2020) 8, 165
arXiv:1911.00003

This work gives a good overview of quantum simulations of gauge theories

Figure: The Hilbert space H of a quantum many body system (represented here by a 3D box) is exponentially large, since it is the tensor product of the Hilbert spaces of the constituents. Gauge symmetry allows to identify a smaller space, called the physical Hilbert space HP. This is the subspace spanned by those states that fulfill all the local constraints imposed by the gauge symmetry and is represented by a membrane inside H.


Abstract
Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented - a classical simulation approach - applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed.