Quantum Algorithms and Methods

Some selected publications

Mini review of error mitigation and dimensional expressivity analysis
Towards Quantum Simulations in Particle Physics and Beyond on Noisy Intermediate-Scale Quantum Devices
Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Manuel Schneider, Paolo Stornati, Xiaoyang Wang
Published in Phil.Trans.A.Math.Phys.Eng.Sci. 380 (2021) 2216, 20210062
arXiv:2110.03809

This work is an invited article in the Philosophical Transaction of the Royal Physical Society

Figure: (Left:) Representation of the tangent space (illustrated in yellow) of the manifold M spanned by the tangent vector (blue arrow). (Right:) Energy histogram for the transverse field Ising model for 2048 experiments, N=4 qubits, and couplings J=-1, h=1. For each of the experiments, we used 2048 shots and a homogeneous bit-flip probability p=0.05 for all qubits. The vertical dashed green line indicates the true ground-state energy, the solid orange line the theoretical prediction for the noisy measurement outcomes, and the dashed black line a fit to the data.


Abstract
We review two algorithmic advances that bring us closer to reliable quantum simulations of model systems in high energy physics and beyond on noisy intermediate-scale quantum (NISQ) devices. The first method is the dimensional expressivity analysis of quantum circuits, which allows for constructing minimal but maximally expressive quantum circuits. The second method is an efficient mitigation of readout errors on quantum devices. Both methods can lead to significant improvements in quantum simulations, e.g., when variational quantum eigensolvers are used.

Dimensional expressivity analysis of quantum circuits
Dimensional Expressivity Analysis of Parametric Quantum Circuits
Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Paolo Stornati
Published in Quantum 5 (2021) 422
arXiv:2011.03532

This work allows to identify superfluous gates and hence reduces noise in quantum simulations

Figure: Success of gradient based variational quantum simulation to continually deform the known ground state of a Hamiltonian H(0) to the target Hamitonian H(1). The VQS ground state energy (blue solid line) coincides the analytic ground state (orange dashed line). The first excited state energy (green dash-dotted line) is shown as well. The inset shows the relative deviation of the VQS ground-state energy from the exact result.


Abstract
Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, we develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

An efficient readout error mitigation method
Measurement Error Mitigation in Quantum Computers Through Classical Bit-Flip Correction
Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Paolo Stornati, Xiaoyang Wang
Published in Phys.Rev.A 105 (2022) 6, 062404
arXiv:2007.03663

our way of doing readout error mitigation

Figure: Mean value (blue dots) and standard deviation (orange triangles) of the absolute error after applying the correction procedure (filled symbols) and without it (open symbols) as a function of the number of shots s. The solid green line corresponds to a power law fit to all our data points for the mean absolute error, the red dashed line to fit including the lowest four number of shots. Different panels correspond to single-qubit data obtained on quantum hardware (a) ibmq london and (b) ibmq burlington.


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.

Formulating and testing a θ-term in 3+1 dimensions
Investigating a (3+1)D Topological θ-Term in the Hamiltonian Formulation of Lattice Gauge Theories for Quantum and Classical Simulations
Angus Kan, Lena Funcke, Stefan Kühn, Luca Dellantonio, Jinglei Zhang, Jan F. Haase, Christine A. Muschik, Karl Jansen
Published in Phys.Rev.D 104 (2021) 3, 034504
arXiv:2105.06019

building the ground to study topological terms in 3+1 dimensions

Figure: Low-lying spectrum as a function of θ for β = 0.3. The ground state and the first excited state show an avoided level-crossing at θ =0.333 indicating the occurence of the expected phase transition.


Abstract
Quantum technologies offer the prospect to efficiently simulate sign-problem afflicted regimes in lattice field theory, such as the presence of topological terms, chemical potentials, and out-of-equilibrium dynamics. In this work, we derive the (3+1)D topological θ-term for Abelian and non-Abelian lattice gauge theories in the Hamiltonian formulation, paving the way towards Hamiltonian-based simulations of such terms on quantum and classical computers. We further study numerically the zero-temperature phase structure of a (3+1)D U(1) lattice gauge theory with the θ-term via exact diagonalization for a single periodic cube. In the strong coupling regime, our results suggest the occurrence of a phase transition at constant values of θ, as indicated by an avoided level-crossing and abrupt changes in the plaquette expectation value, the electric energy density, and the topological charge density. These results could in principle be cross-checked by the recently developed (3+1)D tensor network methods and quantum simulations, once sufficient resources become available.

Classical splitting a quantum circuit
Classical Splitting of Parametrized Quantum Circuits
Cenk Tüysüz, Giuseppe Clemente, Arianna Crippa, Tobias Hartung, Stefan Kühn, Karl Jansen
To be published
arXiv:2206.09641

Testing a classical splitting for classification and VQE

Figure: All types of ansätze used. (a) An N-qubit generic ansatz consisting of L layers of the parametrized unitary U are separated in to k = N/m many m-qubit ansätze. This ansatz will be referred to as the classically split (CS) ansatz. The standard ansatz can be recovered by setting m = N. (b) Extended classically split (ECS) ansatz. This is an extension to the CS ansatz. First L layers of the ansatz consists of k = N/m many m qubit U blocks. Then, T layers of N qubit V layers are applied. (c) A simple ansatz that consists of RY rotation gates and CX gates connected in a ladder layout. (d) Hardware Efficient Ansatz (HEA) that is used to produce the quantum dataset. Parameters of the first column of U3 gates are sampled from a uniform distribution. (e) EfficientSU2 ansatz with full entangler layers.


Abstract
Barren plateaus appear to be a major obstacle to using variational quantum algorithms to simulate large-scale quantum systems or replace traditional machine learning algorithms. They can be caused by multiple factors such as expressivity, entanglement, locality of observables, or even hardware noise. We propose classical splitting of ansätze or parametrized quantum circuits to avoid barren plateaus. Classical splitting is realized by splitting an N qubit ansatz to multiple ansätze that consists of O(log N) qubits. We show that such an ansatz can be used to avoid barren plateaus. We support our results with numerical experiments and perform binary classification on classical and quantum datasets. Then, we propose an extension of the ansatz that is compatible with variational quantum simulations. Finally, we discuss a speed-up for gradient-based optimization and hardware implementation, robustness against noise and parallelization, making classical splitting an ideal tool for noisy intermediate scale quantum (NISQ) applications.

Feynman graphs on a quantum computer
Variational quantum eigensolver for causal loop Feynman diagrams and acyclic directed graphs
Giuseppe Clemente, Arianna Crippa, Karl Jansen, Selomit Ramírez-Uribe, Andrés E. Rentería-Olivo, Germán Rodrigo, German F. R. Sborlini, Luiz Vale Silva
To be published
arXiv:2210.13240

Testing a classical splitting for classification and VQE

Figure: Representative topologies used for the performance comparison of Grover algorithm and VQE implementations. We considered the following configurations: (A) two eloops with four vertices; (B) three eloops with four vertices; and (C) four eloops with five vertices. Also, we include the complete set of four-eloop six-vertex topologies: (D) t-channel; (E) s-channel; and (F) u-channel.


Abstract
We present a variational quantum eigensolver (VQE) algorithm for the efficient bootstrapping of the causal representation of multiloop Feynman diagrams in the Loop-Tree Duality (LTD) or, equivalently, the selection of acyclic configurations in directed graphs. A loop Hamiltonian based on the adjacency matrix describing a multiloop topology, and whose different energy levels correspond to the number of cycles, is minimized by VQE to identify the causal or acyclic configurations. The algorithm has been adapted to select multiple degenerated minima and thus achieves higher detection rates. A performance comparison with a Grover's based algorithm is discussed in detail. The VQE approach requires, in general, fewer qubits and shorter circuits for its implementation, albeit with lesser success rates.