arXiv subjects

Christian B. Mendl

Publications and source records attributed to Christian B. Mendl.

At least 55 records · Page 3Linked to original sources

arXiv2024

Quantum and classical correlations in shrinking algorithms for optimization

Understanding the benefits of quantum computing for solving combinatorial optimization problems (COPs) remains an open research question. In this work, we extend and analyze algorithms that solve COPs by recursively shrinking them. The algorithms leverage correlations between variables extracted from quantum or classical subroutines to recursively simplify the problem. We compare the performance of the algorithms equipped with correlations from the quantum approximate optimization algorithm (QAOA) as well as the classical linear programming (LP) and semi-definite programming (SDP) relaxations. This allows us to benchmark the utility of QAOA correlations against established classical relaxation algorithms. We apply the recursive algorithm to MaxCut problem instances with up to a hundred vertices at different graph densities. Our results indicate that LP outperforms all other approaches for low-density instances, while SDP excels for high-density problems. Moreover, the shrinking algorithm proves to be a viable alternative to established methods of rounding LP and SDP relaxations. In addition, the recursive shrinking algorithm outperforms its bare counterparts for all three types of correlations, i.e., LP with spanning tree rounding, the Goemans-Williamson algorithm, and conventional QAOA. While the lowest depth QAOA consistently yields worse results than the SDP, our tensor network experiments show that the performance increases significantly for deeper QAOA circuits.

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arXiv2024

Ground-State Preparation of the Fermi-Hubbard Model on a Quantum Computer with 2D Topology via Quantum Eigenvalue Transformation of Unitary Matrices

Quantum computing holds immense promise for simulating quantum systems, a critical task for advancing our understanding of complex quantum phenomena. One of the primary goals in this domain is to accurately approximate the ground state of quantum systems. The Fermi-Hubbard model, particularly, is of profound interest due to its implications for high-temperature superconductivity and strongly correlated electron systems. The quantum eigenvalue transformation of unitary matrices (QETU) algorithm offers a novel approach for ground state estimation by utilizing a controlled Hamiltonian time evolution operator, circumventing the resource-intensive block-encoding required by previous methods. In this work, we apply the QETU algorithm to the $2 \times 2$ Fermi-Hubbard model, presenting circuit simplifications tailored to the model and introducing a mapping to a 9-qubit grid-like hardware architecture inspired by fermionic swap networks. We investigate how the selection of a favorable hardware architecture can benefit the circuit construction. Additionally, we explore the feasibility of this method under the influence of noise, focusing on its robustness and practical applicability.

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arXiv2024

Predicting interacting Green's functions with neural networks

Strongly correlated materials exhibit complex electronic phenomena that are challenging to capture with traditional theoretical methods, yet understanding these systems is crucial for discovering new quantum materials. Addressing the computational bottlenecks in studying such systems, we present a proof-of-concept machine learning-based approach to accelerate Dynamical Mean Field Theory (DMFT) calculations. Our method predicts interacting Green's functions on arbitrary two-dimensional lattices using a two-step ML framework. First, an autoencoder-based network learns and generates physically plausible band structures of materials, providing diverse training data. Next, a dense neural network predicts interacting Green's functions of these physically-possible band structures, expressed in the basis of Legendre polynomials. We demonstrate that this architecture can serve as a substitute for the computationally demanding quantum impurity solver in DMFT, significantly reducing computational cost while maintaining accuracy. This approach offers a scalable pathway to accelerate simulations of strongly correlated systems and lays the groundwork for future extensions to multi-band systems.

cond-mat.str-el↗

arXiv2024

Block encoding of matrix product operators

Quantum signal processing combined with quantum eigenvalue transformation has recently emerged as a unifying framework for several quantum algorithms. In its standard form, it consists of two separate routines: block encoding, which encodes a Hamiltonian in a larger unitary, and signal processing, which achieves an almost arbitrary polynomial transformation of such a Hamiltonian using rotation gates. The bottleneck of the entire operation is typically constituted by block encoding and, in recent years, several problem-specific techniques have been introduced to overcome this problem. Within this framework, we present a procedure to block-encode a Hamiltonian based on its matrix product operator (MPO) representation. More specifically, we encode every MPO tensor in a larger unitary of dimension $D+2$, where $D = \lceil\log(χ)\rceil$ is the number of subsequently contracted qubits that scales logarithmically with the virtual bond dimension $χ$. Given any system of size $L$, our method requires $L+D$ ancillary qubits in total, while the number of one- and two-qubit gates decomposing the block encoding circuit scales as $\mathcal{O}(L\cdotχ^2)$.

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arXiv2024

Quantum-Classical Computing via Tensor Networks

Circuit knitting offers a promising path to the scalable execution of large quantum circuits by breaking them into smaller sub-circuits whose output is recombined through classical postprocessing. However, current techniques face excessive overhead due to a naive postprocessing method that neglects potential optimizations in the circuit structure. To overcome this, we introduce qTPU, a framework for scalable hybrid quantum-classical processing using tensor networks. By leveraging our hybrid quantum circuit contraction method, we represent circuit execution as the contraction of a hybrid tensor network (h-TN). The qTPU compiler automates efficient h-TN generation, optimizing the balance between estimated error and postprocessing overhead, while the qTPU runtime supports large-scale h-TN contraction using quantum and classical accelerators. Our evaluation shows orders-of-magnitude reductions in postprocessing overhead, a $10^4\times$ speedup in postprocessing, and a 20.7$\times$ reduction in overall runtime compared to the state-of-the-art Qiskit-Addon-Cutting (QAC).

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arXiv2024

On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond

The contraction cost of a tensor network depends on the contraction order. However, the optimal contraction ordering problem is known to be NP-hard. We show that the linear contraction ordering problem for tree tensor networks admits a polynomial-time algorithm, by drawing connections to database join ordering. The result relies on the adjacent sequence interchange property of the contraction cost, which enables a global decision of the contraction order based on local comparisons. Based on that, we specify a modified version of the IKKBZ database join ordering algorithm to find the optimal tree tensor network linear contraction order. Finally, we extend our algorithm as a heuristic to general contraction orders and arbitrary tensor network topologies.

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arXiv2024

Efficient Encodings of the Travelling Salesperson Problem for Variational Quantum Algorithms

Routing problems are a common optimization problem in industrial applications, which occur on a large scale in supply chain planning. Due to classical limitations for solving NP-hard problems, quantum computing hopes to improve upon speed or solution quality. Several suggestions have been made for encodings of routing problems to solve them with variational quantum algorithms. However, for an end user it is hard to decide a priori which encoding will give the best solutions according to their needs. In this work, we investigate different encodings for the Travelling Salesperson Problem. We compare their scaling and performance when using the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver and provide a clear guide for users when to choose which encoding. For small instances, we find evidence that the permutation encoding can yield good results since it does not suffer from feasibility issues.

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arXiv2024

PyTreeNet: A Python Library for easy Utilisation of Tree Tensor Networks

In recent years, tree tensor network methods have proven capable of simulating quantum many-body and other high-dimensional systems. This work is a user guide to our Python library PyTreeNet. It includes code examples and exercises to introduce the library's functions and familiarise the reader with the concepts and methods surrounding tree tensor networks. PyTreeNet implements all the tools required to implement general tree tensor network methods, such as tensor decompositions and arbitrary tree structures. The main focus is on the time evolution of quantum systems. This includes an introduction to tree tensor network states and operators and the time-evolving block decimation and time-dependent variational principle. The library's capabilities are showcased with the example of a modified transverse field Ising model on tree structures that go far beyond the ability of common state vector methods.

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arXiv2024

Reducing Mid-Circuit Measurements via Probabilistic Circuits

Mid-circuit measurements and measurement-controlled gates are supported by an increasing number of quantum hardware platforms and will become more relevant as an essential building block for quantum error correction. However, mid-circuit measurements impose significant demands on the quantum hardware due to the required signal analysis and classical feedback loop. This work presents a static circuit optimization algorithm that can substitute some of these measurements with an equivalent circuit with randomized gate applications. Our method uses ideas from constant propagation to classically precompute measurement outcome probabilities. Our proposed optimization is efficient, as its runtime scales polynomially on the number of qubits and gates of the circuit.

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arXiv2024

State Diagrams to determine Tree Tensor Network Operators

This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches $3$.

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arXiv2024

SCIM MILQ: An HPC Quantum Scheduler

With the increasing sophistication and capability of quantum hardware, its integration, and employment in high performance computing (HPC) infrastructure becomes relevant. This opens largely unexplored access models and scheduling questions in such quantum-classical computing environments, going beyond the current cloud access model. SCIM MILQ is a scheduler for quantum tasks in HPC infrastructure. It combines well-established scheduling techniques with methods unique to quantum computing, such as circuit cutting. SCIM MILQ can schedule tasks while minimizing the makespan, i.e., the time that elapses from the start of work to the end, improving on average by 25%. Additionally, it reduces the noise in the circuit by up to 10%, increasing the outcome's reliability. We compare it against an existing baseline and show its viability in an HPC environment.

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arXiv2024

Quantum-Informed Recursive Optimization Algorithms

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We demonstrate the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow (depth $p=1$) circuits of the quantum approximate optimization algorithm (QAOA), solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor available online on Amazon Braket to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics, such as simulated annealing and greedy algorithms, even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms, highlighting the potential of QIRO. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a blueprint for designing a broader class of hybrid quantum-classical algorithms for combinatorial optimization.

quant-ph↗

arXiv2024

Evaluating Ground State Energies of Chemical Systems with Low-Depth Quantum Circuits and High Accuracy

Solving electronic structure problems is considered one of the most promising applications of quantum computing. However, due to limitations imposed by the coherence time of qubits in the Noisy Intermediate Scale Quantum (NISQ) era or the capabilities of early fault-tolerant quantum devices, it is vital to design algorithms with low-depth circuits. In this work, we develop an enhanced Variational Quantum Eigensolver (VQE) ansatz based on the Qubit Coupled Cluster (QCC) approach, which demands optimization over only $n$ parameters rather than the usual $n+2m$ parameters, where $n$ represents the number of Pauli string time evolution gates $e^{-itP}$, and $m$ is the number of qubits involved. We evaluate the ground state energies of $\mathrm{O_3}$, $\mathrm{Li_4}$, and $\mathrm{Cr_2}$, using CAS(2,2), (4,4) and (6,6) respectively in conjunction with our enhanced QCC ansatz, UCCSD (Unitary Coupled Cluster Single Double) ansatz, and canonical CCSD method as the active space solver, and compare with CASCI results. Finally, we assess our enhanced QCC ansatz on two distinct quantum hardware, IBM Kolkata and Quantinuum H1-1.

quant-ph↗

arXiv2023

Universality and classification of elementary thermal operations

Elementary thermal operations are thermal operations that act non-trivially on at most two energy levels of a system at the same time. They were recently introduced in order to bring thermal operations closer to experimental feasibility. A key question to address is whether any thermal operation could be realized via elementary ones, that is, whether elementary thermal operations are universal. This was shown to be false in general, although the extent to which elementary thermal operations are universal remained unknown. Here, we characterize their universality in both the sense described above and a weaker one, where we do not require them to decompose any thermal operation, but to be able to reproduce any input-output pair connected via thermal operations. Moreover, we do so for the two variants of elementary thermal operations that have been proposed, one where only deterministic protocols are allowed and one where protocols can be conditioned via the realization of a random variable, and provide algorithms to emulate thermal operations whenever their elementary counterparts are (weakly or not) universal. Lastly, we show that non-deterministic protocols reproduce thermal operations better than deterministic ones in most scenarios, even when they are not universal. Along the way, we relate elementary thermal operations to random walks on graphs.

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arXiv2023

Multithreaded parallelism for heterogeneous clusters of QPUs

In this work, we present MILQ, a quantum unrelated parallel machines scheduler and cutter. The setting of unrelated parallel machines considers independent hardware backends, each distinguished by differing setup and processing times. MILQ optimizes the total execution time of a batch of circuits scheduled on multiple quantum devices. It leverages state-of-the-art circuit-cutting techniques to fit circuits onto the devices and schedules them based on a mixed-integer linear program. Our results show a total improvement of up to 26 % compared to a baseline approach.

quant-ph↗

arXiv2023

Riemannian quantum circuit optimization for Hamiltonian simulation

Hamiltonian simulation, i.e., simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation (TEBD) algorithm.

quant-ph↗

arXiv2023

Trotter error with commutator scaling for the Fermi-Hubbard model

We derive higher-order error bounds with small prefactors for a general Trotter product formula, generalizing a result of Childs et al. [Phys. Rev. X 11, 011020 (2021)]. We then apply these bounds to the real-time quantum time evolution operator governed by the Fermi-Hubbard Hamiltonian on one-dimensional and two-dimensional square and triangular lattices. The main technical contribution of our work is a symbolic evaluation of nested commutators between hopping and interaction terms for a given lattice geometry. The calculations result in explicit expressions for the error bounds in terms of the time step and Hamiltonian coefficients. Comparison with the actual Trotter error (evaluated on a small system) indicates that the bounds still overestimate the error.

quant-ph↗

arXiv2023

Quantum Wasserstein GANs for State Preparation at Unseen Points of a Phase Diagram

Generative models and in particular Generative Adversarial Networks (GANs) have become very popular and powerful data generation tool. In recent years, major progress has been made in extending this concept into the quantum realm. However, most of the current methods focus on generating classes of states that were supplied in the input set and seen at the training time. In this work, we propose a new hybrid classical-quantum method based on quantum Wasserstein GANs that overcomes this limitation. It allows to learn the function governing the measurement expectations of the supplied states and generate new states, that were not a part of the input set, but which expectations follow the same underlying function.

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