Abstract State-of-the-art cosmological simulations on classical computers are limited by time, energy, and memory usage. Quantum computers can perform some calculations exponentially faster than classical computers, using exponentially less energy and memory, and may enable extremely large simulations that accurately capture the whole dynamic range of structure in the universe within statistically representative cosmic volumes. However, not all computational tasks exhibit a “quantum advantage.” Quantum circuits act linearly on quantum states, so nonlinearities (e.g., self-gravity in cosmological simulations) pose a significant challenge. Here we outline one potential approach to overcome this challenge and solve the (nonlinear) Schrödinger–Poisson equations for the evolution of self-gravitating dark matter, based on a hybrid quantum–classical variational algorithm framework (Lubasch et al.). We demonstrate the method with a proof-of-concept mock quantum simulation, envisioning a future where quantum computers will one day lead simulations of dark matter.
Abstract State-of-the-art cosmological simulations on classical computers are limited by time, energy, and memory usage. Quantum computers can perform some calculations exponentially faster than classical computers, using exponentially less energy and memory, and may enable extremely large simulations that accurately capture the whole dynamic range of structure in the universe within statistically representative cosmic volumes. However, not all computational tasks exhibit a “quantum advantage.” Quantum circuits act linearly on quantum states, so nonlinearities (e.g., self-gravity in cosmological simulations) pose a significant challenge. Here we outline one potential approach to overcome this challenge and solve the (nonlinear) Schrödinger–Poisson equations for the evolution of self-gravitating dark matter, based on a hybrid quantum–classical variational algorithm framework (Lubasch et al.). We demonstrate the method with a proof-of-concept mock quantum simulation, envisioning a future where quantum computers will one day lead simulations of dark matter.
Cosmology is in an era of rapid discovery especially in areas related to dark energy, dark matter and inflation. Quantum cosmology treats the cosmology quantum mechanically and is important when quantum effects need to be accounted for, especially in the very early Universe. Quantum computing is an emerging new method of computing which excels in simulating quantum systems. Quantum computing may have some advantages when simulating quantum cosmology, especially because the Euclidean action of gravity is unbounded from below, making the implementation of Monte Carlo simulation problematic. In this paper we present several examples of the application of quantum computing to cosmology. These include a dark energy model that is related to Kaluza-Klein theory, dark matter models where the dark sector is described by a self interacting gauge field or a conformal scalar field and an inflationary model with a slow roll potential. We implement quantum computations in the IBM QISKit software framework and show how to apply the Variational Quantum Eigensolver (VQE) and Evolution of Hamiltonian (EOH) algorithms to solve the Wheeler-DeWitt equation that can be used to describe the cosmology in the mini-superspace approximation. We find excellent agreement with classical computing results and describe the accuracy of the different quantum algorithms. Finally we discuss how these methods can be scaled to larger problems going beyond the mini-superspace approximation where the quantum computer may exceed the performance of classical computation.
Quantum computing has become increasingly popular in recent years, since it has the potential to outperform classical computation, at least in specific tasks. In principle, this advantage could help many fields of physics research, such as cosmology. Numerical simulations of cosmological observables have become more and more complex and demanding in terms of resources. Quantum computers might help and this thesis is devoted to develop this idea within the NISQ era and its limitations. The work focuses both on introducing quantum algorithms in existing classical codes and on developing a hybrid code to simulate the evolution of DM. The first analysis substitutes the Fast Fourier Transform with the Quantum Fourier Transform to compute the matter power spectrum and shows that the current generation of quantum computers is not enough to produce physically relevant results. The second analysis considers the development of a hybrid classical-quantum code to solve Schr\"{o}dinger-Poisson equations in one dimension. Results in ideal simulators prove that the code can be used to find the correct solution, while real quantum computers allow recovering part of the expected signal only at the price of introducing approximations.
We propose a hybrid quantum-classical algorithm for the simulation of real-time dynamics in interacting quantum field theories coupled to classical fields, focusing on the self-consistent estimation of semiclassical backreaction. By discretizing space and time, we construct an iterative protocol that simulates the Trotterized dynamics of the quantum fields subject to the dynamical classical fields. By estimating certain quantum expectation values through a set of projective measurements, we source the equations of motion of the classical fields, and solve them numerically to feed them forward to the quantum simulation in an iterative self-consistent loop. Semiclassical backreaction is relevant in various fields of physics, particularly in cosmology, where quantum matter fluctuations affect the gravitational field dynamics, and a controlled renormalization must be carefully considered to get a sensible continuum limit. We benchmark our algorithm in this context, focusing on scalar-tensor theories of modified gravity exhibiting a chameleon mechanism, such that a light classical scalar field driving cosmic acceleration becomes massive in high-density regions, effectively screening any possible yet unobserved fifth force. By focusing on numerically tractable regimes, we explicitly show the convergence and robustness of our algorithm when considering the continuum limit and the effect of quantum shot noise. Our work paves the way for future experiments exploring other non-tractable regimes, including non-perturbative interactions of the quantum fields and how these can change backreaction and the gravitational dynamics.
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to n=12 Ansatz qubits, computing PDE solutions with 2n grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than 220) of grid points, but also a practical number of gates per circuit and circuit shots.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.
Cosmological simulations describing the evolution of density perturbations of a self-gravitating collisionless dark matter (DM) fluid in an expanding background provide a powerful tool to follow the formation of cosmic structures over wide dynamic ranges. The most widely adopted approach, based on the N-body discretization of the collisionless Vlasov-Poisson (VP) equations, is hampered by an unfavorable scaling when simulating the wide range of scales needed to cover at the same time the formation of single galaxies and of the largest cosmic structures. On the other hand, the dynamics described by the VP equations is limited by the rapid increase of the number of resolution elements (grid points and/or particles) which is required to simulate an ever growing range of scales. Recent studies showed an interesting mapping of the six-dimensional + 1 (6D+1) VP problem into a more amenable 3D+1 nonlinear Schr\"odinger-Poisson (SP) problem for simulating the evolution of DM perturbations. This opens up the possibility of improving the scaling of time propagation simulations using quantum computing. In this paper, we introduce a quantum algorithm for simulating the Schr\"odinger-Poisson (SP) equation by adapting a variational real-time evolution approach to a self-consistent, nonlinear, problem. To achieve this, we designed a novel set of quantum circuits that establish connections between the solution of the original Poisson equation and the solution of the corresponding time-dependent Schr\"odinger equation. We also analyzed how nonlinearity impacts the variance of observables. Furthermore, we explored how the spatial resolution behaves as the SP dynamics approaches the classical limit ($\ensuremath{\hbar}/m\ensuremath{\rightarrow}0$) and discovered an empirical logarithmic relationship between the required number of qubits and the scale of the SP equation ($\ensuremath{\hbar}/m$). This entire approach holds the potential to serve as an efficient alternative for solving the Vlasov-Poisson (VP) equation by means of classical algorithms.