# Quantum advantage from energy measurements of many-body quantum systems

Leonardo Novo1, Juani Bermejo-Vega2,3,4, and Raúl García-Patrón1

1Centre for Quantum Information and Communication, Ecole Polytechnique de Bruxelles, CP 165, Université Libre de Bruxelles, 1050 Brussels, Belgium
3Carlos I Institute of Theoretical and Matter Physics, University of Granada, Granada, Spain
4Free University of Berlin, Berlin, Germany

### Abstract

The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state – a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be $\textit{exponentially fast-forwarded}$, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.

In recent years, we have seen an incredible progress in the development of quantum computers and simulators. They are reaching a scale that allows us to investigate questions about quantum matter beyond the reach of common simulation methods using standard (classical) computers. But how can we be sure that a quantum device outperforms any classical computer at a certain task? The challenge here is that, for any candidate task where quantum computers may show an advantage, it is hard to rule out the existence of superior classical algorithms that have not yet been discovered.

In our work, we focus on showing quantum advantage for the task of measuring the energy of a quantum system or, more precisely, of sampling outcomes of measurements of a Hamiltonian on a quantum state. This is a fundamental task in quantum mechanics that allows us to learn about the Hamiltonian describing a given quantum system which determines, for example, how it evolves in time or its properties at a given temperature. Inspired by recent results about the complexity of sampling from quantum circuits, we give strong evidence that quantum devices are much faster than classical computers at simulating this measurement process.

One of the main parameters describing an energy measurement is its resolution i.e., how accurately we can measure the energy. We show that, for certain families of Hamiltonians, there are efficient quantum energy measurement protocols that reach a very high resolution, unlikely to be reached by classical simulations. Importantly, this quantum advantage still holds even if the quantum measurement is affected by a limited amount of noise, opening up the possibility for implementations in near-term noisy quantum devices. We also give complexity-theoretic arguments showing that even more coarse-grained energy measurements, with much lower resolution, should be hard to simulate with classical computers. Namely, we show that if a classical computer were able to approximately simulate any such measurement, it would also be able to do any quantum computation efficiently, which is believed to be impossible.

Overall, we believe our work contributes towards strengthening the belief that no present or future classical algorithm can beat quantum devices for the task of simulating quantum physics. Furthermore, it can potentially inspire new theoretical and experimental demonstrations of quantum advantage for other physically-motivated problems.

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### Cited by

[1] Sriram Akella, Kishore Thapliyal, H. S. Mani, and Anirban Pathak, "Dynamics of single-mode nonclassicalities and quantum correlations in the Jaynes–Cummings model", Journal of the Optical Society of America B 39 7, 1829 (2022).

[2] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational fast forwarding for quantum simulation beyond the coherence time", npj Quantum Information 6, 82 (2020).

[3] A. Roggero, "Spectral-density estimation with the Gaussian integral transform", Physical Review A 102 2, 022409 (2020).

[4] Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman, "The importance of the spectral gap in estimating ground-state energies", arXiv:2007.11582.

[5] Jirawat Tangpanitanon, Supanut Thanasilp, Ninnat Dangniam, Marc-Antoine Lemonde, and Dimitris G. Angelakis, "Expressibility and trainability of parametrized analog quantum systems for machine learning applications", Physical Review Research 2 4, 043364 (2020).

[6] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Fault-tolerant quantum speedup from constant depth quantum circuits", arXiv:2005.11539, Physical Review Research 2 3, 033444 (2020).

[7] Jirawat Tangpanitanon, Supanut Thanasilp, Marc-Antoine Lemonde, Ninnat Dangiam, and Dimitris G. Angelakis, "Quantum supremacy in driven quantum many-body systems", arXiv:2002.11946.

The above citations are from Crossref's cited-by service (last updated successfully 2022-07-05 15:37:38) and SAO/NASA ADS (last updated successfully 2022-07-05 15:37:39). The list may be incomplete as not all publishers provide suitable and complete citation data.