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 beexponentially 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.
Highlights
Impressive recent developments in experimental quantum physics are enabling the manipulation of manybody quantum systems of larger and larger sizes
We provide complexity theoretic evidence that an efficient classical simulation of energy measurements should not be possible, and discuss how the latter provides a suitable test of quantum advantage for suitable resolution and sampling error regimes
Due to their relevance in describing physical systems, we focus on measurements of k-local Hamiltonians acting on n qubits i.e., Hamiltonians of the form H = j Hj where each term Hj acts on k qubits, for constant k ∈ O(1)
Summary
Impressive recent developments in experimental quantum physics are enabling the manipulation of manybody quantum systems of larger and larger sizes. It is of utmost importance to put statements about quantum advantage on rigorous mathematical ground This has been the subject of several recent works which demonstrate, based on strong complexity-theoretic evidence, that there are certain tasks that can be performed efficiently by quantum devices for which an efficient classical algorithm cannot exist. We give an explicit example of a simple family of Hamiltonians (e.g., with nearest neighbour interactions on the 2D square lattice) for which energy measurements are hard to simulate on classical computers, yet, should be relatively feasible to measure on a near-term quantum device, which is able to approximately sample from 2D circuits of commuting gates [24] For this example, the correct functioning of the quantum measurement device can be efficiently verified using existing fidelity-witness methods [24], if reliable single-qubit measurements are available. We believe this concept can be of relevance outside the scope of this work and may lead to new investigations of speed-ups with respect to classical algorithms
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