We introduce the concept of an eigenstate witness and use it to find energies of quantum systems with quantum computers.

Efficiently characterising quantum systems [1-3], verifying operations of quantum devices [4-6] and validating underpinning physical models [7,8], are central challenges for the development of quantum technologies [9-11] and for our continued understanding of foundational physics [12,13]. Machine-learning enhanced by quantum simulators has been proposed as a route to improve the computational cost of performing these studies [14, 15]. Here we interface two different quantum systems through a classical channel -a silicon-photonics quantum simulator and an electron spin in a diamond nitrogen-vacancy centre -and use the former to learn the latter's Hamiltonian via Bayesian inference. We learn the salient Hamiltonian parameter with an uncertainty of approximately 10 −5 . Furthermore, an observed saturation in the learning algorithm suggests deficiencies in the underlying Hamiltonian model, which we exploit to further improve the model itself. We go on to implement an interactive version of the protocol and experimentally show its ability to characterise the operation of the quantum photonic device. This work demonstrates powerful new quantum-enhanced techniques for investigating foundational physical models and characterising quantum technologies.In science and engineering [16,17], physical systems are approximated by simplified models to allow the comprehension of their essential features. The utility of the model hinges upon the fidelity of the approximation to the actual physical system, and can be measured by the consistency of the model predictions with the real experimental data. However, predicting behaviour in the exponentially large configuration space of quantum systems is known to be intractable to classical computing machines [18,19]. A fundamental question therefore naturally arises: How can underpinning theoretical models of quantum systems be validated?To address this question, quantum Hamiltonian learning (QHL) was recently proposed [14,15] as a technique that exploits classical machine learning with quantum simulations to efficiently validate Hamiltonian models and verify the predictions of quantum systems or devices. QHL is tractable in cases in which other known methods fail because quantum simulation is exponentially faster than existing techniques [18][19][20] for simulating broad classes of complex quantum systems [21][22][23][24]. Our experimental demonstration of QHL uses a programmable silicon-photonics quantum simulator, shown in Figs. 1a,b, to learn the electron spin dynamics of a negatively charged nitrogen-vacancy (NV − ) centre in bulk diamond, shown in Figs. 1c,d. We further demonstrate an interactive QHL protocol that allows us to characterise and verify single-qubit gates using other trusted gates on the same quantum photonic device. arXiv:1703.05402v1 [quant-ph]

Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor's factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, nonfault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a silicon quantum photonic device. The approach is verified to be well suited for prethreshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.