Hamiltonian Learning and Certification Using Quantum Resources

Abstract

In recent years quantum simulation has made great strides culminating in experiments that operate in a regime that existing supercomputers cannot easily simulate. Although this raises the possibility that special purpose analog quantum simulators may be able to perform computational tasks that existing computers cannot, it also introduces a major challenge: certifying that the quantum simulator is in fact simulating the correct quantum dynamics. We provide an algorithm that, under relatively weak assumptions, can be used to efficiently infer the Hamiltonian of a large but untrusted quantum simulator using a trusted quantum simulator. We illustrate the power of this approach by showing numerically that it can inexpensively learn the Hamiltonians for large frustrated Ising models, demonstrating that quantum resources can make certifying analog quantum simulators tractable.Quantum information processing promises to dramatically advance physics and chemistry by providing efficient simulators for the Schrödinger or Dirac equations [1][2][3]. This is important because conventional methods are inefficient, scaling exponentially in the number of interacting subsystems. Consequently, quantum simulations beyond a few tens of interacting particles are generally believed to be beyond the limitations of conventional supercomputers. This inability to simulate large quantum systems means that important questions in condensed matter, such as the shape of the phase diagram for the Fermi-Hubbard model, remain open. Analog quantum simulation raises the possibility that special purpose analog devices may be able to address such problems using current or near-future hardware [4][5][6]. A major objection to this avenue of inquiry is that analog simulators are not necessarily trustworthy [7,8] and certification of them is not known to be efficient. Without such certification, an analog simulator can at best only provide hints about the answer to a given computational question. A resolution to this problem is therefore essential if analog quantum simulators are to compete on an even footing with classical supercomputers.An important first step towards a resolution is provided in [9], where it is shown that quantum systems with local time-independent Hamiltonians can be efficiently characterized given ensemble readout. However, the method is not generally applicable, can be expensive and is not known to be either error robust or stable in cases where single shot measurements are used. A number of machine learning and statistical inference methods [10][11][12][13][14][15][16][17] have been recently introduced to address similar problems in metrology or Hamiltonian learning. In the context of Hamiltonian learning, such ideas have are known to be error-robust and lead to substantial reductions in the cost of high-precision Hamiltonian inference [15], albeit at the price of sacrificing the efficient scaling exhibited by [9].We overcome these challenges by providing a robust method that can be used to characterize unknown Hamiltonians by uni...

Self-Guided Quantum Tomography

“…In the same way as for states, SGQT can be used to find quantum channels where randomized benchmarking [10,11] can be used to efficiently estimate the fidelity to certain classes of unitaries. Finally, we note that to further mitigate the issues of complexity, it may become viable in the future to aid the estimation of the distance measure with quantum resources [12,13], such as the swap test [14].…”

Experimental quantum hamiltonian learning using a silicon photonic chip and a nitrogen-vacancy electron spin in diamond

“…However, the impossibility to efficiently predict the behaviour of complex quantum models on classical machines makes this challenge to be intractable to classical approaches. Quantum Hamiltonian Learning (QHL) [1,2] combines the capabilities of quantum information processing and classical machine learning to allow the efficient characterisation of the model of quantum systems. In QHL the behaviour of a quantum Hamiltonian model is efficiently predicted by a quantum simulator, and the predictions are contrasted with the data obtained from the quantum system to infer the system Hamiltonian via Bayesian methods.…”

Adaptive identification of coherent states

“…Bayesian inference techniques have been developed, e.g, for phase [14,17,23,31,42,43], state [44,45], and Hamiltonian [46][47][48][49][50][51] estimation. For certain one-parameter estimation problems it is possible to perform local Bayesian optimization of the measurement settings analytically [14,23,31,48,49]. For a larger number of unknown parameters, however, finding optimal measurement settings adaptively becomes generally analytically intractable.…”