www.nature.com/npjqi ARTICLE OPEN IBM Q Experience as a versatile experimental testbed for simulating open quantum systems Guillermo García-Pérez 1,2,4*, Matteo A. C. Rossi 1,4 and Sabrina Maniscalco1,3 The advent of noisy intermediate-scale quantum (NISQ) technology is changing rapidly the landscape and modality of research in quantum physics. NISQ devices, such as the IBM Q Experience, have very recently proven their capability as experimental platforms accessible to everyone around the globe. Until now, IBM Q Experience processors have mostly been used for quantum computation and simulation of closed systems. Here, we show that these devices are also able to implement a great variety of paradigmatic open quantum systems models, hence providing a robust and flexible testbed for open quantum systems theory. During the last decade an increasing number of experiments have successfully tackled the task of simulating open quantum systems in different platforms, from linear optics to trapped ions, from nuclear magnetic resonance (NMR) to cavity quantum electrodynamics. Generally, each individual experiment demonstrates a specific open quantum system model, or at most a specific class. Our main result is to prove the great versatility of the IBM Q Experience processors. Indeed, we experimentally implement one and two-qubit open quantum systems, both unital and non-unital dynamics, Markovian and non-Markovian evolutions. Moreover, we realise proof-of-principle reservoir engineering for entangled state generation, demonstrate collisional models, and verify revivals of quantum channel capacity and extractable work, caused by memory effects. All these results are obtained using IBM Q Experience processors publicly available and remotely accessible online. 1234567890():,; npj Quantum Information (2020) 6:1 ; https://doi.org/10.1038/s41534-019-0235-y INTRODUCTION from being completely understood and, in fact, it is peppered with The theory of open quantum systems studies the dynamics of unanswered questions of deep nature. quantum systems interacting with their surroundings.1–3 In its most The increasing ability to coherently control an ever increasing general formulation, it allows us to describe the out-of-equilibrium number of individual quantum systems, together with the 6 properties of quantum systems, it provides a theoretical framework discovery of quantum coherence in complex biological systems, to assess the quantum measurement problem, and it gives us the has brought to light several scenarios in which the Markovian tools to investigate, understand and counter the deleterious effects assumption fails. This has in turn given rise to a proliferation of of noise on quantum technologies. For these reasons its range results on the characterisation of memory effects and non- 7–10 of applicability is extremely wide, from solid state physics to Markovian dynamics. Interestingly, the cross-fertilisation of quantum field theory, from quantum chemistry and biology to ideas from quantum information theory and open quantum quantum thermodynamics, from foundations of quantum theory systems has led to a new understanding of memory effects in – to quantum technologies. terms of information backflow,11 16 namely a partial return of Generally, the dynamics of open quantum systems are quantum information previously lost from the open system due to described in terms of a master equation, i.e., the equation of the interaction with the environment. motion for the reduced density operator describing the quantum Experimental results on quantum reservoir engineering, includ- state of the system. Master equations are either phenomenolo- ing the possibility to design desired forms of non-Markovian – gically postulated or derived microscopically from a Hamiltonian dynamics,17 25 naturally lead to the question of whether or not model of quantum system plus environment. Contrarily to the memory effects are useful for quantum technologies, in the sense – case of closed quantum systems, where the equation of motion of constituting a resource for certain tasks.15,26 31 This question describing the state dynamics is the Schrödinger equation, the has not yet been satisfactorily answered. Even more remarkably, a general form of the master equation for an open quantum system complete understanding of non-Markovianity is still missing, as is not known. Only under certain assumptions, known as the clearly illustrated in the insightful review of ref. 10 Born–Markov approximation, one can derive a general equation in Given the considerations above, it is not surprising that in the so called Lindblad form, able to describe the physical recent years a number of experiments have been proposed and evolution of quantum states.1–5 When these assumptions are realised to verify paradigmatic open quantum system models and not satisfied, e.g., for strong system–environment interaction and/ test some of their predictions. Examples are numerous: the or long-living environmental correlations, we enter the intricate milestone experiment on the decoherence of a Schrödinger cat (and somewhat fuzzy) reign of non-Markovian dynamics. This state with trapped ions is one of the first examples of engineered consideration already illustrates how, despite its indubitable environment,32 followed by the open system quantum simulator foundational nature, open quantum systems theory is still far of ref. 33 The latter is also the first experimental realisation of an 1QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland. 2Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turun Yliopisto, Finland. 3QTF Centre of Excellence, Center for Quantum Engineering, Department of Applied Physics, Aalto University School of Science, FIN-00076 Aalto, Finland. 4These authors contributed equally: Guillermo García-Pérez, Matteo A. C. Rossi. *email: ggaper@utu.fi Published in partnership with The University of New South Wales G. García-Pérez et al. 2 idea that shifted our perspective about environmental noise. family Φt of completely positive and trace preserving (CPTP) maps, 34,35 ρ Φ ρ ρ Following a proposal by Vertsrate et al., experimentalists known as the dynamical map: SðtÞ¼ t Sð0Þ, with Sð0Þ the proved that, by engineering certain types of Markovian master initial state. The equation of motion for the state of the system is equation, one may actually create entangled states as stationary the master equation and, if the dynamical map is invertible, can be asymptotic states of the dynamics, therefore turning dissipation written in a time-local form and decoherence from enemies to allies of quantum 33,36,37 ρ_ ρ ; technologies. SðtÞ¼Lt SðtÞ (1) In optical platforms, simulators of Markovian open quantum where Lt is the time-dependent generator of the dynamics systems have been used to prove the existence of interesting Z phenomena, such as sudden death of entanglement38,39 and t 40,41 Φ τ ; sudden transition from quantum to classical decoherence. In t ¼ T exp Lτ d (2) the same platform, experiments have shown how to engineer 0 19 collisional models, wherein the microscopic interaction between with T the chronological ordering operator, and Φ0 ¼ I. Under system and environment is obtained through a sequence of rather general conditions,1 the generator can be written in the collisions between the open quantum system and one or more form ancillae, the latter collectively describing the environment.42 More P 17–19,21,23,24 43 ρ ; ρ γ ρ y 1 y ; ρ : recently, experiments in linear optics, cavity QED, Lt SðtÞ¼Ài½HS SðtÞ þ k kðtÞ Vk SðtÞVk À 2 fVkVk SðtÞg NMR22 and trapped ions25 successfully demonstrated the engi- neering of non-Markovian open quantum systems and monitored (3) the Markovian to non-Markovian crossover. Also complex In the equation above, the first term on the r.h.s. describes the quantum networks have been proposed as new systems for unitary dynamics, with HS the system Hamiltonian, and the second 44 reservoir engineering of arbitrary spectral densities, and a term, the dissipative dynamics induced by the interaction with the bosonic implementation with optical frequency combs has been environment, with γ ðtÞ and V the decay rates and jump 45 k k presented. operators, respectively. If the decay rates are positive and Most of the experiments until now realised for simulating open γ γ constant, i.e., kðtÞ 0, the dynamical map is a semigroup, quantum systems rely on the idea of analogue quantum simulator, Φtþt0 ¼ Φt Φt0 , and we refer to the dynamics as semigroup that is a quantum system whose dynamics resemble those of Markovian. Extending this definition, it is nowadays common to 1234567890():,; another quantum system that we wish to study and understand. say that the dynamics are Markovian whenever all the decay rates In contrast, a digital quantum simulator is a gate-based quantum γ kðtÞ are positive at all times. In this case, the dynamical map computer which can be used to simulate any physical system, if satisfies the property of CP-divisibility, namely Φt ¼ Φt;s Φs, with suitably programmed.46,47 Φt;s a two-parameter family of CPTP maps. Non-Markovian Theoretical and experimental research on open systems digital dynamics occur, instead, whenever at least one of the decay fl 48–53 quantum simulators is only now starting to ourish. While, in rates becomes negative for a certain interval of time. In this case, principle, general algorithms for digital simulation of open Φ 48,50,53 the intermediate map t;s is not CP anymore and the dynamics is quantum systems have been theoretically investigated, non-CP-divisible. their experimental implementation poses several challenges, since In the following subsections, we present experiments run on the physical quantum gates depend on the experimental platform the IBM Q Experience processors simulating different types of and the circuit decomposition needs to be optimised in view of open quantum systems dynamics, both Markovian and non- gates and measurement errors as well as qubit connectivity.