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. Markovian. We begin by considering an example of Markovian Therefore, the existence of general algorithms for implementing semigroup master equation and dynamical evolution. theoretically universal open quantum system simulators does not guarantee the practical implementability in a realistic experiment on a given platform. Markovian reservoir engineering In this paper, we demonstrate that a careful circuit decomposi- For decades, noise induced by the environment has been tion allows us to experimentally implement a vast number of considered the archetype enemy of quantum technologies. This fundamental open quantum systems models for one and two is because very often the interaction between a quantum system qubits. Not only are we able to generate different classes of open and its surroundings leads to the fast disappearance of quantum quantum dynamics, namely, unital (e.g., pure dephasing dynami- properties, notoriously coherences and entanglement, playing a cal maps), non-unital (e.g., amplitude damping dynamical maps), key role in providing quantum advantage. This point of view phase covariant, and non-phase covariant (e.g., Pauli dynamical drastically changed as soon as physicists demonstrated that maps), but also we can explore the Markovian to non-Markovian appropriate manipulation of an artificial environment (quantum crossover, including the recently discovered examples of essen- reservoir engineering) would allow one to steer the open system tial16 and eternal54,55 non-Markovianity. We implement a recently towards, e.g., a maximally entangled state,33,36 hence turning proposed non-Markovianity witness56 and we use our simulator to upside down the perspective of the environment as an enemy. prove the non-monotonic behaviour of quantum channel Following the lines of ref., 33 in this subsection we experimen- capacity15 and extractable work,31 with implications to quantum tally simulate a semigroup Markovian master equation for a two- communication and quantum thermodynamics. qubit open system having as asymptotic stationary state the Bell Overall, our results clearly prove that even small quantum state ψÀ ¼ p1ffiffi ð 01 À 10 , where we indicate with 0 and 1 ji 2 ji jiÞ ji ji processors are versatile and robust testbeds for verifying a the computational basis of each qubit and we use the notation number of theoretical open quantum systems results and ji01 ¼ ji0 1ji1 2. This allows us to prepare a maximally entangled predictions, therefore paving the way to both new discoveries state as a result of the dissipative open system dynamics. and a deeper understanding of one of the most fascinating and fi Each of the four Bell states is uniquely determined as an fundamental elds of quantum physics. ð1Þ ð2Þ eigenstate with eigenvalues ± 1 with respect to σz σz and ð1Þ ð2Þ ðiÞ ðiÞ σx σx , where σx and σz , with i ¼ 1; 2, are the x and z Pauli RESULTS operators of qubit 1 and 2. The dissipative dynamics that pumps We consider an open quantum system represented by a density two qubits from an arbitrary initial state into the Bell state jiψÀ is ρ operator S. The dynamics of the open system are described by a realised by the composition of two channels that pump from the
npj Quantum Information (2020) 1 Published in partnership with The University of New South Wales G. García-Pérez et al. 3
Fig. 1 Simulation of the Markovian reservoir engineering. Every plot compares the measured overlap between the state of the system and the four Bell states (dots) with the corresponding analytical prediction for the ZZ pump (a), the XX pump (b) and their consecutive application (c). The pumps are applied to an initial maximally mixed state. The first two pumps clearly reduce the populations of the þ1 eigenspace of the corresponding stabiliser operator, and their composition results on a probability for the jiψÀ state around 0.94. Throughout the paper, the error bars indicating statistical errors are not shown, as they would be indistinguishable from the markers given the large number of experimental runs (see Methods).
ð1Þ ð2Þ Sudarshan–Lindblad form.4 Interestingly, contrarily to the micro- þ1 into the À1 eigenspaces of the stabiliser operators σz σz scopic derivation of the Markovian master equation, this approach and σð1Þ σð2Þ. x x does not rely on Born–Markov approximation, but it automatically Specifically, we consider the two p-parametrised families of leads to a CPTP dynamical semigroup. Also non-Markovian master Φ ρ ρ y ρ y CPTP maps zz S ¼ E1z SE1z þ E2z SE2z, with equations can be introduced in the collisional model picture, for ffiffiffi p Ið1Þ σð2Þ 1 I σð1Þ σð2Þ ; example by allowing for the ancillae to interact with each other in E1z ¼ p x þ z z fi 57–59 2 a well de ned manner. 1 ð1Þ ð2Þ We implement experimentally an exemplary model of dynamics E ¼ I À σ σ (4) 16 2z 2 z z displaying the property of essential non-Markovianity, following ffiffiffiffiffiffiffiffiffiffiffi 60 p ð1Þ ð2Þ the lines of ref. As research in the field of non-Markovian open þ 1 À p 1 I þ σ σ ; 2 z z quantum systems progressed, more refined definitions of memory 16 Φ ρ ρ y ρ y effects started to appear in the literature. In ref. a hierarchical and xx S ¼ E1x SE1x þ E2x SE2x, where E1x and E2x have the same classification of non-Markovianity was introduced, generalising ð2Þ form of E1z and E2z in Eq. (4), provided that we replace σx with the notion of CP-divisibility. In particular, the dynamical map is ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ Φ σz and σz σz with σx σx . said to be P-divisible if the two-parameter family t;s is positive. By changing the parameter 0 p 1 we simulate different This is, of course, a weaker condition than CP-divisibility, since types of open quantum system dynamics. For p 1, the repeated there exist maps that are P-divisible but not CP-divisible, while all application of, e.g., Φzz generates a master equation of Lindblad CP-divisible maps are also P-divisible. We say that the dynamics is form with jump operator operator V ¼ 1 Ið1Þ σð2Þ essentially non-Markovian if it is non-P-divisible, while it is weakly 2 x non-Markovian if it is non-CP-divisible but P-divisible. ð1Þ ð2Þ À I þ σz σz . For p ¼ 1, the map Φxx Φzz generates jiψ We consider n ancillae initially prepared in the classically ρ 1 n n n n for any initial state. correlated state corr ¼ 2 ðji0 hj0 þ ji1 hj1 Þ. The collision In Fig. 1, we show the action of the dissipative pumping maps between the system qubit and the kth environmental ancilla is Φ Φ Φ Φ igτσz xx, zz and their composition xx zz as a function of p, for a described by the unitary operator Uk ¼ e ji0 khjþ0 τσ maximally mixed initial state. We compare the theoretical Àig z e ji1 khj1 , where g is the coupling strength. The dynamical prediction (dashed lines) with the experimental data. Our results map after n ¼ t=τ collisions is given by show a very good agreement between theory and experiment for Φ ρ ¼ Tr ½U ÁÁÁU U ðρ ρ ÞUy Uy ÁÁÁUy the implementation of both the two families of maps and their t S E n 2 1 S corr 1 2 n (5) 2 τ ρ 2 τ σ ρ σ : composition. In the latter case, the results are more sensitive to ¼ cos ðng Þ S þ sin ðng Þ z S z errors because of the larger depth of the circuit implementing the composition of maps. Details on the circuit implementations and We compare this dynamics to the case in which the fi environmental ancillae are prepared in the uncorrelated state on their optimisation with respect to the speci c qubit n connectivity are given in Methods. jiþ . In this case, the dynamics is given by ΦðþÞρ ¼ 1 ð1 þ cosnð2gτÞÞρ t S 2 S (6) Collisional model and essential non-Markovianity 1 n τ σ ρ σ : þ 2 ð1 À cos ð2g ÞÞ z S z Our second example of an open quantum system simulator deals τ π= with a class of models known as collisional models. These describe In the weak-coupling case (when g < 4) the map in Eq. (6) the interaction between a quantum system and its environment in gives rise to Markovian dynamics. In contrast, the map in Eq. (5) π= terms of consecutive collisions between the system, in our case a alternates intervals, with periodicity 2, in which it is P-divisible τ π= qubit, and a sequence of environmental qubits (ancillae) in a given (0 Published in partnership with The University of New South Wales npj Quantum Information (2020) 1 G. García-Pérez et al. 4 Fig. 2 Simulation of a collisional model. The red triangles show the collisions with the ancillae prepared in a classically correlated state (red triangles), while the blue dots show the case of a separable state. In both cases g ¼ 1, τ ¼ π=6 (weak-coupling regime). The red and blue dashed curves show, respectively, the theoretical predic- tions of Eqs. (5) and (6) for time t ¼ ngτ. Up to 7 collisions were simulated: the depth of the circuits grows with the number of collisions, causing increasing decoherence, but the oscillations due to the essential non-Markovianity are clearly visible in the case of correlated ancillae. are required to swap the ancillae that have interacted with the qubit with fresh ones: there is a small probability that the swap fails and the system interacts again with the same ancilla. This can introduce memory effects, as shown, e.g., in refs. 58,59 Markovian and non-Markovian dissipative dynamics The dynamical maps of Eqs. (5) and (6) are purely dephasing, since Fig. 3 Simulation of the amplitude damping channel. a Markovian they affect only the coherences of the qubit. In this section we dynamics (R ¼ 0:2). b Non-Markovian dynamics (R ¼ 100). Blue simulate an exactly solvable dissipative open quantum system triangles show the measured populations of the excited state as a known as the Jaynes–Cummings or generalised amplitude function of time, whereas orange circles stand for the non- Markovianity witness, given by the probability of observing the damping model. The microscopic derivation of the master ϕþ equation can be found in textbooks (see, e.g., ref. 1). The master system and an ancilla, initially prepared in the state ji, in that same state as the channel acts on the system (see Methods for equation is given by further details). In b, the experimental results clearly show a non- dρ 1 monotonic behaviour, hence confirming the non-Markovian char- S ðtÞ¼γðtÞ σ ρ ðtÞσ À fσ σ ; ρ ðtÞg ; (7) dt À S þ 2 þ À S acter of the dynamical map. Dashed lines show the theoretical predictions for both quantities. where σ ± are the raising and lowering operators and the time- dependent decay rate γðtÞ has the following analytical form c_ ðtÞ case the dynamical map is not CP-divisible, and therefore γðtÞ¼À2R 1 ; (8) c1ðtÞ non-Markovian. In Fig. 3a, b we show, as example of Markovian and non-Markovian dynamics, the evolution of the excited state with  ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffiffi population of an initially excited state for two values of the Àλt=2 λt c1ðtÞ¼c1ð0Þe cosh 1 À 2R parameter R, corresponding to Markovian (R ¼ 0:2) and non- ÀÁ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi i λ p (9) Markovian (R ¼ 100) dynamics, respectively. The figures show þ pffiffiffiffiffiffiffiffi1 sinh t 1 À 2R : 1À2R 2 the monotonic decay of the excited state population in the former The equation above is obtained assuming that the environment case, while in the latter case the population oscillates since the is a bosonic zero-temperature reservoir with Lorentzian spectral qubit exchanges information and energy with the central mode of density the Lorentzian peak, resonant with the qubit’s Bohr frequency. As done for the other open quantum systems simulators, we 1 γ λ2 JðωÞ¼ 0 ; (10) compare the experimental data with theoretical predictions, π ω ω 2 λ2 2 ð 0 À Þ þ finding a good agreement. with ω0 the qubit frequency, γ an overall coupling strength, and λ We also implement an experimentally friendly witness of non- 0 56 the half height width of the Lorentzian profile. The time- Markovianity which was recently introduced in ref. and stems dependent coefficient c1ðtÞ defined in Eq. (9) crucially depends from the spectral properties of the dynamical map. The witness is γ =λ on the ratio R ¼ 0 between the coupling strength and the based on the behaviour of an initial maximally entangled state of width of the spectrum. Also, this coefficient fully determines the qubit and auxiliary ancilla, and requires only the measurement of qubit dynamics as one can see from the following expression σ σ ! the expectation values of local observables such as i i, 2 (i x; y; z). Non-monotonic behaviour of the witness as a function c t cÃc t ¼ ρ j 1ð Þj 0 1ð Þ : SðtÞ¼ (11) of time signals non-Markovianity. Other witnesses of non- à 2 11 c0c1ðtÞ 1 Àjc1ðtÞj Markovianity (such as the BLP measure introduced in ref. ) From Eq. (9) a straightforward calculation shows that the time- could be implemented, and may be more efficient at detecting dependent decay rate γðtÞ,defined in Eq. (8), takes negative memory effects, but they generally require a larger number of values for certain time intervals whenever 2R 1. In this measurements or optimisation over initial states. npj Quantum Information (2020) 1 Published in partnership with The University of New South Wales G. García-Pérez et al. 5 CP-divisible for all times t. More precisely, we use this experimental simulation to demonstrate a phenomenon predicted in ref., 31 namely the presence of oscillations in the extractable work. This shows an application of open quantum system simulation on the IBM Q Experience processors to fields other than quantum information theory, specifically quantum thermo- dynamics for the example here considered. DISCUSSION The results presented in the previous section highlight how few- qubit NISQ devices publicly available on the cloud already provide sufficient robustness and reliability to implement experimentally a Fig. 4 Simulation of the depolarising channel. Tomography of a single-qubit density matrix initially prepared in the state number of open quantum systems dynamics, both Markovian and ψ π= π= iπ=4 non-Markovian, both unital (dephasing collisional model) and ji0 ¼ cos 80jiþ sin 8e ji1 , for various values of the para- meter p (cf. Eq. (12)). non-unital (amplitude damping, depolarising and Pauli channels). We have simulated all the paradigmatic open quantum systems In Fig. 3, we plot the dynamics of the entanglement witness for models typically used to demonstrate physical phenomena the amplitude damping channel. In Fig 3b, the witness clearly induced by the presence of the environment, its consequences fi shows oscillatory behaviour, and therefore properly signals the for quantum technological applications, but also possible bene ts presence of memory effects. The circuits implementing both the in the spirit of reservoir engineering. As an example of a amplitude damping model and the non-Markovianity witness are fundamental study on open quantum systems, we have measured presented and discussed in Methods. the dynamics of a non-Markovianity witness and showed that it correctly signals the presence of memory effects for time-local amplitude damping dynamics. Depolarising and Pauli channels In this section, we build on these results to substantiate our The depolarising channel is one of the most common models of claim of the IBM Q Experience being a versatile testbed for the qubit decoherence due to its nice symmetry properties. We can experimental verification of physical effects due to the open describe it by stating that, with probability 1 À p the qubit remains character of the dynamics. Specifically, we focus on two intact, while it becomes maximally mixed with probability p. This applications: non-Markovian quantum channel capacity, of poten- mixing can be modelled through a bit flip error, described by the tial use in quantum communication, and extractable work action of σx, a phase flip error, described by the action of σz,or dynamics, relevant in quantum thermodynamics. both, described by the action of σy. The dynamical map of a Let us first consider a typical setting in quantum information Markovian open quantum system subjected to depolarising noise processing and communication: Alice and Bob are at the opposite can be written as ends of a quantum channel, the former sending information X 3 pðtÞ (classical or quantum) and the latter receiving it. The maximum Φtρ ¼ 1 À pðtÞ ρ þ σiρ σi; (12) amount of information that can be reliably transmitted along a S 4 S 4 S i noisy quantum channel is known as the channel capacity. fi fi where i ¼ x; y; z and pðtÞ¼1 À eÀγt, with γ the Markovian Speci cally, we consider the quantum capacity Q de ned as the decay rate. limit to the rate at which quantum information can be reliably sent In Fig. 4, we plot the qubit density matrix elements for various down a quantum channel. values of p, comparing the experimental data with the theoretical In the following, we focus on the time-local amplitude damping prediction. For exemplary purposes, we choose a specific initial model introduced in the previous section. In this case, the state possessing non-zero coherences, but we have verified, by quantum channel capacity can be calculated exactly and takes the form15 repeating the experiment for different initial states, that the hi agreement observed between experiment and theory is indepen- 2 2 QðΦtÞ¼ max H2ðjc1ðtÞj pÞÀH2ð½1 Àjc1ðtÞj pÞ ; (14) dent from the initial state chosen. As for the other simulated p2½0;1 models, we postpone the discussion on the circuit implementa- 2 for jc ðtÞj > 1=2 and QðΦ Þ¼0 otherwise. In the equation above, tion, the readout, and the error mitigation strategy to the Methods 1 t H is the binary Shannon entropy and c ðtÞ is given by Eq. (9). section. 2 1 One may be led into believing that the quantum channel Let us now introduce the most general single-qubit open capacity decreases, and in some case disappears, for increasing quantum system model, namely the time-dependent Pauli lengths of the noisy channel, due to the cumulative destructive channel. The master equation in this case takes the form effect of decoherence. This, however, only holds for Markovian ρ X 15 d S 1 noise, as theoretically demonstrated in ref. Indeed, due to ðtÞ¼ γ ðtÞ½σiρ ðtÞσi À ρ ðtÞ : (13) dt 2 i S S memory effects, the quantum channel capacity may take again i non-zero values after disappearing for a certain finite length of the γ γ We note that for iðtÞ¼ , we recover the Markovian channel. In Fig. 5a, we experimentally demonstrate this effect, depolarising channel. Generally, the dynamics described by the comparing the measured behaviour of QðΦtÞ with the theoretical master equation above is not phase-covariant,61 except for the prediction for an amplitude damping model with R ¼ 100, γ γ case in which xðtÞ¼ yðtÞ. Moreover, since the decay rates may R ¼ 200, and R ¼ 400. take negative values, conditions for complete positivity must be We now consider an example of interest in the field of quantum imposed, and they are given in terms of a set of inequalities thermodynamics, specifically related to the presence of memory involving all the three decay rates, as one can see, e.g., from ref. 16 effects in the open system dynamics. In ref., 31 the connection In the Discussion section we present the simulation of a specific between information backflow and thermodynamic quantities form of time-dependent Pauli channel proposed in ref. 54 and was established by means of a non-Markovianity indicator based used as an example of eternal non-Markovianity, i.e., an open on the quantum mutual information. More specifically, it was quantum system dynamics for which the dynamical map is non- proven that there exists a link between memory effects, as Published in partnership with The University of New South Wales npj Quantum Information (2020) 1 G. García-Pérez et al. 6 We now further specify our analysis to the eternal non- γ γ λ= Markovianity model, for which xðtÞ¼ yðtÞ¼ 2 and γ ω ω = zðtÞ¼À tanhð tÞ 2. We compare the dynamics with the case γ γ λ= γ ω ω = fi in which xðtÞ¼ yðtÞ¼ 2 and zðtÞ¼ tanð tÞ 2. The rst γ dynamics is eternally non-Markovian, since zðtÞ < 0 at all times, while the second one is non-CP-divisible but not eternally non- γ Markovian, since zðtÞ takes both positive and negative values. This comparison allows us to distinguish between quantum and classical memory effects.31 In Fig. 5b, we show the experimental results and the theoretical prediction for the dynamics of the extractable work for the two cases here considered. Despite the presence of experimental imperfections, clear oscillations of the extractable work are visible in one case, but completely absent in the other one, therefore showing for the first time not only the impact of memory effects on a thermodynamic quantity such as Wex, but also the subtle origin of these oscillations. To conclude, it is worth mentioning that experiments on open quantum systems performed in dedicated experimental labora- tories in different platforms such as, e.g., trapped ions, linear optical systems, superconducting qubits, NMR, NV centres in diamonds and cavity QED, generally reach higher precision and fidelity than those reported in our paper. However, this often occurs at the expense of increased specificity in terms of the models that can be simulated. Moreover, these experiments are accessible only to the researchers working in the given laboratory. Open source small quantum processors already are, however, a Fig. 5 Applications in quantum communication and thermody- namics. a Quantum channel capacity as a function of time for three reality and they have opened the door of experimental physics to 2 quantum researchers who do not have either specificexperimental different values of R calculated from Eq. (14), where jc1ðtÞj has been fi approximated by the ratio between the population of the excited training or suf cient economic resources to set up a laboratory. We state at time t and at t ¼ 0 in order to account for the deviations in therefore envisage a rapid change in the way in which research is the initial state preparation. b Rescaled extractable work conducted in a field which has been until now dominantly WexðtÞ=kTln2 as a function of time for the eternally non-Markovian theoretical, namely the study of open quantum system dynamics. γ γ λ= γ ω ω = Pauli channel, with xðtÞ¼ yðtÞ¼ 2 and zðtÞ¼À tanh t 2, Perhaps, in a not too far future, this will blur the line separating λ ω 1 with ¼ 1 and ¼ 2 (green circles) and for the non-CP-divisible theoretical and experimental research, and an increasing number of γ γ λ= γ ω ω = λ Pauli channel xðtÞ¼ yðtÞ¼ 2 and zðtÞ¼ tan t 2, with ¼ experiments will be remotely programmable by using a simple 0:1 and ω ¼ 2. The extractable work has been evaluated using Eq. interface and language, in the true spirit of a quantum simulator. (15) from a full two-qubit tomography of system S and memory ancilla M. METHODS indicated by oscillations in the quantum mutual information and The results presented in this paper have been obtained on the IBM Q the non-monotonic behaviour of the extractable work Wex. Experience processors freely available online: two 5-qubit machines, To understand this connection, let us recall that in order to link ibmqx2 and ibmq_vigo, and a 14-qubit machine, ibmq_16_mel- 63 non-Markovianity with the evolution of thermodynamical quan- bourne. The circuits have been written using IBM’sQiskit, a Python-based tities one needs to describe not only the system S, whose programming language for the IBM Q Experience. Each circuit has been run for 8192 shots (the maximum allowed by the devices) to gather information content we are interested in, but also an observer or statistics from the measurements. When needed, one- and two-qubit full memory M. Interestingly, quantum mechanical correlations state tomography has been obtained by using the tools provided in Qiskit, between system and observer may lead to the exciting possibility performing a maximum-likelihood reconstruction of the density matrix.64 62 of extracting work while erasing information on the system. In the following, we give detailed explanations of the various circuits Following ref., 31 we consider the case of a qubit S subjected to implemented to produce the results presented in the paper. We first make a Pauli dynamical map, see master Eq. (13), and an isolated qubit a few general considerations on the sources of error and on how to devise acting as memory, M. If the system and memory are prepared in a circuits with high fidelity on the IBM Q Experience devices. maximally entangled state, work can be extracted during the Given the large number of experimental runs allowedpffiffiffiffi by the devices, the errors due to statistical fluctuations, of order Oð1= NÞ0:011, are too information erasure by using the initial entanglement, as – 62 small for errorbars to be distinguishable from markers in Figs 1 5, and thus predicted in ref. In this framework, the extractable work takes they are not shown. The discrepancy between the experimental points and 31 the form the theoretical prediction, on the other hand, comes from systematic errors induced by various factors in the hardware implementation. First, the Φ ρ Φ ρ : ρ ; WexðtÞ¼½n À Hð t SÞþIð t S MÞ kTln2 (15) qubits undergo relaxation and decoherence due to external noise. This error becomes more relevant when increasing the depth of the circuit. Φ ρ with n the number of qubits in S, Hð t SÞ the von Neumann Second, the gates have errors due to cross-talk and unwanted interactions Φ ρ : ρ entropy of the evolved state of the system, Ið t S MÞ the between qubits when addressing them with pulses. In particular, the error quantum mutual information quantifying the amount of total rate of CNOT gates is about 10 times larger than single-qubit unitaries. correlations between system S and memory M, k the Boltzmann Finally, there is a readout error affecting the quantum measurement, constant and T the temperature of the reservoir used for the although this can be mitigated to some extent, as we discuss later on. The error rates and noise parameters on IBM Q devices are characterised on a erasure. For the open system here considered, revivals of daily basis using randomised benchmarking techniques. Since there are extractable work are due to the interplay between memory various, interdependent sources of noise, modelling and predicting the effects, witnessed by the quantum mutual information, and deviations in the experimental data is a non-trivial matter, beyond the entropy dynamics, as dictated by Eq. (15). scope of this paper. In this work, all quantum channels have been npj Quantum Information (2020) 1 Published in partnership with The University of New South Wales G. García-Pérez et al. 7 Fig. 6 Circuits implementing the reservoir engineering protocol. a ZZ pump, b XX pump and c ZZ and XX pump. The circuits were run on ibmqx2. Qubits s1 and s2 (corresponding to q2 and q1 on the device) are the system qubits, while qubits aXX and aZZ (q4 and q0) are the environment ancillae for the two maps. State preparation and measurement are not shown in the circuits above. decomposed into circuits bearing in mind some general guidelines we explain the resulting circuits, let us mention that, since the IBM Q targeted at minimising the aforementioned inaccuracies. Experience devices are not equipped with the reset operation, we must Since the IBM Q Experience devices are universal quantum computers, use a different ancilla for every pump. they enable the implementation of any unitary transformation of their The way we map the eigenspace information into an ancilla is by first applying a CNOT gate between the system qubits. Suppose that qubits s constituent qubits; once a quantum circuit is provided for its execution, it is pffiffiffi1 ’ ± compiled into an equivalent circuit involving only the machine s basis and s2 are initially in some Bell state, for instance, ji¼ðϕ ji00 ±11jiÞ= 2. gates, that is, those realisable experimentally. In the case of the IBM Q A CNOT gate controlled by s1 transforms the state intoji ± ji0 . Instead, Experience devices, these include any single-qubit rotation, whereas the jiψ ± would be transformed intoji ± ji1 . Hence, we see that the only multi-qubit gates are CNOTs. However, if the circuit requires a multi- ð1Þ ð2Þ information regarding the σx σx eigenspace (namely, the sign) is qubit gate among qubits that are not physically connected, the contained in the state of s1 after the transformation, whereas the one corresponding gate will be replaced with a longer circuit in which the corresponding to the σð1Þ σð2Þ is in qubit s . Now, let us consider the states are swapped to neighbouring qubits in the compiled circuit. Since z z 2 σð1Þ σð2Þ every swap gate includes a minimum of three CNOT gates, and these circuit implementing the z z pump in Fig. 6a. To map the introduce considerable noise to the execution, it is crucial to assign the eigenspace information into the environment ancilla aZZ, we apply a relevant qubits involved in the simulation (e.g., system, environment and CNOT controlled by the relevant qubit, s2. After these two gates (and ancillae) to the machine’s qubits so that the number of CNOT gates considering that the initial state of the ancilla isji 1 ), aZZ will be in stateji 1 ± ± between disconnected qubits is minimised. Furthermore, the devices are if the initial state of the system is jiϕ andji 0 if it is jiψ . Therefore, the calibrated daily and the errors of the basis gates are reported. This conditional rotation gate only acts in the former case, while it does not information can also be taken into account in the qubit assignment, as modify the state in the latter. The angle of the controlled rotation, in turn,ffiffiffi fi θ p using the CNOT gates with smaller errors is preferable. controls the ef ciency of the pump p via the relation ¼ 2 arcsin p. In addition to the gate errors, the qubits’ readout errors characterising Finally, the last two CNOT gates simply revert the mapping part of the ð1Þ ð2Þ the discrepancies between the qubit state and the measurement outcome circuit. The working principle of the σx σx pump (Fig. 6b) is essentially probabilities are also provided. In the IBM Q Experience devices, there are the same. However, we need to add an extra Hadamard gate to transform considerable differences in the readout errors of the different qubits, so the state of s1 before mapping the information to the ancilla aXX. As for the this information should also be taken into account in the qubit-assignment composite pump, we can simply concatenate the two circuits. Notice that process: if possible, it is preferable to assign the system (and any auxiliary in the direct concatenation there would be two consecutive CNOTs ancillae whose measurement is required) to low readout-error qubits, while between the system qubits, which can be removed. qubits with large readout errors can still be used to simulate the Regarding the measurement process, the IBM Q Experience platform only environment or other ancillae that need not be measured. In any case, enables measurement in the computational basis. Hence, to assess the Qiskit provides post-processing error-mitigation tools that generally probabilities for each of the Bell states, we need to change basis by applying improve the experimental results under the package ignis. To do so, again a combination of CNOT and Hadamard to the system qubits, so jiϕþ is we first prepare all possible basis statesji 0 ÁÁÁ0 ; ji0 ÁÁÁ1 ; ¼ ; ji1 ÁÁÁ1 and mapped intoji 00 , etc. Again, notice that this would result in repeated measure their corresponding outcome probabilities (notice that it is only consecutive gates, the effect of which amounts to identity, so they can be necessary to include the qubits whose measurement outcomes need to be removed from the circuits. Finally, in Fig. 1, we show the results starting from corrected). Once these are known, they can be used to correct any other the maximally mixed state. To do so, we simulated the circuits preparing the experimental result by finding, via likelihood maximisation, the experi- system in four different initial conditions,ji 00 ,01ji,10jiandji 11 . mental outcome that is most congruent with the observed measurement deviations. All the data shown in the paper have been mitigated as Collisional model described above, with the only exception of the collisional model with the The circuit used to implement the collisional model described in the ancillae prepared in the separable state; the reason why we have not section “Collisional model and essential non-Markovianity” is depicted in mitigated those results is that, due to the qubit swapping, the Fig. 7. Initially, the system and ancillae are prepared in the desired initial measurements are performed on different physical qubits. state, then each collision is applied, and finally the qubit is measured in the igτσ Àigτσ σx basis. The collision unitary U ¼ e z ji0 hj0 þ e z ji1 hj1 can be Reservoir engineering implemented by means of a rotation RzðgτÞ around the Z-axis between In ref., 33 the authors provide the circuits for the implementation of the two CNOT gates. Bell-state pumping. However, these are composed of gates that are natural When the ancillae are prepared in the state ρ n n n n = to the trapped-ions platform used in that work, so their direct corr ¼ðji0 hj0 þ ji1 hj1 Þ 2, we can simplify the circuit by noticing implementation on the IBM Q Experience devices would result in far too that each ancilla is in the maximally mixed state, and the action of U does long circuits. Therefore, we propose a different set of circuits that follow not affect its state. We thus onlypffiffiffi need three ancillae prepared in the GHZ ψ = the same basic working principles, but have been designed specifically state jiGHZ ¼ðji000 þ jiÞ111 2: by tracing out the third one we are left ρð2Þ = keeping in mind the characteristics of the IBM Q Experience platform. with the two-qubit state E ¼ðji00 hjþ00 ji11 hjÞ11 2, and then we can The pumping circuits proposed in ref. 33 are composed of four parts. keep colliding with either of the two ancillae. First, the relevant information regarding the state of the system (that is, The circuit with the ancillae in the separable state was implemented and whether the system is in the þ1 or the À1 eigenspaces of the stabiliser run on the device ibmq_16_melbourne (14 qubits), in order to have a operators) is mapped into an ancilla. Second, the state of the system is larger number of ancillae for the collisions. The system qubit was chosen to modified depending on the state of the ancilla. Third, the mapping circuit have the highest connectivity (three qubits) and smallest readout error. is reversed. At this stage, the system has been pumped, but if the ancilla is The connectivity layout of the computer does not allow for direct collisions to be used again for a new pumping cycle, it needs to be reset, which is with more than three ancillae, and swaps between qubits are required, the fourth step. We follow these same lines, designing circuits that perform increasing the errors in the simulation and possibly introducing memory these same steps while minimising the number of gates involved. Before effects (as discussed in the Results section). Published in partnership with The University of New South Wales npj Quantum Information (2020) 1 G. García-Pérez et al. 8 Fig. 9 Circuit implementation of the depolarising channel. The circuit was run on ibmqx2. Qubit q2 is used as the system qubit, as it is the only one that can be targeted with three CNOTs. The ancillae a1; a2 and a3 are initially in the state ji0 and rotated around the y θ 1 direction by an angle ðpÞ¼2 arccosð1 À 2pÞ. They then act as Fig. 7 Circuits implementing the collisional model. The top circuit controls for controlled-X,-Y and -Z gates, respectively. shows the case of the collision with three ancillae in the separable jiþþþ state, prepared by means of Hadamard gates. Each collision suggested in ref., 56 we need not use an extra CNOT gate to project on the consists of a rotation around Z between two CNOTs. A final Bell basis; instead, we can take advantage of the fact that jiϕþ hjϕþ ¼ measurement in the X basis is performed in order to measure the ðI I þ σx σx À σy σy þ σz σzÞ=4 and measure the corresponding coherence of the system qubit. The bottom circuit shows how, in the local observables. Figure 8 shows the circuit corresponding to the case of ancillae prepared in the correlated state, we can effectively measurement of σx σx. simulate the mappffiffiffi with three ancillae prepared in the GHZ state = ðji000 þ jiÞ111 2, by colliding alternately with two of them. Notice Depolarising channel that, in principle, we could have always a collision with the same fi ancilla. This, however, would cause the compiler to remove The depolarising channel de ned in Eq. (12) can be implemented, for any value of p pðtÞ2½0; 1, with the circuit shown in Fig. 9. Three ancillary consecutive CNOTs and join the rotations into a single gate, making ψ θ= θ= the circuit trivial. The top circuit was run on ibmq_16_melbourne, qubits are prepared in a state ji¼θ cos 20jiþsin 21ji, and are used while the bottom circit was run on ibmqx2. as controls for, respectively, a controlled-X (CNOT), a controlled-Y and a controlled-Z rotation. This way, each gate will be applied with a probability sin2θ=2. The rotation angle θ must be chosen so that each of the gates is applied with probability p. Notice that applying X and then Y, but not applying Z is equivalent (up to global phases) to just applying Z, and so on. The resulting equation that binds θ to p is thus θ θ θ θ p sin2 cos4 þ sin4 cos2 ¼ ; (16) 2 2 2 2 4 θ 1 with solution ðpÞ¼2 arccosð1 À 2pÞ. Pauli channel At a specific time instant t, the Pauli channel described by the master Eq. (13) can be written as X3 ρ σ ρσ ; Fig. 8 Circuit for the amplitude damping channel with non- Eð Þ¼ pi i i (17) Markovianity witness. The circuit was run on ibmq_vigo. Qubits i¼0 P q1, q2 and q3 were used as system, environment and witness ancilla, θ with 0 pi 1 and ipi ¼ 1. The depolarising channel is a special case of respectively. Setting ¼ arccos c1ðtÞ, the channel acting on the = the Pauli channel where p1 ¼ p2 ¼ p3 ¼ p 4. system qubit simulates the amplitude damping dynamics. The two One could think to implement the channel by generalising the circuit of Hadamard gates rotate the state of the qubits in order to measure Fig. 9, specifically, by allowing for the three ancillae to be prepared in σ σ σ σ y the observable x x. To measure y y, the combination S H different states. One can see, however, that this cannot be done in the σ σ can be used instead, whereas no gates are required for z z. most general case. For example, the eternally non-Markovian channel presented in the Discussion section is not implementable in this way. It is possible to implement the general Pauli channel with just two fi Amplitude damping channel qubits, if we allow them to be in an entangled state. The rst qubit acts as the control for a controlled-X (CNOT) gate, and the second one for a The circuit in Fig. 8 implements the amplitude damping channel with the controlled-Y. Notice that, as remarked in the previous section, applying non-Markovianity witness. For an arbitrary pure state of the system both X and Y is effectively equivalent to applying a Z gate. ψ α β jis ¼ ji0 s þ ji1 s, and setting the state of the environment to vacuum The state jiψ of the ancillae needed for the Pauli channel can be ji0 e, the two gates between the system and environment qubits transform implemented by the circuit in Fig. 10, parametrised by the three angles α β θ β θ θ ; θ ; θ the joint state into ji0 sji0 e þ cos ji1 sji0 e þ sin ji0 sji1 e. Therefore, 1 2 3 identifying the statesji 0 s andji 1 s with the ground and excited states ψ θ ji¼ðc1c2c3 þ s1s2s3Þji00 þðc1c2s3 À s1s2c3Þji01 respectively, and by choosing ¼ arccos c1ðtÞ, the reduced state of the (18) system becomes Eq. (11). þðc1s2c3 À s1c2s3Þji10 þðs1c2c3 þ c1s2s3Þji11 The non-Markovianity witness for a channel Φ proposed in ref. 56 is t where c cos θ and s sin θ . based on the dynamical behaviour of the entanglement between the i i i i The angles θi can be found by solving the following system of equations: system and an ancilla initially prepared in a maximally entangled state. 8 þ þ þ þ 2 2 Namely, this quantity is defined as f Φ ¼hϕ jI Φ ½jϕ ihϕ jjϕ i. Hence, > p 00 ψ c c c s s s t > 0 ¼jh j ij ¼ð 1 2 3 þ 1 2 3Þ implementing this quantity requires preparing the state jiϕþ between the < ψ 2 2 p1 ¼jh01j ij ¼ðc1c2s3 À s1s2c3Þ system and an ancilla, which can be achieved using a Hadamard and a : (19) > p ¼jh10jψij2 ¼ðc s c À s c s Þ2 CNOT gate, applying the dynamical map to the system and measuring the :> 2 1 2 3 1 2 3 þ ψ 2 2 probability of finding the joint system and ancilla state in jiϕ .As p3 ¼jh11j ij ¼ðs1c2c3 þ c1s2s3Þ npj Quantum Information (2020) 1 Published in partnership with The University of New South Wales G. García-Pérez et al. 9 18. Chiuri, A., Greganti, C., Mazzola, L., Paternostro, M. & Mataloni, P. Linear optics simulation of quantum non-Markovian dynamics. Sci. Rep. 2, 17520 (2012). 19. Cuevas, Á. et al. All-optical implementation of collision-based evolutions of open quantum systems. Sci. Rep. 9, 3205 (2019). 20. Yu, S. et al. Experimental investigation of spectra of dynamical maps and their relation to non-Markovianity. Phys. Rev. Lett. 120, 060406 (2018). 21. Bernardes, N. K. et al. Experimental observation of weak non-Markovianity. Sci. Rep. 5, 17520 (2015). 22. Bernardes, N. K. et al. High resolution non-Markovianity in NMR. Sci. Rep. 6, 33945 (2016). Fig. 10 Circuit implementation of a general Pauli channel on 23. Cialdi, S. et al. All-optical quantum simulator of qubit noisy channels. Appl. Phys. ibmqx2. Qubit q2 is used as the system qubit because it can be Lett. 110, 081107 (2017). targeted with two CNOTs and has the smallest readout error. The 24. Liu, Z.-D. et al. Experimental implementation of fully controlled dephasing ancillae are initially in the state ji0 and rotated around y with the dynamics and synthetic spectral densities. Nat. Commun. 9, 3453 (2018). angles θ1; θ2 and θ3 found by solving the system in Eq. (19), and they 25. Wittemer, M., Clos, G., Breuer, H.-P., Warring, U. & Schaetz, T. Measurement of act as controls for a controlled X and a controlled Y. quantum memory effects and its fundamental limitations. Phys. Rev. A 97, 020102 (2018). The above system of equations allows for multiple analytical solutions 26. Liu, B.-H. et al. Efficient superdense coding in the presence of non-Markovian whose expressions are too cumbersome to be reported here. The choice of noise. EPL 114, 10005 (2016). the solution to use in each case depends on a number of factors, such as 27. Rossi, M. A. C., Cattaneo, M., Paris, M. G. A. & Maniscalco, S. Non-Markovianity is the gate fidelity for the specific values of the parameters. not a resource for quantum spatial search on a star graph subject to generalized Notice that the circuit for the Pauli channel can be used also for the percolation. Quantum Meas. Quantum Metrol. 5,40–49 (2018). depolarising channel, instead of the three-qubit circuit of Fig. 9, but it 28. Laine, E.-M., Breuer, H.-P. & Piilo, J. Nonlocal memory effects allow perfect tele- proves to be less accurate on the IBM Q Experience devices, presumably portation with mixed states. Sci. Rep. 4, 4620 (2014). because of the required entanglement between the two qubits. 29. Chin, A. W., Huelga, S. F. & Plenio, M. B. Quantum metrology in non-Markovian environments. Phys. Rev. Lett. 109, 233601 (2012). 30. Addis, C., Ciccarello, F., Cascio, M., Palma, G. M. & Maniscalco, S. Dynamical DATA AVAILABILITY decoupling efficiency versus quantum non-Markovianity. New J. Phys. 17, 123004 (2015). All data supporting the findings of this study are available from the authors upon 31. Bylicka, B., Tukiainen, M., Chruściński, D., Piilo, J. & Maniscalco, S. Thermodynamic request. The code to reproduce all the results is available online at https://github. power of non-Markovianity. Sci. Rep. 6, 27989 (2016). com/matteoacrossi/ibmq_open_quantum_systems. 32. Myatt, C. J. et al. Decoherence of quantum superpositions through coupling to engineered reservoirs. Nature 403, 269–273 (2000). Received: 21 June 2019; Accepted: 6 December 2019; 33. Barreiro, J. T. et al. An open-system quantum simulator with trapped ions. Nature 470, 486–491 (2011). 34. Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633–636 (2009). 35. Poyatos, J. F., Cirac, J. I. & Zoller, P. Quantum reservoir engineering with laser REFERENCES cooled trapped ions. Phys. Rev. Lett. 77, 4728–4731 (1996). 1. Breuer, H.-P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford 36. Barreiro, J. T. et al. Experimental multiparticle entanglement dynamics induced by University Press, 2007). decoherence. Nat. Phys. 6, 943–946 (2010). 2. Weiss, U. Quantum Dissipative Systems. (World Scientific, Singapore, 2011). 37. Krauter, H. et al. Entanglement generated by dissipation and steady state 3. Rivas, Á. & Huelga, S. F. Open Quantum Systems. (Springer Berlin, Heidelberg, entanglement of two macroscopic objects. Phys. Rev. Lett. 107, 080503 (2011). 2012). 38. Almeida, M. P. et al. Environment-induced sudden death of entanglement. Sci- 4. Gorini, V. Completely positive dynamical semigroups of n-level systems. J. Math. ence 316, 579–582 (2007). Phys. 17, 821 (1976). 39. Yu, T. & Eberly, J. H. Sudden death of entanglement. Science 323, 598–601 (2009). 5. Lindblad, G. On the generators of quantum dynamical semigroups. Commun. 40. Mazzola, L., Piilo, J. & Maniscalco, S. Sudden transition between classical and Math. Phys. 48, 119–130 (1976). quantum decoherence. Phys. Rev. Lett. 104, 200401 (2010). 6. Huelga, S. & Plenio, M. Vibrations, quanta and biology. Contemp. Phys. 54, 41. Xu, J.-S. et al. Experimental investigation of classical and quantum correlations 181–207 (2013). under decoherence. Nat. Commun. 1, 7 (2010). 7. Rivas, Á., Huelga, S. F. & Plenio, M. B. Quantum non-Markovianity: characteriza- 42. Ziman, M. & Bužek, V. All (qubit) decoherences: complete characterization and tion, quantification and detection. Rep. Prog. Phys. 77, 094001 (2014). physical implementation. Phys. Rev. A 72, 022110 (2005). 8. Breuer, H.-P., Laine, E.-M., Piilo, J. & Vacchini, B. Colloquium: non-Markovian 43. Deléglise, S. et al. Reconstruction of non-classical cavity field states with snap- dynamics in open quantum systems. Rev. Mod. Phys. 88, 021002 (2016). shots of their decoherence. Nature 455, 510–514 (2008). 9. de Vega, I. & Alonso, D. Dynamics of non-Markovian open quantum systems. Rev. 44. Nokkala, J., Galve, F., Zambrini, R., Maniscalco, S. & Piilo, J. Complex quantum Mod. Phys. 89, 015001 (2017). networks as structured environments: engineering and probing. Sci. Rep. 6, 26861 10. Li, L., Hall, M. J. & Wiseman, H. M. concepts of quantum non-Markovianity: a (2016). hierarchy. Physics Reports 759,1–51 (2018). 45. Nokkala, J. et al. Reconfigurable optical implementation of quantum complex 11. Breuer, H.-P., Laine, E.-M. & Piilo, J. Measure for the degree of non-Markovian networks. New J. Phys. 20, 053024 (2018). behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 46. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (2009). (1982). 12. Rivas, Á., Huelga, S. F. & Plenio, M. B. Entanglement and non-Markovianity of 47. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). quantum evolutions. Phys. Rev. Lett. 105, 050403 (2010). 48. Wang, H., Ashhab, S. & Nori, F. Quantum algorithm for simulating the dynamics of 13. Lu, X.-M., Wang, X. & Sun, C. P. Quantum fisher information flow and non- an open quantum system. Phys. Rev. A 83, 062317 (2011). Markovian processes of open systems. Phys. Rev. A 82, 042103 (2010). 49. Schindler, P. et al. Quantum simulation of dynamical maps with trapped ions. Nat. 14. Lorenzo, S., Plastina, F. & Paternostro, M. Geometrical characterization of non- Phys. 9, 361–367 (2013). Markovianity. Phys. Rev. A 88, 020102 (2013). 50. Wang, D.-S., Berry, D. W., de Oliveira, M. C. & Sanders, B. C. Solovay-kitaev 15. Bylicka, B., Chruściński, D. & Maniscalco, S. Non-Markovianity and reservoir decomposition strategy for single-qubit channels. Phys. Rev. Lett. 111, 130504 memory of quantum channels: a quantum information theory perspective. Sci. (2013). Rep. 4, 5720 (2014). 51. Lu, H. et al. Experimental quantum channel simulation. Phys. Rev. A 95, 042310 16. Chruściński, D. & Maniscalco, S. Degree of non-Markovianity of quantum evolu- (2017). tion. Phys. Rev. Lett. 112, 120404 (2014). 52. Xin, T., Wei, S.-J., Pedernales, J. S., Solano, E. & Long, G.-L. Quantum simulation of 17. Liu, B.-H. et al. Experimental control of the transition from Markovian to quantum channels in nuclear magnetic resonance. Phys. Rev. A 96, 062303 (2017). non-Markovian dynamics of open quantum systems. Nat. Phys. 7, 931–934 53. Shen, C. et al. Quantum channel construction with circuit quantum electro- (2011). dynamics. Phys. Rev. B 95, 134501 (2017). Published in partnership with The University of New South Wales npj Quantum Information (2020) 1 G. García-Pérez et al. 10 54. Hall, M. J. W., Cresser, J. D., Li, L. & Andersson, E. Canonical form of master AUTHOR CONTRIBUTIONS equations and characterization of non-Markovianity. Phys. Rev. A 89, 042120 G.G.-P., M.A.C.R. and S.M. contributed to the design and implementation of the (2014). research, to the analysis of the results, and to the writing of the paper. 55. Megier, N., Chruściński, D., Piilo, J. & Strunz, W. T. Eternal non-Markovianity: from random unitary to Markov chain realisations. Sci. Rep. 7, 6379 (2017). 56. Chruściński, D., Macchiavello, C. & Maniscalco, S. Detecting non-Markovianity of COMPETING INTERESTS quantum evolution via spectra of dynamical maps. Phys. Rev. Lett. 118, 080404 The authors declare no competing interests. (2017). 57. McCloskey, R. & Paternostro, M. Non-Markovianity and system-environment correlations in a microscopic collision model. Phys. Rev. A 89, 052120 (2014). 58. Lorenzo, S., Ciccarello, F. & Palma, G. M. Class of exact memory-kernel master ADDITIONAL INFORMATION equations. Phys. Rev. A 93, 052111 (2016). Correspondence and requests for materials should be addressed to G.G.-P. 59. Kretschmer, S., Luoma, K. & Strunz, W. T. Collision model for non-Markovian quantum dynamics. Phys. Rev. A 94, 012106 (2016). Reprints and permission information is available at http://www.nature.com/ 60. Filippov, S. N., Piilo, J., Maniscalco, S. & Ziman, M. Divisibility of quantum dyna- reprints mical maps and collision models. Phys. Rev. A 96, 032111 (2017). 61. Holevo, A. A note on covariant dynamical semigroups. Rep. Math. Phys. 32, Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims 211–216 (1993). in published maps and institutional affiliations. 62. del Rio, L., Åberg, J., Renner, R., Dahlsten, O. & Vedral, V. The thermodynamic meaning of negative entropy. Nature 474,61–63 (2011). 63. Gadi A. et al. Qiskit: An Open-source Framework for Quantum Computing (2019). 64. Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the Open Access This article is licensed under a Creative Commons maximum-likelihood quantum state from measurements with additive gaussian Attribution 4.0 International License, which permits use, sharing, noise. Phys. Rev. Lett. 108, 070502 (2012). adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party ACKNOWLEDGEMENTS material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the We thank Dariusz Chruściński and Thomas Bullock for interesting discussions. We article’s Creative Commons license and your intended use is not permitted by statutory acknowledge use of the IBM Q Experience for this work. The views expressed are regulation or exceeds the permitted use, you will need to obtain permission directly fl fi those of the authors and do not re ect the of cial policy or position of IBM or the IBM from the copyright holder. To view a copy of this license, visit http://creativecommons. fi Q team. We acknowledge nancial support from the Academy of Finland via the org/licenses/by/4.0/. Centre of Excellence programme (Project no. 312058 as well as Project no. 287750). G. G.-P. acknowledges support from the emmy.network foundation under the aegis of the Fondation de Luxembourg. © The Author(s) 2020 npj Quantum Information (2020) 1 Published in partnership with The University of New South Wales