Quantum Simulation of Dissipative Processes without Reservoir Engineering

We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of our method providing specific error bounds that quantify itss accuracy.

While every physical system is indeed coupled to an environment 1,2 , modern quantum technologies have succeeded in isolating systems to an exquisite degree in a variety of platforms [3][4][5][6] . In this sense, the last decade has witnessed great advances in testing and controlling the quantum features of these systems, spurring the quest for the development of quantum simulators [7][8][9][10] . These efforts are guided by the early proposal of using a highly tunable quantum device to mimic the behavior of another quantum system of interest, being the latter complex enough to render its description by classical means intractable. By now, a series of proof-of-principle experiments have successfully demonstrated the basic tenets of quantum simulations revealing quantum technologies as trapped ions 11 , ultracold quantum gases 12 , and superconducting circuits 13 as promising candidates to harbor quantum simulations beyond the computational capabilities of classical devices.
It was soon recognised that this endeavour should not be limited to simulating the dynamics of isolated complex quantum systems, but should more generally aim at the emulation of arbitrary physical processes, including the open quantum dynamics of a system coupled to an environment. Tailoring the complex nonequilibrium dynamics of an open system has the potential to uncover a plethora of technological and scientific applications. A remarkable instance results from the understanding of the role played by quantum effects in the open dynamics of photosynthetic processes in biological systems 14,15 , recently used in the design of artificial light-harvesting nanodevices [16][17][18] . At a more fundamental level, an open-dynamics quantum simulator would be invaluable to shed new light on core issues of foundations of physics, ranging from the quantum-to-classical transition and quantum measurement theory 19 to the characterization of Markovian and non-Markovian systems [20][21][22] . Further motivation arises at the forefront of quantum technologies. As the available resources increase, the verification with classical computers of quantum annealing devices 23,24 , possibly operating with a hybrid quantum-classical performance, becomes a daunting task. The comparison between different experimental implementations of quantum simulators is required to establish a confidence level, as customary with other quantum technologies, e.g., in the use of atomic clocks for time-frequency standards. In addition, the knowledge and control of dissipative processes can be used as well as a resource for quantum state engineering 25 .
Facing the high dimensionality of the Hilbert space of the composite system made of a quantum device embedded in an environment, recent developments have been focused on the reduced dynamics of the system that emerges after tracing out the environmental degrees of freedom. The resulting nonunitary dynamics is governed by a dynamical map, or equivalently, by a master equation 1,2 . In this respect, theoretical [26][27][28] and experimental 29 efforts in the simulation of open quantum systems have exploited the combination of coherent quantum operations with controlled dissipation. Notwithstanding, the experimental complexity required to simulate an arbitrary open quantum dynamics is recognised to substantially surpass that needed in the case of closed systems, where a smaller number of generators suffices to design a general time-evolution. Thus, the quantum simulation of open systems remains a challenging task.
In this Letter, we propose a quantum algorithm to simulate finite dimensional Lindblad master equations, corresponding to Markovian or non-Markovian processes. Our protocol shows how to reconstruct, up to an arbitrary finite error, physical observables that evolve according to a dissipative dynamics, by evaluating multi-time correlation functions of its Lindblad operators. We show that the latter requires the implementation of the unitary part of the dynamics in a quantum simulator, without the necessity of physically engineering the system-environment interactions. Moreover, we demonstrate how these multi-time correlation functions can be computed with a reduced number of measurements. We further show that our method can be applied as well to the simulation of processes associated with non-Hermitian Hamiltonians. Finally, we provide specific error bounds to estimate the accuracy of our approach. Consider a quantum system coupled to an environment whose dynamics is described by the von  30 . Nevertheless, Eq. (1) is our starting point, and in the following we show how to simulate this equation regardless of its derivation. Indeed, our algorithm does not need to control any of the approximations done to achieve this equation. We can decompose where H(t) is defined by H S plus a term due to the lamb-shift effect and it may depend on time. Instead,  D t is the dissipative contribution and it follows the Lindblad form 31 where L i are the Lindblad operators modelling the effective interaction of the system with the bath that may depend on time, while t i γ ( ) are nonnegative parameters. Notice that, although the standard derivation of Eq. (1) requires the Markov approximation, a non-Markovian equation can have the same form.
then Eq. (1) corresponds to a completely positive non-Markovian channel 32 . Our approach can deal also with non-Markovian processes of this kind, keeping the same efficiency as the Markovian case. While we will consider the general case t i i γ γ = ( ), whose sign distinguishes the Markovian processes from the non-Markovian ones, for the sake of simplicity we will consider the case H H t ≠ ( ) and L L t i i ≠ ( ) (in the following, we will denote  H t simply as  H ). However, the inclusion in our formalism of time-dependent Hamiltonians and Lindblad operators is straightforward.
One can integrate Eq. (1) obtaining a Volterra equation 33 The first term at the right-hand-side of Eq. (2) corresponds to the unitary evolution of 0 ρ ( ) while the second term gives rise to the dissipative correction. Our goal is to find a perturbative expansion of Eq. (2) in the  D t term, and to provide with a protocol to measure the resulting expression in a unitary way. In order to do so, we consider the iterated solution of Eq. (2) obtaining ρ ( ) has the following general structure: In this way, Eq. (3) provides us with a general and useful expression of the solution of Eq. (1). Let us consider the truncated series in Eq.
where n corresponds to the order of the approximation. We will prove that an expectation value ρ ≡ () corresponding to a dissipative dynamics can be well approximated as In the following, we will supply with a quantum algorithm based on single-shot random measurements to compute each of the terms appearing in Eq. (4), and we will derive specific upper-bounds quantifying the accuracy of our method. Notice that the first term at the right-hand-side of Eq. (4), i.e.   Our next step is to provide a method to evaluate general terms as the one appearing in Eq. (7). The standard approach to estimate this kind of quantities corresponds to measuring the expected value A s Nevertheless, this strategy involves a huge number of measurements, as we need to estimate an expectation value at each chosen time. Our technique, instead, is based on single-shot random measurements and, as we will see below, it leads to an accurate estimate of Eq. (7). More specifically, we will prove that are sampled uniformly and independently. As already pointed out, the integrand in Eq. (7) involves multi-time correlation functions. In this respect, we note that a quantum algorithm for their efficient reconstruction has recently been proposed 34 . Indeed, the authors in Ref. [34] show how, by adding only one ancillary qubit to the simulated system, general time-correlation functions are accessible by implementing only unitary evolutions of the kind  e t H , together with entangling operations between the ancillary qubit and the system. It is noteworthy to mention that these operations have already experimentally demonstrated in quantum systems as trapped ions 35 or quantum optics 6 , and have been recently proposed for cQED architectures 36 . Moreover, the same quantum algorithm allows us to measure single-shots of the real and imaginary part of these quantities providing, therefore, a way to compute the term at the right-hand-side of Eq. (8). Notice that the evaluation of each term A s in Eq. (7), requires a number of measurements that depends on the observable decomposition, see Eq. (6). After specifying it, we measure the real and the imaginary part of the corresponding correlation function. Finally, in the supplemental material 37 we prove that , allowing, hence, to dramatically reduce the resources required by our quantum simulation algorithm. Notice that the required number of measurements to evaluate the order n is bounded by 3 n n Ω , and the total number of measurements needed to compute the correction to the expected value of an observable up the order K is bounded by 3 In the following, we discuss at which order we need to truncate in order to have a certain error in the final result.
So far, we have proved that we can compute, up to an arbitrary order in  D t , expectation values corresponding to dissipative dynamics with a unitary quantum simulation. It is noteworthy that our method does not require to physically engineer the system-environment interaction. Instead, one only needs to implement the system Hamiltonian H. In this way we are opening a new avenue for the quantum simulation of open quantum dynamics in situations where the complexity on the design of the dissipative terms excedes the capabilities of quantum platforms. This covers a wide range of physically relevant situations. One example corresponds to the case of fermionic theories where the encoding of the fermionic behavior in the degrees of freedom of the quantum simulator gives rise to highly delocalized operators 38, 39 . In this case a reliable dissipative term should act on these non-local operators instead of on the individual qubits of the system. Our protocol solves this problem because it avoids the necessity of implementing the Lindblad superoperator. Moreover, the scheme allows one to simulate at one time a class of master equations corresponding to the same Lindblad operators, but with different choices of i γ , including the relevant case when only a part of the system is subjected to dissipation, i.e. 0 i γ = for some values of i.
We shall next quantify the quality of our method. In order to do so, we will find an error bound certifying how the truncated series in Eq. (3) is close to the solution of Eq. (1). This error bound will depend on the system parameters, i.e. the time t and the dissipative parameters i γ . As figure of merit we choose the trace distance, defined by D 10 Our goal is to find a bound for is the series of Eq. (3) truncated at the n-th order. We note that the the following recursive relation holds where t Nt γ = 37 . Here, a discussion on the efficiency of the method is needed. From the previous formula, we can say that our method performs well when M is low, i.e. in case where each Lindblad operators can be decomposed in few Pauli-kind operators. Moreover, as our approach is perturbative in the dissipative parameters i γ , the method is more efficient when i γ are small. Notice that analytical perturbative techniques are not available in this case, because the solution of the unperturbed part is assumed to be not known. Lastly, it is evident that the algorithm is efficient for certain choices of time, and the relevance of the simulation depends on the particular cases. For instance, a typical interesting situation is a strongly coupled Markovian system. Let us assume a site-independent couple parameter g and a dissipative parameter γ. We have that e e Mt 1 12