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Publication# A quantum algorithm for the direct estimation of the steady state of open quantum systems

Nathan Ramusat, Vincenzo Savona

*VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF, *2021

Article

Article

Résumé

Simulating the dynamics and the non-equilibrium steady state of an open quantum system are hard computational tasks on conventional computers. For the simulation of the time evolution, several efficient quantum algorithms have recently been developed. However, computing the non-equilibrium steady state as the long-time limit of the system dynamics is often not a viable solution, because of exceedingly long transient features or strong quantum correlations in the dynamics. Here, we develop an efficient quantum algorithm for the direct estimation of averaged expectation values of observables on the non-equilibrium steady state, thus bypassing the time integration of the master equation. The algorithm encodes the vectorized representation of the density matrix on a quantum register, and makes use of quantum phase estimation to approximate the eigenvector associated to the zero eigenvalue of the generator of the system dynamics. We show that the output state of the algorithm allows to estimate expectation values of observables on the steady state. Away from critical points, where the Liouvillian gap scales as a power law of the system size, the quantum algorithm performs with exponential advantage compared to exact diagonalization.

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Matrice densité

En physique quantique, la matrice densité, souvent représentée par \rho, est un objet mathématique introduit par le mathématicien et physicien John von Neumann permettant de décrire l'é

Algorithme

thumb|Algorithme de découpe d'un polygone quelconque en triangles (triangulation).
Un algorithme est une suite finie et non ambiguë d'instructions et d’opérations permettant de résoudre une classe de

Opérateur d'évolution

En mécanique quantique, l'opérateur d'évolution est l'opérateur qui transforme l'état
quantique au temps t_0 en l'état quantique au temps t résultant
de l'évolution du systèm

It's been a little more than 40 years since researchers first suggested exploiting quantum physics to build more powerful computers. During this time, we have seen the development of many quantum algorithms and significant technological advances to make these devices. As a result, at this point, large-scale quantum computers, capable of providing valuable solutions to complex problems, seem like a certainty, even if still distant.Nonetheless, there is a great gap between the communities of quantum algorithms and quantum devices researchers. On the one hand, algorithm designers most often describe algorithms in a high level of abstraction, using a mixture of natural language, pseudocode, and mathematical formulas---a form deriving asymptotic complexity estimates while shielding researchers from the low-level complexities and restrictions. On the other hand, physical devices only understand algorithms implemented using the primitive low-level abstractions they support.Most programming systems available for quantum computing are intertwined with the quantum circuit model, so developers must implement algorithms in terms of low-level unitary operators. Not surprisingly, the implementation of quantum algorithms on such a low level of abstraction is very time-consuming, error-prone, and results in non-portable programs---given the technological diversity of quantum devices. In this thesis, I study problems related to the compilation of quantum programs, seeking forms of augmenting the expressive power of current frameworks and narrowing the gap between algorithmic research and concrete implementations. I focus on scalability and practicality---in particular, together with theoretical investigations, I developed concrete algorithms that are performant and scalable. The embodiment of my research contribution is a compiler companion library for the synthesis and compilation of quantum circuits called tweedledum.

Many-body open quantum systems are exposed to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. As the exact treatment of such models is generally unfeasible, it is favourable to formulate an approximate description by means of the dynamics of the reduced density operator of the system. When studying models with a weak coupling to a memoryless environment, the existence of such solution is granted and the reduced dynamics is governed by the Lindblad quantum master equation. In recent years, open quantum systems have evolved into a major field of studies. Focus of these studies are the characterization of emergent phenomena and dissipative phase transitions, as well as the ongoing debate about whether quantum computing schemes are still hard to simulate classically --- and thus achieve quantum supremacy --- when in presence of some degree of noise-induced decoherence. However, the challenge posed by this task lies in the complexity of the density matrix that increases exponentially with the system size, and the quest for efficient numerical methods is a research field that is still in its infancy. Here, we first develop a real-time full configuration interaction quantum Monte Carlo technique that stems from a class of methods generally known as projector Monte Carlo. The approach enables the stochastic sampling of the Lindblad time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the non-equilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce statistical error. We demonstrate the efficiency of the approach by applying it to the dissipative two-dimensional XYZ spin-1/2 model on a lattice. As the importance sampling of projector approaches is often combined with variational results, we then introduce a novel method that is based on the variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the Lindblad quantum master equation. We test the effectiveness of the method by modeling the steady state of the dissipative two-dimensional XYZ spin-1/2 model through a dissipative phase transition, and also the real-time dynamics of the dissipative Ising model. In addition, we discuss the application of the developed methods and the open questions of the field.

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to re- trieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbla- dian evolution (spin, fermions, bosons, ... ), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other di- agonalization techniques and retrieves the Li- ouvillian low-lying spectrum even for system sizes for which it would be impossible to per- form exact diagonalization.