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Process tensors describe the influences of environments on open quantum systems. They have two main areas of application: On the one hand, they are used in quantum information theory to characterize environments of open quantum systems by generalizing quantum process tomography to arbitrary multi-time correlation functions.. ^{[1] [2]} On the other hand, process tensors can be used to simulate non-Markovian dynamics in open quantum systems numerically exactly. ^{[3] [4]}

Formally, process tensors are defined for a composite system consisting of a system of interest (S) and its environment (E) on a discretized time grid. ^[1] At each time step k an arbitrary quantum process ${\displaystyle A_{k}}$ may be performed on the system (S). This can be, e.g., a unitary evolution induced by a system Hamiltonian, a quantum measurement, or no operation at all, in which case the operation is the identity map. A process tensor is then defined as the map from the sequence of all operations ${\displaystyle \{A_{n-1};\dots ;A_{1};A_{0}\}}$to the final (generalized) reduced system density matrix ${\displaystyle \rho _{n}}$obtained at the last time step n. ^[1] Thus, the process tensor contains the complete information about the open quantum system that can be revealed by performing experiments on the system (S). For example, if all processes ${\displaystyle A_{k}}$ describe unitary time evolution, the result describes the time evolution of the reduced system in the form of a density matrix. If two of the processes ${\displaystyle A_{i}}$ and ${\displaystyle A_{j}}$ additionally contain projective measurements with operators ${\displaystyle O_{i}}$ and ${\displaystyle O_{j}}$, respectively, the result is the two-time correlation function ${\displaystyle \langle O_{i}(t_{i})O_{j}(t_{j})\rangle }$. Hence, once the process tensor is known, all multitime correlation functions of the open quantum system can be recovered.

Due to the vast amount of information, process tensors as defined above are very big objects generally scaling exponentially with the number of time steps. However, process tensors can be efficiently represented in matrix product operator form ^[1], in which case they are referred to as process tensor matrix product operators (PT-MPOs) ^[5].

The first applications of PT-MPOs for simulating open quantum systems ^[3] were based on the observation that, for the spin-boson model, process tensors are equivalent to the Feynman-Vernon influence functional ^[6]. Thus, known path integral results can be used to obtain expressions for nodes in a tensor network representing the process tensor ^[7]. PT-MPOs are then obtained by contracting this tensor network while keeping MPO form at all times. ^[3] Recent progress has lead to algorithms with quasi- and sublinear scaling with respect to the total number of time steps using a combination of a divide-and-conquer approach and periodic (indefinitely repeatable) PT-MPOs ^[5]. Furthermore, a method scaling linearly in the memory time has been devised ^[8], which also relies on periodic PT-MPOs. PT-MPO-based algorithms have also been described for fermionic problems. ^[9] The algorithm Automated Compression of Environments (ACE) ^[4] enables the construction of PT-MPOs for very general environments, namely those where the environment is composed of independent modes.

A noteworthy feature of PT-MPOs is that multiple PT-MPOs can be put together for the description of open quantum systems coupled to multiple environment while still yielding numerically exact results. ^[4] This enables investigations of complex open quantum systems including the analysis of non-additive cross interaction between different environments ^[10] and collective effects in multi-partite open quantum systems ^[11].

Computer codes for numerical simulation of open quantum systems are freely available, such as OQuPy ^[12] and the ACE code ^[13]

References