Matrix product density operators (MPDOs) are an important class of states with interesting properties. Consequently, it is important to understand how to prepare these states experimentally. One possible way to do this is to design an open system that evolves only towards desired states. A Markovian evolution of a quantum mechanical system can be generally described by a Lindbladian. In this work we develop an algorithm that for a given (small) linear subspace of MPDOs determines if this subspace can be the stable space for some frustration free Lindbladian consisting of only local terms and, if so, outputs a suitable Lindbladian.
Matrix product density operators (MPDOs) are an important class of states with interesting properties. Consequently, it is important to understand how to prepare these states experimentally. One possible way to do this is to design an open system that evolves only towards desired states. A Markovian evolution of a quantum mechanical system can be generally described by a Lindbladian. In this work we develop an algorithm that for a given (small) linear subspace of MPDOs determines if this subspace can be the stable space for some frustration free Lindbladian consisting of only local terms and, if so, outputs a suitable Lindbladian.
Understanding quantum phases of matter is a fundamental goal in physics.For pure states, the representatives of phases are the ground states of locally interacting Hamiltonians, which are also renormalization fixed points (RFPs).These RFP states are exactly described by tensor networks.Extending this framework to mixed states, matrix product density operators (MPDOs) which are RFPs are believed to encapsulate mixed-state phases of matter in one dimension.However, it remains an open question whether, by analogy, such MPDO RFPs can be realized as steady states of local Lindbladians.In this work, we resolve this question by analytically constructing parent Lindbladians for MPDO RFPs.These Lindbladians are local, frustration-free, and exhibit minimal steady-state degeneracy.Interestingly, we find that parent Lindbladians possess a rich structure, including non-commutativity for certain classes of RFPs, distinguishing them from their Hamiltonian counterparts.
Understanding quantum phases of matter is a fundamental goal in physics. For pure states, the representatives of phases are the ground states of locally interacting Hamiltonians, which are also renormalization fixed points (RFPs). These RFP states are exactly described by tensor networks. Extending this framework to mixed states, matrix product density operators (MPDOs) that are RFPs are believed to encapsulate mixed-state phases of matter in one dimension, where nontrivial topological phases have already been shown to exist. However, to better motivate the physical relevance of those states, and in particular the physical relevance of the recently found non trivial phases, it remains an open question whether such MPDO RFPs can be realized as steady states of local Lindbladians. In this work, we resolve this question by analytically constructing parent Lindbladians for MPDO RFPs. These Lindbladians are local, frustration free, and exhibit minimal steady-state degeneracy. Interestingly, we find that parent Lindbladians possess a rich structure that distinguishes them from their Hamiltonian counterparts. In particular, we uncover an intriguing connection between the noncommutativity of the Lindbladian terms and the fact that the corresponding MPDO RFP belongs to a nontrivial phase.
Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the k-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality k≥5, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.
We study whether one can write a Matrix Product Density Operator (MPDO) as the Gibbs state of a quasi-local parent Hamiltonian. We conjecture this is the case for generic MPDO and give supporting evidences. To investigate the locality of the parent Hamiltonian, we take the approach of checking whether the quantum conditional mutual information decays exponentially. The MPDO we consider are constructed from a chain of 1-input/2-output (`Y-shaped') completely-positive maps, i.e., the MPDO have a local purification. We derive an upper bound on the conditional mutual information for bistochastic channels and strictly positive channels and show that it decays exponentially if the correctable algebra of the channel is trivial. We also introduce a conjecture on a quantum data processing inequality that implies the exponential decay of the conditional mutual information for every Y-shaped channel with trivial correctable algebra. We additionally investigate a close but nonequivalent cousin: MPDO measured in a local basis. We provide sufficient conditions for the exponential decay of the conditional mutual information of the measured states and numerically confirm they are generically true for certain random MPDO.
Characterizing the dynamics of open quantum systems at the level of microscopic interactions and error mechanisms is essential for calibrating quantum hardware, designing robust simulation protocols, and developing tailored error-correction methods. Under Markovian noise/dissipation, a natural characterization approach is to identify the full Lindbladian generator that gives rise to both coherent (Hamiltonian) and dissipative dynamics. Prior protocols for learning Lindbladians from dynamical data assumed pre-specified interaction structure, which can be restrictive when the relevant noise channels or control imperfections are not known in advance. In this paper, we present a sample-efficient protocol for learning sparse Lindbladians without assuming any a priori structure. Our protocol is ancilla-free, uses only product-state preparations and Pauli-basis measurements, admits provably stable coefficient reconstruction, and achieves near-optimal time resolution, making it compatible with near-term experimental capabilities. Together, this provides a systematic route to scalable characterization of open-system quantum dynamics, especially in settings where the error mechanisms of interest are unknown.
Entangled possibly mixed states are an essential resource for quantum computation, communication, metrology, and the simulation of many-body systems. It is important to develop and improve preparation protocols for such states. One possible way to prepare states of interest is to design an open system that evolves only towards the desired states. A Markovian evolution of a quantum system can be generally described by a Lindbladian. Tensor networks provide a framework to construct physically relevant entangled states. In particular, matrix product density operators (MPDOs) form an important variational class of states. MPDOs generalize matrix product states to mixed states, can represent thermal states of local one-dimensional Hamiltonians at sufficiently large temperatures, describe systems that satisfy the area law of entanglement, and form the basis of powerful numerical methods. In this work we develop an algorithm that determines for a given linear subspace of MPDOs whether this subspace can be the stable space of some frustration free k-local Lindbladian and, if so, outputs an appropriate Lindbladian. We proceed by using machine learning with networks of quantum channels, also known as quantum neural networks (QNNs), to train denoising post-processing devices for quantum sources. First, we show that QNNs can be trained on imperfect devices even when part of the training data is corrupted. Second, we show that QNNs can be trained to extrapolate quantum states to, e.g., lower temperatures. Third, we show how to denoise quantum states in an unsupervised manner. We develop a novel quantum autoencoder that successfully denoises Greenberger-Horne-Zeilinger, W, Dicke, and cluster states subject to spin-flip, dephasing errors, and random unitary noise. Finally, we develop recurrent QNNs (RQNNs) for denoising that requires memory, such as combating drifts. RQNNs can be thought of as matrix product quantum channels with a quantum algorithm for training and are closely related to MPDOs. The proposed preparation and denoising protocols can be beneficial for various emergent quantum technologies and are within reach of present-day experiments.