Ignacio Cirac

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this paper, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an $N$-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth $T = O(N^{\alpha})$ realizing the MPU, where the constant $\alpha$ depends only on the bulk and boundary tensor and not the system size $N$. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of $C^*$-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally-varying MPUs and show that they can be implemented by a circuit of depth $O(N^{\beta} \, \mathrm{poly}\, D)$ where $\beta \le 1 + \log_2 \sqrt{D}/ s_{\min}$, with $D$ being the bond dimension and $s_{\min}$ is the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

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, where non-trivial 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 non-commutativity of the Lindbladian terms and the fact that the corresponding MPDO RFP belongs to a non-trivial phase.

We consider the preparation of matrix product states (MPS) on quantum devices via quantum circuits of local gates. We first prove that faithfully preparing translation-invariant normal (i.e., short-range correlated) MPS of N sites requires a circuit depth T = Ω(log N). We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error ε in depth T = O[log(N/ε)]. The algorithm has thus the fastest scaling possible for this class of states. We also show that measurement and feedback leads to an exponential speedup of the algorithm to T = O[log log(N/ε)]. Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range non-normal ones, in the same depth. Finally, the algorithm naturally extends to inhomogeneous MPS.