The Density Matrix & Product Operators

Edwin D. Becker

Kohtaro Kato

Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.

Chi-Fang Chen, Kohtaro Kato, Fernando G. S. L. Brandão

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.

Hans Paetz gen. Schieck

Ambroise Müller, Thomas Ayral, Corentin Bertrand

Matrix product density operators (MPDOs) are tensor network representations of locally purified density matrices where each physical degree of freedom is associated to an environment degree of freedom. MPDOs have interesting properties for mixed state representations: guaranteed positivity by construction, efficient conservation of the trace and computation of local observables. However, they have been challenging to use for noisy quantum circuit simulation, as the application of noise increases the dimension of the environment Hilbert space, leading to an exponential growth of bond dimensions. MPDOs also lack a unique canonical form, due to the freedom in the choice of basis for the environment Hilbert space, which leads to a vast variation of bond dimensions. In this work, we present a systematic way to reduce the bond dimensions of MPDOs by disentangling the purified state. We optimize the basis for the environment Hilbert space by performing density matrix renormalization group (DMRG)-like sweeps of local 2-qubit basis optimization. Interestingly, we find that targeting only the disentanglement of the purified state leads to a reduction of the environment dimension. In other words, a compact MPDO representation requires a low-entanglement purified state. We apply our compression method to the emulation of noisy random quantum circuits. Our technique allows us to keep bounded bond dimensions, and thus bounded memory, contrary to previous works on MPDOs, while keeping reasonable truncation fidelities.

Mark Linne