A Knowledge Compilation Map for Quantum Information

Despite their widespread use in quantum computing and physics, the relative strengths and weaknesses of Matrix Product States (MPS), Decision Diagrams (DDs), and Restricted Boltzmann Machines (RBMs) remains poorly understood. We analytically compare the succinctness of these quantum state representations and analyze the complexity of key operations on them. To overcome shortcomings of the tractability measure, we introduce `rapidity' conditions that identify when non-canonical representations efficiently simulate each other. Our results reveal that:

1. Most DD variants are redundant with respect to MPS in a strong sense; MPS is more rapid.
2. Only one DD variant, called LIMDD, and RBM have succinctness incomparable to MPS.
3. LIMDD and RBM seem to achieve this by sacrificing tractability of counting queries, as shown by a metatheorem on the conditional hardness of these queries.