Many-Body Quantum Magic

By Zi-Wen Liu, Andreas J. Winter. PRX Quantum, 2020

195 citations recorded by the source corpus.

Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation. To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic. Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions). Here we present an initiatory study of the magic of genuinely many-body quantum states (with focus on the important case of many qubits) at a quantitative level. We first address the basic question of how magical a many-body state can be, and show that the maximum and typical magic of an $n$-qubit state is essentially $n$, simultaneously for a range of natural resource measures. As a corollary, we show that the resource theory of magic with stabilizer-preserving free operations is asymptotically reversible. In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions. Next, we go on and investigate many-body magic in practical and physical contexts. We first consider Pauli measurement-based quantum computation, in which magic is a necessary feature of the initial resource state. We show that $n$-qubit states with nearly $n$ magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers. We then present an example of analyzing the magic of "natural" condensed matter systems. We apply the Boolean function techniques to derive explicit bounds on the magic of the ground states of certain 2D symmetry-protected topological (SPT) phases, and comment on possible further connections between magic and the quantum complexity of matter.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/18ba262909faeaffc45a49bf9aec009c7cadf725