Many-Body Quantum Magic

Abstract

Magic (nonstabilizerness) 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 that may be strongly entangled, 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 magic of an n-qubit state is essentially n, simultaneously for a range of “good” magic measures. As a corollary, the resource theory of magic has asymptotic golden currency states. We then show that, in fact, almost all n-qubit pure states have magic of nearly n. 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 a variant of measurement-based quantum computation where the client is restricted to Pauli measurements, 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 of physical interest. We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative two-dimensional symmetry-protected topological states, and comment on possible further connections between magic and the quantum complexity of phases of matter.