The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
An accurate calculation of the properties of quantum many-body systems is one of the most important yet intricate challenges of modern physics and computer science. In recent years, the tensor network ansatz has established itself as one of the most promising approaches enabling striking efficiency of simulating static properties of one-dimensional systems and abounding numerical applications in condensed matter theory. In higher dimensions, however, a connection to the field of computational complexity theory has shown that the accurate normalization of the two-dimensional tensor networks called projected entangled pair states (PEPS) is #Pcomplete. Therefore, an efficient algorithm for PEPS contraction would allow to solve exceedingly difficult combinatorial counting problems, which is considered highly unlikely. Due to the importance of understanding two-and three-dimensional systems the question currently remains: Are the known constructions typical of states relevant for quantum many-body systems? In this work, we show that an accurate evaluation of normalization or expectation values of PEPS is as hard to compute for typical instances as for special configurations of highest computational hardness. We discuss the structural property of average-case hardness in relation to the current research on efficient algorithms attempting tensor network contraction, hinting at a wealth of possible further insights into the average-case hardness of important problems in quantum many-body theory.
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