Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of stabilizer rank, which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n-bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components. CONTENTS

The ubiquitous presence of inhibitory interneurons in the thalamus of primates contrasts with the sparsity of interneurons reported in mice. Here, we identify a larger than expected complexity and distribution of interneurons across the mouse thalamus, where all thalamic interneurons can be traced back to two developmental programs: one specified in the midbrain and the other in the forebrain. Interneurons migrate to functionally distinct thalamocrtical nuclei depending on their origin: the abundant, midbrain-generated class populates the first and higher order sensory thalamus while the rarer, forebrain-generated class is restricted to some higher order associative regions. We also observe that markers for the midbrain-born class are abundantly expressed throughout the thalamus of the New World monkey marmoset. These data therefore reveal that, despite the broad variability in interneuron density across mammalian species, the blueprint of the ontogenetic organisation of thalamic interneurons of larger-brained mammals exists and can be studied in mice.

12The proportion and distribution of local inhibitory neurons (interneurons) in the thalamus 13 varies widely across mammals. This is reflected in the structure of thalamic local circuits, 14which is more complex in primates compared to smaller-brained mammals like rodents. 15An increase in the number of thalamic interneurons could arise from addition of novel 16 interneuron types or from elaboration of a plesiomorphic ontogenetic program, common to 17 all mammals. The former has been proposed for the human brain, with migration of 18 interneurons from the ventral telencephalon into higher order thalamus as one of its unique 19 features (Letinic and Rakic, 2001). 20Here, we identify a larger than expected complexity and distribution of interneurons across 21 the mouse thalamus. All thalamic interneurons can be traced back to two developmental 22 programs: one specified in the midbrain and the other in the forebrain. Interneurons migrate 23 to functionally distinct thalamic nuclei, where the midbrain-derived cells populate the sensory 24 thalamus, and forebrain-generated interneurons only the higher order regions. The latter 25 interneuron type may be homologous to the one previously considered to be human-specific, 26while we also observe that markers for the midbrain-born class are abundantly expressed in 27 the primate thalamus. These data therefore point to a shared ontogenetic organization of 28 thalamic interneurons across mammals. 29 30

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