Efficient Magic State Cultivation on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2<

Preparation of high-fidelity logical magic states is crucial for fault-tolerant quantum computation. Among previous attempts to reduce the substantial cost of magic state preparation, magic state cultivation (MSC) [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], a recently proposed protocol for logical <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi mathvariant="normal">T</a:mi> </a:mrow> </a:math> state preparation without magic state distillation, achieves state-of-the-art efficiency. Inspired by this work [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], we propose a MSC procedure that can produce logical <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mrow> <d:mi mathvariant="normal">T</d:mi> </d:mrow> </d:math> states on the rotated surface code at a further reduced cost. To maintain high efficiency throughout our protocol, we design structured codes along with compact circuits bridging between them. More specifically, we construct a code family, the RP code, by putting the rotated surface code on <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mrow> <g:mi mathvariant="double-struck">R</g:mi> <g:mi mathvariant="double-struck">P</g:mi> </g:mrow> <g:mn>2</g:mn> </g:msup> </g:math> (a two-dimensional manifold), as well as two self-dual Calderbank-Shor-Steane codes, named SRP-3 and SRP-5, respectively. In our MSC protocol, we start with a cultivation process, in which a high-fidelity <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mi mathvariant="normal">T</k:mi> </k:mrow> </k:math> state is prepared on a small RP code with distance 3 or 5. Then, to preserve the logical <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mrow> <n:mi mathvariant="normal">T</n:mi> </n:mrow> </n:math> state, we use an efficient and easy-to-decode expansion stage to grow a small RP code to a larger rotated surface code in one syndrome extraction (SE) round. The RP code serves as an efficient transfer station with efficient SE circuits and compact interfaces between the SRP-3 (or SRP-5) code—used in the cultivation process to efficiently verify the correctness of the logical <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mrow> <q:mi mathvariant="normal">T</q:mi> </q:mrow> </q:math> state—and larger rotated surface codes for preserving the prepared logical <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mrow> <t:mi mathvariant="normal">T</t:mi> </t:mrow> </t:math> state. Our MSC protocol utilizes nonlocal connectivity, available on both neutral atom array and ion trap platforms. According to our Monte Carlo sampling results, our MSC protocol requires about an order of magnitude smaller space-time volume to reach a target logical error rate of around <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msup> <w:mn>10</w:mn> <w:mrow> <w:mo>−</w:mo> <w:mn>9</w:mn> </w:mrow> </w:msup> </w:math> compared to the original MSC protocol.