Quantum Magic in early FTQC: From Diagonal Clifford Hierarchy No-Go Theorems to Architecture Design Blueprints

Abstract

We address the circuit-design problem of maximizing quantum magic in early fault-tolerant quantum computing (early FTQC), where logical dynamics natively take the form of alternating Clifford layers and diagonal non-Clifford layers. To render this optimization analytically tractable, we first prove a uniqueness theorem: for operational magic functionals built from Pauli expectation values, the axioms of faithfulness and tensor-product additivity force a R\'enyi-type dependence on the Pauli-spectrum. Leveraging the closed phase-polynomial description of the diagonal Clifford hierarchy, we derive exact Pauli-spectrum expressions and tight bounds for a shallow-layer model. These bounds expose a zero-magic mechanism and prove that maximal magic strictly requires graph-state preconditioning. Consequently, we establish our first no-go theorem: hierarchy level alone cannot universally order operational magic. Extending our framework to the $N$-layer model motivated by the Space-Time Efficient Analog Rotation (STAR) architecture, we obtain an exact iterative update rule for the Pauli spectrum. This yields a second no-go theorem: no state-independent sequence of operations can guarantee monotonic magic improvement. Together, these theorems demonstrate that algebraic gate structures are fundamentally insufficient to dictate resource generation. To overcome this, we reframe early FTQC gate selection as a state-aware, differentiable optimization over continuous analog parameters. Finally, we identify a severe kinematic expressibility bottleneck in architectures restricted to single-qubit $Z$-rotations and show that introducing nonlinear diagonal phases, such as multi-qubit $Z$-rotation, shatters this bottleneck. This provides a fundamental principle for demonstrating early FTQC, establishing scalable magic generation as a foundational benchmark for evaluating early FTQC architectures.