A Memory-Magic Exchange Law in Streaming Clifford+T Compilation

Abstract

A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $α$, committed $T$ gates per bit of memory forgone, for ancilla-free coordinatewise Clifford+$T$ compilation. The Ramanujan bound for the Clifford+$T$ lattice gives $α\ge2$ with explicit constants, the square-root barrier of the spectral method. An elementary determinant method, using that quaternion numerators are lattice points on spheres in both real embeddings of $\mathbb{Q}(\sqrt2)$, counts words near an arbitrary rotation coset below that barrier and gives $α\ge11/5$ asymptotically and unconditionally, and a height dichotomy for the resulting sphere sections raises this to $α\ge17/7$. At Clifford-framed cosets the volume law holds up to subexponential factors: all but a vanishing fraction of $z$-rotations need $T$-count $(3-o(1))\log_2(1/\varepsilon)$, and processes whose committed pieces are close to Clifford-framed $z$-rotations, including per-rotation pipelines, have $α\ge3-o(1)$, which a fractional-passthrough family attains under the Ross-Selinger typical-cost hypothesis. Under an equidistribution conjecture supported by exhaustive enumeration to $T$-count 22, $α=3$ in general and memory should be shed in whole rotations. The bounds hold even when the phases cancel to the identity; side information enters through a conditional entropy; probabilistic mixing halves the costs but not the rate. With clean ancillas and a phase-gradient catalyst, table lookups batched across coordinates drive the rate to $O(1/\log\log(1/\varepsilon))$, so the constant-rate law is specific to coordinatewise synthesis.