Reducing the cost of ancilla purification for Majorana quantum computation

Abstract

Braiding Majorana zero modes does not by itself provide a universal set of quantum gates. Additional operations can be supplied through ancillary states, whose preparation need not have the same protection as braiding. Here, we consider purification of the non-Gaussian ancilla used for an entangling gate in Bravyi's scheme. We keep his elementary braid-and-fusion comparison and change how its outputs are combined. An additional layer compares the product of the three stabilizer signs. The first accepted errors giving an incorrect output then require three faulty input blocks rather than two. As a result, the preparation error decreases cubically instead of quadratically, and the asymptotic raw-copy overhead is reduced. We also construct a preliminary sequence in which a processed block is paired with a fresh copy from the same noisy source. For independent inputs supported in the even-parity sector and ideal purification operations, this sequence establishes a sufficient distillability fidelity of 59.95 percent after free Gaussian twirling. At a common source fidelity, it reduces the expected cost of preparing an input for the recurrence by more than seven orders of magnitude relative to the balanced construction studied here. Finally, we examine imperfect purification operations in an effective syndrome-noise model. The additional layer leaves the leading operation-error coefficient unchanged, although a nonzero error floor remains.

The paper studies purification of Bravyi’s eight-Majorana non-Gaussian ancilla, which supplies an entangling Clifford gate; a separate resource is needed for non-Clifford gates. It retains the braid-and-fusion two-copy comparison and changes its composition, using Gaussian frame changes to select stabilizer products while retaining unequal syndrome populations between layers. A fourth layer compares the product of three stabilizers on two independently prepared three-layer outputs. Accepted incorrect outputs then require three faulty raw blocks rather than two, giving $\epsilon_{\mathrm{out}}=(384/343)\epsilon^3+O(\epsilon^4)$. The isotropic recurrence’s unstable fidelity fixed point decreases from $0.61560748$ to $0.60471167$, and its asymptotic expected-copy overhead becomes $O([\log(1/\delta)]^{\log_3 16})$, versus exponent $3$ previously. Integer fault enumerators, exact recurrence polynomials, and root counting establish these claims. A twelve-stage initialization sequence includes one comparison between a processed block and a fresh source copy. Exact rational verification certifies distillability for independent even-parity inputs with target overlap $F\geq0.5995$, after Gaussian twirling. At $F=0.5997$, its expected seed cost is about $1.04\times10^6$ copies, versus $1.76\times10^{13}$ for the stated balanced benchmark; neither construction is proved optimal. Near the certified endpoint, subsequent purification remains costly: the prescribed schedule needs about $2.78\times10^{20}$ copies to reach infidelity $10^{-6}$. An effective syndrome-noise model gives the same leading operation-error floor, $17\lambda/7$, for both recurrences, not a hardware threshold. Costs assume perfect storage and exclude routing and timing. Arbitrary correlated inputs and unconditioned parity noise are not covered. Distillability for the isotropic family remains unresolved between the convex-Gaussian boundary $F=1/2$ and $0.5995$.