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arXiv:2610.37991

Reducing the cost of ancilla purification for Majorana quantum computation

Piotr {Ć}wikli{ń}ski, Łukasz Fojcik, Michał Studzi{ń}ski

quant-phcond-mat.str-elSubmitted Oct 27, 2026• Announced Oct 29, 2026• 0 citations
v1 · Oct 27, 2026v2 · Oct 30, 2026
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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.

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21 pages, 3 figures

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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.
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Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For nnn copies of a ddd-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most εεε using O~(n)\widetilde{O}(n)O(n) T gates, where the tilde-OOO notation suppresses polylogarithmic factors in nnn, ddd, and 1/ε1/ε1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<121−1/nε<\frac{1}{2}\sqrt{1-1/n}ε<21​1−1/n​ requires at least 2n−22n-22n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
Adiabatic braiding of Majorana zero modes can be used for topologically protected quantum information processing. While extremely robust to many environmental perturbations, these systems are vulnerable to noise with high-frequency components. Ironically, slower processes needed for adiabaticity allow more noise-induced excitations to accumulate, resulting in an antiadiabatic behavior that limits the precision of Majorana gates if some noise is present. In a recent publication (2017 Phys. Rev. B 96 075158), fast optimal protocols were proposed as a shortcut for implementing the same unitary operation as the adiabatic braiding in a low-energy effective model. These shortcuts sacrifice topological protection in the absence of noise but provide performance gains and remarkable robustness to noise due to the shorter evolution time. Nevertheless, gates optimized for vanishing noise are suboptimal in the presence of noise. If we know the noise strength beforehand, can we design protocols optimized for the existing unavoidable noise, which will effectively correct the noise-induced errors? We address this question in the present paper, focusing on the same low-energy effective model. We find such optimal protocols using simulated-annealing Monte Carlo simulations. The numerically found pulse shapes, which we fully characterize, are in agreement with Pontryagin's minimum principle, which allows us to arbitrarily improve the approximate numerical results (due to discretization and imperfect convergence) and obtain numerically exact optimal protocols. The protocols are bang–bang (sequence of sudden quenches) for vanishing noise, but contain continuous segments in the presence of multiplicative noise due to the nonlinearity of the master equation governing the evolution. We find that such noise-optimized protocols completely eliminate the above-mentioned antiadiabatic behavior. The final error corresponding to these optimal protocols monotonically decreases with the total time (in three different regimes). A liner fit to 1/ τ indicates extrapolation of the cost function to finite value in the limit. However, quadratic and cubic fits are more suggestive of the cost function extrapolating to zero in the limit of infinite time. Our results set the precision limit of the device as a function of the noise strength and total time.
Quantum gates executed on physical hardware are inevitably degraded by environmental noise. While state purification effectively distills static quantum resources, the dynamic execution of quantum algorithms requires a higher-order approach to mitigate errors on the operations themselves. In this work, we investigate universal unitary purification: the task of utilizing a quantum higher-order operation to partially restore the ideal action of an unknown unitary corrupted by a known noise model. We consider deterministic purification of unknown single-qubit unitaries under independent depolarizing noise with ideal auxiliary operations. We prove that within the indefinite causal order (ICO) framework, no nontrivial 2-slot higher-order operation can universally purify the set of single-qubit unitaries. Overcoming this strict limitation, we establish that a 3-slot parallel architecture provides the minimal realization for nontrivial purification. We analytically derive the optimal average fidelity over all 3-slot ICO strategies and show that a parallel strategy attains it, strictly surpassing trivial strategies through input encoding and joint decoding of the channel outputs and a quantum memory. Furthermore, we provide a concrete quantum circuit construction attaining this optimum. Our results clarify the possibilities and limitations of purifying unknown qubit unitaries under depolarizing noise, offering insights for robust gate design.
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