Topological quantum computation with Majorana zero modes (MZMs) can dispense with physical braiding: multi-qubit Clifford gates are enacted by sequences of fermionic parity measurements. Tran, Bocharov, Bauer, and Bonderson (SciPost Phys. 8, 091 (2020)) formalized the compilation of Clifford gates from such measurements under a device-dependent 'difficulty weight' cost model, and used exhaustive search to find optimal measurement sequences and MZM labeling configurations for small hexon arrays, including a surface-code stabilizer-measurement schedule. Their framework leaves a precise agenda that remains unaddressed in the literature: (i) the search space grows exponentially with sequence length and array size (23n−42^{3n-4} per hexon; roughly 176k 16n−k−1176k\,16^{n-k-1} per hexon pair), and no provably efficient or approximation-guaranteed optimization algorithm — nor a hardness result — is known; (ii) the procrastination and Majorana-Pauli tracking techniques that make compilation efficient rely on measurement outcomes lying in Abelian fusion channels and have no extension to non-Abelian anyon systems, which the source's Final Remarks identifies as a direction for future work. The problem asks for a scalable compilation theory resolving these two linked cores.

Background

In topological quantum computation, quantum information is stored non-locally in the fusion state of non-Abelian anyons so that local noise is suppressed; gates are usually enacted by braiding the anyons around one another. Bonderson, Freedman, and Nayak showed in 2008 that physical braiding can be replaced entirely by measurement: a sequence of topological charge measurements (measurements of the total fusion outcome of a group of anyons) reproduces the braid transformation. For Majorana zero modes — zero-energy, non-Abelian (Ising-type) bound states realizable at the ends of semiconductor-superconductor nanowires — the elementary operation is a fermionic parity measurement of an even number of MZMs, e.g. by charge sensing or interferometry on the superconducting island hosting them. A 'hexon' is a superconducting island containing six MZMs with fixed total parity; its low-energy space is four-dimensional, and with one MZM pair reserved as ancillas of definite parity it encodes a single logical qubit. A 'tetron' is the analogous four-MZM island without ancillas. In such architectures, any multi-qubit Clifford gate (an element of the Clifford group, the gate set that maps Pauli operators to Pauli operators) can be compiled from parity measurements interleaved between islands.

Two strategies manage measurement outcomes. In forced measurement, an unwanted fusion outcome is eliminated by adaptively repeating measurements in different bases until the desired (typically trivial) outcome is obtained, a probabilistic process with a fluctuating number of steps. In procrastination and Majorana-Pauli tracking, every outcome is accepted and its effect is tracked classically; for MZMs, the tracking theorem of Tran et al. guarantees the accumulated deviation from the target gate is always a Pauli operator that can be corrected in software, so the number of physical measurements is deterministic and typically much smaller.

To compare measurement sequences beyond counting measurements, Tran et al. introduced a difficulty weight w(M)=wcncwtntwanaf(N)w(\mathcal{M}) = w_c^{n_c} w_t^{n_t} w_a^{n_a} f(N) for a parity measurement M\mathcal{M} of 2N2N MZMs, counting the number ncn_c of cutter gates (electrostatic gates that split an island) opened, the number ntn_t of tunneling junctions involved, the interferometry-loop area nan_a, and the number of islands NN; the parameters wc,wt,wa,f(N)w_c, w_t, w_a, f(N) are device-dependent and currently only estimated. Brute-force search under this cost model yielded globally optimal single-qubit Clifford sequences (verified exhaustively up to length 9), optimal controlled-Pauli sequences for one-sided and two-sided hexon architectures (the two scalable island layouts considered), optimal MZM labeling configurations (the assignment of computational roles to the six MZMs of each hexon), and an optimized surface-code (a leading topological quantum error-correcting code) stabilizer-measurement schedule using only nearest-neighbor island parity measurements.

The framework's limitations, stated by the authors, frame the open problem. The search space grows exponentially (23n−42^{3n-4} for single-hexon sequences of length nn; roughly 176k 16n−k−1176k\,16^{n-k-1} for two-hexon sequences with kk four-MZM measurements), so brute force 'is not practical in general'; Appendix B offers only unguaranteed reduction heuristics. The tracking and procrastination methods 'can only be applied when the measurement outcomes correspond to fusion channels that are Abelian' (as for MZMs and Ising anyons); for anyons with non-Abelian fusion outcomes one must fall back on forced measurements, and the source's Final Remarks points to extending the framework to other non-Abelian anyon systems and to systems with different topological orders. Noise, by contrast, is not part of the source's compilation agenda: quasiparticle poisoning (erroneous parity flips of an island) is explicitly neglected 'for the discussion in this paper' as a working assumption, and the difficulty weight is itself introduced as 'primarily ... a stand-in for error-rate' until a more accurate picture is obtained through physical experiments. Non-Clifford gates lie outside this compilation problem: universality is expected from measurement-based magic-state injection, as studied by Karzig et al.

Problem Statement

Develop a scalable compilation theory for measurement-only topological quantum computation with Majorana zero modes, replacing the proof-of-principle brute-force framework of Tran et al. (SciPost Phys. 8, 091 (2020)) with rigorous and deployable methods. The difficulty-weight parameters wc,wt,wa,f(N)w_c, w_t, w_a, f(N) are inputs (their experimental determination is a separate experimental program). The problem comprises two linked components: component (1) is relative to the same hexon/tetron measurement-architecture family and cost model w(M)=wcncwtntwanaf(N)w(\mathcal{M}) = w_c^{n_c} w_t^{n_t} w_a^{n_a} f(N) as the source, while component (2), following the source's Final Remarks, extends beyond that family to general non-Abelian anyon systems:

(1) Scalable optimization. Given a target Clifford gate, an array of one-sided/two-sided hexons and tetrons, the allowed 2-MZM, 4-MZM, and multi-island parity measurements, and weight parameters, compute measurement sequences and labeling configurations whose total difficulty weight is optimal or within a proven approximation ratio, with computational cost provably scalable (e.g., polynomial) in sequence length, number of qubits, and array size — or prove that no efficient algorithm with such guarantees exists (e.g., establish computational hardness of the minimal-weight compilation problem).

(2) Non-Abelian extension. Extend procrastination/Majorana-Pauli tracking compilation and its theorem beyond measurement outcomes in Abelian fusion channels, i.e., to non-Abelian anyon systems where projective measurements can yield non-Abelian fusion outcomes, including the associated cost models and compilation strategies — or prove a rigorous obstruction showing tracking cannot eliminate forced measurements in such systems.

Both components serve one objective: a compilation theory that the architect of a Majorana-based quantum computer can apply at array scale with performance guarantees, on the widest class of topological hardware.

References

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