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arXiv:2110.13599 · Physical Review Letters

Fermion-Parity-Based Computation and Its Majorana-Zero-Mode Implementation

Campbell McLauchlan, Benjámin Béri

Published May 5, 2022• 13 citations
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Abstract

Majorana zero modes (MZMs) promise a platform for topologically protected fermionic quantum computation. However, creating multiple MZMs and generating (directly or via measurements) the requisite transformations (e.g., braids) pose significant challenges. We introduce fermion-parity-based computation (FPBC): a measurement-based scheme, modeled on Pauli-based computation, that uses efficient classical processing to virtually increase the number of available MZMs and which, given magic state inputs, operates without transformations. FPBC requires all MZM parities to be measurable, but this conflicts with constraints in proposed MZM hardware. We thus introduce a design in which all parities are directly measurable and which is hence well suited for FPBC. While developing FPBC, we identify the "logical braid group" as the fermionic analog of the Clifford group.

DOI
10.1103/physrevlett.128.180504

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Majorana zero modes (MZMs) promise a platform for topologically protected fermionic quantum computation. However, creating multiple MZMs and generating (directly or via measurements) the requisite transformations (e.g., braids) pose significant challenges. We introduce fermion-parity-based computation (FPBC): a measurement-based scheme, modeled on Pauli-based computation, that uses efficient classical processing to virtually increase the number of available MZMs and which, given magic state inputs, operates without transformations. FPBC requires all MZM parities to be measurable, but this conflicts with constraints in proposed MZM hardware. We thus introduce a design in which all parities are directly measurable and which is hence well suited for FPBC. While developing FPBC, we identify the "logical braid group" as the fermionic analog of the Clifford group.
One of the main challenges for quantum computation is that while the number of gates required to perform a non-trivial quantum computation may be very large, decoherence and errors in realistic quantum architectures limit the number of physical gate operations that can be performed coherently. Therefore, an optimal mapping of the quantum algorithm into the physically available set of operations is of crucial importance. We examine this problem for a measurement-only topological quantum computer based on Majorana zero modes, where gates are performed through sequences of measurements. Such a scheme has been proposed as a practical, scalable approach to process quantum information in an array of topological qubits built using Majorana zero modes. Building on previous work that has shown that multi-qubit Clifford gates can be enacted in a topologically protected fashion in such qubit networks, we discuss methods to obtain the optimal measurement sequence for a given Clifford gate under the constraints imposed by the physical architecture, such as layout and the relative difficulty of implementing different types of measurements. Our methods also provide tools for comparative analysis of different architectures and strategies, given experimental characterizations of particular aspects of the systems under consideration. As a further non-trivial demonstration, we discuss an implementation of the surface code in Majorana-based topological qubits. We use the techniques developed here to obtain an optimized measurement sequence that implements the stabilizer measurements using only fermionic parity measurements on nearest-neighbor topological qubit islands.
We present a topologically protected teleportation protocol based on projective parity measurements between spatially separated Majorana zero modes (MZMs), eliminating the need for dynamic braiding. Unlike conventional teleportation schemes, our method preserves logical information through nonlocal encoding and suppresses decoherence exponentially with Majorana separation. We provide a rigorous mathematical framework that includes six theorems and a lemma, proving fidelity bounds, no entropy increase under ideal QND parity measurement under quantum non-demolition (QND) measurements, and compliance with the no-cloning theorem. We demonstrate that all correction operations lie within the Clifford group, enabling efficient, fault-tolerant implementation. Furthermore, we outline a scalable architecture for multi-qubit teleportation and relate our framework to recent experimental advances in quantum-dot-based Kitaev chains and superconducting nanowire platforms. These results position Majorana-based teleportation as a thermodynamically stable and experimentally viable approach to scalable quantum information transfer. All operations discussed are Clifford-only; achieving universality requires non-Clifford resources and lies outside our scope.
Majorana-based quantum computation seeks to encode information nonlocally in pairs of Majorana zero modes, thereby isolating qubit states from a local noisy environment. In addition to long coherence times, the attractiveness of Majorana-based quantum computing relies on achieving topologically protected Clifford gates from braiding operations. Recent works have conjectured that mean-field BCS calculations may fail to account for nonuniversal corrections to the Majorana braiding operations. Such errors would be detrimental to Majorana-based topological quantum computing schemes. In this work, we develop a particle-number-conserving approach for measurement-based topological quantum computing and investigate the effect of quantum phase fluctuations. We demonstrate that braiding transformations are indeed topologically protected in charge-protected Majorana-based quantum computing schemes.
Quantum error correction protocols protect against local errors by storing logical information non-locally. This poses a challenge: how to design efficient logical gates on the non-local ``hidden'' logical information, and how to implement these gates using the local physical operations. We develop a general description of planar Pauli stabilizer codes and protocols for logical operations in terms of point-like particles called Majorana fermions. Information is stored in the pairwise fermion parities of spatially separated Majorana fermions. The description in terms of Majorana fermions captures not only large distance asymptotics, but also all scales down to the lattice constants. We exploit this locality to densely pack logical information in spacetime. The simplest application is to a static case: dense memory. More importantly, we implement fault-tolerant Majorana motion and leverage this primitive to design braiding-based logical gates. This approach reduces space overhead of logical operations resulting in an improved logical error rate given fixed number of physical qubits. We illustrate a practical use of our approach by designing and benchmarking of 2-qubit Clifford gates. We find numerically that our protocol outperforms lattice surgery in this setting for near-term error rates and realistic device constraints. More generally, introduction of compact motion of Majorana fermions as an efficient computational primitive opens a promising new route for the design of low overhead error correction protocols.
FINDING: Six Majorana zero modes (MZMs) form three qubit pairs whose braiding generates a finite Clifford-group representation — the minimal non-Abelian platform for universal-adjacent topological quantum computation. | MATH: 2n MZMs → n qubits; 6 MZMs → 3 qubits → Hilbert space dim 2³ = 8. Braid group B₆ (or its image) maps to Clifford group C₃ ⊂ U(8). Ising anyon σ braiding: γᵢ → γⱼ, γⱼ → −γᵢ (i≠j), γₖ → γₖ. Majorana algebra {γᵢ,γⱼ} = 2δᵢⱼ. Fusion: σ×σ = 1 + ψ. Braid generator eigenvalues e^{±iπ/8} (topological phase, related to ν=5/2). | CONNECTION: Clifford group order for n qubits = 2^{n²+2n}·∏(4ᵏ−1); for n=3: 2^{15}·(4−1)(16−1)(64−1) = 32768·3·15·63 = 92,897,280. Braid group B₆ has 6 generators σ₁…σ₅ with relations σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ (Yang–Baxter / braid relation) — a cubic algebraic relation, echoing the 3-fold symmetry of A₂ root system and the 2π/3 rotation in SU(3). The 6 MZMs sit at vertices of an octahedron (6 vertices, 8 faces = 2³ Hilbert states); braiding = SO(6) rota Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Kitaev chain, with topological protection scaling exponentially in wire length, enabling topologically fault-tolerant quantum gates. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)−μ(cj†cj−1/2)+Δ(cjcj+1+h.c.)] H = \sum_j [-t(c_j^\dagger c_{j+1} + h.c.) - \mu(c_j^\dagger c_j - 1/2) + \Delta(c_j c_{j+1} + h.c.)] H=∑j​[−t(cj†​cj+1​+h.c.)−μ(cj†​cj​−1/2)+Δ(cj​cj+1​+h.c.)]; Majorana operators γ2j−1=cj+cj† \gamma_{2j-1} = c_j + c_j^\dagger γ2j−1​=cj​+cj†​, γ2j=−i(cj−cj†) \gamma_{2j} = -i(c_j - c_j^\dagger) γ2j​=−i(cj​−cj†​), satisfying {γa,γb}=2δab \{\gamma_a, \gamma_b\} = 2\delta_{ab} {γa​,γb​}=2δab​; topological gap Δtop∼Δ \Delta_{\text{top}} \sim \Delta Δtop​∼Δ (superconducting gap); exponential suppression of coupling between MZMs: ϵ∝e−L/ξ \epsilon \propto e^{-L/\xi} ϵ∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F / \Delta ξ=ℏvF​/Δ is the coherence length; braid group Bn B_n Bn​ generated by exchanges σi \sigma_i σi​ with σi2=−1 \sigma_i^2 = -1 σi2​=−1 (fermionic parity), yielding non-Abelian representation ρ(σi)=e±πγiγi+1/4 \rho(\sigma_i) = e^{\pm \pi \gamma_i \gamma_{i+1}/4} ρ(σi​)=e±πγi​γi+1​/4. | CONNECTION: The braid group Bn B_n Bn​ is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com