Majorana zero modes (MZMs) exhibit a fundamental property of non-Abelian statistics, which makes them potentially useful as ideal quantum bits to manipulate certain topological quantum tasks. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems, and there has been no investigation of their crucial non-Abelian aspects. Here, we experimentally investigate the non-Abelian character of MZMs in the Kitaev chain model using a photonic quantum simulator. Through the Jordan-Wigner transformation, the geometry phases of the braiding of two MZMs can be directly obtained in an optical system. The local phase-error immunity of MZMs is further studied. This work will be useful for the study of quantum non-Abelian statistics and may provide a novel way to investigate topological quantities of quantum systems.
Majorana zero modes (MZMs) exhibit a fundamental property of non-Abelian statistics, which makes them potentially useful as ideal quantum bits to manipulate certain topological quantum tasks. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems, and there has been no investigation of their crucial non-Abelian aspects. Here, we experimentally investigate the non-Abelian character of MZMs in the Kitaev chain model using a photonic quantum simulator. Through the Jordan-Wigner transformation, the geometry phases of the braiding of two MZMs can be directly obtained in an optical system. The local phase-error immunity of MZMs is further studied. This work will be useful for the study of quantum non-Abelian statistics and may provide a novel way to investigate topological quantities of quantum systems.
The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan-Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.
We simulate the non-Abelian exchange of Majorana zero modes (MZMs) on a quantum computer. Rather than utilizing MZMs at the boundaries of quantum Ising chains, which are typically represented as nonlocal operators on a quantum computer, using a Kitaev lattice allows us to exploit a local representation of MZMs. We detail the protocol for braiding two and four MZMs in terms of a spin Hamiltonian, i.e. physical qubit Hamiltonian. Projecting this onto a subspace of states, we extract an effective Hamiltonian which drives a non-Abelian Berry phase. Using several approximations, we construct a set of gates which mimics this accumulation of non-Abelian phase and process this construction on a quantum computer. For two and four MZMs, we realize braiding fidelities of approximately 85 and 47%, respectively.
The realization and manipulation of Majorana zero modes have drawn significant attention for their crucial role in enabling topological quantum computation. Conventional approaches to the braiding of Majorana zero modes rely on adiabatic processes. In this work, using a composite 2-Kitaev-chain system accommodating Majorana zero modes as a working example, we propose a nonadiabatic and non-Abelian geometry phase-based protocol to execute operations on these Majorana zero modes. This is possible by locally coupling the edge sites of both quantum chains with an embedded lattice defect, successfully simulating the braiding operation of two Majorana modes in a highly nonadiabatic fashion. To further enhance the robustness against control imperfections, we apply a multiple-pulse composite strategy to our quantum chain setting for second-order protection of the braiding operations. Our proposal can also support the fast and robust realization of the $\ensuremath{\pi}/8$ gate, an essential ingredient for universal quantum computation. This work hence offers a potential pathway toward the nonadiabatic and fault-tolerant control of Majorana zero modes.
FINDING: Majorana zero modes (MZMs) in topological superconductors yield non-Abelian anyonic statistics, enabling topologically protected quantum error correction via ground-state degeneracy in the Kitaev chain. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)+Δ(cjcj+1+h.c.)−μ(cj†cj−1/2)]. In the topological phase (∣μ∣<2t), unpaired Majorana operators γ1,γ2 satisfy γi†=γi, {γi,γj}=2δij, and the nonlocal fermion f=(γ1+iγ2)/2 gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: σ×σ=1+ψ, with quantum dimension dσ=2. | CONNECTION: The Ising anyon fusion rule σ×σ=1+ψ maps directly to the golden-ratio-adjacent structure of the Fibonacci anyon model (though MZMs are Ising, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting systems that can be viewed as fractionalized or “half” electrons. These quasiparticles obey non-Abelian braiding statistics which is one manifestation of such half-electron character. Due to such non-Abelian braiding property, Majorana zero mode pairs hold promise as potential qubits for topological quantum computation. It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor systems can be induced to host such esoteric Majorana modes as edge states if some precise experimental conditions are satisfied. Because of the relative simplicity of material and experimental requirements to host Majorana modes, there has been a flurry of experimental effort to realize them in semiconductor nanowire systems. While experimental efforts have produced preliminary evidence for the presence of Majorana zero modes in these systems, a thorough confirmation is lacking. The experimental signature in question is the presence of a zero-bias conductance-peak that, while necessary, is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism over the presence of Majorana modes in these experimental systems, it would seem natural to attempt braiding these putative Majorana modes in the near future. In that case, an observation of non-Abelian statistics would provide the necessary and sufficient condition in favor of Majorana presence in the studied experimental systems. This thesis has three distinct parts. First, we assume perfect Majorana modes as given that can be successfully braided. In this case, we calculate the diabatic error due to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit the underlying Majorana modes may be imperfect (the modes are not precisely at zero energy). We study the interplay of dissipation and finite energy splitting of the Majorana modes and study its consequence regarding the probability of successful braiding. Lastly, we propose studying the correlation between independent left and right conductance measurements as a means to distinguish between a topological versus a non-topological the mechanism underlying the observed zero-bias conductance-peak.
FINDING: Majorana zero modes (MZMs) in topological superconductors obey non-Abelian braid statistics, with exponential suppression of coupling between spatially separated modes enabling topologically protected quantum gates. | MATH: The Kitaev chain Hamiltonian H=−t∑ci†ci+1+Δ∑cici+1+h.c.−μ∑ci†ci yields unpaired Majorana operators γ1,γ2 with γi†=γi, {γi,γj}=2δij. Braiding exchanges γi→γj via unitary Uij=exp(4πγiγj), giving non-Abelian representation of the braid group Bn (e.g., U12U23=U23U12). Splitting energy ΔE∝e−L/ξ where ξ=ℏvF/Δ is the coherence length — exponential suppression with wire length L. | CONNECTION: The braid group Bn is generated by transpositions σi satisfying \( \sigma_i\s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com