Negative Energy Splits Unlock More Stable Quantum Bits for Future Computers

Majorana zero modes, crucial components for building topological quantum computers, experience hybridization when their wavefunctions overlap, causing energy level splitting and increasing error rates during quantum gate operations. Cole Peeters, Themba Hodge, and Stephan Rachel, all from the School of Physics at the University of Melbourne, demonstrate a surprising phenomenon, negative hybridization, which effectively reduces the overall hybridization energy and consequently lowers gate errors. Their research reveals that this negative hybridization can suppress errors below the critical threshold required for fault-tolerant quantum computation. Importantly, this intrinsic property of Majorana zero modes offers a pathway to restore functionality to systems utilising imperfect modes, representing a significant advance in the field of quantum information processing.

This counterintuitive phenomenon offers a potential solution to a critical challenge in building stable quantum computers, namely the accumulation of errors during quantum gate operations.

The research demonstrates that negative hybridization, an intrinsic characteristic of Majorana zero modes, can substantially reduce the average hybridization energy of a quantum gate, thereby suppressing errors. Through illustrative examples, researchers show that this negative hybridization can maintain gate fidelities above the crucial fault-tolerance threshold.
This work addresses a long-standing problem in topological quantum computing, where the successful execution of quantum gates depends on carefully controlling the speed at which Majorana modes are manipulated. Existing limitations stem from the hybridization between Majorana modes, caused by the overlap of their wavefunctions, which splits their energy levels and introduces errors.

By demonstrating negative hybridization, the study establishes a pathway to mitigate these errors, even in systems with imperfect Majorana zero modes. The ability to reduce hybridization energy, or even drive it towards zero, effectively extends the permissible timeframe for performing quantum operations without significant degradation of quantum information.
The investigation centres on analysing the Bogoliubov, de Gennes Hamiltonian, revealing that in finite systems, Majorana modes do not exist at precisely zero energy. Researchers quantify the impact of energy splitting on gate performance using fidelity, aiming to achieve values exceeding 99% over extended braid times, a benchmark for demonstrating non-Abelian statistics.

Applying this approach to a specific example, a √Z gate involving two Majorana zero modes on a T-junction, the study shows how negative hybridization suppresses energy splitting and maintains high fidelity. Specifically, the average hybridization energy is calculated by integrating the instantaneous energy splitting over the duration of the gate operation.

Spectroscopic analysis confirms that the energy of Majorana bound states can indeed cross through zero, indicating a swapping of ground state parity. This intrinsic error correction property of Majorana modes promises to restore non-Abelian statistics even in systems constrained by material limitations and imperfect device fabrication, paving the way for more robust and scalable topological quantum computers.

Mitigation of Majorana zero mode hybridisation via symmetric braid decomposition

A detailed analysis of Majorana zero mode (MZM) hybridization underpinned this work, focusing on its detrimental effect on topological quantum gate performance. Researchers addressed the issue of wavefunction overlap, which causes energy level splitting and introduces errors during topological quantum gate operations.

The study demonstrated that the energy splitting of MZMs can become negative, a property exploited to reduce average hybridization energy and improve gate fidelity. Two illustrative examples were constructed to showcase how negative hybridization suppresses gate errors below the 1% threshold. The core methodological innovation involved implementing symmetric braids to correct for hybridization.

Initial braids were decomposed into two halves, denoted B ij and B ij , allowing for the creation of trivial operations, B ij B ij −1 or B ij B ij −1 , that mimic braiding movements without contributing to braiding statistics. A corrected √X gate was simulated, visualizing MZMs at the boundaries between topological and trivial regions defined by positive and negative chemical potentials.

The time-dependent hybridization energy was then plotted to illustrate the impact of the correction protocol. Specifically, the process involved simultaneously ramping the entire system from a positive chemical potential (μ topo ) to a negative one (−μ topo ) at the midpoint of the uncorrected braid (T = 1/6), defining an operation M (+) ij .

Its counterpart, M (−) ij , reversed this transformation, maintaining symmetry and preventing net energy changes. For √Z gates, a more sophisticated procedure was employed to avoid quasiparticle poisoning, detailed in supplementary material. Simulations assessed the fidelity of both √X and √Z gates, comparing uncorrected and corrected braids as a function of total braiding time, T.

The resulting error in fidelity, 1 − F, was plotted, with the red region indicating errors exceeding the 1% threshold. The effectiveness of the correction was demonstrated by achieving errors below 10 −7 , significantly surpassing the requirements for error correction, even when introducing asymmetry (δ = μ + topo + μ − topo ) between positive and negative chemical potentials. These protocols are agnostic to specific system parameters, eliminating the need for prior knowledge of hybridization dynamics.

Negative Majorana energy splitting enables robust topological quantum gates

Researchers demonstrate that the energy splitting of Majorana zero modes can become negative, offering a pathway to reduce hybridization energy during topological quantum gate operations. This negative hybridization suppresses gate errors to levels below the established error threshold in two illustrative examples.

As an inherent characteristic of Majorana zero modes, this negative hybridization allows systems with imperfect Majorana zero modes to regain functionality for quantum information processing. The study details the parameter δ, representing asymmetry between positive and negative chemical potentials, influencing hybridization-dependent error.

Although imperfections reintroduced hybridization, error rates remained significantly lower than uncorrected cases, even with considerable values of δ relative to μtopo. This highlights the robustness of the corrected gates and their ability to maintain functionality despite system variations. A CNOT gate, utilising a 2-qubit system comprised of six Majorana zero modes, was implemented and assessed.

Braiding error, quantified as 1 − F, was substantially reduced through the application of negative hybridization. The fidelity of the corrected CNOT gate was tested across multiple random initial states, revealing a striking improvement over the uncorrected gate. The error associated with static systems, representing hybridization during idle periods, served as a lower bound for the corrected gate error.

Analysis at a braid time of 4000ħ/t showed that corrected and uncorrected CNOT gate errors remained separated by approximately four orders of magnitude across a range of chemical potentials. Simulations were performed on Kitaev chains arranged in T-junction geometries, with the Hamiltonian extended to accommodate multiple vertical legs connected to a horizontal chain. The system incorporated buffer sites and alternating vertical leg lengths to facilitate parity swaps required for braiding operations.

Negative Hybridization Enables Fault-Tolerant Quantum Computation with Majorana Zero Modes

Majorana zero modes, essential for building quantum bits in topological quantum computers, typically experience hybridization that degrades performance through energy level splitting and increased error rates during quantum gate operations. Recent investigations demonstrate that the energy splitting of these Majorana zero modes can become negative, offering a pathway to reduce average hybridization energy and improve gate fidelity.

This negative hybridization effectively suppresses errors, achieving levels below the fault-tolerance threshold in illustrative examples. This intrinsic property of Majorana zero modes allows systems with imperfect components to regain functionality for quantum information processing. Simulations of a two-qubit CNOT gate, utilising six Majorana zero modes, reveal a significant improvement in fidelity when employing this correction method, with corrected gates consistently outperforming uncorrected counterparts across a range of initial states and parameter values.

The observed error reduction extends to static systems, where hybridization occurs even without movement of the Majorana zero modes, establishing a lower bound for potential error mitigation. While acknowledging that simulations were performed on specific Kitaev chains, the authors suggest the potential for tailoring future topological superconductor platforms to directly implement or readily access negative hybridization. This work indicates that fault-tolerant quantum computation based on Majorana zero modes is achievable through the exploitation of negative hybridization, paving the way for more robust and reliable quantum computers.