Encoding a qubit in a trapped-ion mechanical oscillator (1807.01033v1)

Abstract: The stable operation of quantum computers will rely on error-correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator. A powerful encoding is formed from a periodically spaced superposition of position eigenstates. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach. Here, we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped Calcium ion, controlling and measuring the oscillator through coupling to an ancilliary internal-state qubit. We prepare and reconstruct logical states with an average square fidelity of $87.3 \pm 0.7 \%$, and demonstrate a universal logical single qubit gate set which we analyze using process tomography. For Pauli gates we reach process fidelities of $\approx 97\%$, while for continuous rotations we use gate teleportation achieving fidelities of $\approx 89 \%$. The control demonstrated opens a route for exploring continuous variable error-correction as well as hybrid quantum information schemes using both discrete and continuous variables. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum.

• The paper demonstrates the encoding of a qubit in a trapped-ion mechanical oscillator using the GKP code to simplify error correction and reduce hardware complexity.
• It employs state-dependent forces to manipulate displaced squeezed states, achieving an average square fidelity of 87.3% and Pauli gate fidelities around 97%.
• The study lays the groundwork for scalable quantum architectures by outlining protocols for continuous variable error correction and potential multi-qubit interactions.

Encoding a Qubit in a Trapped-Ion Mechanical Oscillator

The paper presents a significant advancement in the domain of quantum computation, particularly focusing on the field of error correction. The authors demonstrate the encoding of a qubit within a trapped-ion mechanical oscillator—a promising alternative to utilizing arrays of many physical qubits. This work leverages the Gottesman-Kitaev-Preskill (GKP) code to encode quantum information using a single higher-dimensional quantum system, a method that offers potential reductions in hardware overhead and simplified control mechanisms.

Key Experiments and Results

The researchers use the periodic superposition of displaced squeezed states to encode qubits within the axial motion of a single ^{40}Ca^+ ion. The primary experimental tool involves state-dependent forces (SDF), allowing for manipulation and measurement of the oscillator. The preparation and characterization of logical states achieved an average square fidelity of $87.3\pm0.7\%$. Furthermore, through process tomography, the authors illustrate a universal logical single-qubit gate set, obtaining process fidelities of approximately $97\%$ for Pauli gates and around $89\%$ for continuous rotations via gate teleportation.

An essential aspect of their work is the experimental realization of the GKP qubit whose logical states are observed to have a grid-like structure in phase space, as revealed by measuring both position and momentum probability densities. Additionally, the implementation of logical operations and the measurement of these states showcase fidelity results that align closely with theoretical expectations, demonstrating robust control over the encoded qubits.

Implications and Future Directions

The demonstrated approach offers a novel route for exploring continuous variable error-correction protocols and hybrid quantum information approaches combining both discrete and continuous variables. The potential impact of this research is wide-ranging, possibly enhancing not only error correction schemes but also quantum sensing applications where simultaneous precision measurements of small displacements in both position and momentum are desired.

Looking to future developments, the work could be expanded to include error correction implementations and the extension of the approach to multiple encoded qubits. Notably, achieving multi-qubit interactions could be facilitated through modes coupling, whether by laser mediation or exploiting the Coulomb force inherent in trapped-ion systems. These advancements set the stage for integrated quantum architectures capable of supporting complex quantum algorithms, reducing errors through fault-tolerant operations.

While achieving fault-tolerant quantum computation with the GKP qubit requires further improvement in state fidelity and phase-space control, the authors' work lays foundational insights into realizing quantum computers robust against decoherence and operational faults.

In conclusion, by demonstrating the practical implementation of a GKP-encoded qubit in a trapped-ion mechanical oscillator, the authors open avenues for more resource-efficient and reliable quantum processing techniques. Future research may build on these findings to enhance the performance and scalability of quantum computation systems, drawing closer to the goals of error-resistant quantum information processing.