Creating entangled logical qubits in the heavy-hex lattice with topological codes (2404.15989v1)

Abstract: Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hex architecture of current IBM Quantum devices. In such cases it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error correcting codes, many qubits remain effectively unused. In this work we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133 qubit IBM Quantum device. Using transversal CX gates and lattice surgery we demonstrate entanglement between these logical qubits with code distance up to $d = 4$ and five rounds of stabilizer measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical $XX$, $YY$, and $ZZ$ observables. With this we verify the violation of Bell's inequality for both the $d=2$ case with post selection featuring a fidelity of $94\%$, and the $d=3$ instance using only quantum error correction.

• The paper presents a novel method for creating entangled logical qubits on the heavy-hex lattice by overlapping the 3CX surface code and the Bacon-Shor code to leverage existing qubit connectivity.
• Key findings include successful execution of transversal CX gates and lattice surgery, enabling measurement of logical entanglement and Bell's inequality violation.
• This approach enhances scalability and fault tolerance by utilizing existing qubit architectures effectively, demonstrating a path towards more resource-efficient quantum computation.

Entangled Logical Qubits on the Heavy-Hex Lattice with Topological Codes

The paper under consideration presents a novel approach to quantum error correction on IBM's quantum devices using the heavy-hex lattice architecture. The authors, Bence Hetényi and James R. Wootton, aim to leverage the unique qubit connectivity constraints of this architecture by employing an innovative method that involves overlapping the 3CX surface code and the Bacon-Shor code. This method facilitates the implementation of entangled logical qubits, harnessing the otherwise underutilized qubits in the heavy-hex lattice.

Overview

The primary contribution of this work lies in turning a limitation of the heavy-hex connectivity into an advantage by running multiple quantum error-correcting codes simultaneously. The authors have successfully demonstrated that it is possible to execute two different topological codes—3CX and Bacon-Shor—on the same set of qubits, thereby enabling fault-tolerant entangling gates and measurements.

Key findings include the ability to apply transversal CX gates and carry out lattice surgery operations seamlessly. This enables measurement of the logical entanglement between qubits and violation of Bell's inequality, showcasing the potential of IBM's 133-qubit system. The experimental execution achieves logical qubit entanglement using CX gates and lattice surgery, reaching agreement with theoretical expectations of error thresholds and scalability.

Numerical Results and Implications

The paper reports success in achieving fidelities up to 94% for $d=2$ code distances with post-selection, and simulations confirm Bell's inequality violations. The logical operations demonstrated maintain non-planar connectivity, allowing for simultaneous measurement of $XX$, $YY$, and $ZZ$ observables. Such results are pivotal as they highlight the compatibility and robustness of the combined code not only in theory but also in practical application under realistic conditions.

The results underscore the potential of utilizing existing qubit architectures to optimize resources, marking a shift towards more versatile and resource-efficient approaches to quantum computation on existing hardware.

Practical and Theoretical Implications

1. Resource Optimization: By showing that unused qubits can be utilized effectively, the paper paves the way for more resource-efficient strategies on current quantum devices.
2. Scalability and Fault Tolerance: The approach enhances scalability and provides a pathway to achieving fault-tolerant quantum computations without significant hardware modifications.
3. Quantum Error Correction Strategies: It introduces a compelling method that others in the field can explore, especially for hardware with similar connectivity constraints.
4. Nonplanar Connectivity in Practical Systems: Demonstrating functional nonplanar connectivity serves as a precursor to implementing more complex quantum error-correcting codes in the future.

Future Directions

Looking ahead, this work lays the groundwork for further exploration into hybrid coding techniques and extending these methods to tackle larger distance codes and more rounds of stabilizer measurements. These advances would be crucial for deploying quantum computers that can perform practical, error-corrected quantum computations. Further research might also explore the integration of these strategies with fast feedback systems, potentially speeding up quantum computation processes.

In conclusion, this paper illustrates an adept application of current quantum hardware capabilities, setting a precedent for future innovations in quantum error correction methods. The effective use of heavy-hex lattice architecture exemplifies a forward-compatible strategy in quantum computing, poised to catalyze further advancements in the quest for scalable, fault-tolerant quantum systems.