Fault-Tolerant Code-Switching Protocols: A Leap Forward for Near-Term Quantum Computing

Quantum computing’s potential is hampered by noise, which limits the accuracy of quantum algorithms. To increase robustness against noise, quantum information is encoded on logical qubits, with quantum error correction (QEC) used to correct errors. However, implementing a universal fault-tolerant (FT) gate set is challenging. Topological color codes are seen as a solution for FT quantum computing, with a combined approach of switching between two and three-dimensional codes. Recent advancements in quantum computing focus on practical encoding of qubits and quantum memory. The development of fault-tolerant code-switching protocols for near-term quantum processors is seen as a promising avenue for universal FT quantum computing.

What are Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors?

Quantum computing is a rapidly evolving field that promises to perform certain computational tasks exponentially faster than any known classical algorithm. However, in the current noisy intermediate-scale quantum (NISQ) era, the accuracy of quantum algorithms is limited by noise. One way to increase their robustness against noise is to encode quantum information on logical qubits, with each logical qubit consisting of multiple physical qubits. On these logical qubits, quantum error correction (QEC) can be performed to correct for errors on physical qubits and recover the initially encoded information.

For physical error rates below a certain threshold, QEC enables practical quantum computing for arbitrarily long times, given suitable fault-tolerant (FT) circuit constructions. FT circuits can be designed using transversal gate operations, which are composed of single-qubit unitaries acting on individual qubits in each encoded block. This prevents potential errors in any of these operations from proliferating uncontrollably.

A discrete set of operations that forms a universal gate set can be used to approximate arbitrary computations on encoded qubits. This requires at least one non-Clifford gate. However, the Eastin-Knill theorem states that no QEC code exists that has a universal transversally encoded and, therefore, FT gate set. This complicates the implementation of a universal FT gate set and poses a key challenge towards error-corrected universal quantum computing.

How are Topological Color Codes Used in Quantum Computing?

Topological color codes are widely acknowledged as promising candidates for fault-tolerant quantum computing. However, neither a two-dimensional nor a three-dimensional topology can provide a universal gate set HTC-NOT with the T-gate missing in the two-dimensional and the H-gate in the three-dimensional case. These complementary shortcomings of the isolated topologies may be overcome in a combined approach by switching between a two and a three-dimensional code while maintaining the logical state.

In this work, resource-optimized deterministic and non-deterministic code-switching protocols for two and three-dimensional distance-three color codes using fault-tolerant quantum circuits based on flag qubits were constructed. Deterministic protocols allow for the fault-tolerant implementation of logical gates on an encoded quantum state, while non-deterministic protocols may be used for the fault-tolerant preparation of magic states.

Taking the error rates of state-of-the-art trapped-ion quantum processors as a reference, a logical failure probability of 3% for deterministic logical gates, which cannot be realized transversally in the respective code, was found. By replacing the three-dimensional distance-three color code in the protocol for magic state preparation with the morphed code introduced in Vasmer and Kubica PRX Quantum 3030319 2022, the logical failure rates were reduced by two orders of magnitude, thus rendering it a viable method for magic state preparation on near-term quantum processors.

What are the Recent Advancements in Quantum Computing?

Recent quantum computing experiments are focused on the practical encoding of qubits on a logical level and investigate the implementation of quantum memory. For example, on trapped-ion systems, the preparation of logical states, error-detecting codes, FT stabilizer readouts in a shuttling-based architecture, and repeated cycles of QEC have been implemented.

In superconducting qubits, logical qubits have successfully been initialized, repeated QEC has been realized, and error-detection codes have been realized. Recently, a distance-five surface-code logical qubit outperformed a distance-three logical qubit, demonstrating an improvement of performance of QEC codes with an increasing number of qubits.

Rydberg atoms are a promising candidate for building quantum processors due to their strong long-range interactions and scalability, and they have shown rapid progress in single and multi-qubit control, as well as are the first elements of quantum error correction. Advancements in other qubit architectures have been reported as well, for example, a three-qubit phase-correcting code in a silicon-based architecture, and fault-tolerant operations on a logical qubit using spin qubits in diamond, among others.

How Does Code-Switching Enable Fault-Tolerant Quantum Computing?

These advancements in practical and scalable implementations of logical qubits enable FT operations on these encoded states and motivate the search for ways to achieve FT universal quantum computing on near-term devices. The FT control of an error-corrected single logical qubit has been demonstrated on a trapped-ion processor, as have logical operations in a distance-two error-detecting surface code on a superconducting architecture and entangling gates between logical qubits.

A universal set of gates was recently implemented for the first time on a seven-qubit Steane code, which is the smallest error-correcting color code using FT circuit constructions with flag qubits. Here, the universal gate set is completed by using a combined approach of switching between a two and a three-dimensional code while maintaining the logical state.

This method of code-switching enables the fault-tolerant and deterministic implementation of a universal gate set under realistic conditions and thereby provides a practical avenue to advance universal fault-tolerant quantum computing and enable quantum algorithms on first error-corrected logical qubits.

What is the Future of Quantum Computing?

The future of quantum computing looks promising with the development of fault-tolerant code-switching protocols for near-term quantum processors. These protocols allow for the fault-tolerant implementation of logical gates on an encoded quantum state and the fault-tolerant preparation of magic states.

With the error rates of state-of-the-art trapped-ion quantum processors as a reference, the logical failure probability of deterministic logical gates can be significantly reduced. This makes it a viable method for magic state preparation on near-term quantum processors.

The advancements in practical and scalable implementations of logical qubits, along with the development of fault-tolerant code-switching protocols, provide a practical avenue to advance universal fault-tolerant quantum computing and enable quantum algorithms on first error-corrected logical qubits. This will undoubtedly revolutionize the field of quantum computing and open up new possibilities for computational tasks.