Implementing topologically ordered time crystals on quantum processors

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In a new study published in Nature Communications, scientists have implemented the topologically ordered time crystal on a quantum processor for the first time.

This marks a new era for quantum technology, since time crystals have been traditionally challenging to combine with topological order. However, achieving this combination adds stability and robustness to the system, a requirement for quantum computing applications.

Time crystals are a recent introduction to science, with the idea first proposed in 2012 by Nobel laureate Frank Wilczek. It is a quantum system that can naturally oscillate between states without the need for a continuous external energy source.

In simple terms, time crystals are a type of material in which the atoms are arranged periodically in time, rather than space, which is how ordinary crystals (like diamonds) are.

The system always remains in the lowest energy state during oscillations, which is the ground level. Time crystals were experimentally confirmed in 2017 and have potential applications in various quantum technologies.

Achieving topological order, or global order, in time crystals can be challenging due to their dynamic nature. The research team aimed to bridge this gap by demonstrating a topologically ordered time crystal.

Phys.org spoke to some of the researchers behind the study, including Dr. Liang Xiang, Wenjie Jiang, Zehang Bao, Assistant Prof. Qiujiang Guo, and Prof. Haohua Wang from Zhejiang University and Associate Prof. Dong-Ling Deng from Tsinghua University.

Speaking of the process of bringing topologically ordered time crystals to life, the researchers said, "The collaboration between theoretical physicists and the experimental team makes this beautiful project happen. The goal is to push forward the understanding of topological order in periodically driven systems."

The uniqueness of time crystals

Time crystals exhibit something called time-translation symmetry breaking. Time translation is a property of systems wherein the system's state does not change over time.

In the case of time crystals, this symmetry is broken since time crystals periodically oscillate between different states. What makes time crystals unique is that they can exhibit periodic motion in time without energy dissipation.

However, they do not violate conservation laws.

After being given an initial push, the system starts oscillating at a frequency lower than the driving force. The oscillations are self-sustained due to many-body interactions, creating collective, synchronized oscillations.

Their lack of equilibrium allows them to maintain motion indefinitely, even in their ground state. This symmetry breaking is a form of non-equilibrium matter which brings out the uniqueness of time crystals.

Topological order

A system can be characterized via two types of properties—local and global.

Local properties are those that depend on a specific environment or position within a system—for example, the spin of an atom or its magnetic moment.

Global properties, on the other hand, depend on the whole system at large. These properties are long-ranged compared to local properties, which are short-ranged. An example of a global property is superconductivity.

Topological order is a type of global property of a system not described by local disturbances or noise. It emerges as a result of long-range entanglement within a system, meaning that changes in one part of the system can affect another part of the system.

Quantum computers are very prone to errors or noise from the environment. These can interact with the system, leading to a loss of coherence. This means the system can no longer maintain superposition and entanglement, which are the core features of a quantum system.

Since topological order is stable against local disturbances, it is of interest for topological quantum computing.

The challenge with introducing topological order in time crystals is their dynamic nature. Topological order requires stability and equilibrium states, which time crystals do not exhibit.

Programmable superconducting qubits

The researchers set up their processor using 18 programmable superconducting transmon qubits arranged in a two-dimensional square lattice. These qubits are more stable than other ones.

The square lattice facilitates qubit interactions, which helps generate the entanglement needed for quantum algorithms and error correction (noise reduction).

The researchers highlighted a key feature of their setup, the surface code.

"The surface code is not only one of the most successful models for quantum error correction, but it also supports exotic topologically ordered phases. The quantum processor developed by the experimental team at Zhejiang University naturally fit the surface code model."

"This makes it an ideal testbed for exploring these elusive topologically ordered states and potentially expanding their boundaries from equilibrium systems to non-equilibrium ones," they explained.

Their processor has a depth of 700 circuit layers and can run 2,300 single-qubit gates and 1,400 two-qubit gates. This demonstrates its ability to handle large and complex computations.

The platform can also simulate four-body interactions, which are needed for modeling complex systems. Further, a neuroevolution algorithm was used to find optimal configurations for quantum circuits, which can improve the efficiency and effectiveness of the computations.

The researchers added, "To meet the stringent requirements of observing the long-lived topological time crystal dynamics, we have made a huge amount of effort to improve the performance of our processor, which has state-of-the-art gate fidelities and qubit coherence times now."

Robust and stable topological time crystal order

The researchers found that the system could maintain stable behavior at a fractional frequency of the driving force, even when dealing with noise, demonstrating its stability. The system also stabilized after being disturbed without any unwanted oscillations.

The system also remained stable under small fluctuations, meaning it can function in noisy environments. When faced with stronger disturbances, the system loses its time-crystal-like behavior.

The team found that the system's topological entanglement matched the one made by theoretical predictions, suggesting its properties are robust and well-defined.

For four-body qubit interactions, the system could successfully measure the interactions with high accuracy. The system could also create and verify quantum states driven periodically, supporting its time-periodic behavior.

The researchers commented on the potential of their system, saying, "We believe that with the growth of the system size, control accuracy, and coherence time, superconducting processors will allow scientists to explore more exotic non-equilibrium phases of matter that are not accessible in natural materials."

They also mention that their experiment has demonstrated all necessary building blocks for implementing Floquet-enriched topological order that hosts dynamical anyon permutation and emergent non-Abelian anyons.

"An observation of such an unconventional phenomenon would also mark an important step in deepening our understanding of exotic non-equilibrium phases," they remarked.

More information: Liang Xiang et al, Long-lived topological time-crystalline order on a quantum processor, Nature Communications (2024). DOI: 10.1038/s41467-024-53077-9

Journal information: Nature Communications

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