Realizing topologically ordered states on a quantum processor (2104.01180v1)

Abstract: The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of $\ln2$, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction.

• The paper demonstrates the experimental realization of the toric code’s ground state on a 31-qubit quantum processor.
• It employs scalable quantum circuits with Hadamard and CNOT gates to efficiently generate and measure topological entanglement entropy near −ln2.
• This work validates anyon braiding statistics via controlled interferometry, underscoring potential for fault-tolerant quantum error correction.

Realizing Topologically Ordered States on a Quantum Processor

The paper "Realizing topologically ordered states on a quantum processor" articulates a significant experimental realization of topological quantum states using a superconducting quantum processor. The authors have reported the preparation of the ground state of the toric code Hamiltonian, a foundational model in topological quantum computation, using a designed quantum circuit on a system of superconducting qubits. This research stands out due to its demonstration of advanced topological phenomena, such as topological entanglement entropy and anyon braiding statistics, within a quantum computational framework. These findings carry substantial implications for the future development of fault-tolerant quantum computation and error correction.

In their experimental setup, the authors utilized a 31-qubit configuration to encode the toric code—a paradigmatic example of a system with $\mathds{Z}_2$ topological order. This is a critical benchmark due to the toric code’s relevance in stabilizer quantum error correction codes. The team implemented a scalable quantum circuit characterized by a sequence of Hadamard and controlled-phase (CNOT) gates, optimized for the architecture of Google's Sycamore processor. The strategic design allowed for efficient generation of the ground state in a manner linearly scalable with the width of the lattice.

Moreover, the research includes a precise measurement of the topological entanglement entropy $S_\text{topo}$, achieving a value close to the theoretical expectation of $-\ln 2$, across diverse subregions. Such measurements are crucial, as $S_\text{topo}$ is a definitive signature of non-trivial topology, illustrating long-range quantum entanglement properties distinct from conventional phases.

Additionally, the paper presents controlled-simulations of anyon interferometry to probe the braiding statistics inherent to the toric code's excitation spectrum. The interferometric schemes deployed involved controlled operations steered via ancillary qubits, enabling the measurement of mutual and exchange statistics of the excitations, corroborating the theoretical predictions of $\pi$ phase shifts due to anyon interactions.

The implications of these results are manifold. Practically, they underscore the potential and challenges of implementing topological quantum error correction schemes. The ground states of two-dimensional topologically ordered systems afford a robust means of encoding quantum information that is inherently protected from local errors. The capacity to prepare and manipulate such states on a quantum processor attests to the burgeoning capability of current quantum technologies to engage with complex, error-resistant quantum computational paradigms.

Theoretically, the exploration of topological quantum states in a programmable platform could drive a deeper understanding of quantum matter and inspire novel computational models. In the broader context of quantum computing, this research could facilitate the development of advanced algorithms exploiting topological properties, which might offer resilience and efficiency enhancements over conventional approaches.

Looking forward, further advancements in coherence times and gate fidelities are pivotal. The work presented offers a blueprint for similar studies on more sophisticated models, including those with non-Abelian anyons, which hold the promise of universal quantum computation through topological means. The experimental realization of topological states on quantum processors presents an essential step toward the practical deployment of quantum systems capable of exploiting the exotic characteristics of topologically ordered phases.