Long-range entangled quantum states are crucial in various fields, including quantum information, condensed matter physics, and high-energy physics. Preparing these states on quantum computers presents a significant challenge because their preparation using unitary dynamics requires extensive circuit depth due to finite Lieb-Robinson velocities that limit the spread of correlations. This limitation restricts the possibilities with near-term quantum devices, particularly due to limited coherence times. The paper introduces a novel approach that leverages mid-circuit measurement and feed-forward to overcome this constraint. By introducing measurements during state preparation, the unitarity assumption is broken, allowing for instantaneous correlation generation across the system. The probabilistic nature of measurements is addressed by employing conditional quantum gates based on measurement outcomes (feed-forward). This shifts the computationally intensive part of the process to the classical channel, effectively reducing the burden on the quantum system. Deterministic preparation of excitation-free states is essential for simulating topologically ordered systems and for quantum error correction protocols. Feed-forward is particularly crucial for the efficient preparation of non-Abelian states requiring multiple layers of measurement. To achieve deterministic, constant-depth preparation of long-range entangled states, high-fidelity, fast feed-forward, mid-circuit measurement, and entangling gates are required, all operating within the coherence time limitations of the platform. While individual components of this approach have been demonstrated, combining them for deterministic creation of long-range entangled states has remained elusive until now.

The authors review existing literature on topological quantum memory, quantum field theory of many-body systems, quantum simulators, and quantum algorithms for state preparation. They highlight the limitations of unitary dynamics for preparing long-range entangled states, emphasizing the need for methods that go beyond unitary evolution. The literature on measurement-based quantum computation, including the use of feed-forward, is also discussed. The challenges of preparing topologically ordered states and the role of anyons in these systems are reviewed. Previous experimental demonstrations of individual elements of the proposed approach are noted. The need for high fidelity and fast operation within the limited coherence times of near-term quantum computers is stressed.

The experiment uses Quantinuum's H1 programmable ion-trap quantum computer, featuring 20 qubits encoded in two hyperfine states of 171Yb ions. The mobile qubit ions are a key advantage, minimizing cross-talk during mid-circuit measurements. The researchers target the ground state of Kitaev's toric code Hamiltonian, representing qubits on a square lattice with periodic boundary conditions. The chosen Hamiltonian realizes Z2 topological order with four ground states. The specific ground state is targeted using a three-step procedure: (1) initializing all ions to |0⟩, (2) measuring Ap operators on selected plaquettes, and (3) applying conditional single-qubit Z gates based on measurement outcomes to annihilate anyons. The measurement of Ap is performed using either an ancilla-assisted method or an ancilla-free method. A heralded state preparation strategy is used, discarding runs with an odd number of anyons. State fidelity is evaluated by measuring expectation values of X and Z stabilizers, and by calculating topological entanglement entropy using a randomized measurement scheme. Anyon dynamics are studied by introducing two defects into the system, allowing for anyon transmutation and braiding interferometry. The transmutation is observed by moving an anyon across the line connecting the defects, resulting in a change of its type. Braiding interferometry confirms the fermionic exchange statistics of the created anyon by measuring a phase shift upon rotation. The experimental setup minimizes cross-talk due to the physical separation of qubits. The methods also detail how the purity of a reduced quantum state is estimated using randomized measurements and how SPAM error mitigation is applied to account for state preparation and read-out errors.

The researchers successfully prepared the toric code ground state with high fidelity, achieving an energy density of -0.929 ± 0.004. The expectation values of logical string operators were close to 1, confirming the preparation of the target logical state. Topological entanglement entropy measurements were consistent with Z2 topological order. The experiment demonstrated anyon transmutation, showing a change in anyon type upon movement across a defect line. Braiding interferometry confirmed the fermionic exchange statistics of an electric-magnetic composite anyon, showing a phase shift of approximately -0.87 ± 0.018. The high fidelity of the results is attributed to the combined use of mid-circuit measurement, feed-forward, and low-error gates. Cross-talk was found to be negligible, highlighting the advantages of the mobile qubit architecture. The ancilla-free and ancilla-based strategies were both evaluated with comparable outcomes. Error mitigation techniques were utilized and their effectiveness was evaluated. Global fidelity with the target state was estimated at 0.80 ± 0.049.

The results demonstrate the feasibility of preparing topologically ordered states on a quantum computer using a constant-depth protocol. The method successfully overcomes the limitations of unitary dynamics for preparing long-range entangled states. The achievement of high fidelity and the observation of anyon dynamics strongly support the theoretical framework. The negligible cross-talk highlights the effectiveness of the chosen hardware architecture and experimental techniques. This work provides a crucial step toward simulating more complex topological systems and exploring their dynamics. The ability to prepare long-range entangled states with minimal quantum resources opens new avenues for quantum simulation and quantum information processing. While initially focusing on Abelian topological orders, the approach extends to non-Abelian topological orders, paving the way for fault-tolerant quantum computing.

This paper presents a significant advance in the field of quantum computation by demonstrating the deterministic, high-fidelity preparation of long-range entangled states on a trapped-ion quantum computer using mid-circuit measurement and feed-forward. The successful creation and manipulation of anyons opens exciting possibilities for future research, including the exploration of more complex topological orders and quantum simulations of challenging many-body systems. Future work could explore the scalability of the approach to larger systems and the application of more sophisticated error mitigation techniques.

The current experiment was conducted on a 20-qubit system. Scaling the approach to larger systems may present challenges. The error mitigation strategies used could be further refined. The fidelity of the state preparation, while high, is not perfect and leaves room for further improvement. Although cross-talk was found to be negligible, further optimization might be possible. The limitations of the current decoder design could also be addressed by implementing more sophisticated decoders.