Lazy Quantum Walks with -qubit Gates Model Fluid Dynamics with Predicted Fidelity

Quantum walks, the quantum mechanical counterparts to classical random walks, offer a powerful new way to model complex systems like fluid dynamics. Steph Foulds and Viv Kendon, both from the University of Strathclyde, investigate how to implement these walks on neutral atom hardware, a promising platform due to its ability to perform operations on multiple qubits simultaneously. The team demonstrates gate sequences and predicts the accuracy of these operations for simplified, or ‘lazy’, quantum walks, which incorporate a resting state. This achievement represents a crucial step towards realising quantum walks capable of simulating the behaviour of fluids, potentially unlocking new insights into this challenging area of physics.

Quantum walks, the quantum analogue to classical random walks, demonstrate potential in modelling fluid dynamics. Neutral atom hardware represents a promising platform for implementing these walks, owing to its capacity to implement native multiqubit gates and dynamically rearrange qubits. Researchers developed gate sequences and predicted the performance of several simplified quantum walks, including ‘lazy’ walks which incorporate a resting state.

Fidelity Measurement of Quantum Walks with Rest States

Scientists developed a rigorous methodology to evaluate the performance of quantum walks on near-term neutral atom quantum computers, focusing on the crucial inclusion of a rest state for simulating fluid dynamics. The study employed pure two-dimensional qubits as inputs, modelling realistic neutral atom qubit systems, and represented composite qubit states using standard state vector notation. Researchers defined state fidelity using the Hellinger distance, a measure comparing probability distributions of ideal and experimentally-obtained states, and expressed this as a value between 0 and 1, where 1 indicates complete agreement. This fidelity measure allows direct comparison with previously published results and provides a quantitative assessment of quantum walk accuracy.

To assess gate performance, the team utilized a limited tomography prescription, calculating gate fidelity by evaluating the overlap between the ideal and effective unitary transformations acting on a specific input state. This approach quantifies the accuracy of individual gate operations within the quantum walk sequence. The methodology involved simulating quantum walks with and without a rest state on ring-shaped qubit arrangements ranging from 4 to 16 nodes, allowing for systematic investigation of system size effects. Researchers meticulously modelled errors arising from multiqubit Rydberg gates, utilizing two-photon adiabatic rapid passage, alongside single and passive error sources, to generate realistic performance predictions.

The study further established a direct link between quantum walks and computational fluid dynamics, demonstrating the equivalence of a single step in a lazy quantum walk to the advection equation governing smoothed-particle hydrodynamics. By tuning the coin operator within the quantum walk, scientists showed the ability to map parameters of the SPH method, highlighting the potential for quantum algorithms to accelerate fluid simulations. The team focused on evaluating the shift operator, a core component of the quantum walk, as a building block for a complete end-to-end PDE solver, paving the way for practical applications in computational physics.

Neutral Atom Quantum Walks Demonstrate High Fidelity

Researchers are achieving significant advances in quantum walks using neutral atom hardware, a promising platform due to its ability to natively manipulate multiple qubits and dynamically rearrange them. The work focuses on developing gate sequences and predicting the fidelity of these operations for various quantum walk scenarios, including ‘lazy’ walks which incorporate a resting state. The team measured fidelity, comparing the probability of measuring a particular state in an ideal, error-free model with the probability in a model incorporating errors. This fidelity, bounded between 0 and 1, provides a quantitative measure of the accuracy of the quantum walk implementation, with a value of 1 indicating perfect agreement between the ideal and error-affected models.

The study defines gate fidelity using a limited tomography prescription, revealing how closely the implemented gate approximates the ideal gate, with values closer to 1 signifying higher accuracy. Experiments involved encoding the position of the ‘walker’ using binary code on qubits, allowing for the simulation of walks on rings of varying sizes. The team developed circuits to perform these walks, utilizing shift operators that move the walker clockwise or anticlockwise depending on the ‘coin’ state, which determines the direction of movement. For a four-node ring, the system requires specific gate arrangements to enact these shifts, and the complexity scales with the size of the ring.

To introduce ‘laziness’, where the walker has a probability of remaining in the same position, researchers expanded the coin state to two qubits. This allows for control over the probability of resting versus moving, with a 50% probability for each. For an eight-node ring, the system requires a more complex arrangement of gates to accommodate the two-qubit coin and the increased number of positions. The team demonstrated that a two-qubit coin quantum walk on a 2n-node ring requires gates acting on a range of qubits, from one to n+2, depending on the complexity of the walk. The Strathclyde neutral atom hardware, featuring a programmable 2D array of Cesium atoms, provides the physical platform for these experiments. The system utilizes hyperfine states and Rydberg states, coupled via two-photon transitions, to create interactions between atoms and implement multiqubit gates. This allows for precise control over the qubits and the implementation of complex gate sequences necessary for simulating quantum walks.

Neutral Atom Quantum Walks Exceed 99% Fidelity

Researchers have demonstrated the feasibility of implementing quantum walks on neutral atom hardware. Results indicate that with projected gate fidelities of 99. 81% for CZ gates and 99. 54% for CCZ gates, a quantum walk performed on a four-node ring achieves a final state fidelity exceeding 99% after four time steps and remains above 90% after twenty. The study reveals that implementing more complex quantum walks requires the addition of a four-qubit C3Z gate with a projected fidelity of 98.

50% to meet the same fidelity tolerances, either for a walk with a rest state on a four-node ring or a walk without a rest state on an eight-node ring. Analysis of composite fidelities suggests that increasing the native gate set from using gates acting on up to three qubits to those acting on up to four qubits provides a significant advantage, although further increases yield diminishing returns. The researchers conclude that a native four-qubit gate and the ability to rearrange qubits mid-circuit are essential for implementing interesting quantum walks on this hardware. Acknowledging limitations, the authors note that their model did not incorporate error correction, an important consideration for future work. They also suggest exploring position encoding with Gray codes and utilising qutrit coins as potential methods to reduce gate requirements. Future research directions include extending the model to non-linear cases relevant to fluid simulation by entangling the coin qubit with an ancilla, and ultimately, incorporating fault-tolerant protocols for multiqubit gates.