Computing Statistical Properties of Velocity Fields on Current Quantum Hardware

TL;DR

The paper tackles the readout bottleneck in quantum CFD by enabling direct extraction of turbulence statistics from amplitude-encoded velocity fields on near-term quantum hardware. It introduces a measurement pipeline using Hadamard-test-based observables and parallel-circuit strategies to compute central moments and structure functions without full quantum state tomography, facilitated by QESEM error mitigation. Demonstrations on 16-point 1D velocity fields (4 qubits) show that mitigated quantum results align with classical references for both a sine signal and Burgers' turbulence snapshots, highlighting practical viability on IBMQ devices. The work also develops hardware-aware circuit designs for IBM’s heavy-hex topology and discusses scalability, norm dependencies, and the trade-offs between measurement depth and estimator precision, outlining a path toward larger-QPU quantum CFD readouts.

Abstract

Quantum algorithms are gaining attention in Computational Fluid Dynamics (CFD) for their favorable scaling, as encoding physical fields into quantum probability amplitudes enables representation of two to the power of n spatial points with only n qubits. A key challenge in Quantum CFD is the efficient readout of simulation results, a topic that has received limited attention in literature. This work presents methods to extract statistical properties of spatial velocity fields, such as central moments and structure functions, directly from parameterized ansatz circuits, avoiding full quantum state tomography. As a proof of concept, we implement our approach for 1D velocity fields, encoding 16 spatial points with 4 qubits, and analyze both a sine wave signal and four snapshots from Burgers' equation evolution. Using Qedma's error mitigation software QESEM, we demonstrate that such computations achieve high accuracy on current quantum devices, specifically IBMQ's Heron2 system ibm_fez.

Figures (17)

• Figure 1: Top left: Example of the spatiotemporal evolution of the forced Burgers’ equation (random initial condition, $N=16$, $\nu=0.1$), showing the formation and propagation of shock waves. This panel is a subset of the full simulation shown in the top panel of Figure \ref{['fig:moments_over_time']}. Bottom left: Temporal evolution of the corresponding fourth central moment, peaking at $0.8\,T$, where $T$ denotes the final simulation time. Right: Spatial velocity profile at $0.8\,T$, showing a steep velocity gradient.
• Figure 2: Left: Hadamard test. The ancilla is measured in the $X$ basis (denoted by the meter symbol), yielding the Pauli-$X$ expectation value $\textrm{Re}\langle \phi | U | \phi \rangle$. Right: Modified Hadamard test, estimating $\textrm{Re}\langle \phi_1 | \phi_2 \rangle$ where $\ket{\phi_1}, \ket{\phi_2}$ are prepared by applying $U_1,U_2$ on the all-zero state respectively.
• Figure 3: An example of the ansatz architecture featuring a brick wall structure of CNOT and $R_Y$ gates for 4 qubits. An initial layer of $R_Y$ gates is followed by eight layers of CNOT-$R_Y$ blocks.
• Figure 4: Ansatz architecture, shown here for 4 qubits as an example, incorporating adjustments regarding given hardware requirements. Compared to the original ansatz architecture, it includes an initial layer of Hadamard gates, rotations only on every second qubit within the eight CNOT-$R_Y$ ansatz layers and a final layer of rotations on every qubit.
• Figure 5: Example of qubit positioning on IBM QPU connectivity. The circuit qubits are shown in light gray, the adjacent auxiliary qubits are shown in dark gray.