Quantum compilation framework for data loading

TL;DR

The paper presents an automated, approximation-aware quantum data-loading framework that jointly optimizes state preparation and diagonal encoding under a prescribed error budget $\varepsilon$, using a resource-estimation workflow built on PennyLane. It supports a broad set of methods (e.g., MPS, QROM, FSL, Walsh transforms) and enables hybrid, divide-and-conquer loading to minimize quantum resources. Two key innovations—compact block-encoding of $d$-diagonal matrices and an exact, QSP-based block encoding for kinetic energy operators—extend the framework's applicability to more complex Hamiltonians. Across Gaussian states, quantum chemistry, and CFD, the framework discovers non-obvious, structure-aware strategies that yield orders-of-magnitude reductions in gate counts and measurement overhead, demonstrating practical potential for resource-constrained quantum hardware.

Abstract

Efficient encoding of classical data into quantum circuits is a critical challenge that directly impacts the scalability of quantum algorithms. In this work, we present an automated compilation framework for resource-aware quantum data loading tailored to a given input vector and target error tolerance. By explicitly exploiting the trade-off between exact and approximate state preparation, our approach systematically partitions the total error budget between precision and approximation errors, thereby minimizing quantum resource costs. The framework supports a comprehensive suite of state-of-the-art methods, including multiplexer-based loaders, quantum read-only memory (QROM) constructions, sparse encodings, matrix product states (MPS), Fourier series loaders (FSL), and Walsh transform-based diagonal operators. We demonstrate the effectiveness of our framework across several applications, where it consistently uncovers non-obvious, resource-efficient strategies enabled by controlled approximation. In particular, we analyze a computational fluid dynamics workflow where the automated selection of MPS state preparation and Walsh transform-based encoding, combined with a novel Walsh-based measurement technique, leads to resource reductions of over four orders of magnitude compared to previous approaches. We also introduce two independent advances developed through the framework: a more efficient circuit for d-diagonal matrices, and an optimized block encoding for kinetic energy operators. Our results underscore the indispensable role of automated, approximation-aware compilation in making large-scale quantum algorithms feasible on resource-constrained hardware.

Figures (13)

• Figure 1: Workflow of the automated compilation framework. Given a vector $\vec{\alpha}$ and an error tolerance $\varepsilon$, a weight $w$ is selected by taking different values of a grid partition and distributing the total error between the precision error and the approximation error. Once $w$ is fixed, the optimal hyperparameters of the different algorithms are calculated, and the resources are estimated using the PennyLane resource estimation framework. The value of $w$ is then updated, and finally, we return the most efficient method found previously.
• Figure 2: Example of hybrid state preparation. The histogram (left) reveals structural patterns in the target state. Our recursive method applies different controlled routines (SP$_1$, SP$_2$, SP$_3$) in the quantum circuit (right) to efficiently prepare each region.
• Figure 3: Block encoding of d-diagonal matrices. (a) Compact restructuring of the diagonal block encoding by unifying the quantum in-place Adders into a single Adder. Here we also use $D_k$ as the block encoding of the $k$-th diagonal. (b) Reconstruction using the 64 most significant Walsh coefficients, where the x-axis and y-axis show the diagonal's index and value, respectively.
• Figure 4: Block encoding circuit for the kinetic operator. The operators $\hat{T}_x$, $\hat{T}_y$, and $\hat{T}_z$ are prepared using a quantum signal processing approach as described in the Appendix \ref{['sec:QSP']}.
• Figure 5: State preparation of Gaussian states using FSL. (a) Efficient preparation of a discretized Gaussian state over 11 qubits using the Fourier Series Loader (FSL). Only 8 Fourier coefficients are required to achieve an approximation error of $\varepsilon = 1\mathrm{e}{-4}$. (b) Trade-off for a 14-qubit Gaussian state ($\sigma=0.9$). The $x$-axis shows the fraction of the total error budget $\varepsilon$ assigned to the preparation error, $w=\varepsilon_p/\varepsilon$ (displayed in %). The $y$-axis reports the estimated T-gate count. Each curve corresponds to a different total error $\varepsilon \in \{5\mathrm{e}{-1}, 5\mathrm{e}{-2}, 5\mathrm{e}{-3}, 5\mathrm{e}{-4}\}$. For a fixed $\varepsilon$, increasing the allocation to $\varepsilon_p$ (thereby decreasing the approximation error $\varepsilon_a=(1-w)\varepsilon$) illustrates the trade-off between preparation and approximation resources.