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arXiv:2605.18877

Logical Resource Estimation for Quantum State Preparation with Compilation

Diyi Liu, Hanyu Wang, Shuchen Zhu, Jason Cong, Wibe A. de Jong, Di Fang, Zhen Huang, Costin Iancu, Chao Yang

quant-phSubmitted May 15, 2026• Announced May 19, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum state preparation is a fundamental primitive in quantum algorithms for encoding classical data into quantum amplitudes. We compare the cost of preparing general $n$-qubit states with real amplitudes using two common paradigms: rotation-based methods, based on controlled rotations, and sampling-based methods, based on a structured representation of the target state. Although these approaches are often theoretically compared using CNOT count and $T$-count, their relative performance in total gate count remains less well understood practically. We compare representative rotation-based and sampling-based methods using $T$-count and total gate count, and analyze how compilation overhead affects their relative performance. We also develop a software package for compiling state preparation circuits, designed as a practical subroutine for more general quantum computations. Numerical experiments on resource states and quantum states related to quantum chemistry, condensed matter physics, and simulation via Magnus expansion over a range of target accuracies $ε$ support the analysis. Our results show that sampling-based methods achieve asymptotically lower $T$-count and retain an overall advantage after accounting for total gate count and compilation overhead.

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Quantum state preparation is a fundamental primitive in quantum algorithms for encoding classical data into quantum amplitudes. We compare the cost of preparing general nnn-qubit states with real amplitudes using two common paradigms: rotation-based methods, based on controlled rotations, and sampling-based methods, based on a structured representation of the target state. Although these approaches are often theoretically compared using CNOT count and TTT-count, their relative performance in total gate count remains less well understood practically. We compare representative rotation-based and sampling-based methods using TTT-count and total gate count, and analyze how compilation overhead affects their relative performance. We also develop a software package for compiling state preparation circuits, designed as a practical subroutine for more general quantum computations. Numerical experiments on resource states and quantum states related to quantum chemistry, condensed matter physics, and simulation via Magnus expansion over a range of target accuracies εεε support the analysis. Our results show that sampling-based methods achieve asymptotically lower TTT-count and retain an overall advantage after accounting for total gate count and compilation overhead.
Quantum state preparation is the process of producing a target quantum state that will be used as the input to a quantum circuit. Many quantum algorithms require an input state where amplitudes, phases, or basis probabilities encode problem data, and the cost of preparing this state can be significant in gate count and circuit depth. This review summarizes common goals, assumptions, and methods for quantum state preparation, with emphasis on preparing states from classical vectors, probability distributions, and feature data used in quantum machine learning. We organize approaches by the information they load and by the resources they require, including gate count, circuit depth, qubit overhead, and classical preprocessing. It also compares exact and approximate preparation procedures, and discusses how precision targets affect cost. The review highlights links between families of methods, typical sources of resource estimates, and criteria that help match a preparation method to a task and hardware constraints.
Quantum amplitude estimation can reduce the sampling cost of rare-event probability estimation, but applying it to correlated Ising graphical models is limited by the difficulty of preparing the target distribution and building a practical event oracle. This work explores two approximate strategies for mitigating these challenges. We introduce a sample-free state-preparation method combining loopy belief propagation with the Chow--Liu algorithm. The resulting tree approximation is compiled into a quantum circuit with linear gate count and depth, and its accuracy is evaluated across graph families spanning different topologies, coupling strengths, and coupling signs. We also construct a structural oracle that evaluates threshold rules with reversible Boolean gates. Using a twenty-node supply-chain disruption model as a case study, we compare maximum likelihood amplitude estimation against four classical Monte Carlo baselines. Under the fixed-depth schedule used throughout this work, the quantum estimator has the same asymptotic error scaling as the classical methods but achieves lower estimation error by a constant factor. This reduction narrows when amplitude-encoding queries replace raw shots as the resource metric. We separate statistical error from the deterministic errors caused by approximate state preparation and oracle construction, and identify the requirements for achieving an improvement beyond a constant factor.
Practical realizations of quantum computers are poised to deliver outstanding computational capabilities far beyond the reach of any classical supercomputer. While classical systems are reliably implemented using CMOS technology, the fabrication of quantum hardware has not yet reached a commercially viable level of maturity. Nowadays, many different technologies are being developed as competing candidates to build the first error-corrected quantum computer. Among the software resources required to operate such a system, quantum compilers translate a high-level description of a quantum algorithm into low-level, technology-dependent quantum instructions. Many quantum algorithms, including Shor's and Grover's algorithms, require to compute some classical logic functions on the quantum computer. The first part of this thesis deals with the problem of compiling quantum circuits that perform such classical Boolean functions, called oracles, into a fault-tolerant set of instructions. Oracle circuits often demand many resources in terms of the number of qubits and gates. The focus is on compiling quantum circuits starting from multi-level logic networks representing large Boolean functions while minimizing the resource footprint of the obtained quantum circuits. At the same time, the trade-off between memory resources and the number of operations typical of such a compilation process is explored. As part of the effort in reducing the resources required to perform a given quantum program, I also present some quantum circuit optimization techniques. The researched algorithms leverage data structures and techniques borrowed and adapted from classical logic synthesis, e.g., SAT solvers, LUT mapping, and multi-level logic networks. I implemented all the compilation algorithms presented in this thesis as part of open-source projects. In particular, I develop and maintain caterpillar---a C++ header-only library dedicated to the quantum compilation of oracles and quantum memory management. The second part of this thesis describes how to equip quantum programming language compilers with automatic accuracy management. Despite the availability of many of such languages, resource estimation of quantum algorithms does not yet support taking approximation errors into account. This general methodology is applicable to any programming language and I demonstrate its integration into the Q# compiler. The technique consists of providing language support to ease the accuracy-aware programming task. The user can define accuracy parameters that will be automatically optimized according to a constraint and a cost function directly generated from the source code. In the presented practical evaluations, the constraint function is the overall allowed accuracy, while the cost function is application-dependent and related to the number of gates. During the optimization process, such functions are evaluated hundreds of times. For this reason, we extract them as (near-)symbolic expressions, whose evaluation time does not depend on the quantum algorithm size. The algorithms and the methodologies presented in this thesis are part of a widespread effort of the research community to build a complete and efficient software stack to program and control the first practical universal quantum computer.
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.
Quantum state preparation is an important class of quantum algorithms that is employed as a black-box subroutine in many algorithms, or used by itself to generate arbitrary probability distributions. We present a novel state preparation method that utilizes less quantum computing resource than the existing methods. Two variants of the algorithm with different emphases are introduced. One variant uses fewer qubits and no controlled gates, while the other variant potentially requires fewer gates overall. A general analysis is given to estimate the number of qubits necessary to achieve a desired precision in the amplitudes of the computational basis states. The validity of the algorithm is demonstrated using a prototypical problem of generating Ising model spin configurations according to its Boltzmann distribution.
The preparation of nnn-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when mmm ancillary qubits are available. The reduction in complexity is due to the use of a single operator ΛΛΛ for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.