State preparation and block-encoding are essential subroutines in quantum computing. The former provides basic encoding of quantum states, while the latter transforms classical data into a matrix representation within a quantum circuit. Some quantum advantages are built on the assumption that the subroutine– block encoding has been compiled in the quantum circuit, and this derives a problem of how to efficiently compile a block encoding. The resource trade-offs of block encoding, such as circuit size, subnormalization factor, compilation complexity (both time and space), and robustness against errors are central to its efficiency. In this work, the Binary Tree Block-Encoding (BITBLE) protocol is introduced, which optimizes these trade-offs. For a classical matrix in C2n× 2n, our approach reduces the compilation time toO(n22n) using n ancilla qubits, achieving superior resource trade-offs compared to existing methods. Numerical experiments further reveal that the approach outlined in BITBLE enhances compilation efficiency, resource scalability, and robustness against single-qubit gate errors in various standard data encoding tasks. Moreover, all algorithms are available as open-source.
State preparation and block-encoding are essential subroutines in quantum computing. The former provides basic encoding of quantum states, while the latter transforms classical data into a matrix representation within a quantum circuit. Some quantum advantages are built on the assumption that the subroutine– block encoding has been compiled in the quantum circuit, and this derives a problem of how to efficiently compile a block encoding. The resource trade-offs of block encoding, such as circuit size, subnormalization factor, compilation complexity (both time and space), and robustness against errors are central to its efficiency. In this work, the Binary Tree Block-Encoding (BITBLE) protocol is introduced, which optimizes these trade-offs. For a classical matrix in C2n× 2n, our approach reduces the compilation time toO(n22n) using n ancilla qubits, achieving superior resource trade-offs compared to existing methods. Numerical experiments further reveal that the approach outlined in BITBLE enhances compilation efficiency, resource scalability, and robustness against single-qubit gate errors in various standard data encoding tasks. Moreover, all algorithms are available as open-source.
Block-encoding operators are one of the essential components in quantum algorithms based on Quantum Signal Processing. Their gate complexity largely determines the overall gate complexity of the full algorithm. Using variational methods, we compile single-ancilla block-encoding unitaries with near-optimal resource requirements for a large range of input matrices. We find that the number of variational parameters in the parameterized quantum circuit approaches the number of free parameters in the input matrices, depending on whether they are real, complex and/or hermitian. Additionally, symmetries present in the input matrix can be incorporated into the ansatz circuit, reducing the parameter count further and enhancing optimizability. While performing a variational compilation of block-encodings ceases to be computationally feasible for large system sizes, the constructed operators can be used as components of larger block-encodings via a linear combination of block-encodings.
Abstract How to efficiently encode classical data is a fundamental task in quantum computing. While many existing works treat classical data encoding as a black box in oracle-based quantum algorithms, their explicit constructions are crucial for the efficiency of practical algorithm implementations. Here, we unveil the mystery of the classical data encoding black box and study the Clifford + T complexity in constructing several typical quantum access models. For general matrices (even including sparse ones), we prove that sparse-access input models and block-encoding both require nearly linear circuit complexities relative to the matrix dimension. We also give construction protocols achieving near-optimal gate complexities. On the other hand, the construction becomes efficient with respect to the data qubit when the matrix is a linear combination of polynomial terms of efficiently implementable unitaries. As a typical example, we propose improved block-encoding when these unitaries are Pauli strings. Our protocols are built upon improved quantum state preparation and a select oracle for Pauli strings, which hold independent values. Our access model constructions provide considerable flexibility, allowing for tunable ancillary qubit numbers and offering corresponding space-time trade-offs.
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 and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general n-qubit state, we improve the C-NOT count of the Plesch-Brukner algorithm (2011) from (23/24)×2n to (11/12)×2n. For block encoding, our single-ancilla protocol for 2n−1×2n−1 matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term (11/48)×4n. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term (2⌈log2K⌉+(11/12))×2n for a rank-K matrix. This is the first quantum algorithm that encodes matrices using the optimal normalization factor while also allowing the C-NOT count to be adjusted according to the matrix rank. Our approach builds upon the recursive Block-ZXZ decomposition from Krol et al. (2024) and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the z-axis to minimize the use of C-NOT gates.
Efficient state preparation is a challenging and important problem in quantum computing. In this work, we present a recursive state preparation algorithm that combines logarithmic-depth Dicke state circuits with Hamming weight encoders for efficiently preparing ``leaf-separable" quantum states. The algorithm is built on binary partition trees, generalized weight distribution blocks (gWDBs), and leaf-level encoders. We evaluate the performance of the algorithm by numerically simulating it on randomly generated target states with between 4 and 15 qubits. Compared to general state preparation approaches which require O(2n) CX gates, our algorithm achieves a circuit depth of O(klogkn+2k) and uses O(n(k+2k)) two-qubit gates, where k<n denotes the subtree size. We also compare implementations of the algorithm with and without the use of ancilla qubits, providing a detailed analysis of the trade-offs in circuit depth and two-qubit gate counts. These results contribute to scalable state preparation for quantum algorithms that require structured inputs such as Dicke or near-Dicke states.
Block encoding severs as an important data input model in quantum algorithms, enabling quantum computers to simulate non-unitary operators effectively. In this paper, we propose an efficient block-encoding protocol for sparse matrices based on a novel data structure, called the dictionary data structure, which classifies all non-zero elements according to their values and indices. Non-zero elements with the same values, lacking common column and row indices, belong to the same classification in our block-encoding protocol's dictionary. When compiled into the {U(2), CNOT} gate set, the protocol queries a 2n×2n sparse matrix with s non-zero elements at a circuit depth of O(log(ns)) , utilizing O(n2s) ancillary qubits. This offers an exponential improvement in circuit depth relative to the number of system qubits, compared to existing methods \cite{clader2022quantum,zhang2024circuit} with a circuit depth of O(n) . Moreover, in our protocol, the subnormalization, a scaled factor that influences the measurement probability of ancillary qubits, is minimized to ∑l=0s0|Al| , where s0 denotes the number of classifications in the dictionary and Al represents the value of the l -th classification. Furthermore, we show that our protocol connects to linear combinations of unitaries (LCU) and the sparse access input model (SAIM) . To demonstrate the practical utility of our approach, we provide several applications, including Laplacian matrices in graph problems and discrete differential operators.