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Hermitian Matrix Function Synthesis without Block-Encoding

Anuradha Mahasinghe, Kaushika De Silva, Xavier Cadet, Peter Chin, Frederic Cadet, Jingbo Wang

TL;DR

We address efficient synthesis of polynomial functions of Hermitian matrices on quantum hardware without block-encoding by leveraging Generalized Quantum Signal Processing (GQSP) and a symmetric expansion of unitary conjugates. The core idea expresses A as $A=\tfrac{1}{2}(U+U^{\dagger})$ with $U=A+i\sqrt{I-A^2}$ and builds $P(A)$ as a symmetric combination of $R_j(U)$ and $R_j(U^{\dagger})$, circumventing block-encoding and LCU overhead. The paper provides a constructive symmetric polynomial expansion, circuit architectures, and a discussion of favorable applicability—notably for sparse Hermitian matrices, graph Laplacians, and low-rank operators—along with extensions to normal matrices and rational-function synthesis. The result is a resource-efficient alternative to traditional block-encoding-based techniques, expanding the toolbox for quantum matrix-function transformations. Overall, the work advances practical quantum circuit design by decoupling polynomial synthesis from block-encoding constraints.

Abstract

Implementing arbitrary functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advance quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix, by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding by expressing the target Hermitian matrix as a symmetric combination of unitary conjugates, enabling polynomial synthesis via GQSP circuits applied to each unitary component. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead, and opens new pathways for quantum algorithm design for functions of Hermitian matrices.

Hermitian Matrix Function Synthesis without Block-Encoding

TL;DR

We address efficient synthesis of polynomial functions of Hermitian matrices on quantum hardware without block-encoding by leveraging Generalized Quantum Signal Processing (GQSP) and a symmetric expansion of unitary conjugates. The core idea expresses A as with and builds as a symmetric combination of and , circumventing block-encoding and LCU overhead. The paper provides a constructive symmetric polynomial expansion, circuit architectures, and a discussion of favorable applicability—notably for sparse Hermitian matrices, graph Laplacians, and low-rank operators—along with extensions to normal matrices and rational-function synthesis. The result is a resource-efficient alternative to traditional block-encoding-based techniques, expanding the toolbox for quantum matrix-function transformations. Overall, the work advances practical quantum circuit design by decoupling polynomial synthesis from block-encoding constraints.

Abstract

Implementing arbitrary functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advance quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix, by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding by expressing the target Hermitian matrix as a symmetric combination of unitary conjugates, enabling polynomial synthesis via GQSP circuits applied to each unitary component. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead, and opens new pathways for quantum algorithm design for functions of Hermitian matrices.

Paper Structure

This paper contains 14 sections, 1 theorem, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Qubitization and Chebyshev Polynomial Encoding
  4. Quantum Signal Processing (QSP)
  5. Generalized Quantum Signal Processing (GQSP)
  6. The quantum algorithm
  7. Symmetric polynomial expansion
  8. Circuit analysis
  9. Contexts of favorable applicability
  10. Tractable square root cases
  11. Contexts where block-encoding is prohibitive
  12. LCU overheads in polynomial synthesis
  13. Outlook and extensions
  14. Conclusion

Key Result

Lemma 1

Let $A \in \mathbb{C}^{n \times n}$ be a Hermitian matrix with $\|A\| \leq 1$. Then for every integer $n \geq 0$, there exists a degree-$n$ polynomial $R_n(x)$ such that where and, the polynomial $R_n(x)$ is given by:

Figures (1)

  • Figure 1: Quantum circuit implementing the polynomial transformation $\sum_{j=0}^d c_j R_j(U)$ via GQSP, controlled by the ancilla qubit. Measurement of the first two qubits projects the data qubit into $P(A)\ket{\psi}$.

Theorems & Definitions (2)

  • Lemma 1: Symmetric Polynomial Expansion of Hermitian Powers
  • proof