To treat a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operations, or gates: this is the quantum description of the problem. These operations are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BE). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hermitian (LCH) or a Linear Combination of Unitary (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. Qubitization constructs a specific query that respects single-qubit behavior when expressed in the appropriate basis. In this work, we extend the notion of qubitization to Hamiltonian matrices used to map the problem of interest. These methods concern almost all the problems implemented on QPUs: from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), or sparse matrices. This work underlines interesting properties associated with the qubitized Hamiltonian basic gate decomposition. It includes the ability to switch from LCH to LCU, to map non-Hermitian problems, and to construct the different quantum circuit primitives (queries) needed for the quantum description of the problem. We also provide a list of qubitized Hamiltonians that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.
To treat a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operations, or gates: this is the quantum description of the problem. These operations are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BE). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hermitian (LCH) or a Linear Combination of Unitary (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. Qubitization constructs a specific query that respects single-qubit behavior when expressed in the appropriate basis. In this work, we extend the notion of qubitization to Hamiltonian matrices used to map the problem of interest. These methods concern almost all the problems implemented on QPUs: from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), or sparse matrices. This work underlines interesting properties associated with the qubitized Hamiltonian basic gate decomposition. It includes the ability to switch from LCH to LCU, to map non-Hermitian problems, and to construct the different quantum circuit primitives (queries) needed for the quantum description of the problem. We also provide a list of qubitized Hamiltonians that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.
This article proposes a formalism which unifies Hamiltonian simulation techniques from different fields. This formalism leads to a competitive method to construct the Hamiltonian simulation with a comprehensible, simple-to-implement circuit generation technique. It leads to a gate decomposition and a scaling different from the usual strategy based on a Linear Combination of Unitaries (LCU) reformulation of the problem. It can significantly reduce the quantum circuit number of rotational gates, multi-qubit gates, and the circuit depth. This method leads to one exact Hamiltonian simulation for each summed term and Trotter step. Each of these Hamiltonian simulation unitary matrices also allows the construction of the non-exponential terms with a maximum of six unitary matrices to be Block-encoded.The formalism is easy to apply to the widely studied High-order Unconstrained Binary Optimization (HUBO) problems, fermionic transition Hamiltonian, and basic finite difference method instances. For the HUBO problems, our implementation exponentially reduces the number of gates for high-order cost functions with respect to the HUBO problem order. The individual electronic transitions are implemented without error for the second-quantization Fermionic Hamiltonian. Finite difference proposed matrix decompositions are straightforward, very versatile, and scale as the state-of-the-art proposals.
We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a foundational primitive in both variational and fault-tolerant quantum algorithms, conventional approaches based on the Pauli basis can require a number of terms that scales quadratically with matrix size. We introduce the Sigma basis, a compact set of simple, non-unitary operators that can better capture sparsity and structure, enabling decompositions with only polylogarithmic scaling in the number of terms. We develop both numerical and semi-analytical methods for computing Sigma basis decompositions of arbitrary matrices. Given this new basis is comprised of non-unitary operators, we leverage the concept of unitary completion to design efficient quantum circuits for evaluating observables in variational quantum algorithms and for constructing block encodings in fault-tolerant quantum algorithms. We compare our method to related techniques like unitary dilation, and demonstrate its effectiveness through several PDE examples, showing exponential improvements in decomposition size while retaining circuit efficiency.
One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for solving PDEs. Furthermore, the algorithm is extended to the multidimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.
An efficient implementation of quantum algorithms is often hindered by the lack of efficient primitives for operators and state preparation. This limits both the ability of near-term quantum hardware to simulate complex problems and the potential of fault-tolerant algorithms to achieve practical quantum advantage. To address this, we propose a framework that maps discrete operators onto compact, structured quantum circuits. The resulting variational circuits can be tailored to the connectivity and long-range interaction of the target hardware. The learning process employs backpropagation together with a cost function that efficiently optimizes unitary operators and non-unitary --dense or sparse-- operators using only a single ancilla qubit for block encoding. Additionally, we introduce a regularization term that reduces the approximation error. The approach is validated for both quantum mechanical and engineering applications. In the former case, we learn propagators that arise in native quantum problems --such as quantum simulation and quantum chemistry-- and achieve improved resource scaling in comparison to standard Suzuki--Trotter expansions. In the latter case, we demonstrate the approach's ability to implement the second-order central finite difference approximation of the Laplace operator --relevant for solving partial differential equations-- while improving upon current error metrics. The final example deals with learning a dense, non-unitary operator that arises in the analysis of inviscid potential flow around an airfoil. These results motivate the extension of the framework to a broader class of operator-synthesis problems beyond the prototypical engineering and quantum applications considered here.
We develop a novel approach for efficiently applying the variational quantum linear solver (VQLS) in the context of structured sparse matri-ces. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. Conventionally, Pauli basis is used for the linear combination of uni-taries (LCU) decomposition of the underlying matrix to facilitate evaluation of the global/local VQLS cost functions. However, the number of LCU terms can scale quadratically with respect to the matrix sizeNin the worst case. Using the heat equation as an example, we show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions. With one additionaln- Toffoli gate (where,n=logN) and one ancilla qubit, our method could provide an exponential time advantage over conventionally used Pauli basis for some applications. We perform a rigorous analysis of our approach and compare with other related concepts in literature including unitary dilation and Bell basis measurement, and discuss its pros/cons.
Implementing polynomial 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 advanced 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 and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. 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, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.