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.
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.
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.
To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators 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 (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (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. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH 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.
Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency ω multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a O(poly(n)) gate complexity, where n is the number of qubits, in the worst-case scenario used for banded matrices, and O(n) when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.
Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only k=poly(n) Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in n. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least 1−δ, for any δ. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in n, k, and log(1/δ). These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.
Linear combination of unitaries (LCU) is a fundamental primitive in quantum algorithms, whose cost is typically governed by the most expensive unitary appearing in the combination. We develop a transducer-based LCU framework that reduces this worst-case dependence to a weighted average query complexity, when the constituent unitaries share access to a common set of primitive oracles.
Consider A=∑jcjUj, where cj>0 and each unitary Uj can be implemented using Cj primitive queries. Given an upper bound a≥∥A∥, our algorithm implements a block-encoding of A/α with rescaling factor α=O(a), using O(Cmax+Cλ/a) primitive queries, where λ=∑jcj, Cmax=maxjCj, and C=∑jcjCj/λ. By comparison, the standard LCU construction requires O(Cmaxλ/a) primitive queries. The improvement can therefore be substantial when costly unitaries have small weights and λ/a is large. As applications, we obtain improved block-encodings of sparse matrices, leading to quantum algorithms for sparse Hamiltonian simulation and quantum linear systems with near-optimal query complexity up to polylogarithmic factors in all relevant parameters.
Our main technique is the transducer framework developed by Belovs, Jeffery, and Yolcu [Quantum, 8:1444 (2024)]. Here we develop a complementary operator-level theory tailored to block-encodings. We identify the resolvent norm K as a key complexity measure for transducer implementation besides the existing catalyst complexity. We show that the transducer action can be converted into an ε-approximate block-encoding using only O(Klog(1/ε)) queries to the transducer, improving the precision dependence from polynomial to logarithmic.
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.