Block encoding of sparse matrices with a periodic diagonal structure

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 $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ 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.

💡 Research Summary

This paper presents a concrete quantum circuit construction for block‑encoding sparse matrices whose main diagonal exhibits a periodic structure. The authors build on the linear combination of unitaries (LCU) framework and introduce a specialized diagonal unitary operator that encodes a complex exponential e^{i ω k} into the computational basis. By decomposing this operator into a product of single‑qubit Z‑rotations, they achieve an implementation that requires only one phase gate per qubit, i.e., O(n) gates for an n‑qubit register.

The core technical contribution is Theorem 1, which shows that a simple three‑step circuit—Hadamard on an ancilla, controlled application of the diagonal unitary V(ω) and its square V(2ω), followed by another Hadamard—exactly block‑encodes the real‑valued diagonal matrix C(ω)=diag(cos(k ω)). The sub‑normalisation factor is α=1 and only a single ancilla qubit is needed. By adding a Pauli‑Y on the ancilla, the sine matrix S(ω)=diag(sin(k ω)) is similarly encoded. The probability of obtaining the desired ancilla outcome depends on the input state amplitudes and the frequency ω, oscillating between 0 and 1 with a characteristic interference pattern.

To handle full sparse banded matrices, the authors express a target matrix M as a linear combination of four unitaries: the periodic diagonal C(ω), left and right cyclic shift permutations L and R, and the identity I. The coefficients α₀,…,α₃ are encoded into a quantum state via a state‑preparation unitary P_REP. The SELECT unitary then applies the appropriate unitary conditioned on the index register. The overall block‑encoding circuit uses three ancilla qubits (one for the diagonal block, two for the LCU state preparation) and has gate complexity O(n²). The dominant cost comes from the controlled versions of U_C(ω) and the shift operators. Controlled V(ω) is realized by adding two Toffoli gates, n additional controlled rotations, and 2n controlled CNOTs. The shift operators can be built from quantum adders; their controlled versions require only O(n) gates when using the clean‑ancilla adder of Gidney (or a dirty‑ancilla variant). The multi‑controlled SELECT operation is optimized using the unary‑iteration technique, reducing its depth to O(J) where J=4 is the number of terms in the LCU.

Complexity analysis shows that for the worst‑case banded matrix the method scales polynomially in n, while for a pure diagonal matrix the scaling is linear O(n). This is a dramatic improvement over generic dense‑matrix block‑encoding schemes that typically scale as O(poly(2ⁿ)) or require quantum RAM. Moreover, when the frequency ω is small (e.g., ω ≤ 2π/N), the diagonal unitary can be approximated by acting only on the most significant O(log 1/ε) qubits, further reducing gate count.

The authors demonstrate the practical relevance of their construction by applying it to the discretized advection‑diffusion‑reaction (ADR) partial differential equation with periodic coefficients. The resulting matrix fits the prescribed structure, allowing the use of quantum singular‑value transformation (QSVT) or quantum phase estimation with optimal scaling. Numerical simulations on a four‑qubit example illustrate the ancilla‑measurement probability p₀ as a function of ω and different input states (uniform superposition, single basis state, generic distribution), confirming the theoretical predictions.

In conclusion, the paper delivers a tailored block‑encoding technique that exploits periodic diagonal patterns, achieving O(n) or O(poly(n)) gate complexity, minimal ancilla overhead, and straightforward extensions to multi‑frequency or more general periodic structures. This advances the toolbox for quantum linear‑algebra subroutines, especially in contexts where the underlying operators arise from discretized PDEs or structured graph Laplacians. Future work may explore extensions to non‑uniform periodicities, error‑robust implementations on near‑term hardware, and integration with higher‑level quantum algorithms for scientific computing.