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Scientific Reports

Quantum circuits from non-unitary sparse binary matrices

Krishnageetha Karuppasamy, Varun Puram, K. M. George, Thomas P. Johnson

Published Jul 1, 2025• 2 citations
Publisher

Abstract

Quantum computing leverages unitary matrices to perform reversible computations while preserving probability norms. However, many real-world applications involve non-unitary sparse matrices, posing a challenge for quantum implementation. This paper introduces a novel method for transforming a class of non-unitary sparse binary matrices into higher-dimensional permutation matrices, ensuring unitarity. Our approach is efficient in both space and time, ensuring practical applicability to large-scale problems. We demonstrate the utility of this transformation in constructing quantum gates and apply the method to model quantum finite state machines (QFSMs) derived from classical deterministic finite automata (DFAs). This work offers a practical pathway for integrating non-unitary transformations into quantum systems, with implications for the many applications that are based on sparse, non-unitary matrices. The significance of this work for automata theory and quantum computation is outlined.

DOI
10.1038/s41598-025-03424-7

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Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [Formula: see text] in a larger unitary transformation [Formula: see text] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [Formula: see text], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by
Arbitrary exponentially large unitaries cannot be implemented efficiently by quantum circuits. However, we show that quantum circuits can efficiently implement any unitary provided it has at most polynomially many nonzero entries in any row or column, and these entries are efficiently computable. One can formulate a model of computation based on the composition of sparse unitaries which includes the quantum Turing machine model, the quantum circuit model, anyonic models, permutational quantum computation, and discrete time quantum walks as special cases. Thus, we obtain a simple unified proof that these models are all contained in BQP. Furthermore, our general method for implementing sparse unitaries simplifies several existing quantum algorithms.
There are several papers that define quantum finite automaton analogous to classical finite automaton. Some papers describe the power of quantum automaton. In general, in quantum automata unitary matrices are associated to input symbols. Even though quantum circuits (or unitary matrices) are shown in specific cases, there is no general method given to construct a circuit or unitary matrix is found in the literature. This research focuses on construction of unitary matrices for each input symbol from the classical definition of a Deterministic Finite State Machine (DFSM). An algorithm is developed to associate a quantum circuit to an input symbol which considered as a mapping of classical DFSM states. IBM quantum composer is used to design circuits and build examples.