In order to explore the possibility of cross-fertilization between quantum computing and neural networks as well as to improve the classification performance of quantum neural networks, this paper proposes an improved Variable Split Shadow Quantum Circuit (VSQC-WOA) model based on the Whale Optimization Algorithm. In this model, we design a strongly entangled local shadow circuit to achieve efficient characterization of global features through local shadow feature extraction and a sliding mechanism, which provides a rich quantum feature representation for the classification task. The gradient is then computed by the parameter-shifting method, and finally the features processed by the shadow circuit are passed to the classical fully connected neural network (FCNN) for processing and classification. The model also introduces the Whale Optimization Algorithm (WOA) to further optimize the weights and biases of the fully connected neural network, which improves the expressive power and classification accuracy of the model. In this paper, we firstly use different localized shadow circuit VSQC models to achieve the binary classification task on the MNIST dataset, and our design of strongly entangled shadow circuits performs the best in terms of classification accuracy. The VSQC-WOA model is then used to multi-classify the MNIST dataset (three classifications as an example), and the effectiveness of the proposed VSQC-WOA model as well as the robustness and generalization ability of the model are verified through various comparison experiments.In order to explore the possibility of cross-fertilization between quantum computing and neural networks as well as to improve the classification performance of quantum neural networks, this paper proposes an improved Variable Split Shadow Quantum Circuit (VSQC-WOA) model based on the Whale Optimization Algorithm. In this model, we design a strongly entangled local shadow circuit to achieve efficient characterization of global features through local shadow feature extraction and a sliding mechanism, which provides a rich quantum feature representation for the classification task. The gradient is then computed by the parameter-shifting method, and finally the features processed by the shadow circuit are passed to the classical fully connected neural network (FCNN) for processing and classification. The model also introduces the Whale Optimization Algorithm (WOA) to further optimize the weights and biases of the fully connected neural network, which improves the expressive power and classification accuracy of the model. In this paper, we firstly use different localized shadow circuit VSQC models to achieve the binary classification task on the MNIST dataset, and our design of strongly entangled shadow circuits performs the best in terms of classification accuracy. The VSQC-WOA model is then used to multi-classify the MNIST dataset (three classifications as an example), and the effectiveness of the proposed VSQC-WOA model as well as the robustness and generalization ability of the model are verified through various comparison experiments.

arXiv:2505.09994May 15, 20250 citations

What Can Be Computed with Indefinite Causal Structure?

Omri Shmueli

Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit. We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in $\textbf{NP}$ . The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit. We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in $\textbf{NP}$. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.

quant-pharXiv:2610.40573Oct 30, 20260 citations

A Theory of Cartesian Arrays (with Applications in Quantum Circuit Verification)

Yu‐Fang Chen, Philipp Rümmer, Wei-Lun Tsai

Abstract We present a theory of Cartesian arrays, which are multi-dimensional arrays with support for the projection of arrays to sub-arrays, as well as for updating sub-arrays. The resulting logic is an extension of Combinatorial Array Logic (CAL) and is motivated by the analysis of quantum circuits: using projection, we can succinctly encode the semantics of quantum gates as quantifier-free formulas and verify the end-to-end correctness of quantum circuits. Since the logic is expressive enough to represent quantum circuits succinctly, it necessarily has a high complexity; as we show, it suffices to encode thek-color problem of a graph under a succinct circuit representation, an NEXPTIME-complete problem. We present an NEXPTIME decision procedure for the logic and report on preliminary experiments with the analysis of quantum circuits using this decision procedure.Abstract We present a theory of Cartesian arrays, which are multi-dimensional arrays with support for the projection of arrays to sub-arrays, as well as for updating sub-arrays. The resulting logic is an extension of Combinatorial Array Logic (CAL) and is motivated by the analysis of quantum circuits: using projection, we can succinctly encode the semantics of quantum gates as quantifier-free formulas and verify the end-to-end correctness of quantum circuits. Since the logic is expressive enough to represent quantum circuits succinctly, it necessarily has a high complexity; as we show, it suffices to encode thek-color problem of a graph under a succinct circuit representation, an NEXPTIME-complete problem. We present an NEXPTIME decision procedure for the logic and report on preliminary experiments with the analysis of quantum circuits using this decision procedure.

Lecture notes in computer scienceJan 1, 202318 citations

Simple Quantum Circuits for Data Classification

Joanna Wiśniewska, Marek Sawerwain

Lecture notes in computer scienceJan 1, 20201 citation

Quantum algorithm for neural network enhanced multi-class parallel classification

Anqi Zhang, Xiaoyun He, Shengmei Zhao

Using the properties of quantum superposition, we propose a quantum classification algorithm to efficiently perform multi-class classification tasks, where the training data are loaded into parameterized operators which are applied to the basis of the quantum state in quantum circuit composed by \emph{sample register} and \emph{label register}, and the parameters of quantum gates are optimized by a hybrid quantum-classical method, which is composed of a trainable quantum circuit and a gradient-based classical optimizer. After several quantum-to-class repetitions, the quantum state is optimal that the state in \emph{sample register} is the same as that in \emph{label register}. %A structure of loading data many times is performed as a quantum version of neural network to improve the expression ability of quantum circuit. For a classification task of $L$ -class, the analysis shows that the space and time complexity of the quantum circuit are $O(L*logL)$ and $O(logL)$ , respectively. The numerical simulation results of 2-class task and 5-class task show that the proposed algorithm has a higher classification accuracy, faster convergence and higher expression ability. The classification accuracy and the speed of converging can also be improved by increasing the number times of applying multi-qubit controlled operators on the quantum circuit, especially for multiple classes classification.Using the properties of quantum superposition, we propose a quantum classification algorithm to efficiently perform multi-class classification tasks, where the training data are loaded into parameterized operators which are applied to the basis of the quantum state in quantum circuit composed by \emph{sample register} and \emph{label register}, and the parameters of quantum gates are optimized by a hybrid quantum-classical method, which is composed of a trainable quantum circuit and a gradient-based classical optimizer. After several quantum-to-class repetitions, the quantum state is optimal that the state in \emph{sample register} is the same as that in \emph{label register}. %A structure of loading data many times is performed as a quantum version of neural network to improve the expression ability of quantum circuit. For a classification task of $L$-class, the analysis shows that the space and time complexity of the quantum circuit are $O(L*logL)$ and $O(logL)$, respectively. The numerical simulation results of 2-class task and 5-class task show that the proposed algorithm has a higher classification accuracy, faster convergence and higher expression ability. The classification accuracy and the speed of converging can also be improved by increasing the number times of applying multi-qubit controlled operators on the quantum circuit, especially for multiple classes classification.

arXiv:2203.04097Mar 8, 20220 citations

\textbf{NP}

. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond

\textbf{BQP}

within the process matrix formalism, even while preserving unitarity.

L

-class, the analysis shows that the space and time complexity of the quantum circuit are

O(L*logL)

and

O(logL)

, respectively. The numerical simulation results of 2-class task and 5-class task show that the proposed algorithm has a higher classification accuracy, faster convergence and higher expression ability. The classification accuracy and the speed of converging can also be improved by increasing the number times of applying multi-qubit controlled operators on the quantum circuit, especially for multiple classes classification.