Fig 1.
An example of quantum circuits.
This shows a circuit that performs a quantum calculation called Grover’s algorithm. In quantum computation, qubits representing |0⟩n as the initial states are prepared and passed through a quantum circuit to obtain the results. The wires mean that the quantum states are passed through as they are, and each gate marked with an alphabet changes the quantum state. The rightmost blocks with pictures of meters mean the measurement of the quantum state.
Fig 2.
First, the gates labeled H (Hadamard gate) are applied to the initial states of |0⟩n to create the |+⟩n state. Next, a gate (VD) consisting of diagonal matrices consisting of a high number of polynomials is applied. Finally, the X measurement is performed by applying again the gates labeled H and then performing the observation.
Fig 3.
The circuit is divided into the first half and the second half, and the dagger of the first half is taken. UIQP(x) represents Eq (8), which consists of layers with Hadamard gates H⊗n and a polynomial number of diagonal matrices VD(x).
Fig 4.
Causal relationship based on the artificial dataset generated from Eq (13).
The ellipses represent the variables, and the arrows represent the directions of causality.
Table 1.
Comparison of the accuracy of qLiNGAM and DirectLiNGAM with Gaussian kernel using the artificial data from the Erdos-Reny graph.
Fig 5.
Experimental results using the UCI Heart Disease Data Set.
The ellipses represent the variables, and the arrows represent the directions of causality. (a) A valid causal structure for UCI Heart Disease Data Set, because of arrows drawn from ‘age’ to ‘chol’ and ‘trestbps’. (b) Not a valid causal structure for UCI Heart Disease Data Set, because of arrows drawn from ‘chol’ and ‘trestbps’ to ‘age’. ‘age’, a variable representing age; ‘chol’, a variable representing serum cholesterol in mg/dl; ‘trestbps’, a variable representing resting blood pressure in mmHg on admission to the hospital.
Fig 6.
Experimental results using the Pima Indians diabetes database.
The ellipses represent the variables, and the arrows represent the directions of causality. ‘insulin’, a variable for insulin concentration 2 h after the oral glucose tolerance test; and ‘glucose’, a variable for blood glucose concentration 2 h after the oral glucose tolerance test.
Fig 7.
Gram matrices for 100 cases in the Pima Indians diabetes database using the variable ‘age’.
Each element of the Gram matrix is normalized from 0 to 1. The stronger the black color, the closer it is to 1. (a) The Gram matrix of the quantum kernel with the number of qubits of 4 and the number of depths of 1. (b) The Gram matrix of the quantum kernel created using ibm_kawasaki with the number of qubits of 4 and the number of depths of 1. (c) The Gram matrix of the quantum kernel with the number of qubits of 5 and the number of depths of 2. (d) The Gram matrix of the quantum kernel created using ibm_kawasaki with the number of qubits of 5 and the number of depths of 2.