Compact quantum kernel-based binary classifier

Classification is a canonical pattern recognition problem that aims to label a data point $\tilde{x}\in {\mathbb{R}}^{N}$ as accurately as possible given a labeled (or training) data set

$$\begin{equation*}\mathcal{D}=\left\{({x}_{0},{y}_{0}),\dots ,({x}_{M-1},{y}_{M-1})\right\}\subset {\mathbb{R}}^{N}\times \left\{0,1,\dots ,L-1\right\}.\end{equation*}$$

Since the training data set includes labels, this task can be addressed via a supervised machine learning technique. Among many machine learning approaches, a kernel method provides a straightforward interpretation of the classification process. It is based on choosing a feature space for the data set such that the classification score is defined as a linear function of the similarity (i.e. kernel) between each training data and the test data. This principle naturally connects quantum computing to kernel-based classification since the quantum Hilbert space can be used as the data feature space with a proper definition of the similarity [15, 24].

In this work, we focus on real-valued data as is common in practical machine learning tasks. Moreover, we focus on binary classification (i.e. L = 2) since a multi-class classification can be constructed with binary classifiers by one versus all or one versus one scheme [32]. Note that in binary classification, the two class labels are often denoted as ±1.

The first step towards utilizing the quantum Hilbert space as the feature space is encoding classical data as a quantum state. Although the best encoding strategy remains an open problem, many previous quantum machine learning algorithms chose to represent a classical vector ${\mathbf{x}}_{j}={({x}_{0j},\dots ,{x}_{(N-1)j})}^{\mathrm{\top }}\in {\mathbb{R}}^{N}$ as probability amplitudes of a quantum state in the following form [12, 21, 22, 25, 33],

$$\begin{equation}\vert {\mathbf{x}}_{j}\rangle {:=}\sum\limits _{i=0}^{N-1}{x}_{ij}\vert i\rangle ,\end{equation} \tag{ 1 }$$

using ⌈log₂(N)⌉ qubits, where the input vector is normalized and have unit length, i.e. ||x_j || = 1. The above form of data encoding is often called amplitude encoding, and it can be generalized for a set of M data points x₁, ..., x_M as

$$\begin{equation}\frac{1}{\sqrt{M}}\,\sum\limits _{j=0}^{M-1}\sum\limits _{i=0}^{N-1}{x}_{ij}\vert i\rangle \otimes \vert j\rangle ,\end{equation} \tag{ 2 }$$

which uses at least ⌈log₂(NM)⌉ qubits.

Hereinafter, we will omit the Kronecker product symbol (⊗) and write |i⟩ ⊗ |j⟩ = |ij⟩ whenever the meaning is clear.

A simple model of quantum classifier for real-valued data was introduced in reference [25], and is nowadays referred to as the HTC. This work presents improvements of the HTC in several aspects, and hence this section briefly reviews the algorithm's structure.

For the HTC, the data set $\mathcal{D}$ is encoded in a quantum state as

$$\begin{equation}\vert \psi \rangle =\frac{1}{\sqrt{2}}\,\sum\limits _{j=0}^{M-1}\sqrt{{a}_{j}}\left(\vert 0\rangle \vert {\mathbf{x}}_{j}\rangle +\vert 1\rangle \vert \tilde{\mathbf{x}}\rangle \right)\vert {y}_{j}\rangle \vert j\rangle ,\end{equation} \tag{ 3 }$$

where |x_j ⟩ and $\vert \tilde{\mathbf{x}}\rangle$ are quantum representatives of the classical training and test data vectors by utilizing an encoding of choice (a feature map). The label y_j ∈ {−1, +1} is transformed to the computational basis of the label qubit with the rule y_j → |(1 − y_j )/2⟩ ∈ {|0⟩, |1⟩}. In the original paper [25], the weights were chosen uniform, i.e. a_j = 1/M ∀ j, while it has been shown [27] that it can be used as a variable to be optimized, similar to the treatment in support vector machines. Principally, the HTC applies a Hadamard-test as measurement scheme: this applies a Hadamard gate to the ancillary qubit in order to interfere training and test data states. This is finalized by measuring an expectation value of a two-qubit observable ${\sigma }_{z}^{(\mathrm{a})}{\sigma }_{z}^{(\mathrm{l})}$ on the ancillary and label qubit, leading up to

$$\begin{equation}\langle \psi \vert {H}^{(\mathrm{a})}{\sigma }_{z}^{(\mathrm{a})}{\sigma }_{z}^{(\mathrm{l})}{H}^{(\mathrm{a})}\vert \psi \rangle =\sum\limits _{j=0}^{M-1}{a}_{j}{y}_{j}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle .\end{equation} \tag{ 4 }$$

The superscripts a and l are the qubits on which the corresponding operator is applied, namely the ancillary and the label qubit, respectively. The connection between the HTC and the field of kernel methods is based on this equation. One can see that the kernel function is $k({\mathbf{x}}_{j},\tilde{\mathbf{x}})=\langle {\mathbf{x}}_{j}\vert \tilde{\mathbf{x}}\rangle$. Consequently, the right-hand side of equation (4) defines the classification score, which is denoted by f. Therefore, HTC assigns a new label to the test data by the rule

$$\begin{equation}\tilde{y}=\text{sgn}\left[\sum\limits _{j=0}^{M-1}{a}_{j}{y}_{j}\langle {\mathbf{x}}_{j}\vert {\mathbf{x}}_{j}\rangle \tilde{\mathbf{x}}\right].\end{equation} \tag{ 5 }$$

The generalization of the interference circuit of the HTC was discussed in reference [34] as a means to show that the Hadamard gate is the optimal choice for minimizing the number of sampling. In this work, we take a step further and utilize the generalized interference circuit to reduce the quantum circuit cost of the HTC.

The generalized interference circuit for the HTC uses R_z (ϕ)R_y (θ₀) in place of the first Hadamard gate, and uses R_y (θ₁) in place of the last Hadamard gate, where R_z (ϕ) = cos(ϕ/2)I − i sin(ϕ/2)σ_z and R_y (θ) = cos(θ/2)I − i sin(θ/2)σ_y are the single-qubit rotation gates. As shown in reference [34], the two-qubit expectation value measured in the generalized interference circuit is

$$\begin{align}\langle {\sigma }_{z}^{(\mathrm{a})}{\sigma }_{z}^{(\mathrm{l})}\rangle =\;& \sum\limits _{j=0}^{M-1}{a}_{j}{y}_{j}(\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_{0})\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_{1})-\mathrm{s}\mathrm{i}\mathrm{n}({\theta }_{0})\mathrm{s}\mathrm{i}\mathrm{n}({\theta }_{1})\\ & \times (\mathrm{c}\mathrm{o}\mathrm{s}(\phi )\text{Re}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle -\mathrm{s}\mathrm{i}\mathrm{n}(\phi )\text{Im}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle )).\end{align} \tag{ 6 }$$

While the original HTC only uses the real part of the inner product (i.e., $\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle$) in the classification algorithm, the generalized interference circuit opens up the possibility of harnessing the imaginary part as well. Since the goal of this paper is to utilize the imaginary part, we set θ₀ = π/2 and θ₁ = −π/2 for simplicity. In this case, equation (6) becomes

$$\begin{equation}\langle {\sigma }_{z}^{(\mathrm{a})}{\sigma }_{z}^{(\mathrm{l})}\rangle =\sum\limits _{j=0}^{M-1}{a}_{j}{y}_{j}(\mathrm{c}\mathrm{o}\mathrm{s}(\phi )\text{Re}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle -\mathrm{s}\mathrm{i}\mathrm{n}(\phi )\text{Im}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle ).\end{equation} \tag{ 7 }$$

The above result can also be obtained by applying a single-qubit rotation gate R_z (ϕ) to the ancilla qubit of the HTC in equation (3).

The main idea in this work is based on the observation that if cos(ϕ) and sin(ϕ) in equation (7) have the same sign, then the real and imaginary parts of the state overlap contributes with opposite sign. Thus by encoding the training data with label +1 (−1) to real (imaginary) part of the probability amplitudes of the quantum state, the binary classification can be done without explicitly preparing the label register. This leads to reducing the number of qubits by one. Note that when the number of training data vectors in two classes is the same, $\mathrm{cos}(\phi )=\mathrm{sin}(\phi )=1/\sqrt{2}$. In section 3.3, we discuss how to control ϕ for imbalanced data set.

With this background, the CAE is introduced to utilize the imaginary part as follows. In CAE, two N-dimensional real vectors ${\mathbf{x}}_{k}^{+}={({x}_{0k}^{+},\dots ,{x}_{(N-1)k}^{+})}^{\mathrm{\top }}$ and ${\mathbf{x}}_{k}^{-}={({x}_{0k}^{-},\dots ,{x}_{(N-1)k}^{-})}^{\mathrm{\top }}$ with the corresponding labels indicated by the superscript (±) are loaded in a quantum state in the following form,

$$\begin{equation}{\vert {\mathbf{x}}_{k}\rangle }_{\mathrm{c}}{:=}\sum\limits _{j=0}^{N-1}({x}_{jk}^{+}+\mathrm{i}{x}_{jk}^{-})\vert j\rangle ,\end{equation} \tag{ 8 }$$

where ${\Vert}{\mathbf{x}}_{k}^{+}{{\Vert}}^{2}+{\Vert}{\mathbf{x}}_{k}^{-}{{\Vert}}^{2}=1$ to satisfy the normalization condition. Note that various scaling methods can be employed to satisfy this condition, and one of the natural ways is to scale each vector so that ${\Vert}{\mathbf{x}}_{k}^{\pm }{\Vert}=1/\sqrt{2}$. We implicitly assume this way of normalizing vectors unless stated otherwise. The above equation shows that two N-dimensional vectors are encoded in ⌈log₂(N)⌉ qubits. The subscript c in equation (8) distinguishes the quantum state from amplitude encoding, and indicates that the classical vector is encoded via CAE. It is important to note that |⋅⟩_c means two data points are encoded in the state vector, whereas the ordinary ket vector without the subscript c means one data point is encoded. In addition, we define two more states as

$$\begin{equation}\vert {\mathbf{x}}_{k}^{\pm }\rangle {:=}\frac{1}{{\Vert}{\mathbf{x}}_{k}^{\pm }{\Vert}}\,\sum\limits _{j=0}^{N-1}{x}_{jk}^{\pm }\vert j\rangle .\end{equation} \tag{ 9 }$$

Then the state overlap between two quantum registers, one encodes an N-dimensional real vector $\tilde{\mathbf{x}}$ via amplitude encoding (equation (1)) and the other encodes two N-dimensional real vectors ${\mathbf{x}}_{k}^{+}$ and ${\mathbf{x}}_{k}^{-}$ via CAE (equation (8)) is

$$\begin{equation}{\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{k}\rangle }_{\mathrm{c}}=\frac{1}{\sqrt{2}}(\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{k}^{+}\rangle +\mathrm{i}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{k}^{-}\rangle ).\end{equation} \tag{ 10 }$$

The compact quantum machine learning algorithm that implements the binary classifier from equation (5) is constructed as follows. We first prepare an initial state that encodes the data set $\mathcal{D}$ as

$$\begin{equation}\vert {\psi }_{i}\rangle =\frac{1}{\sqrt{2}}\,\sum\limits _{j=0}^{\frac{M}{2}-1}\sqrt{{b}_{j}}\left(\vert 0\rangle {\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}+{\text{e}}^{-\text{i}\phi }\vert 1\rangle \vert \tilde{\mathbf{x}}\rangle \right)\vert j\rangle ,\end{equation} \tag{ 11 }$$

where the subscript c indicates that the state is prepared via CAE. It is important to note that ${\sum }_{j=1}^{M/2}{\,b}_{j}=1$, and hence the set of weights is different from that of the HTC which satisfies ${\sum }_{j=0}^{M-1}{\,a}_{j}=1$. For example, for a set of uniform weights, b_j = 2/M and a_j = 1/M. For simplicity, we assume that the number of data with label +1 denoted by M₊ is equal to the number of data with label −1 denoted by M₋. The state above is easier to prepare than the state required in HTC shown in equation (3) since the label information is not explicitly encoded and the relative phase e^−iϕ can be added by applying a single-qubit rotation gate R_z (ϕ) on the ancilla qubit. Moreover, since two training data are encoded in one quantum register, the number of terms in the summation is decreased by a factor of 2, meaning that the dimension of the index register is also decreased by a factor of 2. After the state preparation, the rest of the algorithm only requires a Hadamard gate and the measurement of the ancilla qubit in the σ_z basis. The Hadamard gate interferes the copies of the new input and the training inputs to produce a state

$$\begin{equation}\vert {\psi }_{\mathrm{f}}\rangle =\frac{1}{2}\sum\limits _{j=0}^{\frac{M}{2}-1}\sqrt{{b}_{j}}[\vert 0\rangle \hspace{1pt}({\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}+{\text{e}}^{-\text{i}\phi }\vert \tilde{\mathbf{x}}\rangle )+\vert 1\rangle ({\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}-{\text{e}}^{-\text{i}\phi }\vert \tilde{\mathbf{x}}\rangle )]\vert j\rangle .\end{equation} \tag{ 12 }$$

The probability to measure the ancilla qubit in |0⟩ is given as

$$\begin{equation*}\mathrm{P}\mathrm{r}(0)=\frac{1}{4}\sum\limits _{j=0}^{\frac{M}{2}-1}{b}_{j}(2+{\text{e}}^{-\text{i}\phi }{}_{c}\langle {\mathbf{x}}_{j}\vert \tilde{\mathbf{x}}\rangle +{\text{e}}^{\text{i}\phi }{\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}).\end{equation*}$$

By denoting ${\kappa }_{j}={\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}$, the above equation can be written as

$$\begin{equation}\mathrm{Pr}(0)=\frac{1}{2}\sum\limits _{j=0}^{\frac{M}{2}-1}{b}_{j}\left(1+\mathrm{cos}(\phi )\text{Re}({\kappa }_{j})-\mathrm{sin}(\phi )\text{Im}({\kappa }_{j})\right).\end{equation} \tag{ 13 }$$

Similarly, the probability to measure the ancilla qubit in |1⟩ can be calculated as

$$\begin{equation}\mathrm{Pr}(1)=\frac{1}{2}\sum\limits _{j=0}^{\frac{M}{2}-1}{b}_{j}\left(1-\mathrm{cos}(\phi )\text{Re}({\kappa }_{j})+\mathrm{sin}(\phi )\text{Im}({\kappa }_{j})\right).\end{equation} \tag{ 14 }$$

Therefore, the expectation value of the σ_z operator measured on the ancilla qubit is

$$\begin{equation}\langle {\sigma }_{z}^{(\mathrm{a})}\rangle =\sum\limits _{j=0}^{\frac{M}{2}-1}{b}_{j}\left(\mathrm{cos}(\phi )\text{Re}({\kappa }_{j})-\mathrm{sin}(\phi )\text{Im}({\kappa }_{j})\right).\end{equation} \tag{ 15 }$$

By setting $\mathrm{cos}(\phi )=\mathrm{sin}(\phi )=1/\sqrt{2}$ and using equation (10), we arrive at

$$\begin{equation}\langle {\sigma }_{z}^{(\mathrm{a})}\rangle =\frac{1}{2}\sum\limits _{j=0}^{\frac{M}{2}-1}{b}_{j}\left(\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}^{+}\rangle -\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}^{-}\rangle \right).\end{equation} \tag{ 16 }$$

Since the state overlap for the training data in class +1 (−1) always have the positive (negative) sign, the above equation can be written in its final form as

$$\begin{equation}\langle {\sigma }_{z}^{(\mathrm{a})}\rangle =\frac{1}{2}\sum\limits _{j=0}^{M-1}{b}_{j}^{\prime }{y}_{j}\langle \tilde{\mathbf{x}}\vert {\mathbf{x}}_{j}\rangle ,\end{equation} \tag{ 17 }$$

where ${b}_{m}^{\prime }={b}_{m+M/2}^{\prime }={b}_{m}$ for m = 0, ..., M/2 − 1 and ${\sum }_{j=0}^{M-1}{b}_{j}^{\prime }=2$. This outcome is similar to the classification score in the Hadamard classifier obtained by the two-qubit measurement. The constant factor of 1/2 is attributed to the different normalization conditions between the set of weights b_j and a_j as described below equation (11). We refer to this classifier as compact Hadamard classifier (CHC). The CHC is obtained with a single-qubit measurement implying that one can bypass the standardization of the training data set required in HTC for increasing the post-selection probability. The comparison of quantum circuits for implementing HTC and CHC is depicted in figure 1.

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The reduction of quantum circuit sizes, which is of critical importance in practice, achieved with CHC is as follows. By introducing a controllable relative phase between two computational basis states of the ancilla qubit and using CAE, the number of qubits is reduced by two, one for the label register and another in the index register. Then the quantum operations for encoding the training data set is reduced from M gates controlled by log₂(M) index qubits to M/2 gates controlled by log₂(M) − 1 index qubits. Having one fewer controlled qubit can further reduce the quantum circuit depth by a factor of two [37]. Therefore, the number of operations for encoding the training data set x_j is reduced by a factor of four. The number of operations in CHC is also reduced due to the removal of operations for explicitly encoding the label information in a separate register. The number of gates needed for preparing the label register in HTC is 1 when the number of data belonging to each class is equal since one controlled-NOT gate controlled by one of the index qubits applied to the label qubit can split the Hilbert space into two subspaces with an equal number of 0's and 1's in the label qubit. But the number of operation increases linearly with the difference in the number of data with different labels. For example, if αM and βM data belong to class +1 and −1, respectively, where α + β = 1, then the number of operations necessary to encode label information grows as |α − β|M. Therefore, in total, our compact classifier can reduce the number of operations at least by a factor of four, while the reduction can be larger depending on the label distribution of the given training data set. Furthermore, the two-qubit measurement scheme used in HTC is reduced to single-qubit measurement.

As elucidated above, the number of training data vectors in two classes may be different. In this case, the real or imaginary part of the quantum state is zero for the missing data. The difference in the number of training data vectors in two classes can later be compensated by controlling the weights between the real and imaginary parts of the state overlap by finding ϕ that satisfies

$$\begin{equation*}\frac{\mathrm{sin}(\phi )}{\mathrm{cos}(\phi )}=\frac{{M}_{-}}{{M}_{+}},\end{equation*}$$

where M_± is the number of training data points in ±1 class.

Table 1 presents the classification accuracy of proof-of-concept experiments. We divided Iris and Wine datasets into training and test sets with a ratio of 80/20 and two classes and used the second and third features of the Iris dataset and the two principal components of the Wine dataset. We performed a search among 40 combinations of four training patterns (two from each class) and selected the combination with the best training accuracy. We used these four selected training points to evaluate the test accuracy. The use of two features and four training points allowed us to perform experiments in a small-scale quantum device. The accuracies and standard deviations in table 1 are obtained from 30 repetitions of the random division into training and test sets. The experimental results are from ibmq_manila, a quantum computer with five superconducting qubits available on the IBM quantum cloud service.

Let us denote $\vert {\Psi}({\mathbf{x}}_{j},\tilde{\mathbf{x}})\rangle =(\vert 0\rangle {\vert {\mathbf{x}}_{j}\rangle }_{\mathrm{c}}+{\text{e}}^{-\text{i}\phi }\vert 1\rangle \vert \tilde{\mathbf{x}}\rangle )/\sqrt{2}$. By allowing classical sampling from an ensemble $\left\{{a}_{j},\vert {\Psi}({\mathbf{x}}_{j},\tilde{\mathbf{x}})\rangle \right\}$, where a_j is the probability to choose jth state, we can have the mixed state

$$\begin{equation}\sum\limits _{j=0}^{\frac{M}{2}-1}{a}_{j}\vert {\Psi}({\mathbf{x}}_{j},\tilde{\mathbf{x}})\rangle \langle {\Psi}({\mathbf{x}}_{j},\tilde{\mathbf{x}})\vert .\end{equation} \tag{ 18 }$$

It is easy to verify that the expectation measurement of σ_x operator (equivalent to application of a Hadamard gate followed by σ_z measurement) yields the same outcome as shown in equation (15). In this approach, the index register is unnecessary, and hence we further reduce the number of qubits by ⌈log₂(M)⌉ and the number of gates by O(poly(M)). This the smallest kernel-based binary classifier; one only requires ⌈log₂(N)⌉ qubits for encoding the data set, and a qubit for measurement.

Under certain restrictions, the compact encoding scheme can be applied to the quantum feature mapping framework introduced in reference [23] to store two training data in one quantum register. In principle, this can be done with the state preparation

$$\begin{equation}\vert {\Phi}{({\mathbf{x}}_{k}^{\pm })\rangle }_{\mathrm{c}}=\frac{\vert {\Phi}({\mathbf{x}}_{k}^{+})\rangle +\mathrm{i}\vert {\Phi}({\mathbf{x}}_{k}^{-})\rangle }{\sqrt{2}},\end{equation} \tag{ 19 }$$

with a feature map U_Φ(x)|0⟩^⊗L = |Φ(x)⟩ that maps N-dimensional data to L = O(N) qubits. Of course, the above state must satisfy $\langle {\Phi}({\mathbf{x}}_{k}^{+})\vert {\Phi}({\mathbf{x}}_{k}^{-})\rangle =0$. The feature map also needs to satisfy $\langle {\Phi}(\tilde{\mathbf{x}})\vert {\Phi}({\mathbf{x}}_{k}^{\pm })\rangle \in \mathbb{R}$ for all k. One way to prepare the above state is to apply unitary transformation

$$\begin{equation*}V=\frac{{U}_{{\Phi}}({\mathbf{x}}_{k}^{+})+\mathrm{i}{U}_{{\Phi}}({\mathbf{x}}_{k}^{-})}{\sqrt{2}}\end{equation*}$$

to |0⟩^⊗L . Since V is unitary, it must satisfy ${U}_{{\Phi}}({\mathbf{x}}_{k}^{+}){U}_{{\Phi}}{({\mathbf{x}}_{k}^{-})}^{{\dagger}}-{U}_{{\Phi}}({\mathbf{x}}_{k}^{-}){U}_{{\Phi}}{({\mathbf{x}}_{k}^{+})}^{{\dagger}}=0$. Finding an appropriate feature map that satisfies the above conditions while retaining the quantum advantage is a difficult task and we leave it as an interesting open problem.

Given that the state in equation (19) can be prepared, one can follow the strategy from the previous section. More explicitly, classical sampling from a set

$$\begin{equation*}\left\{\frac{\vert 0\rangle \vert {\Phi}{({\mathbf{x}}_{j}^{\pm })\rangle }_{\mathrm{c}}+{\text{e}}^{-\text{i}\phi }\vert 1\rangle \vert {\Phi}(\tilde{\mathbf{x}})\rangle }{\sqrt{2}}\right\}\end{equation*}$$

with probability a_j followed by the expectation measurement of σ_x operator yields

$$\begin{equation*}\frac{1}{2}\sum\limits _{j=0}^{\frac{M}{2}-1}{a}_{j}(\mathrm{c}\mathrm{o}\mathrm{s}(\phi )\langle {\Phi}(\tilde{\mathbf{x}})\vert {\Phi}({\mathbf{x}}_{j}^{+})\rangle -\mathrm{s}\mathrm{i}\mathrm{n}(\phi )\langle {\Phi}(\tilde{\mathbf{x}})\vert {\Phi}({\mathbf{x}}_{j}^{-})\rangle ),\end{equation*}$$

which is reduced to the classifier of equation (17) when ϕ = π/4.