Variational quantum state discriminator for supervised machine learning

A measurement in quantum mechanics is in general described by a set of positive semi-definite operators $\lbrace E_m\rbrace$ satisfying $\sum_{m = 0}^{l-1} E_m = I$. Here, $m\in\lbrace 0,\ldots,l\rbrace$ labels the possible measurement outcomes, and the probability to obtain m given a density matrix ρ that describes the quantum state is given by $p_m = \textrm{Tr}(E_m \rho)$. Note that the operators can be decomposed as $E_m = K_m^{\dagger}K_m$ where $\lbrace K_m\rbrace$ is known as a set of the Kraus operators. Upon obtaining the outcome m, the quantum state evolves to $K_m\rho K_m^{\dagger}/p_m$. We refer to such measurement procedure with the collection of matrices E_m as an l-element POVM [5].

According to Neumark's theorem, an arbitrary POVM can be implemented with a quantum circuit by applying a unitary operator on an extended system consisting of the target quantum state and an ancillary system and then measuring the latter [15, 16]. In this regard, ancillary qubits are introduced and a multi-qubit unitary gate denoted by $\mathcal{U}$ acts on all qubits. The unitary operation $\mathcal{U}$ on target and ancillary qubits can be described in terms of a set of the Kraus operators $\{K_m\}$ as

$$\begin{align} \mathcal{U}|\psi\rangle_\mathrm{T}|\mathbf{0}\rangle_A = \sum^{l-1}_{m = 0} \left(K_m|\psi\rangle_\mathrm{T}\right)|m\rangle_A, \end{align} \tag{ 1 }$$

where $|\psi\rangle_\mathrm{T}$ is the target quantum state consisting of $n_\mathrm{T}$ qubits and $|\mathbf{0}\rangle_A$ denotes all ancillary qubits in the state $|0\rangle$ (see figure 1(a)). Equation (1) signifies the implementation of the quantum channel of Kraus rank l [17, 18]. Performing a POVM on $|\psi\rangle_\mathrm{T}$ is then embodied by measuring ancillary qubits in the computational basis (e.g. projective measurement in the Z basis), which yields the outcome m with the probability $p(m) = {}_\mathrm{T}\langle\psi|K^\dagger_mK_m|\psi\rangle_\mathrm{T}$. Here, we call the unitary operation $\mathcal{U}$ the POVM circuit.

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Since an arbitrary unitary gate can be parameterized with real-valued parameters, the POVM operator that produces the outcome m can be written as

$$\begin{align} \mathcal{E}_m\left(\Theta\right) = E_m\left(\Theta\right) \otimes |\mathbf{0}\rangle_A\langle\mathbf{0}| = \mathcal{U}^\dagger\left(\Theta\right) \left(I_\mathrm{T}\otimes M_m\right)\,\mathcal{U}\left(\Theta\right), \end{align} \tag{ 2 }$$

where Θ is the parameter vector, $I_\mathrm{T}$ is the identity operator acting on the target qubits, and $M_m = |m\rangle_A\langle m|$ represents the projective measurement on the ancillary qubits in the computational basis. Note that each bit string $z_1z_2\cdots z_{n_A}\in\lbrace 0,1\rbrace^{n_A}$ produced by measuring n_A ancillary qubits is converted into the decimal value $m\in\{0,\cdots, l-1\}$, the outcome index of POVMs.

There are several ways to decompose the multi-qubit quantum circuits such as the cosine–sine decomposition (CSD) [19, 20], the decomposition introduced by Knill [21], and the column-by-column decomposition [22]. For a better understanding of the implementation of POVMs in the quantum circuit, we opt for a proper unitary decomposition, the diagrams of which are illustrated in figure 1. The decomposition of $\mathcal{U}$ for $n_\mathrm{T} = n_A = 2$ is discussed in [9]. We generalize the result therein for any $n_\mathrm{T}$ and n_A based on the CSD and provide its rigorous development in appendix A. Specifically, an arbitrary quantum circuit $\mathcal{U}$ is first decomposed into a set of ($n_\mathrm{T}+1$)-qubit unitary operators W_a , uniformly controlled by a − 1 ancilla qubits as depicted in figure 1(a). A unitary operator W_a is then decomposed into a general $n_\mathrm{T}$-qubit unitary gate U and an uniformly controlled single-qubit rotation around the y-axis of the Bloch sphere, denoted by R_y , is applied to the ath ancillary qubit controlled by all target qubits as shown in figure 1(b). The dimension of the parameter vector Θ from all U and R_y gates is $(l-1)(2^{2n_\mathrm{T}}-1 + 2^{n_\mathrm{T}})$.

Let us scrutinize the circuit structures of figures 1(a) and (b) from the perspective of the Kraus operator. The action of the uniformly controlled R_y gate in figure 1(b) is equivalent to the multiplication of a diagonal matrix by a state vector in the target system. Depending on the state of the ancillary qubit, the diagonal matrix is either $C: = \cos(diag(\Theta_{R}))$ if the ancilary qubit is in the state $|0\rangle_A$, or $S: = \sin(diag(\Theta_R))$ for the state $|1\rangle_A$, where $\Theta_{R} = (\theta_1,\cdots,\theta_{2^{n_\mathrm{T}}})$ is a parameter vector involving $2^{n_\mathrm{T}}$ angles of R_y gates. Thus, a single gate W_a implies the implementation of the Kraus operators of rank 2, $\{CU, SU\}$.

In order to generate the Kraus operator of rank l > 2, an ancillary system of $n_A = \lceil \log_2 l \rceil$ qubits is required and $(a-1)$-fold uniformly controlled W_a gates are applied consecutively as the number of control qubits are increased up to $n_A-1$, as shown in figure 1(a). Each uniformly controlled W_a gate builds up the binary rank Kraus operators that depend on the previous results of the ancillary system. The whole procedure can be figured out by introducing a CSD binary tree (see figure 1(c)) [19]. Then the Kraus operator of rank l can be formulated as

$$\begin{align} K_m = \prod^{n_{A}}_{a = 1}D_{j\left(a\right)}U_{j\left(a\right)}, \end{align} \tag{ 3 }$$

where a diagonal matrix $D_{j(a)}$ becomes either $C_{j(a)}$ if $z_{a} = 0$, or $S_{j(a)}$ if $z_{a} = 1$. Here, the index $j(a): = 2^{a-1}+\sum^{a-1}_{i = 1}z_i 2^{a-1-i}$ is introduced to enumerate the actions of the uniformly controlled W_a gate because it is comprised of a sequence of $(a-1)$-fold controlled gates with different angles (see figure 7 in appendix A). For example, a set of rank-4 Kraus operators, $\{C_2U_2C_1U_1, S_2U_2C_1U_1, C_3U_3S_1U_1, S_3U_3S_1U_1\}$, can be carried by using W₁ and uniformly controlled W₂ gates and one of its elements is determined by the measurement outcome $z_1z_2$.

It is noteworthy that the POVM circuit bears a resemblance to dissipative quantum neural networks (DQNN) [23, 24] in that it entangles ancilla qubits initialized at $|0\rangle$ with the target (input) qubits and only measures the ancilla qubits at the output. However, the POVM circuit is more general, as it can implement arbitrary completely positive trace-preserving maps [15]. For instance, the family of circuits that the POVM circuit can generate also includes cases where qubits within the same layer of the DQNN interact before propagating forward to the next layer, or where information propagates backward to the previous layer.

The implementation of the control part of the uniformly controlled W_a gates can be performed classically as it commutes with measurements, as detailed in the appendix (appendix A). As a result, the quantum gate complexity is primarily determined by the implementation of the W_a gate depicted in figure 1(b). In the implementation of W_a , the complexity of the required number of CNOT gates in an arbitrary $n_\mathrm{T}$-qubit unitary U is $O(2^{2n_\mathrm{T}})$ [20]. Additionally, the uniformly controlled R_y operation involves $2^{n_\mathrm{T}}$ CNOT gates [25]. Thus the total number of CNOT gates in the circuit $\mathcal{U}$ is $O(2^{2n_\mathrm{T}}l)$ for a given value of l. In the case of $n_\mathrm{T} = 2$, the total number of CNOT gates reduces to $3(l-1)$, where three is considered the smallest number of CNOT gates for a universal two-qubit gate [26–28]. To reduce the number of elementary two-qubit gates required within $\mathcal{U}$, it may be desirable to choose a simpler ansatz for $\mathcal{U}$ [10, 29]. However, this simplification entails sacrificing global optimality.

Quantum state discrimination is a protocol that aims to identify a quantum state among a set of a priori completely known candidate states. The POVM framework allows for the discrimination of non-orthogonal states with the optimal success probability which depends on the distance between a priori quantum states. This naturally motivates the minimum-error QSD whose goal is to maximize the success probability for correctly guessing a state.

In preparation for minimum-error QSD, let us consider a set of l different quantum states $\{\rho_n\}^{l-1}_{n = 0}$ with the corresponding a priori probabilities $\{q_n\}_{n = 0}^{l-1}$. When POVMs $\{E_m\}$ are performed on quantum states ρ_n , the probability of the outcome m is $p(m|n) = \textrm{Tr}[E_m\rho_n]$. If appropriate POVMs are taken, one can correctly guess a state ρ_m with a high probability $q_mp(m|m)$. In this context, the minimum-error QSD aims to minimize the error probability

$$\begin{align} p_\mathrm{error} = 1- \sum_{m = 0}^{l-1}q_m\textrm{Tr}\left[\rho_mE_m\right], \end{align} \tag{ 4 }$$

where the last term denotes the success probability of the given POVM [30, 31].

If only two elements of POVMs (l = 2) are implemented for discriminating two different quantum states, one can analytically find the optimal POVMs and the associated minimum probability of the error equation (4), known as the Helstrom bound $\mathcal{B}_H$ in this special case [7]. Starting from equation (4), the minimum error probability can be rewritten as

$$\begin{align} \min p_\mathrm{error} = \frac{1}{2} - \max_{E}\frac{1}{2}\textrm{Tr}\left[E\Lambda\right] \end{align} \tag{ 5 }$$

where $E = E_0-E_1$ and $\Lambda = q_0\rho_0-q_1\rho_1$. By applying the spectral decomposition, one can get $\Lambda = \lambda_+|\lambda_+\rangle\langle\lambda_+| + \lambda_-|\lambda_-\rangle\langle\lambda_-|$ where $\lambda_+$ and $\lambda_-$ denote positive and negative eigenvalues of Λ, repsectively. The optimal POVMs can be obtained as $E_0 = |\lambda_+\rangle\langle \lambda_+|$ and $E_1 = |\lambda_-\rangle\langle\lambda_-|$. The Helstrom bound is then given by $\mathcal{B}_\mathrm{H} : = \frac{1}{2}-\frac{1}{2}\textrm{Tr}|\Lambda|$.

When l is larger than two, the set of POVMs that minimizes equation (4) usually does not have an analytical solution. Instead, the necessary and sufficient conditions are known to be satisfied by a set of the optimal POVMs and are expressed as

$$\begin{align} & E_m\left(q_m\rho_m-q_n\rho_n\right)E_n = 0 \end{align} \tag{ 6 }$$

$$\begin{align} & \sum^{l-1}_{m = 0} q_m E_m\rho_m - q_n\rho_n \unicode{x2A7E} 0 \end{align} \tag{ 7 }$$

for any $m, n = 0,\cdots,l-1$ [7, 32, 33]. It is noted that these conditions are equivalent to each other and closely related to the duality relationship of the semidefinite program [33]. The minimum-error QSD problem can be formulated as the SDP. The classical solver for the SDP, however, requires the number of iterations that grows exponentially with the number of target qubits and polynomially with the number of the POVM elements as $O(2^{\omega n_\mathrm{T}}l^{\omega})$, where ω is the constant ranges from 2.371 88 to 3 depending on the algorithm for matrix multiplication (see appendix B). In addition, the objective function involves computing the outcome probabilities of POVM, and hence computing it classically requires exponential runtime. The aforementioned procedure assumes that the target density matrices are fully known. If not, quantum state tomography, which adds another layer of exponential complexity, is also required. The intractability of minimum-error QSD via classical means motivates the development of quantum algorithms for it. Our method, which is described in the following section, is based on training a parameterized quantum circuit, aiming to realize it on the noisy intermediate-scale quantum (NISQ) devices [34]. Later, we will use the classical solver for the SDP to compare its result with that of the VQSD [35].

The first step towards utilizing the VQA for minimum-error QSD is establishing an appropriate cost function that can discriminate different labeled sets beyond individual quantum states. Let us consider that a set of N quantum states with l labels,

$$\begin{align} \mathcal{D} = \left\{\left(|\Psi_n\rangle,y_n\right)|\, y_n\in\left\{0,\cdots,l-1\right\},\; 1\unicode{x2A7D} n\unicode{x2A7D} N\right\}, \end{align} \tag{ 8 }$$

is partitioned into l disjoint subsets. A subset of quantum states with the same label m can be denoted by $\mathcal{D}_m = \{(|\Psi_n\rangle,y_n)| y_n = m,\,\forall n\in\mathcal{N}_m\}$ with the index set $\mathcal{N}_m = \{n|y_n = m,\,\forall n\in\mathbb{N} \}$.

To discriminate between different subsets $\mathcal{D}_m$ of input states $\{|\Psi_n\rangle_\mathrm{I}\}_n$, one can perform POVMs on either all n_I input qubits or $n_\mathrm{T}$ qubits out of $n_\mathrm{I}$ qubits for each input state $|\Psi_n\rangle_\mathrm{I}$. The latter can be considered as measuring the reduced density matrix of $|\Psi_n\rangle_\mathrm{I}$ on the target subsystem consisting of $n_\mathrm{T}$ target qubits. This reduced density matrix can be written as $\rho_{T,n} = \textrm{Tr}_{\bar{T}}|\Psi_n\rangle_\mathrm{I}\langle\Psi_n|$, where the complement of the target subsystem $\bar{T}$ is traced out.

For given $|\Psi_n\rangle_\mathrm{I}$, a POVM E_m may be performed on the target subsystem, which yields the probability of the outcome $p(m|n) = \textrm{Tr}[E_m \rho_{T,n}] = \,_\mathrm{I}\langle\Psi_n|E_m|\Psi_n\rangle_\mathrm{I}$. For given $\mathcal{D}_m$, it is possible to perform a POVM with the outcome m on quantum states in the same class and obtain the probability of the outcome m, $p(m|\mathcal{D}_m): = \sum_{n\in\mathcal{N}_m}\,_\mathrm{I}\langle\Psi_n|E_m|\Psi_n\rangle_\mathrm{I}/|\mathcal{N}_m|$, where $|\mathcal{N}_m|$ is the cardinality of $\mathcal{N}_m$. This probability can also be viewed as performing E_m on the target subsystem of the mixed state

$$\begin{align} \rho^\prime_{m} = \sum_{n\in\mathcal{N}_m}|\Psi_n\rangle\langle\Psi_n |/|\mathcal{N}_m|, \end{align} \tag{ 9 }$$

which is the uniform mixture of all states in a subset $\mathcal{D}_m$.

As mentioned in the previous section, one can make POVMs on the input states in the quantum circuit. Starting from the error probability of the MED as defined in equation (4), the cost function can be expressed in terms of the parameterized quantum circuit as

$$\begin{align} C\left(\Theta\right) = 1-\sum_{m = 0}^{l-1} q_m p\left(m|\mathcal{D}_m\right) = 1-\sum^{l-1}_{m = 0}\sum_{n\in\mathcal{N}_m}\frac{1}{|\mathcal{D}|}\langle\Psi^0_n| \mathcal{E}_m\left(\Theta\right) |\Psi^0_n\rangle, \end{align} \tag{ 10 }$$

where $|\Psi^0_n\rangle = |\Psi_n\rangle_I|\mathbf{0}\rangle_A$ is a quantum state of all qubits and $q_m = |\mathcal{N}_m|/|\mathcal{D}|$ is an a priori probability for the subset $\mathcal{D}_m$. Minimizing this cost function reduces the error of the label discrimination. This signifies that the VQSD is well-trained from a set of the quantum states and can predict the label of a new quantum state based on the measurement that optimally separates the subsets $\mathcal{D}_m$ according to their label m.

The VQSD algorithm can be constructed as follows. First, a set of the input states $\{|\Psi_n\rangle\}_{n\unicode{x2A7D} N}$ is given and ancillary qubits are prepared in the state $|\mathbf{0}\rangle_A$. Note that the algorithm only requires the label information and the input quantum states does not need to be known. Hence it bypasses the quantum state tomography. Second, the parameterized POVM circuit $\mathcal{U}(\Theta)$ acts on n_T target qubits out of $n_\mathrm{I}$ input qubits, which implies that POVMs can be performed on the reduced density matrix $\rho_{\mathrm{T},n}$ of an input state $|\Psi_n\rangle$. When implementing on NISQ devices, small $n_\mathrm{T}$ is desirable. Note that the number of outcomes obtained from POVMs can be controlled by adjusting the number of ancillary qubits. Third, the outcomes m and the corresponding probabilities $p(m|n)$ of POVMs are obtained from measuring ancillary qubits in the computational basis. Fourth, a cost function equation (10) is constructed from the outcomes m and probabilities $p(m|n)$ and minimized using classical optimizer. The parameter vector Θ in the POVM circuit is then optimized toward synchronizing the outcome of POVMs with the label of the input states. The trained POVM circuit $\mathcal{U}(\Theta_\mathrm{opt})$ with the optimal parameter vector $\Theta_\mathrm{opt}$ serves as a quantum state discriminator. In consequence, an unlabeled input state being inserted, the trained POVM circuit and the measurement on ancillary qubits predicts a label with the highest probability. The VQSD algorithm is illustrated in figure 2.

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We demonstrate the performance of the VQSD via classical simulations. The simulations are carried out using PennyLane [38], and the quantum circuit parameters are updated via gradient descent optimization using adaptive moment estimation [39].

The VQSD is tested with both pure and mixed states. The quantum circuit creates a mixed state by tracing out a subsystem from an entangled state. More specifically, the simulation uses the quantum circuit shown in figure 3, which prepares an entangled state by applying $\mathcal{V}_{\rho_{\zeta}}$ to $|00\rangle$. The reduced density matrix on the second qubit, denoted by ρ_ζ , is the mixed state inserted as the input to the POVM circuit. The mathematical form of a single-qubit mixed state ρ_ζ is then given by

$$\begin{align} \rho_\zeta\left(\varphi\right) \equiv \cos^2\varphi|\zeta_+\rangle\langle\zeta_+| + \sin^2\varphi|\zeta_-\rangle\langle \zeta_-|, \end{align} \tag{ 11 }$$

where $|\zeta_{\pm}\rangle$ denotes the eigenstates of the Pauli matrices σ_ζ with $\zeta\in\{x, y, z\}$ and ϕ parameterizes the mixture weights. The Bloch vector of $\rho_\zeta(\varphi)$ lies along the ζ axis of the Bloch sphere. In order to prepare a mixed state $\rho_\zeta(\varphi)$ as an input in the quantum circuit, we make use of a two-qubit unitary gate $\mathcal{V}_{\rho_\zeta}$ illustrated in figure 3 that comprises R_y , CNOT, Hadamard, and phase shift gates. $R_y(\varphi)$ and CNOT gates generate an entangled state $\cos\varphi|00\rangle+\sin\varphi|11\rangle$. The U_ζ gate for each qubit changes the z basis into the ζ basis, $\cos\varphi|\zeta_{+}\zeta_{+}\rangle+\sin\varphi|\zeta_{-}\zeta_{-}\rangle$, where $U_z = I$, $U_x = H$, and $U_y = HS$. We then proceed with only one qubit after the unitary operation $\mathcal{V}_{\rho_\zeta}$, which signifies that the reduced density matrix of the second qubit, $\rho_2 = \text{Tr}_1[\mathcal{V}_{\rho_\zeta}|00\rangle\langle 00|\mathcal{V}^\dagger_{\rho_\zeta}]$, is subject to QSD.

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Together with the quantum circuit figure 3 of the mixed state equation (11), we prepare four a priori sets of quantum states with equal probabilities: $\{\rho_z(\pi/5), \rho_x(\pi/6)\}$, $\{|0\rangle,|1\rangle,|+\rangle\}$, $\{\rho_z(\pi/5), \rho_x(\pi/6), \rho_y(\pi/8)\}$, and $\{\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle), \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle), |00\rangle, |{++}\rangle \}$, We then construct the parameterized POVM circuit equation (2) and the cost function equation (10) with the condition m = n. By implementing the VQSD, we attain the minimum value of the cost function with the convergence tolerance of around 10⁻⁷. The minimum error of which is depicted in figure 4.

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To verify the correctness of the solution obtained from the VQSD, we compare the two-state discrimination task with the Helstrom measurement, and the three-state discrimination tasks with the SDP solved by a convex optimization tool [35]. The difference between two minimum errors from the classical SDP solver and the VQSD is around 10⁻⁶, which confirms the validity of the VQSD. We also compare our method with the pretty-good measurement (PGM) defined as $\Pi_mml: = q_m\rho^{-1/2}\rho_m\rho^{-1/2}$ with the ensemble $\rho = \sum_{n = 0}^{l-1} q_n\rho_n$, which is a suboptimal l-element POVM when l > 2 [40–42]. This measurement entirely depends on the state preparation and is prescribed to gather the information of a quantum system in a simple way with an analytical expression, although the global optimality is not guaranteed. The error probability from performing the PGMs on each set of quantum states is shown in figures 4(b)–(d), which corresponds to the fact that the optimal POVM is generally not a PGM for the minimum-error QSD task [43]. Simulation results verify that the VQSD finds better measurements than the PGM in all instances.

We now apply the VQSD to classification, a ubiquitous problem in data science that can be solved via supervised learning. The training dataset for l-class classification is the same as $\mathcal{D}$ in equation (8), except $|\Psi_n\rangle$ represents the data embedded as the quantum state. With the training dataset, the VQSD can be trained to perform minimum-error QSD to distinguish $\rho^{\prime}_m$ in equation (9) with $m\in\lbrace 0,\ldots,l-1\rbrace$. Once training is completed, a test datum embedded as a quantum state is fed into the VQSD circuit, and the resulting POVM element corresponds to the predicted label.

To demonstrate the ability of the VQSD for supervised classification, we performed classical simulations using PennyLane. The simulation uses the Iris flower dataset for ternary classification. The dataset consists of four attributes (a four-dimensional data point x_n ) and species (a class label y_n ) of Iris flowers, and can be expressed as $\mathcal{D} = \{(\mathbf{x}_n,y_n)|\, \mathbf{x}_n\in\mathbb{R}^4, y_n\in\{0,1,2\}, n = 1,\ldots, 150\}$. Each class has the same number of data points (50 per class).

Applying quantum machine learning to classical data requires an initial step that encodes the classical data into a quantum state. For this, various approaches exist, such as the amplitude encoding, angle encoding, Hamiltonian encoding, and trainable quantum encoding [8, 44–50]. In this simulation, we modify the instantaneous quantum polynomial encoding that utilizes Ising Hamiltoinan, motivated by Havlíček et al [49], to map a four-dimensional data point into a two-qubit state. The feature-encoding method suggested in [49, 51] maps classical data features to the interaction terms of the Ising Hamiltonian. This leads to using the single-qubit rotation gate $R_Z(\phi) = \exp(\mathrm{i}\phi Z_i)$ and the two-qubit coupling gate $R_{ZZ}(\phi) = \exp(\mathrm{i}\phi Z_iZ_j)$ to embed a two-dimensional input vector into two qubits. In addition to the Z operators, we use X operators because we wish to encode four data features into a two-qubit state $|\Psi_n\rangle = |\Phi(\mathbf{x}_n)\rangle = \mathcal{V}_{\Phi(\mathbf{x}_n)}|00\rangle$. A quantum feature map is then defined as $\mathcal{V}_{\Phi} = (V_{\Phi})^{l_\Phi} \mathrm{e}^{\mathrm{i}\phi_1P_1}$, where

$$\begin{align} V_{\Phi} = \prod^4_{\gamma = 1} \mathrm{e}^{\phi_{\gamma,\gamma+1}P_\gamma \cdot P_{\gamma+1}}\mathrm{e}^{\mathrm{i}\phi_\gamma P_\gamma}, \end{align} \tag{ 12 }$$

φ is the coefficient that contains classical information, $l_\Phi$ is the number of layers, and P_γ is the γth element of the set $\{X{\small\otimes}I, I{\small\otimes}X, Z{\small\otimes}I, I{\small\otimes}Z\}$. We do not insert Hadamard gates between encoding layers because each layer already contains non-commuting terms. The circuit diagram of $\mathcal{V}_\Phi$ is illustrated in in figure 5(a). In the quantum circuit for $\mathcal{V}_\Phi$, we encode data x_n into the coefficients φ by using the encoding function $\phi(\mathbf{x})$. The couplings in the quantum feature map $\mathcal{V}_\Phi$ necessitate the information on correlations among the attributes of a data point. There are a variety of approaches to encode a classical data point x into the coefficients φ. We tested all encoding functions suggested in [49, 51], and found the following two types work the best for Iris data classification.

$$\begin{align} \phi_j\left(\mathbf{x}\right) & = x_j, \;\;\; \phi_{i,j}\left(\mathbf{x}\right) = \frac{\pi}{3\cos\left(x_i\right)\cos\left(x_j\right)} \end{align} \tag{ 13 }$$

$$\begin{align} \phi_j\left(\mathbf{x}\right) & = x_j, \;\;\; \phi_{i,j}\left(\mathbf{x}\right) = \exp\left(\frac{|x_i-x_j|^2}{8/\ln\left(\pi\right)}\right) \end{align} \tag{ 14 }$$

Thus, the simulation results in this section are of those carried out with the above encoding functions.

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Before encoding in the Hilbert space, let the Iris data set be preprocessed by rescaling each attribute into the range $[-\pi/5, \pi/5]$. This rescaling is empirically found to work well in combination with both encoding functions. The rescaling mapping we use here is defined as

$$\begin{align} x^\prime_{n,\alpha} = \frac{\pi}{5}\frac{2x_{n,\alpha} - \left(x_{\min,\alpha} + x_{\max,\alpha}\right)}{x_{\max,\alpha}-x_{\min,\alpha}}, \end{align} \tag{ 15 }$$

where $x_{\min,\alpha}: = \min_n x_{n,\alpha}$ and $x_{\max,\alpha}: = \max_n x_{n,\alpha}$ for each attribute $\alpha\in\{0,1,2,3\}$. Next, the rescaled data point $\mathbf{x}_n^{\prime}$ is embedded into the quantum state $|\Psi_n\rangle = \mathcal{V}_{\Phi(\mathbf{x}^{\prime}_n)}|00\rangle$ using the quantum feature map $\mathcal{V}_\Phi$ and two encoding functions equations (13) and (14). Here, the number of layers of the encoding circuit $\mathcal{V}_\Phi$ is empirically chosen as $l_\Phi = 2$.

In conjunction with this quantum data encoding, we facilitate two different structures of the parameterized quantum circuit in the VQA: the single-qubit POVM circuit ($n_\mathrm{T} = 1$) and the two-qubit POVM circuit ($n_\mathrm{T} = 2$) whose diagrams are shown in figures 6(a) and (b), respectively. The encoding function equation (13) is empirically selected for the former and equation (14) for the latter. In both cases, measuring two ancilla qubits in the computational basis yields at most four outcomes of POVMs.

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To assess the classification performance of the VQSD, we use the stratified five-fold cross-validation method. Namely, we shuffle the Iris data set, split the data set in a 1:4 ratio of test to training data, and perform supervised classification on all five splits. The classification performance of the VQSD is quantified by the average test performance of the five classification instances of the cross-validation process. Two metrics are used to determined the test performance in each instance: the accuracy and the area under the receiver operating characteristic (ROC) curve. The procedure for computing theses metrics in our algorithm is described in the following.

First, when an unlabeled data point $\tilde{\mathbf{x}}$ is given, the quantum state discriminator predicts the corresponding label $\tilde{y} = \tilde{m}(\tilde{\mathbf{x}})$ according to a decision rule. The decision rule chooses the label with the highest probability among all possible outcomes of POVMs, which is given by

$$\begin{align} \tilde{m}\left(\tilde{\mathbf{x}}\right) & = \arg \max_m p\left(m|\tilde{\mathbf{x}}\right), \end{align} \tag{ 16 }$$

where $p(m|\tilde{\mathbf{x}}) = \langle\Phi(\tilde{\mathbf{x}})| \mathcal{E}_m(\Theta_{opt}) |\Phi(\tilde{\mathbf{x}})\rangle$ is the probability of obtaining the outcome m from performing POVMs on $|\Phi(\tilde{\mathbf{x}})\rangle$. For a test data set with known labels $\tilde{\mathcal{D}} = \{(\tilde{\mathbf{x}}_n,\tilde{y}_n)|\, \tilde{\mathbf{x}}_n\in\mathbb{R}^4, \tilde{y}_n\in\{0,1,2\}, \forall n\in\mathbb{N}\}$, one can examine how many labels matches the predictions of the quantum model by evaluating the accuracy $\sum_n\delta_{\tilde{m}(\tilde{\mathbf{x}}_n),\tilde{y}_n}/|\tilde{\mathcal{D}}|$. In the five-fold cross-validation, the mean accuracy is 0.893 for the single-qubit POVM circuit and 0.920 for the two-qubit POVM circuit.

The second measure is the ROC curve and the area under the curve (AUC). Although this measure is designed for binary classification, we can apply it to multi-class classification through a one-vs-rest scheme. After formulating the multi-class classification to the one-vs-rest binary classification, the test data is assigned a label whose probability is higher than a predetermined threshold value. Then the true-positive rate and the false-positive rate can be evaluated at the threshold. The ROC curves are then drawn by shifting the threshold value. The farther the ROC curve from the diagonal line the better the performance of the classifier. Therefore, the performance of the classifier can be quantified by the area under the ROC curve (AUC), which ranges from 0.5 to 1, and the larger the AUC of ROC the better the performance of the classifier. The mean ROC curves of the VQSD obtained from the five-fold cross-validation are shown in figure 6. There are three ROC curves in each plot corresponding to three different combination of the one-vs-rest binary classification. The VQSD classifiers with single-qubit and two-qubit POVM circuits both show excellent classification performance as the mean AUC ranges from 0.97 to 1 with an average of 0.985.