Transforming two-dimensional tensor networks into quantum circuits for supervised learning

TN is a framework that approximates higher-order tensors using the contraction of lower-order tensor. In a tensor diagram, each index of a tensor is represented by a line, and the tensor itself is depicted as a node. The edges between nodes represent the contraction of virtual indices, whose dimension is called bond (or virtual) dimension $D$ and generally bounded as $D \unicode{x2A7D} \chi$ to reduce the computational cost. TN contracts in a certain direction, thus forming some specific TN architectures.

MPS can effectively describe the ground state in one-dimensional quantum spin system, which is due to the fact that MPS can fully capture local entanglement characteristics in the system. As shown in figure 1(a), an MPS with $N$ nodes is described as

$$\begin{align}|{\psi _{MPS}}\rangle = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {\left(A_{{a_1}}^{{s_1}}A_{{a_1}{a_2}}^{{s_2}} \cdots A_{{a_{N - 1}}}^{{s_N}}\right)} } \mathop \otimes \limits_{i = 1}^N |{s_i}\rangle \end{align} \tag{ 1 }$$

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where each third-order tensor $A_{{a_{n - 1}}{a_n}}^{{s_n}}$ has ${a_{n - 1}}$ and ${a_n}$ which are related to the left and right virtual indices and has a physical index ${s_n}$. All the virtual indices are contracted to form a tensor with $N$ physical indices, so it can be used to describe the 1D quantum-many body systems.

However, the applicability of MPS is difficult to extend to higher-dimensional systems. The same problem also occurs in machine learning models based on MPS and its quantum version. For example, when the model involves processing two-dimensional data such as natural images, as shown in figure 1(b), pixels are rearranged into one-dimensional vectors in a certain order to meet the requirements of MPS, such processing loses the original correlation information between pixels. To solve this problem, a natural idea is to construct models using 2D TNs with the same geometric structure as data, the most typical of which is the PEPS [15, 16], which is the high-dimensional generalization of MPS. A PEPS with open boundary on a two-dimensional lattice of size $H \times V = 4 \times 4$ shown in figure 1(c) can be written as

$$\begin{align}|{\psi _{PEPS}}\rangle = \sum\limits_{\left\{ s\right\} } {\mathcal{F}\left(M_{\left(1,1\right)}^{{s_{\left(1,1\right)}}},M_{\left(1,2\right)}^{{s_{\left(1,2\right)}}} \cdots ,M_{\left(H,V\right)}^{{s_{\left(H,V\right)}}}\right)\mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{\left(i,j\right)}}\rangle } \end{align} \tag{ 2 }$$

where $M_{(m,n)}^{{s_{(m,n)}}}$ is a fifth-order tensor with four virtual indices $\alpha ,\beta ,\gamma ,\varepsilon$ (corresponding to the left, up, right, down direction) connecting neighbors and a physical index $s$. $\mathcal{F}$ denotes the contraction of virtual indices of all tensors. However, it is more difficult to simulate PEPS than MPS in both classical or quantum way [10, 28], and the expansion of model is limited by computational cost. Here, we introduce other types of 2D TN states, including SBS, EPS and isoTNS, which are described as PEPS subclasses with certain limitations [25, 29]. They can reduce computational cost and the quantum circuit ansätze inspired by them can efficiently describe 2D TN.

A SBS [21] defines a set $S$ of ordered strings, and a one-dimensional MPS for each string. By covering all nodes on the two-dimensional lattice with overlapping MPS, the correlations of two-dimensional nodes are obtained while retaining the advantages of one-dimensional structure. The description power of SBS depends on the choice of strings. In image learning tasks, two-dimensional lattice is usually covered by strings on multiple rows and columns shown in figure 1(d), and overlapping snake strings can also work, which is called Snake-SBS.

Another efficient method for 2D TN is EPS, or called correlator product states. The basic approach of the EPS is to describe a wave function by product of multiple sub-plaquette tensors in a 2D lattice, and to describe the short-range correlation by overlap of sub-plaquettes whose wave functions are constructed exactly [24]. An EPS composed of $P = 9$ plaquettes with 4 physical indices is shown in figure 1(e). The description power of EPS depends on the size of sub-plaquettes. Larger plaquettes bring higher accuracy but also increase computational cost.

An isoTNS describes a TN state with isometry conditions, which allows the two-dimensional network to be reduced to the canonical form of 1D MPS when contracting rows and columns [25]. For 2D isoTNS, all tensors that make up isoTNS are isometries. The physical index of each tensor has an incoming arrow, and the virtual indices have incoming and outgoing arrows. All arrows point in the direction of the center node (called orthogonality center) or the rows and columns where it is located (called orthogonality hypersurfaces), and the directions of theses arrows are opposite to those of tensor contractions. When the incoming virtual indices and physical index of these tensors contract with the corresponding indices of their complex conjugate tensors, the remaining indices yield the identity. For example, the tensor outside the orthogonality hypersurfaces in figure 1(f) satisfies

$$\begin{align}\sum\limits_{s\gamma \varepsilon } {A_{\alpha \beta \gamma \varepsilon }^s} A_{{\alpha ^{\prime} }{\beta ^{\prime} }\gamma \varepsilon }^{\dagger s} = {I_{\alpha {\alpha ^{\prime} }}}{I_{\beta {\beta ^{\prime} }}}\end{align} \tag{ 3 }$$

where $I$ is the identity and the tensor on the orthogonality hypersurfaces satisfies

$$\begin{align}\mathop {\mathop \sum \nolimits }\limits_{s\beta \gamma \varepsilon } A_{\alpha \beta \gamma \varepsilon }^sA_{\alpha ^{\prime}\beta \gamma \varepsilon }^{\dagger s} = {I_{\alpha \alpha ^{\prime}}}.\end{align} \tag{ 4 }$$

IsoTNS can be more easily implemented on quantum circuit by efficiently moving the orthogonality center to the corner of two-dimensional lattice [25]. Note that all of the above TNs satisfy equation (2). They are applicable to classical computing, and they inspire circuit ansätze in QCL. The most obvious difference is that the quantum gates used to describe the states on quantum circuits are unitary, meaning that their corresponding classical 2D TN tensors are also required to be unitary.

We introduced three classical 2D TNs in the previous section, since our goal is to apply 2D TNs to QCL, the first thing we need to do is to encode them into quantum circuits. In this section we propose 3 different 2D TNQC ansätze and we show how they are generated from classical TNs with rigorous mathematical proofs.

2.2.1. Circuit ansätze

QMPS: Before introducing 2D TNQC ansätze, we first review the 1D MPS circuit ansatz. As shown in figure 2(a), a 16-qubit QMPS can be implemented by sequentially applying a two-qubit unitary to adjacent qubits [30]. Each two-qubit unitary entangles the last qubit obtained from a previous unitary with the next one. Starting from the encoded product state $\mathop \otimes \limits_{i = 1}^N |{\phi _i}\rangle$, QMPS returns a state

$$\begin{align}|{\psi _{QMPS}}\rangle = U_{N - 1}^{\left[2\right]}U_{N - 2}^{\left[2\right]} \cdots U_2^{\left[2\right]}U_1^{\left[2\right]}\mathop \otimes \limits_{i = 1}^N |{\phi _i}\rangle \end{align} \tag{ 5 }$$

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where $U_n^{[2]},n = \{ 1,2,\ldots,N - 2,N - 1\}$ denotes a two-qubit unitary acts on the $n$th and $(n + 1)$th qubit.

The structure of QMPS shows that entanglement only occurs between one-dimensional adjacent qubits, while 2D TNQC ansätze extend it to two dimensions.

QSBS: We construct a 2D lattice of qubits of size $N = H \times V$ to demonstrate the spatial structure of 2D TNQC. The first ansatz shown in figure 2(b) is inspired by SBS (called QSBS), which consists of multiple QMPSs. QSBS first applies a set of vertically oriented QMPSs on different columns in parallel on the lattice, followed by a set of horizontally oriented QMPSs on all rows in the next layer. The QMPSs corresponding to the strings in the same direction are set as canonical form with the same orthogonal direction to generate linear sequential circuits [31]. So QSBS returns a state

$$\begin{align}|{\psi _{QSBS}}\rangle = \prod\limits_{m = 1}^H {U_{\left(m,V - 1\right)}^{\left[2\right]} \cdots U_{\left(m,1\right)}^{\left[2\right]}} \prod\limits_{n = 1}^V {U_{\left(H - 1,n\right)}^{\left[2\right]} \cdots U_{\left(1,n\right)}^{\left[2\right]}} \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \end{align} \tag{ 6 }$$

where $U_{(m,n)}^{[2]}$ is a two-qubit unitary acting on the qubits at $(m,n)$ and $(m + 1,n)$ for vertical QMPSs or $(m,n)$ and $(m,n + 1)$ for horizontal QMPSs of the 2D lattice.

QEPS: The EPS-inspired quantum circuit (called QEPS) sequentially applies a unitary $U_{(m,n)}^{[4]}$ to the four qubits of each $2 \times 2$ plaquette from $(m,n)$ to $(m + 1,n + 1)$ on the lattice. It returns a state

$$\begin{align}|{\psi _{QEPS}}\rangle = \prod\limits_{\tau \in \overrightarrow {\mathcal{S}} } {U_\tau ^{\left[4\right]}} \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \end{align} \tag{ 7 }$$

where $\tau = (m,n) \in \overrightarrow {\mathcal{S}}$, $\overrightarrow {\mathcal{S}}$ denotes the order of unitary gates. Figure 2(c) shows a QEPS consisting of 9 four-qubit unitaries acting on different plaquettes, and the unitaries are applied sequentially in the order ${\mathcal{S}} = ((1,1),(1,2),\ldots(3,2),(3,3))$.

QisoTNS: The isoTNS-inspired quantum circuit (called QisoTNS) starts from the corner of the 2D lattice and applies a three-qubit unitary $U_{(1,1)}^{[3]}$ on a qubit group including qubits located at $(1,1)$, $(2,1)$ and $(2,2)$. Then in the next layer, we apply a three-qubit unitary $U_{(m,n)}^{[3]}$ acting on $(m,n)$, $(m + 1,n)$ and $(m + 1,n + 1)$ in parallel at each location offset by +1 row or column relative to the qubit group in the previous layer. This process is repeated up to the boundary of the 2D lattice. A qubit group beyond the rightmost boundary of the lattice only applies a two-qubit unitary $U_{(m,n)}^{[2]}$ on the qubits located at $(m,n)$ and $(m + 1,n)$. Such a design follows the tensor contraction direction and satisfies isometry conditions in isoTNS. A QisoTNS can be described as

$$\begin{align}|{\psi _{QisoTNS}}\rangle = \prod\limits_{l \in \overrightarrow {\mathcal{L}} } {\prod\limits_{\tau \in {\mathcal{S}_l}} {{U_\tau }} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \end{align} \tag{ 8 }$$

where $l$ represents the layer number of circuit, $\overrightarrow {\mathcal{L}}$ is an ordered set of layers ranging from 1 to $H + V - 2$, and these layers are applied sequentially. ${\mathcal{S}_l}$ represents the unitaries applied in the $l$th layer, which contains $\text{min} (l,H + V + 1 - l)$ unitaries of three-qubit or two-qubit, and these unitaries are applied in parallel. In total, a QisoTNS circuit contains $(H - 1) \times V$ unitaries. For example, a 16-qubit QisoTNS ansatz shown in figure 2(d) has 6 layers, namely, ${\{ U_{(1,1)}^{[3]}\} _1}$, ${\{ U_{(2,1)}^{[3]},U_{(1,2)}^{[3]}\} _2}$, ${\{ U_{(3,1)}^{[3]},U_{(2,2)}^{[3]},U_{(1,3)}^{[3]}\} _3}$, ${\{ U_{(3,2)}^{[3]},U_{(2,3)}^{[3]},U_{(1,4)}^{[2]}\} _4}$, ${\{ U_{(3,3)}^{[3]},U_{(2,4)}^{[2]}\} _5}$, ${\{ U_{(3,4)}^{[2]}\} _6}$.

These circuit ansätze are generated from TNs. Recalling the classical TNs, all the indices are set to 2 so that each tensor that makes up them is described by a unitary gate on the quantum circuit. Tensors in MPS and isoTNS first needs to balance the number of incoming and outgoing indices due to unitary gate constraints, then unitary gates are applied in a specific order according to the adjusted directed MPS and isoTNS diagrams to generate QMPS and QisoTNS having the same tensor diagrams. SBS and EPS provide specific methods to represent a 2D TN by overlapping local tensors in a lattice, which allows us to construct quantum circuits with the same rules, so QSBS and QEPS are inspired by the generating rules rather than specific states, and their corresponding TNs can be described by transforming unitaries back to tensors.

2.2.2. Mathematical proofs

Mathematical proof for QMPS: Now we further prove that ansätze introduced above are generated from TNs. First, a QMPS ansatz is a left (or right)-orthogonal MPS with $D = d = 2$, and each tensor of the MPS satisfies the left (or right)-orthogonal condition, i.e.,

$$\begin{align}\sum\limits_{s{a_n}} {A_{\left(n\right){a_{n - 1}}{a_n}}^s} A_{\left(n\right){a_{n - 1}}^{\prime} {a_n}}^{\dagger s} = {I_{{a_{n - 1}}{a_{n - 1}}^{\prime} }}.\end{align} \tag{ 9 }$$

While two-qubit unitary gates applied in a QMPS can be written as

$$\begin{align}\begin{gathered} U_1^{\left[2\right]} = \sum\limits_{{s_1},{a_1}} {B_{\left(1\right){a_1}{\delta _1}{\delta _2}}^{{s_1}}} |{s_1},{a_1}\rangle \langle {\delta _1},{\delta _2}| \hfill \\ U_n^{\left[2\right]} = \sum\limits_{{s_n},{a_n}} {B_{\left(n\right){a_{n - 1}}{a_n}{\delta _{n + 1}}}^{{s_n}}} |{s_n},{a_n}\rangle \langle {a_{n - 1}},{\delta _{n + 1}}| \hfill \\ U_{N - 1}^{\left[2\right]} = \sum\limits_{{s_{N - 1}}{s_N}} {B_{\left(N - 1\right){a_{N - 2}}{\delta _N}}^{{s_{N - 1}}{s_N}}} |{s_{N - 1}},{s_N}\rangle \langle {a_{N - 2}},{\delta _N}| \hfill \\ \end{gathered} .\end{align} \tag{ 10 }$$

with $1 < n < N - 1$ and $\text{dim} ({a_n}) = \text{dim} ({s_n}) = \text{dim} ({\delta _n}) = 2$ where ${\delta _n}$ is the index connecting $|{\phi _n}\rangle$. Since ${U_n}$ is a unitary, it has $U_n^\dagger {U_n} = {I_{{a_{n - 1}}a_{n - 1}^{\prime} }}{I_{{\delta _{n + 1}}\delta _{n + 1}^{\prime} }}$. Then according to equation (10) and $|{\phi _{n + 1}}\rangle = \sum\nolimits_{{\delta _{n + 1}}} {{\nu _{(n + 1){\delta _{n + 1}}}}} |{\delta _{n + 1}}\rangle$, we have

$$\begin{align}\sum\limits_{{s_n}{a_n}} {\sum\limits_{{\delta _{n + 1}}} {\nu _{{\delta _{n + 1}}}^\dagger B_{\left(n\right)a_{n - 1}^{\prime} {a_n}\delta _{n + 1}^{\prime} }^{\dagger {s_n}}B_{\left(n\right){a_{n - 1}}{a_n}{\delta _{n + 1}}}^{{s_n}}{\nu _{{\delta _{n + 1}}}}} } = {I_{{a_{n - 1}}a_{n - 1}^{\prime} }}\end{align} \tag{ 11 }$$

as shown in figure 3(a) with a tensor diagram. Let $\sum\nolimits_{{\delta _{n + 1}}} {B_{(n){a_{n - 1}}{a_n}{\delta _{n + 1}}}^{{s_n}}{\nu _{(n + 1){\delta _{n + 1}}}}} = A_{(n){a_{n - 1}}{a_n}}^{{s_n}}$, then tensor $A_{(n){a_{n - 1}}{a_n}}^{{s_n}}$ satisfies the isometry condition equation (9). Therefore, each unitary gate can be identified as an MPS tensor. Combining equation (5), we get

$$\begin{align}\begin{aligned} |{\psi _{QMPS}}\rangle & = U_{N - 1}^{\left[2\right]}U_{N - 2}^{\left[2\right]} \cdots U_2^{\left[2\right]}U_1^{\left[2\right]}\mathop \otimes \limits_{i = 1}^N |{\phi _i}\rangle \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {B_{\left(N - 1\right){a_{N - 2}}{\delta _N}}^{{s_{N - 1}}{s_N}} \cdots B_{\left(1\right){a_1}{\delta _1}{\delta _2}}^{{s_1}}|{s_1} \cdots {s_N}\rangle \langle {\delta _1} \cdots {\delta _N}|} } \mathop \otimes \limits_{i = 1}^N \left(\sum\limits_{{\delta _i}} {{\nu _{\left(i\right){\delta _i}}}} |{\delta _i}\rangle \right) \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {\sum\limits_{\left\{ \delta \right\} } {B_{\left(N - 1\right){a_{N - 2}}{\delta _N}}^{{s_{N - 1}}{s_N}}{\nu _{\left(N\right){\delta _N}}} \cdots B_{\left(1\right){a_1}{\delta _1}{\delta _2}}^{{s_1}}{\nu _{\left(2\right){\delta _2}}}{\nu _{\left(1\right){\delta _1}}}\mathop \otimes \limits_{i = 1}^N |{s_i}\rangle } } } \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {A_{\left(N - 1\right){a_{N - 2}}}^{s_{N - 1}^{\prime} } \cdots A_{\left(1\right){a_1}}^{{s_1}}\mathop \otimes \limits_{i = 1}^N |{s_i}\rangle } } = |{\psi _{MPS}}\rangle \\ \end{aligned} \end{align} \tag{ 12 }$$

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where $s_{N - 1}^{\prime} = {s_{N - 1}}{s_N}$. This means that a QMPS is generated from a special MPS. For example, as shown in figure 3(b), a QMPS with $n$ unitaries is generated from a left-orthogonal MPS having $n$ tensors. Each tensor has a physical index of dimension $2$, except for the last tensor which has a physical index of dimension ${2^2}$, this can be seen as a contraction between ${A_{N - 1}}$ and an additional tensor [11]. The dimensions of all indices are 2. The temporal sequence of applying unitaries on quantum circuit is the opposite of the arrows' directions.

Mathematical proof for QSBS: Now back to 2D TNQC ansätze. The unitary gates applied in QSBS are the same as those in QMPS. So according to equation (6), we can get

$$\begin{align} \left| {{\psi _{QSBS}}} \right\rangle & = \prod\limits_{m = 1}^H {U_{\left(m,V - 1\right)}^{\left[2\right]} \cdots U_{\left(m,1\right)}^{\left[2\right]}} \prod\limits_{n = 1}^V {U_{\left(H - 1,n\right)}^{\left[2\right]} \cdots U_{\left(1,n\right)}^{\left[2\right]}} \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \nonumber\\ & = \prod\limits_{m = 1}^H {\sum\limits_{\left\{ {s_{\left(m,\right)}}\right\} } {\sum\limits_{\left\{ a_{\left(m,\right)}^{\prime} \right\} } {B_{\left(m,V - 1\right)a_{\left(m,V - 2\right)}^{horiz}\delta _{\left(m,V\right)}^{\prime} }^{{s_{\left(m,V - 1\right)}}{s_{\left(m,V\right)}}} \cdots B_{\left(m,1\right)a_{\left(m,1\right)}^{horiz}\delta _{\left(m,1\right)}^{\prime} \delta _{\left(m,2\right)}^{\prime} }^{{s_{\left(m,1\right)}}}\mathop \otimes \limits_{j = 1}^V |{s_{\left(m,j\right)}}\rangle \mathop \otimes \limits_{j = 1}^V \langle \delta _{\left(m,j\right)}^{\prime} |} } }\nonumber\\ & \quad \times \prod\limits_{n = 1}^V {\sum\limits_{\left\{ \delta _{\left(,n\right)}^{\prime} \right\} } {\sum\limits_{\left\{ {a_{\left(,n\right)}}\right\} } {A_{\left(H - 1,n\right)a_{\left(H - 2,n\right)}^{vert}}^{\delta _{\left(H - 1,n\right)}^{\prime} \delta _{(H,n)}^{\prime} } \cdots A_{(1,n)a_{(1,n)}^{vert}}^{\delta _{(1,n)}^{\prime} }} } } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |\delta _{(i,j)}^{\prime} \rangle \nonumber\\ & = \sum\limits_{\left\{ s\right\} } \sum\limits_{\left\{ a_{(m,)}^{\prime} \right\} } \sum\limits_{\left\{ {a_{(,n)}}\right\} } \sum\limits_{\left\{ {\delta ^{\prime} }\right\} } B_{(H - 1,V - 1)a_{(H - 1,V - 2)}^{horiz}\delta _{(H - 1,V)}^{\prime} }^{{s_{(H - 1,V - 1)}}{s_{(H - 1,V)}}}B_{(H,V - 1)a_{(H,V - 2)}^{horiz}\delta _{(H,V)}^{\prime} }^{{s_{(H,V - 1)}}{s_{(H,V)}}}A_{(H - 1,V)a_{(H - 2,V)}^{vert}}^{\delta _{(H - 1,V)}^{\prime} \delta _{(H,V)}^{\prime} }, \ldots ,\nonumber\\ & \quad \times B_{(1,1)a_{(1,1)}^{horiz}\delta _{(1,1)}^{\prime} \delta _{(1,2)}^{\prime} }^{{s_{(1,1)}}}A_{(1,1)a_{(1,1)}^{vert}}^{\delta _{(1,1)}^{\prime} }A_{(1,2)a_{(1,2)}^{vert}}^{\delta _{(1,2)}^{\prime} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{(i,j)}}\rangle\nonumber\\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a_{\left(m,\right)}^{\prime} \right\} } {\sum\limits_{\left\{ {a_{\left(,n\right)}}\right\} } {M_{\left(H - 1,V - 1\right)a_{\left(H - 1,V - 2\right)}^{horiz}a_{\left(H,V - 2\right)}^{horiz}a_{\left(H - 2,V\right)}^{vert}}^{{s_{\left(H - 1,V - 1\right)}}{s_{\left(H - 1,V\right)}}{s_{\left(H,V - 1\right)}}{s_{\left(H,V\right)}}}, \ldots ,M_{a_{\left(1,1\right)}^{horiz}a_{\left(1,1\right)}^{vert}a_{\left(1,2\right)}^{vert}}^{{s_{\left(1,1\right)}}}} } } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{\left(i,j\right)}}\rangle \nonumber\\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a_{\left(m,\right)}^{\prime} \right\} } {\sum\limits_{\left\{ {a_{\left(,n\right)}}\right\} } {M_{\left(H - 1,V - 1\right)a_{\left(H - 1,V - 2\right)}^{^{\prime} horiz}a_{\left(H - 2,V\right)}^{vert}}^{s_{\left(H - 1,V - 1\right)}^{\prime} }, \ldots ,M_{a_{\left(1,1\right)}^{horiz}a_{\left(1,1\right)}^{^{\prime} vert}}^{{s_{\left(1,1\right)}}}} } } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{(i,j)}}\rangle = |{\psi _{PEPS}}\rangle\end{align} \tag{ 13 }$$

where ${\delta ^{\prime} }$ denes the indices of the first layer tensors connected with the second layer tensors. ${a^{horiz}}$ and ${a^{vert}}$ are virtual indexes in the horizontal and vertical directions. $s_{(H - 1,V - 1)}^{\prime} = {s_{(H - 1,V - 1)}}{s_{(H - 1,V)}}{s_{(H,V - 1)}}{s_{(H,V)}}$ and $a_{(H - 1,V - 2)}^{^{\prime} horiz} = a_{(H - 1,V - 2)}^{horiz}a_{(H,V - 2)}^{horiz}$ mean that tensor ${M_{(H - 1,V - 1)}}$ has a physical index of dimension ${2^4}$, and it has a virtual index of dimension ${2^2}$ in the horizontal direction. Our proposed 16-qubit QSBS using the same generating rule as SBS but is equivalent to the TN shown in figure 3(c), which indicates that the SBS-inspired circuit is a special PEPS with non-uniform virtual and physical bond dimensions. Moreover, $\sum\nolimits_{\delta _{(i,j)}^{\prime} } {B_{(i,j)a_{(i,j - 1)}^{\prime} a_{(i,j)}^{\prime} \delta _{(i,j + 1)}^{\prime} }^{{s_{(i,j)}}}A_{(i,j + 1){a_{(i - 1,j + 1)}}{a_{(i,j + 1)}}}^{\delta _{(i,j + 1)}^{\prime} }} = M_{(i,j)a_{(i,j - 1)}^{\prime} a_{(i,j)}^{\prime} {a_{(i - 1,j + 1)}}{a_{(i,j + 1)}}}^{{s_{(i,j)}}}$ means that two unitaries (three or four in some cases) from two layers create a PEPS tensor. In addition, since indices have directions, QSBS can actually be further classified as the isoTNS.

Mathematical proof for QEPS: The four-qubit unitary gate used in QEPS can be written as

$$\begin{align}U_\tau ^{\left[4\right]} = \sum\limits_{\left\{ {s^\tau }\right\} ,\left\{ {a^\tau }\right\} } {T_{\left(\tau \right)\left\{ {\delta ^\tau }\right\} \left\{ {a^\tau }\right\} }^{\left\{ {s^\tau }\right\} }} |\left\{ {s^\tau }\right\} ,\left\{ a_\mu ^\tau \right\} \rangle \langle \left\{ a_\eta ^\tau \right\} ,\left\{ {\delta ^\tau }\right\} |\end{align} \tag{ 14 }$$

where $T_{(\tau )\{ {\delta ^\tau }\} \{ {a^\tau }\} }^{\{ {s^\tau }\} }$ is an eighth-order tensor (i.e. a ${2^4} \times {2^4}$ matrix), and $\{ {s^\tau }\} ,\{ {a^\tau }\} ,\{ {\delta ^\tau }\}$ are the physical indices, virtual indices, and indices connecting encoded states $|\phi \rangle$ of this tensor, $\{ a_\mu ^\tau \}$ and $\{ a_\eta ^\tau \}$ are the set of its incoming and outgoing virtual indexes. According to equation (7), we have

$$\begin{align}\begin{aligned} |{\psi _{QEPS}}\rangle & = \prod\limits_{\tau \in \overrightarrow {\mathcal{S}} } {U_\tau ^{\left[4\right]}} \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \\ & = \prod\limits_{\tau \in \overrightarrow {\mathcal{S}} } {\sum\limits_{\left\{ {s^\tau }\right\} ,\left\{ {a^\tau }\right\} } {T_{\left(\tau \right)\left\{ {\delta ^\tau }\right\} \left\{ {a^\tau }\right\} }^{\left\{ {s^\tau }\right\} }} |\left\{ {s^\tau }\right\} ,\left\{ a_\mu ^\tau \right\} \rangle \langle \left\{ a_\eta ^\tau \right\} ,\left\{ {\delta ^\tau }\right\} |\mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V \sum\limits_{\left\{ \delta \right\} } {{\nu _{\left(i,j\right){\delta _{\left(i,j\right)}}}}} |{\delta _{\left(i,j\right)}}\rangle } \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {M_{\left(H - 1,V - 1\right)\left\{ {a^{\left(H - 1,V - 1\right)}}\right\} }^{\left\{ {s^{\left(H - 1,V - 1\right)}}\right\} }, \ldots ,M_{\left(1,1\right)\left\{ {a^{\left(1,1\right)}}\right\} }^{\left\{ {s^{\left(1,1\right)}}\right\} }} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{\left(i,j\right)}}\rangle \\ \end{aligned} \end{align} \tag{ 15 }$$

with $\sum\nolimits_{\{ {\delta ^\tau }\} } {T_{\{ {\delta ^\tau }\} \{ {a^\tau }\} }^{\{ {s^\tau }\} }\{ {\nu _\delta }\} } = M_{\{ {a^\tau }\} }^{\{ {s^\tau }\} }$, which shows that each four-qubit unitary gate creates a tensor on the 2D TN. However, the EPS-inspired circuit is not a PEPS, because, as shown in figure 3(d) where lies the corresponding tensor diagram of QEPS, its virtual bonds are not always between adjacent tensors, QEPS is a special 2D TN with non-vertical connections.

Mathematical proof for QisoTNS: Now moving to QisoTNS, a three-qubit unitary gate outside the orthogonality hypersurfaces is described as

$$\begin{align}U_{\left(m,n\right)}^{\left[3\right]} = \sum\limits_{s\gamma \varepsilon } {D_{\left(m,n\right)\alpha \beta \gamma \varepsilon \delta }^s} |s,\varepsilon ,\gamma \rangle \langle \beta ,\alpha ,\delta |\end{align} \tag{ 16 }$$

where $D_{(m,n)\delta \alpha \beta \gamma \varepsilon }^s$ is a sixth-order tensor (i.e. a ${2^3} \times {2^3}$ matrix), and $\alpha ,\beta ,\gamma ,\varepsilon$ are its virtual indices. Since $U_{(m,n)}^{\dagger [3]}U_{(m,n)}^{[3]} = {I_{\alpha {\alpha ^{\prime} }}}{I_{\beta {\beta ^{\prime} }}}{I_{\delta {\delta ^{\prime} }}}$, we can get

$$\begin{align}\sum\limits_{s\gamma \varepsilon } {\sum\limits_\delta {v_{\left(m + 1,n + 1\right)\delta }^\dagger D_{\left(m,n\right){\alpha ^{\prime} }{\beta ^{\prime} }\gamma \varepsilon \delta }^{\dagger s}D_{\left(m,n\right)\alpha \beta \gamma \varepsilon \delta }^s{v_{\left(m + 1,n + 1\right)\delta }}} } = {I_{\alpha {\alpha ^{\prime} }}}{I_{\beta {\beta ^{\prime} }}}.\end{align} \tag{ 17 }$$

Let $\sum\nolimits_\delta {D_{(m,n)\alpha \beta \gamma \varepsilon \delta }^s{v_{(m + 1,n + 1)\delta }}} = M_{(m,n)\alpha \beta \gamma \varepsilon }^s$, then tensor $M_{(m,n)\alpha \beta \gamma \varepsilon }^s$ satisfies the isometry condition equation (3), which is shown in figure 3(e). For a three-qubit unitary gate on the orthogonality hypersurfaces, $\sum\nolimits_{{\delta _1}{\delta _2}} {D_{\alpha \gamma \varepsilon {\delta _1}{\delta _2}}^s{v_{{\delta _1}}}{v_{{\delta _2}}}} = M_{\alpha \gamma \varepsilon }^s$, and $M_{\alpha \gamma \varepsilon }^s$ satisfies the isometry condition equation (4). In addition, the two-qubit unitary gate in QisoTNS has similar properties to that in QMPS. These lead to the fact that each unitary gate can be viewed as an isoTNS tensor. Combined with equation (8), we have

$$\begin{align}\begin{aligned} |{\psi _{QisoTNS}}\rangle & = \prod\limits_{l \in \overrightarrow {\mathcal{L}} } {\prod\limits_{\tau \in {{\mathcal{S}}_l}} {{U_\tau }} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{\phi _{\left(i,j\right)}}\rangle \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {M_{\left(H - 1,V\right)\left\{ {a^{\left(H - 1,V\right)}}\right\} }^{{s_{\left(H - 1,V\right)}}{s_{\left(H,V\right)}}}, \ldots ,M_{\left(1,1\right)\left\{ {a^{\left(1,1\right)}}\right\} }^{{s_{\left(1,1\right)}}}} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{\left(i,j\right)}}\rangle \\ & = \sum\limits_{\left\{ s\right\} } {\sum\limits_{\left\{ a\right\} } {M_{\left(H - 1,V\right)\left\{ {a^{\left(H - 1,V\right)}}\right\} }^{s_{\left(H - 1,V\right)}^{\prime} }, \ldots ,M_{\left(1,1\right)\left\{ {a^{\left(1,1\right)}}\right\} }^{{s_{\left(1,1\right)}}}} } \mathop \otimes \limits_{i = 1}^H \mathop \otimes \limits_{j = 1}^V |{s_{\left(i,j\right)}}\rangle = |{\psi _{PEPS}}\rangle \\ \end{aligned} \end{align} \tag{ 18 }$$

where $s_{(H - 1,n)}^{\prime} = {s_{(H - 1,n)}}{s_{(H,n)}}$ means that each tensor in the bottom row has a physical index of dimension ${2^2}$. And a QisoTNS is generated from a special PEPS satisfying the isometry conditions, namely, an isoTNS. The 16-qubit QisoTNS circuit is equivalent to an isoTNS with 12 tensors as shown in figure 3(f). Each tensor has a physical index of dimension 2, except tensors in the bottom row, which is similar to the QMPS. The green lines indicate indices connected to the encoded states. Such directed tensor diagram is adapted from the original isoTNS. As mentioned before, to construct a QisoTNS equivalent to isoTNS, each tensor is connected by a different number of green lines to balance indices before generating QisoTNS. In addition, making the upper-left tensor the orthogonality center is easier for generating QisoTNS.

These three types of ansätze are generated from special 2D TNs. QSBS and QisoTNS belong to the same type of TN, and QisoTNS has more tensors and more uniform dimensions of indices. Although QSBS is also a 2D TN, not all its connections are between neighbors. Their TN structures may affect the classification performance.

Note that we only show single-layer circuit ansätze with mathematical proofs in this section, which are formally equivalent to TNs with $D = d = 2$. However, when constructing QCL models, the models can follow the proposal of [11] to use multi-layer circuit ansätze to construct TNs with a larger bond dimension $D$. These circuit structures can also be used in encoders, as we will mention in section 2.4.

Our goal is to apply 2D TN to QCL, now that 2D TN has been transformed into 2D TNQC ansätze, the next thing that needs to be done is to apply these ansätze to QCL, so here we propose a TNQC supervised learning framework shifting the perspective from classical TN to quantum one.

Typically, a classification task learns from a dataset of existing classification labels and establishes a mapping from the input data space to the classification label space. Here, we consider the training data with an $N$-dimensional real number vector ${\mathbf{x}} = ({x_1},{x_2},\ldots,{x_N})$ from a grayscale image, where ${x_i} \in [0,1]$, which represents the normalized value of this pixel. It first needs to map data vectors to a high-dimensional space. A typical way in classical TNML is to use the local feature map $\phi ({x_i}) = \text{cos} (\frac{\pi }{2}{x_i})|0\rangle + \text{sin} (\frac{\pi }{2}{x_i})|1\rangle$ for each element of a vector, then we obtain a global feature map $\Phi ({\mathbf{x}}) = \mathop \otimes \limits_{i = 1}^N {\text{ }}\phi ({x_i})$. This can be implemented on quantum circuit by applying a single-qubit $RY$ rotation on each of the $N$ qubits, which is called AE. Starting from the product state $|0{\rangle ^{ \otimes N}}$, the quantum state after feature mapping is

$$\begin{align}|\Phi \rangle = \mathop \otimes \limits_{i = 1}^N {\text{ }}RY\left({x_i}\right)|0\rangle .\end{align} \tag{ 19 }$$

In order to meet the requirements of the geometric position of a 2D TN, the encoded (or mapped) data is placed on a two-dimensional lattice according to the original positions of pixels, which is expressed as the product of $N$ order-1 tensors.

In TNML, the data tensors after feature mapping are multiplied by a $(N + 1)$-order weight tensor ${W^l}$ containing the label index $l$. The next step is to decompose ${W^l}$ into TN form and optimize the tensor by TN method [8, 17]. Finally, we choose $l$ for which $f({\mathbf{x}}) = {W^l}\Phi ({\mathbf{x}})$ is largest as the label of input ${\mathbf{x}}$. While in our TN-inspired QCL, the weight tensor is constructed by a TN-inspired unitary ${U_{TN}}$ containing trainable parameters, which makes the weight tensor actually a $2N$-order tensor. Then we predict labels based on measurement results of quantum circuit. Usually, the measurement operator set $\{ {\mathcal{M}_i}\}$ with completeness is used for measurement, the index $i$ represents the possible results of measurement. These operators act on the state space of the system to be measured, so a TN-inspired quantum classifier can be interpreted as a contraction of input data tensors, weight tensor, measurement operator tensor and their corresponding conjugate transpose tensor from the TN perspective. After measurement, the circuit will output the probabilities of ${2^N}$ different results, the probability of the result $i$ is

$$\begin{align}{p_i} = \langle \Phi |\mathcal{U}_{TN}^\dagger \left({U_j}\left({\theta _k}\right)\right){\mathcal{M}_i}{\mathcal{U}_{TN}}\left({U_j}\left({\theta _k}\right)\right)|\Phi \rangle \end{align} \tag{ 20 }$$

where $\mathcal{U}_{TN}^{}({U_j}({\theta _k}))$ represents the quantum circuit composed of a set of unitaries ${U_j}$ and their trainable parameters ${\theta _k}$, and the self-adjoint measurement operator ${\mathcal{M}_i}$ is the feature space projection of the observable $\sigma _z^{ \otimes N}$. In our models, given an input ${\mathbf{x}}$, the output logits $E$ can be obtained by taking a linear combination of the square roots of the probabilities of all results, specifically $E({\mathbf{x}},\theta ) = \sum\nolimits_{i = 0}^{{2^{N - 1}} - 1} {\sqrt {{p_i}} } - \sum\nolimits_{i = {2^{N - 1}}}^{{2^N} - 1} {\sqrt {{p_i}} }$, which differs from $f({\mathbf{x}})$ in TNML. The meaning of $E({\mathbf{x}},\theta )$ is to divide the observation bases equally into two sets and calculate the difference value between the sum of amplitude absolute values over the bases in the two sets. The result of the optimization will maximize one of the sets' sum of amplitude absolute values, which also means maximizing the sum of probabilities of this sets (Note that using the sum of amplitude absolute values makes the network converge more stably than that of probabilities). Thus $E({\mathbf{x}},\theta )$ only compares probabilities of two sets that the measurement results occur on. Using a finite number of shots (${\sqrt p _i} = 0$ for some $i$) could also normally evaluate $E({\mathbf{x}},\theta )$ and make it work on real quantum machines or simulators obtaining probabilities of results by repeated measurements. We show this in section 3.4.

Next, the optimization of weight ${U_{TN}}$ follows the QCL framework under the classical-quantum hybrid hardware architecture [1], it feeds the logits $E({\mathbf{x}},\theta )$ back to classical computer, and then use the adaptive moment estimation (Adam) optimization method to adjust the parameters $\theta$ to minimize the difference between the predicted labels and the real ones. In binary classification, the loss function is defined as

$$\begin{align}\mathcal{L} = - \frac{1}{{\left| D \right|}}\sum\limits_{{\mathbf{x}} \in D} {\text{log} } \frac{1}{{1 + \text{exp} \left( - E\left({\mathbf{x}},\theta \right)\right)}}\end{align} \tag{ 21 }$$

where $D$ represents a batch of data, logits $E({\mathbf{x}},\theta )$ is processed by the $sigmoid$ function and then used to calculate the cross entropy with the true labels. In multi-class classification, circuits of the same structure containing different parameters are repeated $k$ times to produce $k$ outputs. Its loss function is

$$\begin{align}\mathcal{L} = - \frac{1}{{\left| D \right|}}\sum\limits_{{\mathbf{x}} \in D} {\text{log} \frac{{\text{exp} {E_c}\left({\mathbf{x}},\theta \right)}}{{\sum\limits_{i = 1}^k {\text{exp} } {E_i}\left({\mathbf{x}},\theta \right)}}} \end{align} \tag{ 22 }$$

where $c$ is the sample label, ${E_i}({\mathbf{x}},\theta )$ is the output of the $i$th circuit, which is used to calculate the probability that the predicted label of ${\mathbf{x}}$ is $c$ through the $softmax$ function. Then we calculate the cross entropy of model's probabilities and data's labels.

Each step of training will return an average loss of the batch of data, and the gradient of loss can be calculated exactly using automatic differentiation software for QCL. The parameters are adjusted by Adam algorithm to generate a new quantum circuit for the next training step. This process is iterated to minimize loss and finally obtain the optimal parameters, which correspond to the optimal weight tensor in TNML. Finally, we predict the label of ${\mathbf{x}}$ as ${\text{sign}}(E({\mathbf{x}},\theta ) - \frac{1}{2})$ in binary classification. In multi-class classification, we choose $i$ for which ${E_i}({\mathbf{x}},\theta )$ is largest as the label of input ${\mathbf{x}}$.

In order to improve the accuracy performance of QCL, in addition to applying 2D TN to QCL, considering the effect of data encoders also matters. Although AE maps data to high-dimensional Hilbert space, the number of qubits it requires is equal to the classical data dimension, which makes it difficult to use complete data due to the constraints of a limited number of qubits on NISQ hardware. We have to use dimensionality reduction methods to pre-process data before training. Besides, the essence of AE is a product state feature map, which is untrainable. This may result in the mapped data not being in the optimal position in the Hilbert space for classification using a TNQC ansatz. A better choice is to use a variational feature map with trainable parameters.

To address above two issues, we propose a data encoding method based on convolutional feature map, called CE, which naturally conforms to the spatial arrangement of 2D TNQC ansätze. As shown in figure 4(a), the input image data with shape ${L_0} \times {L_0}$ is processed through convolutional and pooling layers to generate a third-order feature tensor with dimensions $L \times L \times c$, where $c$ represents the number of convolutional channels. Instead of flattening this feature tensor, we group the data at the same position in different channels into a $c$-dimensional vector to inherit the spatial features of the image. Therefore, it can be regarded as a product state in the space ${c^{L \times L}}$. Next, $c$ single qubit rotations selected from $\{ RY,RZ\}$ are applied on each of the $L \times L$ qubits, so the convolutional features are further transformed into a product state in the Hilbert space of dimension ${2^{L \times L}}$. For 2D image data, adopting a trainable CE method can effectively extract 2D features from multiple channels of the images, which can be continuously adjusted during training to better fit the 2D TNQC classifiers. Meanwhile, the CE reduces the dimensionality of original high-dimensional data to a size that can be used by QNN.

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The key to the effectiveness of CE lies in its parameters being learned jointly with those in TNQC during the training process, resulting in a hybrid classical-quantum architecture. It is more desirable to see QNNs perform without the aid of classical network layers, thus eliminating the possibility that the performance of the classifier comes from classical network layers. Following this idea, we introduce trainable parameters into AE, which is called VAE. As shown in figure 4(b), it begins with $|0\rangle$ on each qubit and sequentially applies two single-qubit rotations $RY$ with angles determined by the data ${x_i}$ and a trainable parameter $\theta$ respectively. This is equivalent to applying an $RY$ gate with a rotation angle ${x_i} + \theta$. Therefore, VAE yields a product state $|\psi \rangle = \mathop \otimes \limits_{i = 1}^N {\text{ }}RY({x_i} + \theta )|0\rangle$, and it is trained together with the 2D TNQC classifier. Note that scaling the normalized data ${\mathbf{x}} = ({x_1},{x_2},\ldots,{x_N})$ before training can further improve its adaptability.

Although VAE incorporates trainable parameters, it still requires $N$ qubits to encode $N$-dimensional data which makes it difficult to make full use of the data's power. Moreover, the linear transformation of VAE limits its adaptive capability. Therefore, we propose a VTNE method to act as a scalar between the number of qubits and the data dimension, and to perform a nonlinear transformation. Specifically, the original image is partitioned into several equally sized 2D image blocks. For each block, a 2D TNQC ansatz layer using special unitaries that we will introduce later is applied to encode it, and these encoding layers are applied to quantum circuit in sequence. This method ensures the expression of spatial correlation in image blocks, and we can also add an adjustment layer between two encoding layers to adjust the correlation between pixel blocks. The two types of layers apply the same kind of 2D TNQC ansatz but with different unitaries, and each of them can be viewed as a 2D TN operator. From this perspective, data of any dimension can be encoded as a TN state [11], and measurement operation on the circuit results in a contraction of the TN state and its conjugate transpose. This encoding method is presented in figure 4(c). In addition, when using VTNE, the encoder circuit architecture and the circuit of QCL ansatz used as trainable weights are inspired by the same TN (1D MPS or a 2D TN). Thus the circuits of VTNE-based QCL models always consist of multiple layers of QMPS or 2D TNQC, and a $L$-layer circuit forms a TN of $D = {2^L}$.

Now, we can integrate our proposed variational encoders and 2D TNQC ansätze on the basis of TNQC supervised learning framework to build several novel 2D TNQC classifiers, which we will show in the section 3.

There are three unitaries used to construct distinct circuit ansätze, and these ansätze can be employed for data encoding or describing weight tensor. For ease of distinction, a TNQC ansatz layer used to describe the weight tensor is referred to as a QNN layer. While in VTNE, TNQC ansätze are also used to construct encoding layers and adjustment layers. Here we introduce unitary implementation in different layers and TNQC ansätze. First, TNQC ansätze in the QNN layer and the adjustment layers use the same unitaries including

$$\begin{align}\begin{gathered} U_n^{\left[2\right]}(\theta ) = CNO{T_{n,n + 1}}\mathop \otimes \limits_{i = n}^{n + 1} {\text{(}}R{Y_i}({\theta _i})), \hfill \\ U_{(m,n)}^{\left[2\right]}(\theta ) = CNO{T_{(m,n),(m + 1,n)}}\mathop \otimes \limits_{i = m}^{m + 1} {\text{(}}R{Y_{(i,n)}}({\theta _{(i,n)}})){\text{ or }}CNO{T_{(m,n),(m,n + 1)}}\mathop \otimes \limits_{j = n}^{n + 1} {\text{(}}R{Y_{(m,j)}}({\theta _{(m,j)}})), \hfill \\ U_{(m,n)}^{\left[3\right]}(\theta ) = CNO{T_{(m + 1,n),(m + 1,n + 1)}}CNO{T_{(m,n),(m + 1,n)}}\mathop \otimes \limits_{(i,j) \in \left\{ (m,n),(m + 1,n),(m + 1,n + 1)\right\} } (R{Y_{(i,j)}}({\theta _{(i,j)}})), \hfill \\ U_{(m,n)}^{\left[4\right]}(\theta ) = \prod\limits_{i = m}^{m + 1} {CNO{T_{(i,n + m - i + 1),(i,n - m + i)}}} \prod\limits_{j = n}^{n + 1} {CNO{T_{(m,j),(m + 1,j)}}} \mathop \otimes \limits_{i = m}^{m + 1} \mathop \otimes \limits_{j = n}^{n + 1} {\text{(}}R{Y_{(i,j)}}({\theta _{(i,j)}})){\text{ }} \hfill \\ \end{gathered} .\end{align} \tag{ 23 }$$

Figure 4(d) shows all three multi-qubit unitaries used in different circuit ansätze in these two layers, which consist of single-qubit $RY$ rotations containing trainable parameters and $CNO{T_{i,j}}$ gates. Where $i$ is control qubit and $j$ is target qubit, it is set to always $i < j$ in two-qubit and three-qubit unitaries to ensure that CNOT gates are in the same direction in all unitaries. The four-qubit unitary is required to apply CNOT gate counterclockwise from the upper left corner of the local $2 \times 2$ lattice.

Second, in the encoding layers, we use special unitaries to construct TNQC ansätze compared to the origin ones in QNN or adjustment layer, including

$$\begin{align}\begin{gathered} {U_{enc}}_{(n)}^{\left[2\right]}(\theta ,x) = U_n^{\left[2\right]}(\theta )\mathop \otimes \limits_{i = n}^{n + 1} {\text{(}}R{Y_i}({x_i})), \hfill \\ {U_{enc}}_{(m,n)}^{\left[2\right]}(\theta ,x) = U_{(m,n)}^{\left[2\right]}(\theta )\mathop \otimes \limits_{i = m}^{m + 1} {\text{(}}R{Y_{(i,n)}}({x_{(i,n)}})){\text{ or }}U_{\left[m,n\right]}^{(2)}(\theta )\mathop \otimes \limits_{j = n}^{n + 1} {\text{(}}R{Y_{(m,j)}}({x_{(m,j)}})), \hfill \\ {U_{enc}}_{(m,n)}^{\left[3\right]}(\theta ,x) = U_{(m,n)}^{\left[3\right]}(\theta )\mathop \otimes \limits_{(i,j) \in \left\{ (m,n),(m + 1,n),(m + 1,n + 1)\right\} } (R{Y_{(i,j)}}({x_{(i,j)}})), \hfill \\ {U_{enc}}_{(m,n)}^{\left[4\right]}(\theta ,x) = U_{(m,n)}^{\left[4\right]}(\theta )\mathop \otimes \limits_{i = m}^{m + 1} \mathop \otimes \limits_{j = n}^{n + 1} {\text{(}}R{Y_{(i,j)}}({x_{(i,j)}})){\text{ }} \hfill \\ \end{gathered} .\end{align} \tag{ 24 }$$

To be specific, each special unitary for encoding applies an $RY$ rotation on every qubit prior to the original unitary being applied, with the angle of $RY$ being the corresponding pixel value ${x_i}$ on the 2D lattice location of the qubit. And the $RY$ rotations of different encoding unitaries applied on the same qubit use the same pixel as angle. Figure 4(e) illustrates three multi-qubit unitaries used for VTNE.

Classical neural networks can easily perform multi-class classification tasks, given that our goal is to compare the accuracy performance of quantum and classical classifiers on a fair track, limiting the classification task to binary classification is obviously not fair to classical classifiers. However, quantum classifiers often need to perform multi-class classification tasks with the help of adding a classical dense layer or an MLP classifier, forming the so-called classical-quantum hybrid classifiers. That's not a good thing because it is difficult to determine which part of the network really plays a role. The classical layer can learn and classify on its own, and sometimes using only the classical part of the hybrid model even outperforms the full hybrid classifier. Using the probabilities of the quantum measurement results or the expectation of each qubit as the logits for classification, we can perform multi-class classification without resorting to classical neural networks, but we will get a poor classification accuracy.

In order to enable quantum classifiers to efficiently perform multi-class classification tasks without the help of classical networks, we propose a parallel quantum machine learning method called PQNs for multi-class classification. For a $k$-class classification task, we create $k$ quantum circuits having the same architecture. Each circuit has independent parameters for learning and iteration. They can be considered as $k$ 'quantum nodes' of the classifier. These 'quantum nodes' can be executed in parallel on multiple different quantum machines. For an identical input, they will generate $k$ outputs after measurement, which are then used as logits to calculate loss according to equation (22), and parameters in all $k$ quantum nodes are then simultaneously optimized according to the gradient calculated by loss. Such quantum nodes act similarly to the output neurons in the last layer of the MLP. Note that such a multi-class classifier is still composed of pure QNNs.

In this section, we construct 16 QCL classifiers by combining 4 encoding methods and 4 circuit ansätze under the TNQC supervised learning framework. Among them, encoder baseline and ansätze baseline are the existing QMPS (also known as hardware-efficient ansatz) and basic angle encoding methods, respectively. To answer RQ1 and RQ2, we use a simulator to run quantum classifiers to perform binary classification tasks. The simulation results are used to analyze whether 2D TN and variational encoding help improve accuracy, and to analyze performance differences between classifiers. All simulations below are repeated 10 times with randomly initialized condition.

The number of pixels encoded by the 4 encoding methods is not the same while using the same number of qubits. In order to ensure the fairness of the comparison, we first use AE and VAE on 16 qubits to encode data and test the performance of different ansätze, because they encode the same number of pixels. The simulations are based on MNIST dataset which is a handwritten digital image dataset containing a training set of 60 000 samples and a test set of 10 000 samples, and each sample is a $28 \times 28$ grayscale image and belongs to one of 10 classes. We choose three binary classification tasks of increasing difficulty including 01, 27, and 49 classification tasks. These images are resized to $4 \times 4$ using area-based resampling method, which allows them to retain two-dimensional connections between pixels and be encoded to 16 qubits. All simulations use identical hyperparameters, the batch size is 100 and Adam optimizer is used with an initial learning rate of $\lambda = 0.01$ and a decay rate of $\alpha = 0.1$. The learning rate is decayed after 15 epochs, and the total number of training epochs is 30.

Table 2 shows the mean test accuracy and standard deviation of 10 simulations with randomly initialized parameters in different tasks. Here, the 'Ansatz' column describes the network architecture. QMPS with AE is chosen as a baseline for the simulations, and all ansätze use the same number of layers fairly in the sense of TN to compare their performance. The 'Encoding' column describes the encoding method. The bold values indicate the best result for each classificaiton task.

Table 2. Binary classification accuracy on MNIST dataset.

Ansatz	Encoding	0 or 1	2 or 7	4 or 9
QMPS	AE	96.86 $\pm$ 0.64	93.48 $\pm$ 0.61	80.31 $\pm$ 0.37
QMPS	VAE	97.11 $\pm$ 0.88	94.86 $\pm$ 1.34	80.72 $\pm$ 1.57
QSBS	AE	97.34 $\pm$ 0.43	96.32 $\pm$ 0.19	81.09 $\pm$ 0.95
QSBS	VAE	98.47 $\pm$ 0.20	96.56 $\pm$ 0.45	83.14 $\pm$ 1.05
QEPS	AE	97.45 $\pm$ 0.40	96.62 $\pm$ 0.25	81.81 $\pm$ 0.67
QEPS	VAE	98.34 $\pm$ 0.29	96.89 $\pm$ 0.50	82.56 $\pm$ 0.77
QisoTNS	AE	96.73 $\pm$ 0.59	96.54 $\pm$ 0.21	80.79 $\pm$ 0.94
QisoTNS	VAE	97.28 $\pm$ 0.95	96.89 $\pm$ 0.54	80.42 $\pm$ 0.93

The results lead to following conclusions. First, 2D TNQC ansätze can improve the accuracy performance of QCL since 2D TNQC classifiers outperform 1D QMPS classifier with the same encoder in all three tasks of varying difficulty. For instance, in the '2 or 7' classification task, QisoTNS classifier using AE achieves an accuracy 3.06% higher than that of QMPS. Its advantage stems from the fact that the two-dimensional entanglements can capture the correlations between adjacent pixels and image's overall structural information. From the perspective of TN, 2D TN has a higher-dimensional entanglement than 1D MPS with the same bond dimension, which enables it to represent a larger subspace in the Hilbert space. This property endows the 2D TNQC classifier with a larger solution space for learning global optimal parameters. Second, QCL benefits from variational encoding methods since VAE achieves higher accuracy than AE in the same classifier in almost all cases (except QisoTNS), especially in simpler or harder classification tasks. This illustrates the effectiveness of adaptive feature map. Effective improvements can be achieved by simply adding trainable bias to individual data when encoding. Additionally, there are performance differences among 2D TNQC classifiers with the same encoder. This is not only because these ansätze are generated from different TNs, but also because the specific implementations of unitaries that make up the ansätze are also different.

Quantum classifiers using AE and VAE require O(n) qubits to encode n pixels, so we have to compress the images significantly, resulting in a loss of information, which clearly prevents them from being the best classifiers. Also, it may be more effective to extract higher dimensional adaptive features than VAE to transform individual data into adaptive features separately. So next we use CE and VTNE based classifiers on 16 qubits to conduct the 49-classification task which is the most difficult one in 3 tasks done above. Note that in the simulations using CE, the images are not resized, while in the simulations using VTNE, all images are resized to $12 \times 12$ to prevent the circuit from being too deep. So we are not going to compare these two encodings to each other. The simulations using CE share the same hyperparameters as the AE simulations. While in VTNE simulations, epoch is set to 10, the batch size is 50, and the learning rate starts to decay from the 8th epoch.

Figure 5 display the average accuracy and loss on training dataset and test dataset during the training process with 10 different random parameter initializations. Analyzing the results, we observe that with the aid of CE, all QC classifiers achieve improved accuracy at least 16.26% owing to the availability of larger dimensions of data and efficient nonlinear transformation. With the same bond dimension, 2D TNQC still performs slightly better than 1D QMPS. Specifically, the test accuracy of QMPS reaches 99.09%, to which QEPS is similar (99.17%), and QSBS has the highest test accuracy of 99.40%. So CE-QSBS is the best classifier in classical-quantum hybrid networks.

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We also notice that CE-QisoTNS stabilizes at 100% in accuracy and has the lowest train loss during the training stage, but it suffers from overfitting with a test accuracy of only 99.32%. In order to explore the cause of overfitting and figure out whether CE-QisoTNS has more expressive power comparing to other TNs, we test the overfitting of the model by using L2 regularization at the CE and QisoTNS layers of the network respectively. After training, the training and test accuracy and the difference value Δ between them are as following table 3.

Table 3. Accuracy and difference value of CE-QisoTNS when using L2 regularization at different layers.

Regularization layer	None	CE	QisoTNS	CE&QisoTNS
Training accuracy	100%	99.87%	99.93%	99.64%
Test accuracy	99.32%	99.38%	99.30%	99.35%
Difference value ${{\Delta }}$	0.68%	0.49%	0.63%	0.29%

From the results, it can be found that the accuracy difference of the classifiers with CE regularization layer is reduced to a certain extent compared to the model without regularization (overfitting reduction), and the reduction ${{{\Delta }}_{None}} - {{{\Delta }}_{CE}}$ is greater than the reduction ${{{\Delta }}_{None}} - {{{\Delta }}_{QisoTNS}}$ in difference value of the classifiers with the QisoTNS regularization layer compared to the model without regularization, which indicates that the overfitting of the model mainly occurs in the CE layer.

The reason for this situation may be that the classical layer continues to learn after the quantum layer convergence. All our classical quantum hybrid models will have overfitting, and the sequence of their overfitting depends on the convergence speed of the quantum layer. For example, our results show that the training accuracy of CE-QMPS reaches 100% at the 37th epoch, while the training accuracy of CE-QSBS reaches 100% at the 51epoch, this situation is not observed with CE-QEPS. But their test accuracies do not improve further. This shows that after the convergence of the quantum layer, the classical CE layer still adjust the features adaptively to the quantum classifier and finally achieve 100% training accuracy.

According to the above inference, we believe that QisoTNS has certain advantages in the convergence speed. In order to further test whether it has advantages in accuracy, regularization is used on both CE and QisoTNS layers. ${{{\Delta }}_{CE\& QisoTNS}}$ decreases significantly, which indicates that overfitting is mitigated, but the test accuracy of the model only increases slightly, which is still not enough to exceed CE-QSBS. Therefore, CE-QSBS is still the best hybrid classifier in accuracy.

In the models using VTNE, the 2D TNQC classifiers still have higher performance. The accuracy of QisoTNS classifier is 98.69%, higher than the other three classifiers including QSBS(98.63%), QEPS(98.20%) and QMPS (97.63%). This represents a 17.9% improvement in accuracy compared to AE-QisoTNS. We can see that QEPS performs slightly less well than the other two 2D TNQCs. This could be due to the fact that both QisoTNS and QSBS can be identified as isoTNS (or PEPS), they have similar structures. While QEPS is not a PEPS, its TN has non-vertical indices. Ansätze with similar TN structures perform similarly. Considering that the dimension of the data used is only 12 $\times$ 12, the performance of the VTNE classifier is already excellent. Among them, VTNE-QisoTNS is the best classifier in quantum models.

We test the performance of the QisoTNS classifier on all MNIST pairwise subset classification tasks and achieve a test accuracy of over 99% for almost all classifications. In order not to lose generality, we also perform the same evaluation simulations with another Fashion-MNIST dataset, which is a grayscale image dataset of clothing. It has the same size, format, number of classes, and dataset partitioning rules as the MNIST dataset but is more challenging. The best test accuracy of all pairwise classification tasks is shown in figures 5(e) and (f).

In summary, for RQ1 & 2, we have:

Insight 1. 2D TNQC can improve the accuracy of QCL models, and the variational encoding methods are more effective than the ordinary methods. Under the premise of using the same number of qubits, QisoTNS using TN variational encoding improves the accuracy of AE-QMPS baseline model by 18.38% in 49 classification tasks. VTNE-QisoTNS is the best quantum classifier, CE-QSBS is the best classical-quantum hybrid classifier.

We want to explore whether the models' accuracy performances are affected by some TN features of ansätze. Following the proposal of [11], we use deep quantum circuit to increase the bond dimension of a TN state to explore the relationship between accuracy and bond dimension. Specifically, starting from the encoded quantum state $\mathop \otimes \limits_{i = 1}^N |{\phi _i}\rangle$, we apply a series of layers $\{ {U_t}\} (t = 1,2,\ldots,L)$ having the same structure, and finally form a TN of $D = {2^L}$. In this process, each layer ${U_t}$ is a TN operator with $D = 2$, and all operators are contracted to form a TN with a larger bond dimension. Therefore, we can see that compared to classical 2D TNs, quantum computers require only ${\text{lo}}{{\text{g}}_2}D$ layers of circuits to construct TNs of bond dimension $D$, which alleviates the memory bottleneck problem that exists in classical TNs [11] and allows the construction of larger scale 2D TNs. We choose VTNE-QisoTNS as an example to measure the performance of the classifier under different virtual bond dimensions $D$ of single ansatz used in adjustment layers by increasing the number of layers of TNQC, where the physical bond dimension of ansatz is 2.

As shown in table 4, even if the ansatz virtual bond dimension is 0 (without adding ansatz), the model can still achieve 97.88% accuracy, which is higher than VTNE-QMPS, because the VTNE encodes the data into an isoTNS. As the bond dimension increases, the total parameters of adjustment layers and depth of single ansatz also increase. When $D = {2^2}$, VTNE-QisoTNS reaches the best accuracy, but the accuracy decreases as virtual bond dimension continues to increase, which also happens in classical TNML [19]. This situation is also similar to classical TNML [19]. Situation above shows the accuracy performance of model can be affected by some TN features of ansatz, and increasing the virtual bond dimension of ansatz appropriately can improve the accuracy performance, but this will also increase the circuit depth and the training difficulty. It is difficult to simulate the circuit with $D > {2^4}$ in the simulator limited by memory. Note one will not have this issue for quantum computations.

Table 4. Performance of VTNE-QisoTNS under different virtual bond dimensions of single-layer ansatz.

$D$	0	2	${2^2}$	${2^3}$	${2^4}$	${2^5}$
Parameters	0	132	264	396	528	—
Ansatz depth	0	17	34	51	68	—
Test accuracy	97.88%	98.69%	98.83%	98.70%	98.61%	—

In summary, for RQ3, we have:

Insight 2. The accuracy performance of the model is affected by some TN features of the ansatz. The accuracy of the model will first increase and then decrease as the virtual bond dimension of ansatz increases.

In order to answer RQ4 and enable quantum classifiers to efficiently perform multi-class classification tasks without the help of classical networks, we combine PQN and VTNE-QisoTNS to construct a multi-class classifier, the model accuracy performance is tested and compared with classical neural networks. The principle of this quantum multi-class classifier is that for an input, the same backup is passed to $k$ QNNs with the same architecture but different trainable parameters, resulting in $k$ results, and the serial number of the largest of the $k$ results is selected as its label. We select all handwritten images containing digits 0–3 from the MNIST dataset. All images are resized to a size of $8 \times 8$ to enable normal simulation of the quantum classifier. Such processing is due to the memory limitations of classical simulators, while quantum computers can use full-size images to further improve the model performance, and they support multi-classification tasks with more categories. For comparison, we choose the CNN classifier which contains a convolutional layer, a max pooling layer and a fully connected layer with 512-64-4 neurons. The convolution layer contains three $3 \times 3$ kernels with a stride of 1, and it is activated by ReLU. The pooling layer has a pool size of $2 \times 2$. The batch size of 10 random initialization training is set to 10, and Adam optimizer is used with a learning rate of 0.01 until the fifth epoch, after which the learning rate is decreased to 0.001.

Figure 6(a) shows the performance of CNN classifier and VTNE-QisoTNS classifier on the same test set. The VTNE-QisoTNS classifier achieves lower test loss and gives a higher test accuracy of up to 99.18%, outperforming the CNN classifier which has a test accuracy of 98.79%. This means that 2D TNQC classifier has certain advantages in performance compared to classical simple classifiers when using identical inputs. In figures 6(b) and (c), we provide the confusion matrices of two classifiers on test set, which show their specific differences in classification.

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In summary, for RQ4 & 5, we have:

Insight 3. Based on parallel quantum machine learning method for multi-class classification, quantum classifiers can perform multi-class classification effectively. We use the best VTNE-QisoTNS quantum classifier to build its multi-class version. Compared to classical simple classifiers, the best quantum classifiers can achieve better accuracy performance on a fair track with the same inputs.

Tensorcircuit simulator can only simulate up to 30 quantum qubits on our hardware and cannot support sufficiently deep circuits (e.g. QisoTNS ansatz with D = ${2^5}$). It means that it is hard to further expand the size of 2D TNs and carry out larger-scale tasks using simulator (the largest picture size in our VTNE simulations is only 12 $\times$ 12). And it is difficult to simulate a TN with a higher bond dimension. Some models are very time-consuming using simulators. Therefore, long training time with simulators, difficulty in scaling task sizes and simulating TNs with high bond dimensions are all motivations for running these ansätze on quantum computers. It is necessary to use quantum computers in order to construct larger scale 2D TNQCs for building QNN models with larger data input dimensions and higher accuracy to reduce training time and promote their practical benefits. Therefore, the requirements for evaluating the effectiveness and usability of our methods and models are not only improvements in accuracy performance, but also that the models can function on real quantum machines as mentioned in RQ6. Noise exists on current NISQ machines, so algorithms that can truly execute on them should be able to be noise resilient. QCL is expected to be noise-tolerant for implementation on near-term noisy hardware. In this experiment, we test the impact of noise on the performance of the proposed model by simulating thermal relaxation noise. Subsequently, we train the classifier using a backend that simulates IBM Quantum's real hardware noise model. Finally, the trained model is deployed on a real quantum computer. Due to the limited number of qubits on quantum hardware, here we use the VTNE-QisoTNS classifier for the classification task on 01 dataset of size $3 \times 3$ as a minimal example for testing. While further achievement of practical benefits of 2D TNQC quantum models requires NISQ devices with further improvements in qubit numbers and fidelity.

Thermal relaxation describes a non-unitary evolution of a high-energy state system that spontaneously releases energy towards ground state, which originates from the energy exchange between physical qubits and environment. Thermal relaxation noise causes the quantum system to transition from a pure state to a mixed state. For the state $\rho$ of a quantum system, the model describing noise (called channel $\varepsilon$) can be expressed as $\varepsilon (\rho ) = \sum\nolimits_k {{E_k}\rho E_k^\dagger }$ using Kraus operator, where $\{ {E_k}\}$ are Kraus operators, which need to satisfy the completeness condition $\sum\nolimits_k {E_k^\dagger } {E_k} = I$, and $\varepsilon (\rho )$ is the quantum state after evolution. The thermal relaxation channel $\varepsilon$ can be implemented by applying the dephasing channel [35] ${\varepsilon _d}$ and the amplitude damping channel [35] ${\varepsilon _a}$ after each unitary, i.e. $\varepsilon (\rho ) = {\varepsilon _a}({\varepsilon _d}(\rho ))$. The Kraus operators of the amplitude damping channel are

$$\begin{align}{E_0} = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{\sqrt {1 - {p_a}} } \end{array}} \right],{E_1} = \left[ {\begin{array}{*{20}{c}} 0&{\sqrt {{p_a}} } \\ 0&0 \end{array}} \right].\end{align} \tag{ 25 }$$

While the Kraus operators of the dephasing channel are

$$\begin{align}{E_1} = \left[ {\begin{array}{*{20}{c}} {\sqrt {1 - {p_d}} }&0 \\ 0&{\sqrt {1 - {p_d}} } \end{array}} \right],{E_2} = \left[ {\begin{array}{*{20}{c}} {\sqrt {{p_d}} }&0 \\ 0&{ - \sqrt {{p_d}} } \end{array}} \right].\end{align} \tag{ 26 }$$

The parameters ${p_a}$ and ${p_d}$ are used to characterize the strength of a quantum channel, with larger values indicating stronger channel effects and faster degradation of the fidelity of quantum information. In addition, the thermal relaxation channel can be characterized by the unitary gate duration ${T_g}$, the coherence time ${T_1}$ and dephasing time ${T_2}$ of the qubits. Specifically, ${p_a}$ and ${p_d}$ can be expressed as

$$\begin{align}{P_a} = 1 - {e^{ - \frac{{{t_g}}}{{{T_1}}}}},{P_d} = \frac{1}{2} \times \left(1 - {e^{ - \left(\frac{{{t_g}}}{{{T_2}}} - \frac{{{t_g}}}{{2{T_1}}}\right)}}\right).\end{align} \tag{ 27 }$$

Our noise model applies thermal relaxation channel to each qubit of each unitary gate in the circuit. To assess the model's practical performance on real quantum hardware, we fix the duration of each three-qubit gate at 450 ${\text{ns}}$ and each two-qubit gate at 250 ${\text{ns}}$, and set ${T_2}$ to $0.7 \times {T_1}$. We test the model's performance with ${T_1}$ ranging from 30 $\mu {\text{s}}$ to 210 $\mu {\text{s}}$ in 20 $\mu {\text{s}}$ increments, which represents varying levels of noise impact from high to low. The parameter settings are in line with the real situation of current quantum devices. The classifier used for testing includes a VTNE layer and a 2D TNQC layer, its specific circuit is shown in figure 7(a).

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Figure 7(b) shows the model's performance on 01 classification task, we see that the best test accuracy decreases as ${T_1}$ decreases, which is due to the increasing noise. As a benchmark, the mean test accuracy reaches 99.76% under ideal simulated condition. At ${T_1} = 210\mu {\text{s}}$, ${p_a} = 0.0021$ and ${p_d} = 0.0010$ can be calculated at this time, the test accuracy reaches 99.08%; while at ${T_1} = 130\,\mu {\text{s}}$, which is the current average level of hardware on IBM Quantum [36], the test accuracy is 98.28%, decreasing by only 1.48% compared to the ideal state. This indicates that the model can work on current quantum hardware. As the noise increases, at ${T_1} = 30\,\mu {\text{s}}$, for the thermal relaxation noise with ${p_a} = 0.0149$ and ${p_d} = 0.0069$, the model's accuracy remains at 95.35%, proving that the model has some level of noise resilience.

Further, we demonstrate the usability of the model on a real quantum machine. Training a model directly with a quantum machine is difficult, with system scheduling and shutting down network connections leading to interruptions in the training process. Therefore, we used Qiskit to train the model in the FakeNairobi backend simulated with the noise model based on the real hardware ibmq_nairobi in IBM Quantum and deployed the classifier to the ibmq_nairobi quantum computer. Considering the limited resources of quantum computing and in order to follow the general rules of machine learning, we randomly selected 120 data to construct a mini-dataset and divided it into training and test set with a ratio of 5:1. Such a setup reduces the information leakage and thus more accurately reflects to the performance of the model. A test set of 20 examples is used to determine accuracy. Using only 100 shots per execution to train for our circuits having ${2^9}$ possible results, our model achieves 100% test accuracy on FakeNairobi backend. This shows that our framework and logits $E({\mathbf{x}},\theta )$ can work using repeated measurements with finite shots (although 100 < ${2^9}$). However, to make the result more stable, the model we finally used on the real machine was trained using 8192 shots per execution. For each test example in a real quantum computer, the circuit is set to 8192 shots to obtain results. Due to the limited number of available qubits (only 7), we adopt the method proposed in [5] and use measure and reset operations to prepare a qubit-efficient scheme for the circuit in figure 7(a). To be specific, the qubits ${q_0}$ and ${q_3}$ in the original circuit perform the measurement operation immediately after applying the last unitary acting on them respectively and are reset to $|0\rangle$, they will be reused as ${q_8}$ and ${q_2}$. The specific circuit is shown in figure 7(c). The circuit correctly predicts the class of 16 out of 20 test data, which is not as expected. One reason for this is that the reset operation has a long duration of up to 5696 ns, making it the longest operation on the hardware, which brings thermal relaxation noise beyond expectations. Another reason is that the noise model of the simulation backend is not exactly the same as the noise on a real quantum computer. The trained model has good robustness to the noise of the simulator (achieving a test accuracy of 100%), but it cannot fully absorb the effects of other different noises.

To mitigate these two problems, we use the method proposed in [4, 5] to repeat experiments on quantum circuits and predict labels, one could take a majority vote from 501 experimental results of each test example to obtain the most probable label.

As shown in table 5, by voting after multiple experiments, all the test samples are correctly classified compared with only 16 labels correctly predicted by a single experiment (the bold values indicate incorrectly predicted samples). Figure 7(d) shows the correct classification probability of a single experiment for each of the 20 test samples, and the average is 0.731. The final classification accuracy of test samples is 100%.

In summary, for RQ6, we have:

Insight 4. 2D TNQC classifiers have some level of noise resilience, which enables them to run and function on real quantum machines.