Tensor-network discriminator architecture for classification of quantum data on quantum computers

We demonstrate the use of matrix product state (MPS) models for discriminating quantum data on quantum computers using holographic algorithms, focusing on the problem of classifying a translationally invariant quantum state based on $L$ qubits of quantum data extracted from it. We detail a process in which data from single-shot experimental measurements are used to optimize an isometric tensor network, the isometric tensors are compiled into unitary quantum operations using greedy compilation heuristics, parameter optimization on the resulting quantum circuit model removes the postselection requirements of the isometric tensor model, and the resulting quantum model is inferenced on either product state (single-shot measurement) or entangled quantum data. We demonstrate our training and inference architecture on a synthetic dataset of six-site single-shot measurements from the bulk of a one-dimensional transverse field Ising model (TFIM) deep in its antiferromagnetic and paramagnetic phases. We find that increasing the bond dimension of the tensor-network model, amounting to adding more ancilla qubits to the circuit representation, improves both the average number of correct classifications across the dataset and the single-shot probability of correct classification. We experimentally evaluate models on Quantinuum's H1-2 trapped ion quantum computer using entangled input data modeled as translationally invariant, bond dimension 4 MPSs across the known quantum phase transition of the TFIM. Using linear regression on the experimental data near the transition point, we find predictions for the critical transverse field of $h=0.962$ and 0.994 for tensor-network discriminators of bond dimension $\chi =2$ and $\chi =4$, respectively. These predictions compare favorably with the known transition location of $h=1$ despite training on data far from the transition point. Our techniques identify families of short-depth variational quantum circuits in a data-driven and hardware-aware fashion and robust classical techniques to precondition the model parameters, and can be adapted beyond machine learning to myriad applications of tensor networks on quantum computers, such as quantum simulation and error correction.

Figure 1

MPS linear classifier model represented by tensor-network graphical notation. (a) A linear model MPS classifier is obtained by introducing a label tensor ${\mathbb{P}}^{\ell }$ on one of the bonds of an MPS structure. (b) The overlap of the MPS classifier wave function defined in (a) with a test wave function $|{\psi }_{\mathrm{test}}\rangle$ defines an indicator function $f_{\ell }\left(\right|{\psi }_{\mathrm{test}}\rangle )$ that can be optimized for classification performance. (c) The action of the linear MPS model can be interpreted as an encoding of the test state information into the bond space to define an operator $\mathbb{M}\left(\right|{\psi }_{\mathrm{test}}\rangle )$ shown graphically, followed by classification using the indicator tensor as ${\mathrm{argmax}}_{\ell }\mathrm{Tr}({\mathbb{P}}^{\ell †}\mathbb{M}\left(\right|{\psi }_{\mathrm{test}}\rangle ))$.

Figure 2

Diagram of tensor-network discriminator architecture, with operation read left to right as in quantum circuit diagrams. First, the operator ${\widehat{U}}_R$ creates an initial state in the ${log}_2\chi$ bond qubits, starting from the vacuum state. Next, this bond state is coupled with a physical qubit via the operator ${\widehat{U}}_G$ and the physical qubit state discarded and reset $N_b$ times. This creates a bond state that can be interpreted as a feature space representation of the density matrix obtained by tracing out half of the infinite system. This bond state is now combined with the input state $|\mathbf{x}\rangle$ a single qubit at a time via the operator ${\widehat{U}}_D$, and the physical qubit measured and reinitialized after each application. For clarity, this operation is drawn as though the test state is a product state, but the process also applies to any entangled input state stored in an $L$-qubit register. Finally, the resulting bond degrees of freedom which have been conditioned on the observation of the state $|\mathbf{x}\rangle$ are operated on by ${\widehat{U}}_C$, and the measurement of ${log}_2{N}_C$ of these qubits defines a binary representation of the predicted class label.

Figure 3

Training of discriminator architecture. (a) Tensor-network diagram for the construction of ${\rho }_{\ell \ell }$ from the tensors $R,G,D$, and $C$, as in algorithm 1. (b) Compilation of the unitary embeddings of $R,G,D$, and $C$ results in a quantum circuit sampling from the distribution ${\rho }_{\ell ,\ell }$ with postselection in the $|0\rangle$ state indicated by tags. (c) Example compiled unitary for ${\widehat{U}}_G$ evaluated at the optimal parameters for the postselected circuit in (b), here shown with two bond qubits. Gates connecting qubit wires are cnots, and the single-qubit rotations have the matrix representation ${\widehat{R}}_y\left(\theta \right)\equiv \left(\begin{array}{cc}cos\frac{\theta }{2}& -sin\frac{\theta }{2}\\ sin\frac{\theta }{2}& cos\frac{\theta }{2}\end{array}\right)$. (d) Circuit ansatz for model without postselection, in which the compiled circuit architectures are the same as the postselected model [e.g., panel (c) for ${\widehat{U}}_G]$, but with different values for the parameters.

Figure 4

Inference of an MPS with the discriminator architecture. The diagram shows inference of an iMPS test state (lower half of diagram) with unitaries ${\widehat{U}}_{G^{\prime }}$ and ${\widehat{U}}_{R^{\prime }}$ using the discrimination circuit with unitaries ${\widehat{U}}_R,{\widehat{U}}_G,{\widehat{U}}_D$, and ${\widehat{U}}_C$ (top half of diagram). The test state generates its half-infinite density matrix using $N_b^{\prime }$ iterations of the transfer operator encoded by ${\widehat{U}}_{G^{\prime }}$ applied to the boundary state generated by ${\widehat{U}}_{R^{\prime }}$, while the discriminator generates the feature space half infinite density matrix using $N_b$ iterations of ${\widehat{U}}_G$ applied to the boundary state generated by ${\widehat{U}}_R$ on a different bond register. The test state generator then generates a single qubit of the test state at a time, this state is fed into the discriminator to condition its bond register, and the physical qubit is reset. After $L$ qubits have been generated of the state, the test state bond qubits are discarded, the discriminator bond qubits are fed into the classifier circuit ${\widehat{U}}_C$, and the measurement of the class register qubits defines the predicted class label $\ell$.

Figure 5

Dependence of single-shot probability of success on bond dimension. Histograms of the single-shot probability of correct classification $P_{\mathrm{SS}}$ over the training set are shown for $\chi =2,4,8$. As the bond dimension increases, the average probability of correct classification also increases.

Figure 6

Predicted phases from the tensor-network discriminator and classical benchmark; product state input data. The probability that a classifier predicts the PM class, averaged over the test data, is shown as a function of the transverse field $h$. The inset shows a zoom around the true phase transition point, $h=1$, demonstrating that all classifiers have a bias of a few percent.

Figure 7

Predicted phases from the tensor-network discriminator and classical benchmark; entangled input data. The probability of a classifier returning a call in the PM class for a single run of the circuit is shown as a function of the transverse field $h$. As the bond dimension of the classifier increases, the probability of correct classification increases. The performance of all classifiers is relatively unchanged as the bond dimension of the iMPS generating the test data is increased from ${\chi }_i=4$ to ${\chi }_i=8$.

Figure 8

Performance of $\chi =2$ model on entangled data. The top panel shows the predictions of the $\chi =2$ tensor-network discriminator evaluated on entangled iMPS input data with ${\chi }_i=4$, with the triangles corresponding to the mean of experimental results on the H1 machine and the circles corresponding to simulations of the circuit execution in the absence of hardware noise. All points correspond to 100 shots, and error bars are 90% Wilson confidence intervals. The bottom panel is a zoom around the transition point, together with the linear regression model predicting a transition at $h=$ 0.962.

Figure 9

Performance of $\chi =4$ model on entangled data. The top panel shows the predictions of the $\chi =4$ tensor-network discriminator evaluated on entangled iMPS input data with ${\chi }_i=4$, with the triangles corresponding to the mean of experimental results on the H1 machine and the circles corresponding to simulations of the circuit execution in the absence of hardware noise. All points correspond to 100 shots, and error bars are 90% Wilson confidence intervals. The bottom panel is a zoom around the transition point, together with the linear regression model predicting a transition at $h=$ 0.994.