Adiabatic Encoding of Pre-trained MPS Classifiers into Quantum Circuits

TL;DR

It is proved that training qMPS-classifiers from scratch on a certain artificial dataset is exponentially hard due to barren plateaus, but the adiabatic encoding circumvents this issue and additional numerical experiments on binary MNIST confirm its robustness.

Abstract

Although Quantum Neural Networks (QNNs) offer powerful methods for classification tasks, the training of QNNs faces two major training obstacles: barren plateaus and local minima. A promising solution is to first train a tensor-network (TN) model classically and then embed it into a QNN.\ However, embedding TN-classifiers into quantum circuits generally requires postselection whose success probability may decay exponentially with the system size. We propose an \emph{adiabatic encoding} framework that encodes pre-trained MPS-classifiers into quantum MPS (qMPS) circuits with postselection, and gradually removes the postselection while retaining performance. We prove that training qMPS-classifiers from scratch on a certain artificial dataset is exponentially hard due to barren plateaus, but our adiabatic encoding circumvents this issue. Additional numerical experiments on binary MNIST also confirm its robustness.

Figures (9)

• Figure 1: (a) A qMPS-circuit with 2-qubit gates. (b) A qMPS-classifier without postselection. The purple circles represent the embedded input data $\phi(\vec{x})$, rectangles represent 2-qubit gates, and the trash bins indicate the discarding of the measurement outcomes. (c) Correspondence between MPS-classifiers and qMPS-classifiers with postselection. The triangle is the isometry $V$. $\vert 0 \rangle$ indicates the outcome on which we perform postselection.
• Figure 2: (a) Illustration of a qMPS-classifier with postselection (left), a qMPS-classifier without postselection (right), and a weighted qMPS ansatz (middle). The green boxes labeled $M$ represent measurements on each qubit. (b) Schematic of how the loss landscape changes as $W_1$ is gradually increased from 0 to 1.
• Figure 3: Adiabatic encoding framework for the first-qubit trigger dataset. Here, "success rate per gate" denoted by $P_{\text{srpg}}$ is the average success probability at a each two-qubit gate, which can be calculated from $\mathrm{Tr}[\tilde{\tau}]$ at each $k$-th gate. The overall circuit's success probability $P_{\text{success}}$ can be calculated by $P_{\text{success}} = P_{\text{srpg}}^{{L-1}}$.
• Figure 4: (a) Direct training of the qMPS-classifier suffers from barren plateaus and local minima. (b) Adiabatic encoding successfully circumvents these issues. In the same manner, $P_{\text{success}}$ can be calculated by $P_{\text{success}} = P_{\text{srpg}}^{{256 - 1}}$. Both (a) and (b) are for training dataset.
• Figure 5: (a) Graph of $\Delta\rho_L$ vs. $L$. (b) Graph of $\frac{\partial \Delta\rho_L}{\partial U_L}$ vs. $L$. We calculated the variance and mean by sampling 5000 random initialization.

Theorems & Definitions (12)

• Definition 1: First-Qubit Trigger Dataset
• Lemma 1: Existence of Perfect Classifiers for the First-Qubit Trigger Dataset
• proof
• Theorem 1: Exponential Indistinguishability under Haar-Random Initialization
• Theorem 2: Absence of Barren Plateaus in Classical MPS with Stacked Identity Initialization
• Theorem 3: Exponential postselection Cost for Embedding MPS-Classifiers into qMPS-Classifiers
• Lemma 2
• proof
• Lemma 3
• proof