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Quantum Deep Equilibrium Models

Philipp Schleich, Marta Skreta, Lasse B. Kristensen, Rodrigo A. Vargas-Hernández, Alán Aspuru-Guzik

TL;DR

QDEQ is not only competitive with comparable existing baseline models, but also achieves higher performance than a network with 5 times more layers, demonstrating that the QDEQ paradigm can be used to develop significantly more shallow quantum circuits for a given task, something which is essential for the utility of near-term quantum computers.

Abstract

The feasibility of variational quantum algorithms, the most popular correspondent of neural networks on noisy, near-term quantum hardware, is highly impacted by the circuit depth of the involved parametrized quantum circuits (PQCs). Higher depth increases expressivity, but also results in a detrimental accumulation of errors. Furthermore, the number of parameters involved in the PQC significantly influences the performance through the necessary number of measurements to evaluate gradients, which scales linearly with the number of parameters. Motivated by this, we look at deep equilibrium models (DEQs), which mimic an infinite-depth, weight-tied network using a fraction of the memory by employing a root solver to find the fixed points of the network. In this work, we present Quantum Deep Equilibrium Models (QDEQs): a training paradigm that learns parameters of a quantum machine learning model given by a PQC using DEQs. To our knowledge, no work has yet explored the application of DEQs to QML models. We apply QDEQs to find the parameters of a quantum circuit in two settings: the first involves classifying MNIST-4 digits with 4 qubits; the second extends it to 10 classes of MNIST, FashionMNIST and CIFAR. We find that QDEQ is not only competitive with comparable existing baseline models, but also achieves higher performance than a network with 5 times more layers. This demonstrates that the QDEQ paradigm can be used to develop significantly more shallow quantum circuits for a given task, something which is essential for the utility of near-term quantum computers. Our code is available at https://github.com/martaskrt/qdeq.

Quantum Deep Equilibrium Models

TL;DR

QDEQ is not only competitive with comparable existing baseline models, but also achieves higher performance than a network with 5 times more layers, demonstrating that the QDEQ paradigm can be used to develop significantly more shallow quantum circuits for a given task, something which is essential for the utility of near-term quantum computers.

Abstract

The feasibility of variational quantum algorithms, the most popular correspondent of neural networks on noisy, near-term quantum hardware, is highly impacted by the circuit depth of the involved parametrized quantum circuits (PQCs). Higher depth increases expressivity, but also results in a detrimental accumulation of errors. Furthermore, the number of parameters involved in the PQC significantly influences the performance through the necessary number of measurements to evaluate gradients, which scales linearly with the number of parameters. Motivated by this, we look at deep equilibrium models (DEQs), which mimic an infinite-depth, weight-tied network using a fraction of the memory by employing a root solver to find the fixed points of the network. In this work, we present Quantum Deep Equilibrium Models (QDEQs): a training paradigm that learns parameters of a quantum machine learning model given by a PQC using DEQs. To our knowledge, no work has yet explored the application of DEQs to QML models. We apply QDEQs to find the parameters of a quantum circuit in two settings: the first involves classifying MNIST-4 digits with 4 qubits; the second extends it to 10 classes of MNIST, FashionMNIST and CIFAR. We find that QDEQ is not only competitive with comparable existing baseline models, but also achieves higher performance than a network with 5 times more layers. This demonstrates that the QDEQ paradigm can be used to develop significantly more shallow quantum circuits for a given task, something which is essential for the utility of near-term quantum computers. Our code is available at https://github.com/martaskrt/qdeq.

Paper Structure

This paper contains 21 sections, 1 theorem, 43 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Deep Equilibrium Models
  4. Quantum Models and Variational Algorithms for Classification
  5. A Quantum Deep Equilibrium Model
  6. Experiments
  7. Datasets
  8. Architecture
  9. Models and baselines
  10. Results
  11. MNIST-4
  12. MNIST-10 and FashionMNIST-10
  13. CIFAR-10
  14. Conclusion and open problems
  15. Existence of fixed points for quantum model families
  16. ...and 6 more sections

Key Result

Theorem 3

Let $\mathcal{E}_i(\cdot) = U(\theta^{(i)})(\cdot)U^\dagger(\theta^{(i)})$ be a channel corresponding to the PQC at depth $i$. Additionally, we define an map $R': \mathbb{C}^{2^Q\times 2^Q} \to \mathbb{R}^K$ that describes performing measurements of expectation values with respect to an ensemble of Then, a sequence of such layers can be replicated exactly by an input-injected, weight-tied network

Figures (4)

  • Figure 1: Instance of a Deep Equilibrium Model using a quantum model family. A black-box root-finding method is used to determine the model function's equilibrium state.
  • Figure 2: Circuit used for classification with up to 4 classes, following wang_quantumnas_2022. Blue circuits correspond to input, purple and red shades to parametrized and trainable gates and grey to fixed gates. The RandomLayer has on average 12.5 = 50/4 two-qubit gates (CNOTs).
  • Figure 3: Extension of the circuit presented in wang_-chip_2022 to a scenario of 10 qubits. Similarly to the tensor-network inspired circuits in dilip2022data, we repeat the four-qubit stencils in a staircase manner.
  • Figure 4: Plots of the relation between the single-qubit overlaps $\left| \left< 0 \right| S_{\mathbf{z}_k}^{(1) \dagger} S_{\mathbf{z}'_k}^{(1)} \left| 0 \right> \right|$ and the norm $\norm{\mathbf{z}_k-\mathbf{z}_k'}^2$ for 3000 pairs of random vectors in $\mathbb{R}^4$. Note that all of the points lie above the line corresponding to $1-\sin\left(\norm{\mathbf{z}_k-\mathbf{z}_k'}^2 \right)$.

Theorems & Definitions (2)

  • Definition 1: Family of Quantum Model Functions
  • Theorem 3: Universality of weight-tied quantum models, in analogy to Theorem 3 in bai_deep_2019