Geometric quantum machine learning of BQP$^A$ protocols and latent graph classifiers

TL;DR

This Letter considers Simon's problem for learning properties of Boolean functions, and shows that this can be related to an unsupervised circuit classification problem, thus discovering an example of BQP$^A\neq$BPP protocol with respect to some dataset (oracle $A$).

Abstract

Geometric quantum machine learning (GQML) aims to embed problem symmetries for learning efficient solving protocols. However, the question remains if (G)QML can be routinely used for constructing protocols with an exponential separation from classical analogs. In this Letter we consider Simon's problem for learning properties of Boolean functions, and show that this can be related to an unsupervised circuit classification problem. Using the workflow of geometric QML, we learn from first principles Simon's algorithm, thus discovering an example of BQP$^A\neq$BPP protocol with respect to some dataset (oracle $A$). Our key findings include the development of an equivariant feature map for embedding Boolean functions, based on twirling with respect to identified bitflip and permutational symmetries, and measurement based on invariant observables with a sampling advantage. The proposed workflow points to the importance of data embeddings and classical post-processing, while keeping the variational circuit as a trivial identity operator. Next, developing the intuition for the function learning, we visualize instances as directed computational hypergraphs, and observe that the GQML protocol can access their global topological features for distinguishing bijective and surjective functions. Finally, we discuss the prospects for learning other BQP$^A$-type protocols, and conjecture that this depends on the ability of simplifying embeddings-based oracles $A$ applied as a linear combination of unitaries.

Figures (4)

• Figure 1: Simon's algorithm based on black-box quantum oracles $\hat{\mathcal{U}}_f$ that encode $N$-bit Boolean functions over $2N$ quantum registers. Due to quantum parallelism the measured samples $X \sim p(x)$ contain information about a hidden bitstring $s$. This is obtained from classical post-processing, and defines whether the function is 1:1 or 2:1.
• Figure 2: (a) Workflow for 2:1 and 1:1 function classification based on GQML, where data are mapped from the function to quantum space by equivariant embedding, and probability distributions are studied based on invariant observables. This is equivalent to Simon's algorithm (b).
• Figure 3: Results for the classical post-processing of equivariantly embedded functions. We used unsupervised approaches to classification corresponding to kernel PCA (a); k-means clustering (b); and one-class SVM with linear kernels (c). In (d) we show how the classification performance (F$_1$ score) increases with the number of shots.
• Figure 4: Visualization of function learning process as a latent-space graph classification. (a) Qualitatively different directed computational hypergraphs that emerge from 2:1 and 1:1 functions. Each vertex is a basis state ($N=6$), and directed edges induced by the action of oracle $\hat{U}_f$ point to output states. (b) Degree shown for every vertex in the directed hypergraph. For $f_{1:1}$ functions this is flat, while for $f_{2:1}$ it is non-uniform. (c) Graph Betti number of the zeroth order, $\beta_0$, showing that classification can be performed based on topological properties.

Theorems & Definitions (1)

• Conjecture 1