Topological data analysis on noisy quantum computers

TL;DR

NISQ-TDA is presented, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems.

Abstract

Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical algorithms for computing TDA are exorbitant, and quickly become impractical for high-order characteristics. Quantum computers offer the potential of achieving significant speedup for certain computational problems. Indeed, TDA has been purported to be one such problem, yet, quantum computing algorithms proposed for the problem, such as the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi, require fault-tolerance qualifications that are currently unavailable. In this study, we present NISQ-TDA, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.

Figures (9)

• Figure 1: Results from real hardware of Laplacian applications (using measure and reset projections): A. Circuit depth versus the number of vertices for degree $m=1$ and $3$; (B., C. and D.) Histograms of the probability measurements as obtained from the hardware (right, magenta bars) and from a simulator (left, blue bars) for three different datasets namely, an edge (2 vertices), a square (4 vertices), and a cube (8 vertices). $\phi$ defines the null state, and 'X' denotes the probability mass with incorrect flag readings.
• Figure 2: Results from noisy simulations: A. Mean error surface as a function of the noise levels in (1-qubit, 2-qubits) gates and (number of vertices $n$, circuit depth). B. Mean and the variance of the Betti number estimated as a function of the number of random vectors $\mathrm{n_{v}}$ with $n=8$ vertices, degree $m=5$ and the noise-level: $(0.001, 0.01)$.
• Figure 3: The armed quantum computing advantage race: a set of seven criteria that an aspired quantum algorithm should satisfy (left), and the end-to-end quantum computational model (right), where the funnels symbolize the narrow input and output bottlenecks and the boxes represent the other desired properties.
• Figure 4: A: $k$-simplices or $k$-chains (fully connected sets of $k$+1 data points) shown for $k=0,1,2,3$. B: Point cloud of raw data (left); Points can be connected using any arbitrary distance metric $\varepsilon$ (middle), i.e., edges are inserted between points that are within $\varepsilon$ of each other (alternatively, the data could already come with edge information); higher-order $k$-simplices are created for every $k$-clique (right). C. Input data represented as a graph or network, or is given as (D) an arbitrary complex. E: Persistent homology, where the top region shows the edge connections and simplices at different scales, and the bars represent the formation and cessation of connected components ($H_0$) and 2D holes ($H_1$); The number of bars at a given scale $\varepsilon$ equals the Betti numbers $\beta_0$ and $\beta_1$, respectively. F. Chain complex and homology: Sequence of chain groups connected by boundary operators that map $k$-chains to their boundaries.
• Figure 5: Projection onto the simplicial complex $P_{\Gamma}$: Example circuit diagram with $n=4$ vertices (and six edges)

Theorems & Definitions (6)

• Theorem 1
• Remark 1: Time Complexity Discrepancy
• Lemma 1
• Lemma 2: Chebyshev Minimizing Polynomial musco2015randomized
• Proposition 1
• proof