Learning functions of quantum states with distributed architectures

TL;DR

The paper develops and benchmarks distributed Quantum Extreme Learning Machines (QELMs) for learning functions of quantum states directly from data using solely computational-basis projective measurements. It introduces four architectures—single three-layer, spatial multiplexing, multiple injections, and a distributed entangled design—and derives resource-scaling bounds tying measurement outcomes and reservoir dimensions to the target class (linear vs nonlinear). Numerical experiments with a three-qubit input and an ergodic Ising-like reservoir demonstrate that linear targets are achievable with scalable, parallelizable SM, while nonlinear targets (polynomial, Rényi entropy, entanglement) require increasing numbers of interacting subsystems, with the distributed architecture offering a hardware-efficient route by distributing entangled reservoirs. The results illuminate how architectural choices map to the class of learnable quantum properties, providing a practical framework for quantum property learning on near-term devices and guiding future experimental implementations and comparisons with tomography- and shadow-based approaches.

Abstract

Distributed architectures are gaining prominence in quantum machine learning as a means to overcome hardware limitations and enable scalable quantum information processing. In this context, we analyze the design and performance of distributed Quantum Extreme Learning Machine (QELM) architectures for learning functions of quantum states directly from data, restricting measurements to easily implementable projective measurements in the computational basis. The aim is to determine which schemes can effectively recover specific properties of input quantum states, including both linear and nonlinear features, while also quantifying the resource requirements in terms of measurements and reservoir dimensionality. We compare standard three-layer QELM with a spatially multiplexed architecture composed of multiple independent three-layer units for linear (quantum) tasks, showing a linear reduction in resource requirements per unit. For nonlinear properties, the study examines the multiple-injection architecture and introduces a novel distributed design that incorporates entanglement between subsystems within a spatially multiplexed framework, evaluating its performance through the reconstruction of complex nonlinear quantities such as polynomial targets, Rényi entropy, and entanglement measures. Our results demonstrate that the distributed design enables the reconstruction of higher-order nonlinearities by increasing the number of interacting subsystems with reduced resources, rather than increasing the size of an individual reservoir, providing a scalable and hardware-efficient route to quantum property learning.

Figures (14)

• Figure 1: Schematic setup of the architectures. On the left are the architectures for reconstructing linear properties: (a) and (b). On the right are the architectures for reconstructing nonlinear properties: (c) and (d).
• Figure 2: Qubit requirements across different architectures for a three-qubit input. (a) Comparison between the single three-layer (S3L) and spatially multiplexed (SM) architectures for linear property reconstruction. In S3L, $n_{res}$ denotes the number of qubits in the reservoir, and $n_{tot}$ the total number of qubits used. In SM, $n_{res}$ represents the number of qubits in a single reservoir and $n_{unit}$ the total number of qubits per unit, including the input. (b) Comparison of the multiple-injections (MI) and distributed (D) architectures for nonlinear property reconstruction. In MI, $n_{res}$ represents the number of qubits in the reservoir, while $n_{tot}$ denotes the total number of qubits, including the input. In D, $n_{res}$ represents the number of qubits in a single reservoir, $n_{res}^{tot}$ denotes the total number of qubits across all reservoirs, and $n_{tot}$ represents the overall number of qubits, including all considered inputs.
• Figure 3: Reconstruction error in estimating $Tr[ (\sigma_x \otimes \sigma_x) \rho ]$ for a 2-qubit input using the spatially multiplexed architecture, evaluated for varying numbers of reservoirs ($n$) and qubits per reservoir ($n_{res}$). The error is depicted as a function of the number of available PVM outcomes obtained from measurements in the computational basis $\Pi_z$, which changes depending on the number of reservoirs and the number of qubits in each. The coupling parameter is sampled from $J \in [-1,1]$ and h=1. Each point represents the mean over 100 experimental runs, with bars indicating the standard deviation. The inset shows the mean values on a logarithmic scale, highlighting the point at which perfect reconstruction is achieved.
• Figure 4: Reconstruction error in estimating the purity of a 1-qubit input using all the architectures: single three-layer (S3L), spatially multiplexed (SM), multiple-injections (MI), and distributed (D). The number of qubits in the reservoirs is selected so that all architectures yield the same number of PVM outcomes, with SM and D using 2 reservoirs each and MI performing 2 input injections to enable the reconstruction of second-order nonlinearities. The measurement basis, parameter choices, and statistical representation are the same as in Figure \ref{['fig:smex']}.
• Figure 5: Reconstruction error in estimating polynomial targets of the form $Tr[O\rho^k]$ for a 1-qubit input, using the distributed architecture with a varying number of reservoirs. The number of qubits in each reservoir is adjusted according to the total number of reservoirs used. The top panel shows results for $O=I$ and the bottom panel for $O=\sigma_x$, with degree $k=2,3,4,5$ in each case. The measurement basis, parameter choices, and statistical representation are the same as in Figure \ref{['fig:smex']}.