Machine learning of quantum data using optimal similarity measurements

TL;DR

A sample-optimal, hardware-efficient protocol for estimating quantum similarity -- the state overlap -- using bosonic quantum interference, and establishes joint overlap measurements as a scalable pathway to efficient quantum data analysis and a practical building block for network-integrated quantum machine learning.

Abstract

Quantum machine learning seeks a computational advantage in data processing by evaluating functions of quantum states, such as their similarity, that can be classically intractable to compute. For quantum advantage to be possible, however, it is essential to bypass costly characterisation of individual data instances in favour of efficient, direct similarity evaluation. Here we demonstrate a sample-optimal, hardware-efficient protocol for estimating quantum similarity -- the state overlap -- using bosonic quantum interference. The sample complexity of this approach is independent of the system dimension and is information-theoretically optimal up to a constant factor. Experimentally, we implement the scheme on \emph{Prakash-1}, a quantum computing platform based on a fully programmable integrated photonic processor. By preparing and interfering qudit states on the chip to directly extract their overlap, we demonstrate classification and online learning of quantum data with high accuracy in realistic noisy experiments. Our results establish joint overlap measurements as a scalable pathway to efficient quantum data analysis and a practical building block for network-integrated quantum machine learning.

Figures (10)

• Figure 1: Quantum machine learning (QML) with similarity measurements. (a) QML can process quantum states that encode classical data, though it also shows potential in learning native quantum data, such as the output states from fault-tolerant quantum computers, analogue quantum simulators, quantum sensors or information passed through large-scale quantum networks. A hierarchy of measurements exists to measure the similarity between two unknown quantum states via: (b) Global measurement on $N$ copies of each state; (c) Sequential joint measurement using multi-mode bosonic interference on single copy of each state, derived and demonstrated in this work; (d) Distributed local measurements on $N$ copies of each state separately followed by classical post-processing, an example of which we derive for CV states. Within this hierarchy, the interference-based joint measurement combines optimal sample complexity with experimental accessibility. For application in QML, we demonstrate the measurement as evaluation of (e) kernel functions for quantum data classification and (f) cost function for online learning. PNRD: photon-number-resolving detector.
• Figure 2: An illustration of the fully-programmable $\text{Si}_3\text{N}_4$ integrated photonic processor at the heart of the quantum computing platform Prakash-1. A photonic integrated circuit (PIC), whose temperature is maintained by a thermo-electric cooling (TEC) unit, employs a square mesh of symmetric Mach-Zehnder interferometers (MZIs), with heating elements acting as thermo-optical phaseshifters in each arm of the MZIs. Collectively, the MZIs plus ten stand-alone on-chip phaseshifters (not used in this work and not shown) can implement an arbitrary $10\times10$ unitary transformation according to the Bell-scheme (see inset). The dashed line encloses the MZIs that are tuned in the experiment. The grey, teal, and blue areas indicate the MZIs used to route the photon pair, prepare the qudits, and perform the multi-mode interference, respectively. The output eight modes are fibre-coupled and routed to superconducting nanowire single-photon detectors (SNSPDs).
• Figure 3: Quantum data classification. (a) Schematic of experimental routine for kernel function evaluation. (b-d) Classification results for linearly separable data, spherically separable data and overlapping data respectively. The top panels visualise the test datasets. Each dataset contains $100$ datapoints, which are three-dimensional vectors $\bm{x}=[\theta_1, \theta_2, \theta_3]$ represented by points in 3D space. The colour of each point indicates its groundtruth label, and is plotted with a circle marker if it is correctly classified, or a cross marker if it is misclassified. The bottom panels show the experimentally estimated overlap values (colour bar) between the test data and training data for each dataset. The datapoints are sorted by their labels for visualisation purposes.
• Figure 4: Quantum data online learning. (a) Schematic of experimental routine for cost function evaluation. (b) Infidelity between the learned qudit state and the target qudit state against iteration number. Ten random target states are iteratively learned starting from random initial states, with $N=10^2$ state copies used per overlap evaluation (light blue dashed lines). The median infidelity across the ten learning tasks is shown by the dark blue solid line. The experiment is repeated for different number of state copies per overlap evaluation, $N=10^3$ and $N=10^4$, with median performance shown in green and purple dashed lines respectively. The shaded regions indicate the interquartile range (i.e. between the 25% and 75% quantiles) across simulations over $500$ randomised learning tasks, with median shown by the solid yellow line. (c) Optimisation paths for three example experiments in (b) with $N=10^2$, visualised in the three-dimensional space of qudit phases. The red star markers plot the target point, the blue circle markers plot the initial point of each path, and the yellow triangle markers plot the optimal point found after $500$ iterations.
• Figure 5: An illustration of the experimental setup. A Ti:Sapphire laser produces 775 nm wavelength, $\sim$1 ps long pulses, at 80 MHz, which are spectrally filtered to 0.9 nm bandwidth (full-width at half-maximum) using a pair of angle-tuned bandpass filters (BPF). The pulses are coupled into a 10 mm long ppKTP waveguide that is phase-matched for type-II SPDC. The residual pump is filtered using a long-pass filter (LPF) and a polarising beamsplitter (PBS) separates signal and idler pulses along separate paths, each containing a HWP and a BPF with a matched bandwidth to eliminate spectral correlations between signal and idler. The photon pair is delivered to the fully programmable $\text{Si}_3\text{N}_4$ PIC via polarisation-maintaining (PM) fibre. The thermo-optic phaseshifters on the PIC are programmed by a multi-channel source-measure unit (SMU), and a thermoelectric cooler (TEC) maintains the temperature of the PIC at $28^{\circ}\text{C}$. The eight output channels of the PIC are coupled to PM fibre, enabling detection by superconducting nanowire single-photon detectors (SNSPDs).

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem S1
• proof
• Theorem S2
• proof
• Definition S4
• Lemma S5
• Theorem S6