Quantum learning advantage on a scalable photonic platform

TL;DR

This work presents a photonic implementation of a quantum-enhanced protocol for learning the probability distribution of a multimode bosonic displacement process and demonstrates that even with non-ideal, noisy entanglement, a significant quantum advantage can be realized in continuous-variable quantum systems.

Abstract

Recent advancements in quantum technologies have opened new horizons for exploring the physical world in ways once deemed impossible. Central to these breakthroughs is the concept of quantum advantage, where quantum systems outperform their classical counterparts in solving specific tasks. While much attention has been devoted to computational speedups, quantum advantage in learning physical systems remains a largely untapped frontier. Here, we present a photonic implementation of a quantum-enhanced protocol for learning the probability distribution of a multimode bosonic displacement process. By harnessing the unique properties of continuous-variable quantum entanglement, we obtain a massive advantage in sample complexity with respect to conventional methods without entangled resources. With approximately 5 dB of two-mode squeezing -- corresponding to imperfect Einstein--Podolsky--Rosen (EPR) entanglement -- we learn a 100-mode bosonic displacement process using 11.8 orders of magnitude fewer samples than a conventional scheme. Our results demonstrate that even with non-ideal, noisy entanglement, a significant quantum advantage can be realized in continuous-variable quantum systems. This marks an important step towards practical quantum-enhanced learning protocols with implications for quantum metrology, certification, and machine learning.

Figures (13)

• Figure 1: Quantum entanglement-enhanced learning with photons. (a) Conventional learning of a channel. A multi-mode probe state is sent through a channel, followed by a measurement of the probe state to extract the information about the channel. (b) Conventional learning of a multi-time physical process, where the measurement settings are allowed to be adaptive within a sample. We show that for both (a) and (b), a fundamental entanglement-free complexity bound applies to the required sample overhead for the learning task. (c) Quantum entanglement-enhanced learning of a multi-time process. The probe state is allowed to be entangled with an external memory state. The joint measurement of both states makes overcoming the classical complexity limit possible. (d) Implementation of quantum learning with squeezed light. Two-mode squeezing is generated by interfering the outputs of two optical parametric oscillators (OPOs). One of the spatial modes is temporally-multiplexed and used as the probe state while the other is used as the memory. The physical process to be learned is a phase-space random displacement. A Bell measurement between the corresponding modes of the probe and memory states works to extract the information. (e) We realize the displacement process by mixing a frequency-shifted coherent state into the probe. The frequency-shifted coherent state is shaped by two electro-optic modulators, an intensity modulator (IM) and a phase modulator (PM).
• Figure 2: Reconstruction of a physical process. (a) Experimentally reconstructed characteristic function $\tilde{\lambda}({\beta})$ of an $n=16$-mode three-peak process (defined in Supplementary Material, Definition S4) with fixed parameters using entanglement-free strategies, compared with the true characteristic function $\lambda({\beta})$. The lines (shadings) show the average outcome ($1\sigma$ standard deviation) of 100 runs of the reconstruction task using different numbers of samples. (b) Same as above, but using entangled probe states. Here the number of modes is $n=30$ and we always use $10^5$ samples for the same task. (c) Required number of samples to $\epsilon$-close reconstruct $\lambda({\beta})$ of the three-peak process along the $\beta_0$ direction with a success probability of $1-\delta = 2/3$, versus the number of modes. The points are determined from experimental results, and the $1\sigma$ standard deviation error bars are smaller than the data points. The solid lines are log-linear fits. The gray dashed line is the sample-complexity lower bound that applies to any entanglement-free strategy that can learn all processes in a large family which includes the process we studied. (d) Probability of achieving an $\epsilon$-close reconstruction of the $-4.78dB$, 30-mode characteristic function for various directions in the dual space. The shading highlights the proximity to the displacement direction $\beta_0$. Each probability is computed using $N=1472$ samples---same as required for an $\epsilon$-close reconstruction in (c). The dashed line indicates the target probability of $1-\delta$.
• Figure 3: Hypothesis testing. (a) The objective is to distinguish whether a displacement process belongs to the three-peak family with an unknown parameter or the Gaussian family. (b) An example of the separation of the estimator, $\tilde{\lambda}(\beta=\gamma)$, for two types of 40-mode displacement processes using different amounts of squeezing. In the noiseless case, the value is expected to be 0.5 (0) for three-peak (Gaussian) channel. (c) Sample complexity for achieving $2/3$ success probability in a $\kappa=0.2$ hypothesis test, estimated with varying amounts of squeezing. The solid and dashed lines indicate the classical complexity bound for achieving the same success probability and the exponential fit, respecively. The shading indicates the region excluded by the classical complexity bound (see Supplementary Material Sec. VB for more details). (d) Inset: the measured probability of winning the $\kappa=0.2$ hypothesis testing game for different numbers of modes, using $10^5$ samples and various amounts of squeezing. Solid lines represent a pessimistic estimation of success probability derived from the Hoeffding bound Oh2024. Main: Minimum sample complexity for any conventional strategy to achieve the same success probability as reported in the inset, calculated according to the classical complexity bound, and the corresponding sample collection time at a 1 MHz/mode rate. Error bars represent the $1\sigma$ standard deviation from a 25-step sequential sampling. The shaded region indicates the existence of a quantum advantage.
• Figure S1: Full experimental setup for quantum-enhanced learning with squeezed light. Two-mode squeezed vacuum (TMSV) is created by combining the outputs of two degenerate OPOs with orthogonal polarizations using a polarization beam splitter (PBS), and then Hadamard-mixing the polarizations with a half-wave plate (HWP). A coherent state at the correct sideband frequency is created by intensity (IM) and phase (PM) modulating a classical laser beam, and the displacement operation is induced by mixing this coherent state on a tapping mirror with one polarization modes of the TMSV. The modulators are driven by an arbitrary waveform generator (AWG) to implement the desired random displacement process. A Bell measurement is performed by reverting the polarization mixing of the two OPOs and interfering them with the two polarization modes of a strong classical beam (local oscillator, LO). A function generator (FG) generates the lock--measure signal, turning off the locking beam during the measurement period and turning it back on during the locking period using a fiber-coupled switch. This signal is also used as a trigger signal for the data acquisition digitizer and the AWG. A second function generator produces the Pound--Drever--Hall sidebands (PDH) on the locking beam by using an electro-optical phase modulator (EOM).
• Figure S2: A zero-span noise power measurement of the homodyne detector's output, acquired on a spectrum analyzer centering on the $3.8MHz$ frequency sideband. The trace is recorded during a sweeping time of $17.5ms$ starting at the end of the locking period. The min-hold squeezing is from releasing the LO phase lock and capturing the lowest noise power possible. The increase of squeezing at the first $2ms$ of the trace is due to the response time of the MEMS switch shutting down the locking beam.

Theorems & Definitions (10)

• Definition 1: Probability distribution of a process
• Definition 2: Characteristic function
• Definition 3: Quadrature operators
• Theorem 1: Unbiased characteristic function estimator
• Theorem 2
• Theorem 3
• Definition 4: three-peak process
• Definition 5: Gaussian process
• Theorem 4
• Definition 6: $(\epsilon, \delta)$-close reconstuction