Quantum reservoir computing with a single nonlinear oscillator

Realizing the promise of quantum information processing remains a daunting task given the omnipresence of noise and error. Adapting noise-resilient classical computing modalities to quantum mechanics may be a viable path towards near-term applications in the noisy intermediate-scale quantum era. Here, we propose continuous variable quantum reservoir computing in a single nonlinear oscillator. Through numerical simulation of our model we demonstrate quantum-classical performance improvement and identify its likely source: the nonlinearity of quantum measurement. Beyond quantum reservoir computing, this result may impact the interpretation of results across quantum machine learning. We study how the performance of our quantum reservoir depends on Hilbert space dimension, how it is impacted by injected noise, and briefly comment on its experimental implementation. Our results show that quantum reservoir computing in a single nonlinear oscillator is an attractive modality for quantum computing on near-term hardware.

Figure 1

Input-output scheme for nonlinear oscillator RC. A sinusoidal input signal $u\left(t\right)$ at frequency ${\omega }_u$ is up-converted to a double-sideband modulation of a carrier tone centered at the nonlinear oscillator resonance frequency $\mathrm{\Omega }$, where $\mathrm{\Omega }\gg {\omega }_u$. This is input to the $P$ quadrature of the oscillator, and the $X$ quadrature is continuously measured by monitoring the output field of the oscillator. The circulator at the input/output port of the nonlinear oscillator separates the propagation of the input and output signals.

Figure 2

Quantum and classical reservoir performance. As a function of training set size, average root mean square (RMS) error on the test set for the quantum nonlinear oscillator reservoirs (QRC) with Hilbert space dimension $d=12$, and the classical nonlinear oscillator reservoirs (CRC). The average results are for 101 reservoirs with Gaussian distributed parameters around the mean values described in the main text. The reservoir output is sampled at 100 equally spaced points in time, and the test set size is 5000. One standard deviation for the QRC performance is shown in the shaded blue region. The RMS error for the QRC and CRC with best and worst case performance (determined independently for each training set size) is also shown.

Figure 3

Hilbert space dimension effect on QRC performance. (a) Average RMS error as a function of training set size for quantum nonlinear oscillator reservoirs with various Hilbert space dimension. One standard deviation for 101 reservoirs with Gaussian distributed parameters is shown in the shaded regions. (b),(c) Average RMS error as a function of Hilbert space dimension, for (b) various training set sizes and (c) for a training set size of 30, which is near optimal. In (c) one standard deviation is shown in the shaded area.

Figure 4

Reservoir performance for qubit and classical models. As a function of training set size, average root mean square (RMS) error on the test set for a qubit reservoir, either with a single measured expectation value (QRC) or full tomography (full QRC), the hidden variable model for a qubit as a reservoir (HVRC), and classical nonlinear oscillator reservoirs with single quadrature (CRC) or both quadrature (full CRC) measurements. Reservoir parameter set, measurement sampling, and test set size are the same as in Fig. 2. One standard deviation for the QRC and full QRC performance is shown in the shaded regions.

Figure 5

Performance spread for qubit and classical models. As a function of training set size, spread in root mean square (RMS) error, cf. Eq. (15), on the test set for a qubit reservoir, either with a single measured expectation value (QRC) or full tomography (full QRC), the hidden variable model for a qubit as a reservoir (HVRC), and classical nonlinear oscillator reservoirs with single quadrature (CRC) or both quadrature (full CRC) measurements. Output sampling and test set size the same as in Figs. 2 and 4.

Figure 6

Quantum and classical reservoir performance with noise. Root mean square (RMS) error on the test set for the quantum nonlinear oscillator reservoirs (QRC) with Hilbert space dimension $d=12$, and the classical nonlinear oscillator reservoirs (CRC), as a function of noise standard deviation in units of the input drive amplitude ($\sigma /\alpha$) for noise injected in either the output or the input. These results are for a single set of reservoir parameters described in the main text. Output sampling is as in Fig. 2, the training set size is 10, and the test set size is 5000 for output noise and 100 for input noise.