Single-shot Quantum State Classification via Nonlinear Quantum Amplification

TL;DR

This work tackles single-shot quantum state discrimination by exploiting nonlinear quantum amplification within a realistic superconducting readout chain. It introduces end-to-end optimization of a two-SNAIL system (squeezer and analyzer) and task-specific performance metrics, notably the mean separation $\Delta\boldsymbol{\mu}$ and Fisher discriminant $D_F$, to identify operating regimes where nonlinear processing surpasses linear, quantum-limited limits. Using stochastic master equations, cumulant-based TEOMs, and a weak-nonlinearity perturbative analysis, the authors show how joint tuning of pump and drive phases and amplitudes can map higher-order moment information into observable first-order statistics, achieving near-unity discrimination fidelities in favorable points. The results imply practical nonlinear amplifiers could enable high-fidelity, low-power qubit readout and motivate a broader end-to-end optimization framework for nonlinear quantum sensing and information processing, including extensions to higher-order moments beyond Gaussian approximations.

Abstract

Quantum amplifiers are intrinsically nonlinear systems whose performance limits are set by quantum mechanics. In quantum measurement, amplifier operation is conventionally optimized in the linear regime by maximizing signal-to-noise ratio, an objective that is well-suited to parameter estimation but is typically insufficient for more general tasks such as arbitrary quantum state discrimination. Here we show that single-shot quantum state classification can benefit from operating a quantum amplifier outside the linear regime, when the measurement chain is optimized end-to-end for a task-specific cost function. We analyze a realistic superconducting readout architecture that includes state preparation, cryogenic nonlinear amplification, and room-temperature detection with finite noise. By introducing performance metrics tailored to state discrimination, we identify operating regimes in which nonlinear amplification provides a measurable advantage and clarify the trade-offs that ultimately limit classification fidelity. Our results propose the utility of practical nonlinear quantum amplifiers for quantum state discrimination, and are the first step in a broader research program aimed at developing a general framework for end-to-end, resource-limited optimization of nonlinear quantum amplifiers for such quantum information processing applications.

Figures (4)

• Figure 1: (a) The composite two-SNAIL system for state generation and discrimination. The pump applied to the squeezer (pump 1) sets which input state is generated, (1-blue) or (2-orange). Two additional drives at the pump frequency $\omega_p$ (pump 2) and at $\omega_p/2$ (signal drive with strength $\eta_{d,2}$) act on the analyzer and impact the overall classification performance. The two SNAIL amplifiers are identical except that the analyzer has a non-zero Kerr nonlinearity ($\Lambda$) (b) Intracavity quadrature covariance matrix elements and quadrature measurements for two input states with the analyzer pump off ($g_2 = 0$). (c) When the analyzer is on, the output-state means coincide for linear operation ($\eta_{d,2} \approx 0$) and the classification fidelity is 0.84, but are separated for nonlinear operation with a sufficiently-large signal drive ($\eta_{d,2} \gg 0$) and the classification fidelity is 1 (this is the same operating point as in Fig. 2(d-2)).
• Figure 2: Variation of (a) normalized mean separation and (b) normalized Fisher discriminant versus pump strength $g_2$ (y-axis) and signal drive strength $\eta_{d,2}$ (x-axis). The vertical and horizontal black dashed lines in (a) indicate the cross sections shown in insets (c) and (d), respectively. The white dashed line marks the pump value giving 20 dB analyzer gain under the stiff-pump approximation. (c) Normalized mean separation versus $g_2$, along with measured quadratures for the two quantum-state types at two pump operating points shown in the insets below. The classification fidelity for instances 1 and 2 are 0.855 and 0.995 respectively. (d) Normalized mean separation versus $\eta_{d,2}$, with measured quadratures at three signal drive operating points shown in the insets below. The classification fidelity for instances 1, 2 and 3 are 0.845, 1 and 0.9 respectively. The data shown is obtained by solving Eqs. \ref{['eq-app-teoms']} and the data shown in the insets of (c) and (d) is obtained by solving Eqs. \ref{['eq-app-steoms']} with 100 sample trajectories for each type of input state with integration time 800/$\kappa_2$.
• Figure 3: Variation of (a) normalized mean separation and (b) normalized Fisher discriminant versus pump phase $\phi_2$ (y-axis) and signal drive phase $\phi_{d,2}$ (x-axis). Inset (c) shows the normalized mean separation at fixed $\phi_{d,2}=0$ versus $\phi_2$, with inset (d) displaying corresponding measured quadratures for the two quantum state types at the operating points marked in (c). The classification fidelity for instances 1, 2, 3 and 4 are 0.81, 1, 0.785 and 1 respectively. Inset (e) illustrates the $\phi_1$ values on the unit circle generating type-1 and type-2 states (blue and orange), with the black arrow indicating $\phi_2$ at the chosen operating point. Inset (f) shows normalized mean separation at fixed $\phi_2=0$ versus $\phi_{d,2}$, and inset (g) shows the measured quadratures at the operating points marked in (f). The classification fidelity for instances 1, 2, 3 and 4 are 0.84, 0.99, 0.87 and 0.995 respectively. Inset (h) depicts the anti-squeezing (amplification) direction set by the fixed pump phase ($\phi_2 = 0$), aligned with the type-1 state (blue), with the black arrow marking the corresponding $\phi_{d,2}$. The data shown is obtained by solving Eqs. \ref{['eq-app-teoms']} and the data shown in the insets (d) and (g) is obtained by solving Eqs. \ref{['eq-app-steoms']} with 100 sample trajectories for each type of input state with integration time 800/$\kappa_2$.
• Figure 4: Normalized mean separation and the Fisher discriminant versus pump phase for different classical noise strengths (darker colors indicate larger noise). Black dashed lines mark the mean separation maxima, and grey dashed lines mark the Fisher discriminant maxima in the zero classical noise case. Colored dashed lines indicate the Fisher discriminant maxima for nonzero noise, using matching colors to the corresponding solid curves.