Direct estimation of arbitrary observables of an oscillator

Abstract

Quantum harmonic oscillators serve as fundamental building blocks for quantum information processing, particularly in the context of the bosonic circuit quantum electrodynamics (cQED) platform. Conventional methods for extracting oscillator properties rely on predefined analytical gate sequences to access a restricted set of observables or resource-intensive tomography processes. Here, we introduce the Optimized Routine for Estimation of any Observable (OREO), a numerically optimized protocol that maps the expectation value of arbitrary oscillator observables onto that of an ancillary qubit. We demonstrate OREO in a bosonic cQED system as a means to efficiently measure phase-space quadratures and their higher moments, directly obtain faithful non-Gaussianity ranks, and effectively achieve state preparation independent of initial conditions in the oscillator. These results position OREO as a valuable tool for direct and efficient information extraction from bosonic quantum states, unlocking new possibilities for measurement, control, and state preparation in continuous-variable quantum information processing.

Figures (12)

• Figure 1: Observable mapping. (a) The Optimize Routine for Estimation of any Observable (OREO) maps the expected value of any target observable $\langle \hat{O}\rangle$ of a harmonic oscillator state onto the z-axis of a qubit's Bloch vector $\langle\hat{\sigma}_z\rangle$. (b) The cQED setup and protocol, which consists of a numerically optimized pulse $\hat{U}_{\text{m}}$ applied to the oscillator-qubit system, followed by a qubit readout. A step-by-step implementation guide is included in the Supplemental Material sm.
• Figure 2: Direct estimation of polynomials of phase-space quadratures. a) Application of OREO to directly obtain the values of $\langle \hat{x} \rangle$, $\langle \hat{x}^2 \rangle$, $\langle \hat{p}+\hat{x}^2 \rangle$, and $\langle (\hat{p}+\hat{x}^2)^2 \rangle$ of a variable coherent state $|\alpha\rangle$. The dashed lines indicate one-dimensional cuts in the data plotted in (b), which quantitatively agree with the theoretical expected values (solid curves).
• Figure 3: Non-Gaussianity rank. (a) Reconstructed Wigner of Fock state $|3\rangle$. (b) Measurement of $\langle\hat{\Pi}_3 + \lambda\hat{\Pi}_4 \rangle$ ($\hat{\Pi}_n \equiv |n\rangle\langle n|$) with OREO is used to infer the rank under photon loss for a variable wait time and parameter $\lambda$. The theoretical threshold is shown as the lower boundary of the blue shaded region, above which one infers rank $3$ non-Gaussianity. (c) Reconstructed Wigner for $(|0\rangle + |3\rangle)/\sqrt{2}$, prepared with strong induced dephasing drive. (d) Measurement of $\langle \hat{\Xi}_3 + \lambda \hat{\Pi}_4\rangle$ ($\hat{\Xi}_n \equiv |n\rangle\langle0| + |0\rangle\langle n|$) with OREO under both strong and weak dephasing drives. Both in panels (b) and (d), the outcomes obtained via OREO agree with values derived from the full density matrix reconstruction (dotted lines) as well as full simulations with decoherence (dashed lines).
• Figure 4: Projection to a target oscillator state. a) The protocol that extends OREO, which involves a second numerically optimized pulse $\hat{U}_{\text{r}}$ that reverts the mapping done by $\hat{U}_{\text{m}}$. The resulting oscillator states after successful projections are characterized using Wigner tomography. (b)-(c) The reconstructed Wigner of the initial and final states of the cavity after the projection into the state $|\psi\rangle =(|0\rangle + |4\rangle)/\sqrt{2}$ with the cavity prepared in $|0\rangle$ (b) and in a thermal state with $n_{\text{th}} = 0.24$ (c). The projected oscillator states in (b) and (c) have similar fidelities of $0.84(3)$ and $0.85(2)$, respectively, to the ideal binomial state and non-Gaussianity coherence rank $4$sm despite the differences in initial conditions.
• Figure 5: Hadamard test. Here, $\hat{H}_{\text{a}}$ denotes the Hadamard gate and the control unitary is written as $|g\rangle\langle g|\otimes \mathbb{I} + |e\rangle \langle e |\otimes \hat{U}$ on the qubit-oscillator's space.