Classical shadow tomography for continuous variables quantum systems

Abstract

In this article we develop a continuous variable (CV) shadow tomography scheme with wide ranging applications in quantum optics. Our work is motivated by the increasing experimental and technological relevance of CV systems in quantum information, quantum communication, quantum sensing, quantum simulations, quantum computing and error correction. We introduce two experimentally realisable schemes for obtaining classical shadows of CV (possibly non-Gaussian) quantum states using only randomised Gaussian unitaries and easily implementable Gaussian measurements such as homodyne and heterodyne detection. For both schemes, we show that $N=O\left(\mathrm{poly}\right(\frac{1}{\epsilon },\log \left(\frac{1}{\delta }\right),M_n^{r+\alpha },\log (m)\left)\right)$ samples of an unknown $m$-mode state $\rho$ suffice to learn the expected value of any $r$-local polynomial in the canonical observables of degree $\alpha$, both with high probability $1-\delta$ and accuracy $\epsilon$, as long as the state $\rho$ has moments of order $n>\alpha$ bounded by $M_n$. By simultaneously truncating states and operators in energy and phase space, we are able to overcome new mathematical challenges that arise due to the infinite-dimensionality of CV systems. We also provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes, by leveraging recent energy-constrained entropic continuity bounds. Finally, we provide numerical evidence of the efficiency of our protocols in the case of CV states of relevance in quantum information theory, including ground states of quadratic Hamiltonians of many-body systems and cat qubit states. We expect our scheme to provide good recovery in learning relevant states of 2D materials and photonic crystals.

References in corpus (57)

• Quantum information with continuous variables
• Encoding a qubit in an oscillator
• General Benchmarks for Quantum Repeaters
• Fundamental Limits of Repeaterless Quantum Communications
• Enhancing the sensitivity of the LIGO gravitational wave detector by using squeezed states of light
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Continuous-variable optical quantum state tomography
• Quantum simulation of the Dirac equation
• User-friendly tail bounds for sums of random matrices
• Dynamically protected cat-qubits: a new paradigm for universal quantum computation
• Reconstruction of non-classical cavity field states with snapshots of their decoherence
• Observation of quantum state collapse and revival due to the single-photon Kerr effect
• Realization of a quantum walk with one and two trapped ions
• Progressive field-state collapse and quantum non-demolition photon counting
• Fundamental rate-loss tradeoff for optical quantum key distribution
• 10-qubit entanglement and parallel logic operations with a superconducting circuit
• New class of quantum error-correcting codes for a bosonic mode
• A Schrodinger Cat Living in Two Boxes
• Photon State Tomography for Two-Mode Correlated Itinerant Microwave Fields
• Provably efficient machine learning for quantum many-body problems
• Quantum simulation of the Klein paradox with trapped ions
• Quantum Capacities of Bosonic Channels
• Quantum limited amplification and entanglement in coupled nonlinear resonators
• Sample-optimal tomography of quantum states
• Repetition Cat Qubits for Fault-Tolerant Quantum Computation
• Converse bounds for private communication over quantum channels
• Squeezing with a flux-driven Josephson parametric amplifier
• Quantum-enhanced sensing of a mechanical oscillator
• Optical synthesis of large-amplitude squeezed coherent-state superpositions with minimal resources
• Homodyne tomography characterization and nonlocality of a dual-mode optical qubit
• Observing Topological Invariants Using Quantum Walk in Superconducting Circuits
• Distillation of the two-mode squeezed state
• Full reconstruction of a 14-qubit state within four hours
• Experimental Preparation of High NOON States for Phonons
• Direct characteristic-function tomography of quantum states of the trapped-ion motional oscillator
• Resonance fluorescence from an artificial atom in squeezed vacuum
• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• A quantum parametric oscillator with trapped ions
• Quantum states on Harmonic lattices
• Narrow Bounds for the Quantum Capacity of Thermal Attenuators
• Decoherence Dynamics of Complex Photon States in a Superconducting Circuit
• Non-Gaussian operations on bosonic modes of light: Photon-added Gaussian channels
• Landau levels in strained two-dimensional photonic crystals
• Schwartz operators
• A unified approach to infinite dimensional integration
• Vacuum Measurements and Quantum State Reconstruction of Phonons
• Sensing Atomic Motion from the Zero Point to Room Temperature with Ultrafast Atom Interferometry
• Quantitative tomography for continuous variable quantum systems
• Exact solution for the quantum and private capacities of bosonic dephasing channels
• Convergence rates for the quantum central limit theorem
• Optimized Tomography of Continuous Variable Systems Using Excitation Counting
• All phase-space linear bosonic channels are approximately Gaussian dilatable
• From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
• Quantum tomography in position and momentum space
• Efficient measurement schemes for bosonic systems
• State estimation in quantum homodyne tomography with noisy data
• Precision Bounds on Continuous-Variable State Tomography using Classical Shadows

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