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arXiv:2512.13146

Practical Homodyne Shadow Estimation

Ruyu Yang, Xiaoming Sun, Hongyi Zhou

quant-phSubmitted Dec 15, 2025• Announced Dec 16, 2025• 0 citations
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Abstract

Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to $n_{\mathrm{max}}$ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as $\mathcal{O}(n_{\mathrm{max}}^4)$, improving upon previous $\mathcal{O}(n_{\mathrm{max}}^{13/3})$ bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.

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12 pages, 2 figures

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Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to nmaxn_{\mathrm{max}}nmax​ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as O(nmax4)\mathcal{O}(n_{\mathrm{max}}^4)O(nmax4​), improving upon previous O(nmax13/3)\mathcal{O}(n_{\mathrm{max}}^{13/3})O(nmax13/3​) bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.
Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.
Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called “classical shadows,” with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon-number-resolving, and photon-parity protocols. To reach a desired precision on the classical shadow of an N -photon density matrix with high probability, we show that homodyne detection requires order O(N4+1/3) measurements in the worst case, whereas photon-number-resolving and photon-parity detection require O(N4) measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental analyses of homodyne tomography match closely with our theoretical predictions. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements. Published by the American Physical Society 2024
Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. For multimode continuous-variable (CV) systems, the infinite-dimensional Hilbert space poses an additional challenge. In this work, we develop a CV-DIPE protocol using only local homodyne measurements at shared phase settings and classical communication. Under a known bound on the mean energy per mode, we derive explicit finite-sample guarantees and sufficient copy counts for estimating the full inner product to prescribed accuracy and confidence. The energy constraint yields an inverse-square truncation-bias bound, while Gaussian structure strengthens the tail control to exponential decay and improves the resulting accuracy scaling. Numerical simulations with Gaussian and non-Gaussian states illustrate the balance between truncation bias and statistical error and its dependence on mode number and energy. Our results enable independent CV platforms to benchmark their state preparations against one another, even when neither state admits a tractable classical description.
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 thatN=O(poly (1 /ϵ, log (1/δ),Mr+αn, log(m) )) samples of an unknownm-mode state ρ suffice to learn the expected value of anyr-local polynomial in the canonical observables of degree α, both with high probability 1 - δ and accuracy ϵ, as long as the state ρ has moments of ordern> α bounded byMn. 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.
Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While 333-design protocols achieve optimal worst-case performance, the minimal number of measurement bases required for such optimality has remained open. Here we prove that Θ(d2)Θ(d^2)Θ(d2) measurement bases are both necessary and sufficient for worst-case optimal shadow estimation and construct an explicit basis family. In stark contrast, any state 222-design already suffices for average-case optimality: the mean squared shadow norm of normalized observables is bounded by a universal constant, and we prove strong concentration for Haar-random states, yielding constant sample complexity for generic pure-state fidelity estimation. Easily implementable 222-designs -- from mutually unbiased bases, cyclic measurements, or shallow O(log⁡n)\mathcal{O}(\log n)O(logn)-depth circuits -- enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires Θ(d2)Θ(d^2)Θ(d2) bases, whereas average-case performance requires only Θ(d)Θ(d)Θ(d) bases, with broad implications for quantum information theory and near-term experiments.
Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for K times after the same unitary evolution, and repeats this procedure for M rounds of random sampled unitary. As a result, there are M K times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable O . We find that in addition to the shadow-norm ‖ O ‖ s h a d o w introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm ‖ O ‖ X s h a d o w . For both random Pauli and Clifford measurements, we analyze and show the upper bounds of ‖ O ‖ X s h a d o w . In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.