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arXiv:2211.07578 · IEEE Transactions on Information Theory

Classical Shadow Tomography for Continuous Variables Quantum Systems

Simon Becker, Nilanjana Datta, Ludovico Lami, Cambyse Rouzé

Published Jan 26, 2024• 32 citations
arXivHTMLPDF

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 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.

DOI
10.1109/tit.2024.3357972

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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.
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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
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.
This paper reviews recent advances in quantum learning theory for continuous-variable (CV) systems. Quantum learning theory investigates how to extract classical information from quantum systems as efficiently as possible. CV systems are ubiquitous in nature and in quantum technologies, as they describe bosonic and quantum-optical systems. While quantum learning theory for finite-dimensional systems has been extensively studied, the corresponding theory for CV systems has only recently begun to develop; here we provide a concise review. We focus on the following questions: what is the minimum number of copies (the sample complexity) required to learn a non-Gaussian state, possibly under energy constraints? What is the sample complexity for learning Gaussian states? How does the performance of CV state learning depend on non-Gaussianity? How can one test whether a state is Gaussian or far from the set of Gaussian states? And how can Gaussian processes be learned efficiently? Central to these topics, we also review several bounds on the trace distance between CV states in terms of their covariance matrices, which may be of independent interest. Overall, this work summarises selected developments in tomography of CV systems and highlights a selection of open problems.
Recent results have established dramatic advantages in learning properties of quantum states when a quantum computer is available to process or jointly measure multiple copies of the unknown quantum state. Learning tasks can be accomplished with exponentially fewer copies of the state when compared to optimized classical learning strategies that are restricted to measuring one copy of the state at a time. While these results were established in abstract settings and for artificial learning tasks, they motivate the application of quantum computers to imaging and sensing of weak electromagnetic fields since these settings are ultimately concerned with the learning of unknown quantum states. In this work we apply these new results in quantum learning to the problem of learning Gaussian states of the electromagnetic field, which are germane since they describe most fields used in imaging and sensing. In order to connect with quantum learning theory, we consider the transduction of an nnn-mode Gaussian state into a register of qubits on a quantum computer followed by optimized measurements on these qubits to extract the parameters defining the original Gaussian state. We rigorously bound the number of copies of the Gaussian state required to achieve worst-case additive error in parameter estimates. The scaling of this bound with nnn is exponentially better than naïve strategies for characterizing Gaussian states and matches recently derived bounds for characterization of Gaussian states using continuous-variable (CV) classical shadows. In addition, our bound has a polynomially better dependence on the energy of the multimode Gaussian state compared to the CV shadows protocol.
Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.