Learning quantum state properties is both a fundamental and practical problem in quantum information theory. Classical shadows have emerged as an efficient method for estimating properties of unknown quantum states, with rigorous statistical guarantees, by performing randomized measurement on few copies of the state. With the advent of photonic technologies, formulating efficient learning algorithms for such platforms comes out as a natural problem. Here, we introduce a practical classical shadow protocol for learning photonic quantum states via randomized passive linear optical transformations and photon-number measurement. We provide rigorous theoretical guarantees showing that our scheme is sample- and time-efficient for measuring physical observables of interest. We experimentally demonstrate our photonic classical shadow protocol on both a twelve-mode and a twenty-four-mode integrated quantum processing unit, and showcase its versatility with five different applications, including Hamiltonian measurement and learning complex photonic states.
Learning quantum state properties is both a fundamental and practical problem in quantum information theory. Classical shadows have emerged as an efficient method for estimating properties of unknown quantum states, with rigorous statistical guarantees, by performing randomized measurement on few copies of the state. With the advent of photonic technologies, formulating efficient learning algorithms for such platforms comes out as a natural problem. Here, we introduce a practical classical shadow protocol for learning photonic quantum states via randomized passive linear optical transformations and photon-number measurement. We provide rigorous theoretical guarantees showing that our scheme is sample- and time-efficient for measuring physical observables of interest. We experimentally demonstrate our photonic classical shadow protocol on both a twelve-mode and a twenty-four-mode integrated quantum processing unit, and showcase its versatility with five different applications, including Hamiltonian measurement and learning complex photonic states.
Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes. As part of our technical analysis we obtain ε-approximate relative-error k-designs over the n-qubit symplectic group in depth O(nk+log(1/ε)), which is optimal with respect to n, and may be of some independent interest.
This thesis presents developments in the theory and application of quantum learning.The first two chapters fall under the head of classical shadow tomography, a randomized measurement technique which estimates a large number of expectation values effectively in parallel. In the first chapter on this topic, we extend classical shadow state tomography to the characterization of quantum channels. We prove that with a simple protocol involving randomized measurement of randomly generated input states evolved through channel, one can estimate many channel properties in parallel, with rigorous guarantees on the sample complexity required. In the next chapter, we bring together finite-depth classical shadows, derandomization, and tensor network simulation to present an algorithm for tailoring a sequence of measurements to a set of observables of interest, demonstrating the effectiveness of our technique for a number of tasks in quantum chemistry and simulation. The second part of this thesis presents a step toward applying ideas from quantum learning to quantum error correction. We investigate the use of prior information on qubit error rates in the decoding task, showing that prior information can significantly change decoder performance when a small number of qubits have high error rates. We present a protocol to learn the location of error-prone sites from decoding data in real time, and then apply this in turn to develop in situ protocol for calibration of gates while processing logical information.
A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)NPAHAX1745-247310.1038/s41567-020-0932-7], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict M different functions of a state with only O(log_{2}M) measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.
Classical shadows (CS) have emerged as a powerful way to estimate many properties of quantum states based on random measurements and classical post-processing. In their original formulation, they come with optimal (or close to) sampling complexity guarantees for generic states and generic observables. Still, it is natural to expect to even further lower sampling requirements when equipped with a priori knowledge regarding either the underlying state or the observables. Here, we consider the case where such knowledge is provided in terms of symmetries of the unknown state or of the observables. Criterion and guidelines for symmetric shadows are provided. As a concrete example we focus on the case of permutation invariance (PI), and detail constructions of several families of PI-CSs. In particular, building on results obtained in the field of PI quantum tomography, we develop and study shallow PI-CS protocol. Benefits of these symmetric CS are demonstrated compared to established CS protocols showcasing vastly improved performances.
Given many copies of an unknown quantum state ρ, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that ρ has an eigenstate ∣ϕ⟩ with (unknown) eigenvalue λ>1/2, the goal is to learn a (classical shadows style) classical description of ∣ϕ⟩ which can later be used to estimate expectation values ⟨ϕ∣O∣ϕ⟩ for any O in some class of observables. We consider the sample-complexity setting in which generating a copy of ρ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue λ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when λ is sufficiently close to 1, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.
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 n-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 n 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.