Learning the properties of dynamical quantum systems underlies applications ranging from nuclear magnetic resonance spectroscopy to quantum device characterization. A central challenge in this pursuit is the learning of strongly interacting systems, where conventional observables decay quickly in time and space, limiting the information that can be learned from their measurement. In this work, we introduce a new class of observables into the context of quantum learning---the out-of-time-order correlator---which we show can substantially improve the learnability of strongly interacting systems by virtue of displaying informative physics at large times and distances. We identify two general scenarios in which out-of-time-order correlators provide a significant learning advantage: (i) when experimental access to the system is spatially restricted, for example, via a single ``probe'' degree of freedom, and (ii) when one desires to characterize weak interactions whose strength is much less than the typical interaction strength. We numerically characterize these advantages across a variety of learning problems, and find that they are robust to both read-out error and decoherence. Motivated by these physical scenarios, we introduce several learning tasks---including Clifford tomography, and learning the connectivity of an unknown unitary---in which out-of-time-order experiments have a provable exponential advantage over any learning protocol involving only time-ordered operations.
Learning the properties of dynamical quantum systems underlies applications ranging from nuclear magnetic resonance spectroscopy to quantum device characterization. A central challenge in this pursuit is the learning of strongly interacting systems, where conventional observables decay quickly in time and space, limiting the information that can be learned from their measurement. In this work, we introduce a new class of observables into the context of quantum learning---the out-of-time-order correlator---which we show can substantially improve the learnability of strongly interacting systems by virtue of displaying informative physics at large times and distances. We identify two general scenarios in which out-of-time-order correlators provide a significant learning advantage: (i) when experimental access to the system is spatially restricted, for example, via a single ``probe'' degree of freedom, and (ii) when one desires to characterize weak interactions whose strength is much less than the typical interaction strength. We numerically characterize these advantages across a variety of learning problems, and find that they are robust to both read-out error and decoherence. Motivated by these physical scenarios, we introduce several learning tasks---including Clifford tomography, and learning the connectivity of an unknown unitary---in which out-of-time-order experiments have a provable exponential advantage over any learning protocol involving only time-ordered operations.
Which properties of a quantum many-body system are operationally accessible is a central question underlying spectroscopy, thermodynamics, and quantum information science. Conventional response theory answers this question within a system-only paradigm: one perturbs and measures the matter itself, obtaining susceptibility built from causally ordered nested commutators. Here we show that coherently controlled quantum probes, when measured at the end, define a strictly larger operational learning framework beyond that accessible from response theory. We establish this through a quantum-circuit description that unifies spectroscopy, probe microscopy, and probe-based quantum technologies within a common operational framework, from which we develop quantum protocols for learning many-body properties from probe readout only. This advantage arises because the reduced dynamics of quantum probes generically encode anti-commutator and mixed-order correlators of the target; therefore, measurements on the probe provide access to fluctuations, non-equilibrium structure, and entanglement entropy that are in general not accessible through response functions or a single probe alone. Moreover, we demonstrate that entangled probes can access many-body properties such as von Neumann entropy. We prove that the required probe resources scale with the complexity of the target correlations rather than with the size of the many-body system. Quantum probes are therefore not merely more sensitive sensors but provide a new way to learn many-body properties distinct from those of tomography or quantum simulation.
We establish that there are properties of quantum many-body dynamics which are efficiently learnable if we are given access to out-of-time-order correlators (OTOCs), but which require exponentially many operations in the system size if we can only measure time-ordered correlators. This implies that any experimental protocol which reconstructs OTOCs solely from time-ordered correlators must be, in certain cases, exponentially inefficient. Our proofs leverage and generalize recent techniques in quantum learning theory. Along the way, we elucidate a general definition of time-ordered versus out-of-time-order experimental measurement protocols, which can be considered as classes of adaptive quantum learning algorithms. Moreover, our results provide a theoretical foundation for novel applications of OTOCs in quantum simulations.
Quantum platforms can realize many-body dynamics beyond classical simulation yet complete readout remains intractable: the cost of extracting accessible information scales exponentially with system size. Classical shadows and Bell sampling offer scalable, multi-observable estimation from randomized or entanglement-assisted measurements. Here we aim to push these ideas beyond static snapshots to dynamical correlators, including out-of-time-ordered correlators (OTOCs) and two-point functions. In particular, we introduce the notion of the shadow of an operator, defined as the classical shadow of the vectorized time-evolved operator. Pauli operator-shadows enable simultaneous estimation of all local OTOCs, while Clifford operator-shadows enable efficient simultaneous estimation of all two-point correlators. Alternatively, Bell sampling allows one to simultaneously compute all diagonal OTOCs. We also prove information-theoretic lower bounds for learning OTOCs, fully characterizing their query complexities in many cases, and yielding exponential separations that formalize when the vectorized approach provides measurement-efficiency advantages.
The ability of quantum computers to directly manipulate and analyze quantum states stored in quantum memory allows them to learn about aspects of our physical world that would otherwise be invisible given a modest number of measurements. Here we investigate a new learning resource, which could be available to quantum computers in the future—measurements on the unknown state accompanied by its complex conjugate ρ ⊗ ρ ∗ . For a certain shadow tomography task, we surprisingly find that measurements on only copies of ρ ⊗ ρ ∗ can be exponentially more powerful than measurements on ρ ⊗ K , even for large K . This expands the class of provable exponential advantages using only a constant overhead quantum memory, or minimal quantum memory, and we provide a number of examples where the state ρ ∗ is naturally available in both computational and physical applications. In addition, we precisely quantify the power of classical shadows on single copies under a generalized Clifford ensemble and give a class of quantities that can be efficiently learned. The learning task we study in both the single copy and quantum memory settings is physically natural and corresponds to real-space observables with a limit of bosonic modes, where it achieves an exponential improvement in detecting certain signals under a noisy background. We quantify a new and powerful resource in quantum learning, and we believe the advantage may find applications in improving quantum simulation, learning from quantum sensors, and uncovering new physical phenomena. Published by the American Physical Society 2024
The dynamics of quantum many-body systems is characterized by quantum observables that are reconstructed from correlation functions at separate points in space and time1–3. In dynamics with fast entanglement generation, however, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. To circumvent this limitation and enable access to relevant dynamics in experimental systems, repeated time-reversal protocols have been successfully implemented4. Here we experimentally measure the second-order out-of-time-order correlators (OTOC(2))5–18 on a superconducting quantum processor and find that they remain sensitive to the underlying dynamics at long timescales. Furthermore, OTOC(2) manifests quantum correlations in a highly entangled quantum many-body system that are inaccessible without time-reversal techniques. This is demonstrated through an experimental protocol that randomizes the phases of Pauli strings in the Heisenberg picture by inserting Pauli operators during quantum evolution. The measured values of OTOC(2) are substantially changed by the protocol, thereby revealing constructive interference between Pauli strings that form large loops in the configuration space. The observed interference mechanism also endows OTOC(2) with high degrees of classical simulation complexity. These results, combined with the capability of OTOC(2) in unravelling useful details of quantum dynamics, as shown through an example of Hamiltonian learning, indicate a viable path to practical quantum advantage. Experimental measurements of high-order out-of-time-order correlators on a superconducting quantum processor show that these correlators remain highly sensitive to the quantum many-body dynamics in quantum computers at long timescales.
Quantum resources can dramatically reduce the data required to learn physical systems, with exponential improvements in sample complexity demonstrated in several quantum learning tasks. However, these advantages have typically relied on quantum resources that scale with problem complexity, most notably increasing system dimension or entanglement. This raises a fundamental question: can exponential quantum learning advantages arise within a fixed, unentangled quantum system? In this paper, we address this question by considering the learning of an unknown random-displacement distribution whose complexity is determined not by the dimensionality of the physical system, but by the Fourier resolution of its features. We show that the quantum-limited noise of vacuum probes progressively obscures high-frequency features, leading to an exponential growth in sample complexity. We establish an information-theoretic lower bound for arbitrary classical-state probes and show that squeezing overcomes this classical limit by extending the accessible Fourier bandwidth. Experimentally, we demonstrate an exponential reduction in sample complexity using squeezed vacuum probes for both binary hypothesis testing and characteristic-function reconstruction. Our results show that a single bosonic mode can exhibit exponential quantum learning advantages without entanglement or an increase in system size, identifying accessible Fourier bandwidth as a fundamentally distinct resource for quantum-enhanced learning.