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Classical shadows herald remarkable opportunities for resource-efficient quantum estimation. Although superficially disconnected from traditional inference methods, we show how classical shadows fit under a larger umbrella of least squares regularization, revealing tradeoffs with related methods.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis $\mathcal{W}$ after it has evolved under a unitary transformation randomly sampled from a chosen distribution $\mathcal{U}$. In this work we study the case where $\mathcal{U}$ corresponds to either local or global orthogonal Clifford gates, and $\mathcal{W}$ consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Quantum algorithms exploiting real-time evolution under a target Hamiltonian have demonstrated remarkable efficiency in extracting key spectral information. However, the broader potential of these methods, particularly beyond ground-state calculations, is underexplored. In this work, we introduce the framework of multiobservable dynamic mode decomposition (MODMD), which combines the observable dynamic mode decomposition (DMD), a measurement-driven eigensolver tailored for near-term implementation, with classical shadow tomography. MODMD leverages random scrambling in the classical shadow technique to construct, with exponentially reduced resource requirements, a signal subspace that encodes rich spectral information. Notably, we replace typical Hadamard-test circuits with a protocol designed to predict low-rank observables, thereby broadening the use of classical shadow tomography for predicting many low-rank observables. We establish theoretical guarantees on the spectral approximation from MODMD, taking into account distinct sources of error. In the ideal case, we prove that the spectral error scales as exp (−Δ𝐸𝑡 max ), where Δ𝐸 is the Hamiltonian spectral gap and 𝑡 max is the maximal simulation time. This analysis provides a rigorous justification of the rapid convergence observed across simulations. To demonstrate the utility of our framework, we consider its application to fundamental tasks, such as determining the low-lying, i.e., ground or excited, energies of representative many-body systems. Our work paves the path for efficient designs of measurement-driven algorithms on near-term and early fault-tolerant quantum devices.
We introduce a classical shadow tomography scheme based on dual-unitary brick-wall circuits termed "dual-unitary shadow tomography" (DUST). For this we study operator spreading and Pauli weight dynamics in one-dimensional qubit systems, evolved by random two-local dual-unitary gates arranged in a brick-wall structure, ending with a final measurement layer. We do this by deriving general constraints on the Pauli weight transfer matrix and specializing to the case of dual-unitarity. We first show that dual-unitaries must have a minimal amount of entropy production. Remarkably, we find that operator spreading in these circuits have a rich structure resembling that of relativistic quantum field theories, with massless chiral excitations that can decay or fuse into each other, which we call left- or right-movers. We develop a mean-field description of the Pauli weight in terms of $\rho(x,t)$, which represents the probability of having nontrivial support at site $x$ and depth $t$ starting from a fixed weight distribution. We develop an equation of state for $\rho(x,t)$, and simulate it numerically using Monte Carlo simulations. Lastly, we demonstrate that the fast-thermalizing properties of dual-unitary circuits make them better at predicting large operators than shallow brick-wall Clifford circuits. Our results are robust to finite-size effects due to the chirality of dual-unitary brick-wall circuits.
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.
The ability to extract general laws from a few known examples depends on the complexity of the problem and on the amount of training data. In the quantum setting, the learner's generalization performance is further challenged by the destructive nature of quantum measurements that, together with the no-cloning theorem, limits the amount of information that can be extracted from each training sample. In this paper we focus on hybrid quantum learning techniques where classical machine-learning methods are paired with quantum algorithms and show that, in some settings, the uncertainty coming from a few measurement shots can be the dominant source of errors. We identify an instance of this possibly general issue by focusing on the classification of maximally entangled vs. separable states, showing that this toy problem becomes challenging for learners unaware of entanglement theory. Finally, we introduce an estimator based on classical shadows that performs better in the big data, few copy regime. Our results show that the naive application of classical machine-learning methods to the quantum setting is problematic, and that a better theoretical foundation of quantum learning is required.
Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error 𝜖 in a simulation of time 𝑇 using a 𝑝 th-order product formula with extrapolation, circuit depths of 𝑂(𝑇 1+1/𝑝 polylog (1/𝜖)) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the 𝑇 complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.
Numerical methods enabling simulation of scattered and partially reflected infrasonic propagation paths produced by interaction with fine-scale structure in the middle atmosphere have been implemented in the infraGA ray tracing software. This capability enables simulation of ensonification in the classical stratospheric “shadow zone” that has been observed during the Humming Roadrunner and LSECE surface explosion campaigns as well as in other data sets. In the case of LSECE, a pair of stations roughly 140 kilometers east of the source location observed arrivals with celerities (horizontal group velocities) slightly slower than observed stratospheric paths at similar azimuths. The arrivals exhibited increasing trace velocity later in the wavetrain indicating a steepening of the arrival path for longer or slower propagation paths. Simulation of partially reflected paths using the updated infraGA software methods finds good agreement between observed and predicted infrasonic ensonification at these locations within the stratospheric shadow zone. Further development of the partial reflection physics and comparison with other data sets is needed to more robustly understand how such anomalous infrasonic signals can be predicted; however, the demonstration of this capability is a promising first step in such analyses.
Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by [Huggins et al ., Nature (London) 603 , 416 (2022)] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford shadows to determine the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling postprocessing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [Commun. Math. Phys. 404 , 629 (2023)], which removes the aforementioned exponential bottleneck. We observe from experiments on quantum hardware that the use of Matchgate shadows in QC-QMC is inherently noise robust. We show that this noise resilience has a more subtle origin than in the case of Clifford shadows. Nevertheless, we find that classical postprocessing, while asymptotically efficient, requires hours of runtime on thousands of classical CPUs for even the smallest chemical systems, presenting a major challenge to the scalability of the algorithm.
We study the identification of quantum phases of matter, at zero temperature, when only part of the phase diagram is known in advance. Following a supervised learning approach, we show how to use our previous knowledge to construct an observable capable of classifying the phase even in the unknown region. By using a combination of classical and quantum techniques, such as tensor networks, kernel methods, generalization bounds, quantum algorithms, and shadow estimators, we show that, in some cases, the certification of new ground states can be obtained with a polynomial number of measurements. An important application of our findings is the classification of the phases of matter obtained in quantum simulators, e.g. cold atom experiments, capable of efficiently preparing ground states of complex many-particle systems and applying simple measurements, e.g. single qubit measurements, but unable to perform a universal set of gates.
We present a shadow-tomography-enhanced nonorthogonal quantum eigensolver (NOQE) for more efficient and accurate electronic structure calculations on near-term quantum devices. By integrating shadow tomography into the NOQE, the measurement cost scales linearly rather than quadratically with the number of reference states, while also reducing the required qubits and circuit depth by half. This approach enables extraction of all matrix elements via randomized measurements and classical postprocessing. We analyze its sample complexity and show that, for small systems, it remains constant in the high-precision regime, while for larger systems, it scales linearly with the system size. We further apply shadow-based error mitigation to suppress noise-induced bias without increasing quantum resources. Demonstrations on the hydrogen molecule in the strongly correlated regime achieve chemical accuracy under realistic noise, showing that our method is both resource-efficient and noise-resilient for practical quantum chemistry simulations in the near term.
In this and companion papers, we show that quantum field theories with gauge symmetries permit a broader class of classical dynamics than typically assumed. In this article, we show that the quantization of electromagnetism permits the existence of classical electric field states that do not obey Gauss’s law. These states are gauge invariant and their time evolution can be consistently described using the Schrödinger equation. The time evolution of these states is such that at the classical level, the full set of Maxwell’s equations would appear to hold, with the physical effects of these states being attributable to an auxiliary, static “shadow” charge density with no internal degrees of freedom. This density could affect the dynamics of charged particles in our universe and it may thus be of observational interest.
Solar shadow calculations are needed in building energy modeling and performance simulation of PV systems installed on roofs or facades of buildings. We present a quantum computing approach for calculation of building surface sunlit fractions by recasting solar visibility as a binary optimization problem solved by quantum annealing. Each triangulated surface centroid is encoded as a binary qubit indicating sunlit or shaded status. Geometric visibility constraints are derived from the Möller-Trumbore intersection algorithm and converted into a constrained quadratic binary model compatible with contemporary quantum annealers. The coefficients were embedded to D-Wave quantum computer. To demonstrate feasibility, we conducted a case study in San Francisco for a target building with 52 triangles and roughly 2700 nearby triangles within 50 m evaluated at representative winter and summer solar positions. The results demonstrated that quantum annealing can reliably calculate and distinguish sunlit from shaded surfaces. Quantum samples achieved average accuracy exceeding 92.4 %, with the aggregate surface-level agreement approaching 99.9 %. The outputs of quantum computers agreed closely with classical algorithms, indicating practical feasibility and promising scalability. Finally, the hourly sunlit fractions of building surfaces can be obtained for urban energy modelling. This is the first study to apply quantum computing to the solar shadow and building surface sunlit calculation. It introduces a new paradigm that differs fundamentally from traditional approaches.
Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.
In this work we study the properties of dissipatively stabilized steady states of noisy quantum algorithms, exploring the extent to which they can be well approximated as thermal distributions, and proposing methods to extract the effective temperature T. We study an algorithm called the relaxational quantum eigensolver (RQE), which is one of a family of algorithms that attempt to find ground states and balance error in noisy quantum devices. In RQE, we weakly couple a second register of auxiliary ‘shadow’ qubits to the primary system in Trotterized evolution, thus engineering an approximate zero-temperature bath by periodically resetting the auxiliary qubits during the algorithm’s runtime. Balancing the infinite temperature bath of random gate error, RQE returns states with an average energy equal to a constant fraction of the ground state. We probe the steady states of this algorithm for a range of base error rates, using several methods for estimating both T and deviations from thermal behavior. In particular, we both confirm that the steady states of these systems are often well-approximated by thermal distributions, and show that the same resources used for cooling can be adopted for thermometry, yielding a fairly reliable measure of the temperature. These methods could be readily implemented in near-term quantum hardware, and for stabilizing and probing Hamiltonians where simulating approximate thermal states is hard for classical computers.
Shadow molecular dynamics provide an efficient and stable atomistic simulation framework for flexible charge models with long-range electrostatic interactions. Shadow molecular dynamics simulations are driven by approximate “shadow” Born–Oppenheimer potentials for which the exact charges and forces are directly accessible without relying on costly (and approximate) iterative solvers. While previous implementations have been limited to atomic monopole charge distributions, we extend this approach to flexible multipole models. We derive detailed expressions for the shadow energy functions, potentials, and force terms, explicitly incorporating monopole–monopole, dipole–monopole, and dipole–dipole interactions. In our formulation, both atomic monopoles and atomic dipoles are treated as extended dynamical variables alongside the propagation of the nuclear degrees of freedom. We demonstrate that introducing the additional dipole degrees of freedom preserves the stability and accuracy previously seen in monopole-only shadow molecular dynamics simulations. In addition, we present a shadow molecular dynamics scheme where the monopole charges are held fixed while the dipoles remain flexible. Our extended shadow dynamics provide a framework for stable, computationally efficient, and versatile molecular dynamics simulations involving long-range interactions between flexible multipoles. This is of particular current interest in combination with machine-learned interatomic potentials, including long-range electrostatic interactions.
Experimentally measured reduced density matrices (RDMs) often violate constraints that ensure they represent N-electron states—known as N-representability conditions—because of statistical and hardware noise. In this work, we present a correlated purification framework based on semidefinite programming to restore the accuracy of a noisy, unphysical two-electron RDM (2-RDM). The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the correction to the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank corrections, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for ground states, it can also be applied to excited and nonstationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.