Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via "shallow shadows," a recently proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth t. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing ("twirling") circuit-namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially contiguous Pauli operators of weight k, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth t∼log(k), guarantees an exponential gain in sample complexity over the t=0 protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our Letter connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly optimized protocols for learning different properties of quantum states.
We provide practical and powerful schemes for learning properties of a quantum state using a small number of measurements. Specifically, we present a randomized measurement scheme modulated by the depth of a random quantum circuit in one spatial dimension. This scheme interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. We focus on the regime where depth scales logarithmically in the system size and provide evidence that this retains the desirable sample complexity properties of both extremal schemes while also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We achieve our findings by bringing together tools from shadow estimation, random circuits, and tensor networks.
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 W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and 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 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.
We develop a classical shadow tomography protocol utilizing the randomized measurement scheme based on hybrid quantum circuits, which consist of layers of two-qubit random Clifford unitary gates mixed with single-qubit random Pauli measurements. Unlike conventional protocols that perform all measurements by the end of unitary evolutions, our protocol allows measurements to occur at any space-time position throughout the quantum evolution. We provide a universal classical post-processing strategy to approximately reconstruct the original quantum state from intermittent measurement outcomes given the corresponding random circuit realizations over repeated experiments. We investigated the sample complexity for estimating different observables at different measurement rates of the hybrid quantum circuits. Our result shows that the sample complexity has an optimal scaling at the critical measurement rate when the hybrid circuit undergoes the measurement-induced transition.
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown
d-dimensional quantum state
ρ and a known set of observables
{Oi}i=1m, the goal is to estimate expectation values
{tr(Oiρ)}i=1m to accuracy
ε in
Lp-norm, using possibly adaptive measurements that act on
O(polylog(d)) number of copies of
ρ at a time. We focus on the regime where
ε is below an instance-dependent threshold.
Our main contribution is an instance-optimal characterization of the sample complexity as
Θ~(Γp/ε2), where
Γp is a function of
{Oi}i=1m defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with
L∞-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form
∑i=1mαiOi with
∥α∥q=1 (where
q is dual to
p) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of
Θ(Γpob/ε2). We then show
Θ~(Γp/ε2) is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably,
Γ∞ob=Γ∞. In both cases, allowing
c-copy measurements improves the sample complexity by at most
Ω(1/c).
Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.