The classical shadows protocol is an efficient strategy for estimating properties of an unknown state ρ using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds f_{th} at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n≥2), and (iv) provide a meaningful upper bound t_{max} on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.
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.
The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.
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.
We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local-basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown
d-dimensional quantum state
ρ and a known collection of observables
E1,…,EM, the goal is to estimate all expectation values
{Tr(ρEi)}i=1M to additive accuracy
ε with probability at least
1−δ.
An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on
M, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity
O(ε2log(M)log(M/δ)), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright.
Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.