Résultats de recherche pour « Efficient Classical Shadow Tomography »

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.

Classical shadow tomography serves as a potent tool for extracting numerous properties from quantum many-body systems with minimal measurements. Nevertheless, prevailing methods yielding optimal performance for few-body operators necessitate the application of random two-qubit gates, a task that can prove challenging on specific quantum simulators such as ultracold atomic gases. In this work, we introduce an alternative approach founded on the dynamics of many-body localization, a phenomenon extensively demonstrated in optical lattices. Through an exploration of the shadow norm – both analytically, employing a phenomenological model, and numerically, utilizing the TEBD algorithm – we demonstrate that our scheme achieves remarkable efficiency comparable to shallow circuits or measurement-induced criticality, resulting in a significant improvement in the exponential exponent compared to the previous classical shadow protocol. Our findings are corroborated through direct numerical simulations encompassing the entire sampling and reconstruction processes. Consequently, our results present a compelling methodology for analyzing the output states of quantum simulators.

Shadow tomography aims to build a classical description of a quantum state from a sequence of simple random measurements. Physical observables are then reconstructed from the resulting classical shadow. Shadow protocols which use single-body random measurements are simple to implement and capture few-body observables efficiently, but do not apply to systems with fundamental number conservation laws, such as ultracold atoms. We address this shortcoming by proposing and analyzing a new local shadow protocol adapted to such systems. The "All-Pairs" protocol requires one layer of two-body gates and only $\textrm{poly}(V)$ samples to reconstruct arbitrary few body observables. Moreover, by exploiting the permutation symmetry of the protocol, we derive a linear time post-processing algorithm. We provide a proof-of-principle reference implementation and demonstrate the reconstruction of 2- and 4-point functions in a paired Luttinger liquid of hardcore bosons.

Given copies of a quantum state $ρ$, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision $ε$. We say that a shadow tomography protocol is triply efficient if it is sample- and time-efficient, and only employs measurements that entangle a constant number of copies of $ρ$ at a time. The classical shadows protocol based on random single-copy measurements is triply efficient for the set of local Pauli observables. This and other protocols based on random single-copy Clifford measurements can be understood as arising from fractional colorings of a graph $G$ that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of $G$ with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all $n$-qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample-efficient schemes are provably impossible using only single-copy measurements. Finally, we give a shadow tomography protocol that compresses an $n$-qubit quantum state into a $\text{poly}(n)$-sized classical representation, from which one can extract the expected value of any of the $4^n$ Pauli observables in $\text{poly}(n)$ time, up to a small constant error.

Computing inner products of the form tr(AB), where A is a d-dimensional density matrix [with tr(A)=1, A≥0] and B is a bounded-norm observable [Hermitian with tr(B^{2})≤O[poly(logd)] and tr(B) known], is fundamental across quantum science and artificial intelligence. Classically, both computing and storing such inner products require O(d^{2}) resources, which rapidly becomes prohibitive as d grows exponentially. In this Letter, we introduce a quantum approach based on qudit classical shadow tomography, significantly reducing computational complexity from O(d^{2}) down to O[poly(logd)] in typical cases and at least to O[dpoly(logd)] in the worst case. Specifically, for n-qubit systems (with n being the number of qubit and d=2^{n}), our method guarantees efficient estimation of tr(ρO) for any known stabilizer state ρ and arbitrary bounded-norm observable O, using polynomial computational resources. Crucially, it ensures constant-time classical postprocessing per measurement and supports qubit and qudit platforms. Moreover, classical storage complexity of A reduces from O(d^{2}) to O(mlogd), where the sample complexity m is typically exponentially smaller than d^{2}. Our results establish a practical and modular quantum subroutine, enabling scalable quantum advantages in tasks involving high-dimensional data analysis and processing.

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.

We construct and extensively study a Brownian generalization of the Gaussian Unitary Ensemble (BGUE). Our analysis begins with the non-equilibrium dynamics of BGUE, where we derive explicit analytical expressions for various one-replica and two-replica variables at finite $N$ and $t$. These variables include the spectral form factor and its fluctuation, the two-point function and its fluctuation, out-of-time-order correlators (OTOC), the second Rényi entropy, and the frame potential for unitary 2-designs. We discuss the implications of these results for hyperfast scrambling, emergence of tomperature, and replica-wormhole-like contributions in BGUE. Next, we investigate the low-energy physics of the effective Hamiltonian for an arbitrarily number of replicas, deriving long-time results for the frame potential. We conclude that the time required for the BGUE ensemble to reach $k$-design is linear in $k$, consistent with previous findings in Brownian SYK models. Finally, we apply the BGUE model to the task of classical shadow tomography, deriving analytical results for the shadow norm and identifying an optimal time that minimizes the shadow norm, analogous to the optimal circuit depth in shallow-circuit shadow tomography.

We show how shadows can be efficiently generated in differentiable rendering of triangle meshes. Our central observation is that pre-filtered shadow mapping, a technique for approximating shadows based on rendering from the perspective of a light, can be combined with existing differentiable rasterizers to yield differentiable visibility information. We demonstrate at several inverse graphics problems that differentiable shadow maps are orders of magnitude faster than differentiable light transport simulation with similar accuracy -- while differentiable rasterization without shadows often fails to converge.

We propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of an n-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all k-RDM elements to additive precision ϵ requires on the order of (n/k)k^{3/2}log(n)/ϵ^{2} repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for k≥2, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2-5 times more repetitions.

Classical shadow tomography, harnessing randomized informationally complete (IC) measurements, provides an effective avenue for predicting many properties of unknown quantum states with sample-efficient precision. Projections onto $2^n+1$ mutually unbiased bases (MUBs) are widely recognized as minimal and optimal IC measurements for full-state tomography. We study how to use MUBs circuits as the ensemble in classical shadow tomography. For the general observables, the variance to predict their expectation value is shown to be exponential to the number of qubits $n$. However, for a special class termed as appropriate MUBs-average (AMA) observables, the variance decreases to $poly(n)$. Additionally, we find that through biased sampling of MUBs circuits, the variance for non-AMA observables can again be reduced to $poly(n)$ with the MUBs-sparse condition. The performance and complexity of using the MUBs and Clifford circuits as the ensemble in the classical shadow tomography are compared in the end.