Search Results for « Efficient Classical Shadow Tomography »

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 poly(V) samples to reconstruct arbitrary few body observables. Moreover, by exploiting the permutation symmetry of the protocol, we derive a linear time postprocessing algorithm which applies to both hardcore bosons and spinless fermions in any spatial dimension. We provide a proof-of-principle reference implementation and demonstrate the reconstruction of two- and four-point functions in a paired Luttinger liquid of hardcore bosons.

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.

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 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.

Quantum process tomography is a powerful tool for understanding quantum channels and characterizing the properties of quantum devices. Inspired by recent advances using classical shadows in quantum state tomography [H.-Y. Huang, R. Kueng, and J. Preskill, .], we have developed ShadowQPT, a classical shadow method for quantum process tomography. We introduce two related formulations with and without ancilla qubits. ShadowQPT stochastically reconstructs the Choi matrix of the device allowing for an classical evaluation of the device on arbitrary inputs with respect to arbitrary outputs. Using shadows, we then show how to compute overlaps, generate all k -weight reduced processes, and perform reconstruction via Hamiltonian learning. These latter two tasks are efficient for large systems as the number of quantum measurements needed scales only logarithmically with the number of qubits. A number of additional approximations and improvements are developed, including the use of a pair-factorized Clifford shadow and a series of postprocessing techniques that significantly enhance the accuracy for recovering the quantum channel. We have implemented ShadowQPT using both Pauli and Clifford measurements on the IonQ trapped ion quantum computer for quantum processes up to n=4 qubits, and we achieved good performance. Published by the American Physical Society 2024

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 singlecopy 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 poly(n )-sized classical representation, from which one can extract the expected value of any of the 4n Pauli observables in poly(n ) time, up to a small constant error.

Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called “classical shadows,” with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon-number-resolving, and photon-parity protocols. To reach a desired precision on the classical shadow of an N -photon density matrix with high probability, we show that homodyne detection requires order O(N4+1/3) measurements in the worst case, whereas photon-number-resolving and photon-parity detection require O(N4) measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental analyses of homodyne tomography match closely with our theoretical predictions. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements. Published by the American Physical Society 2024

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 introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.