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The randomized measurement toolbox

Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, Peter Zoller

arXiv:2203.11374v1quant-ph

TL;DR

Characterizing complex many-qubit states from classical measurement data remains difficult even as programmable quantum platforms create and control such states. The paper reviews randomized measurement protocols for turning repeated measurements into classical data to estimate state properties, and concludes that many physically relevant features survive this conversion.

  • Problem

    Characterizing complex quantum systems from measured classical data remains challenging as programmable platforms create and control complex states.

    Source: §I · p. 1

  • Method

    The paper reviews protocols that map repeated randomized measurements of a multiqubit state to succinct classical data for later analysis of properties of interest.

    Source: §I · p. 2

  • Results

    Some reviewed applications reanalyze data originally collected for a different purpose, illustrating that measurement data can support later investigations.

    Source: §I · p. 2

  • Takeaways & Limitations

    The reviewed applications show that many physically relevant features of a complex state can survive its conversion into classical data.

    Source: §I · p. 2

    The quantum-to-classical conversion unavoidably discards a vast amount of information about the state.

    Source: §I · p. 2

Abstract

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Increasingly sophisticated programmable quantum simulators and quantum computers are opening unprecedented opportunities for exploring and exploiting the properties of highly entangled complex quantum systems. The complexity of large quantum systems is the source of their power, but also makes them difficult to control precisely or characterize accurately using measured classical data. We review recently developed protocols for probing the properties of complex many-qubit systems using measurement schemes that are practical using today's quantum platforms. In all these protocols, a quantum state is repeatedly prepared and measured in a randomly chosen basis; then a classical computer processes the measurement outcomes to estimate the desired property. The randomization of the measurement procedure has distinct advantages; for example, a single data set can be employed multiple times to pursue a variety of applications, and imperfections in the measurements are mapped to a simplified noise model that can more easily be mitigated. We discuss a range of use cases that have already been realized in quantum devices, including Hamiltonian simulation tasks, probes of quantum chaos, measurements of nonlocal order parameters, and comparison of quantum states produced in distantly separated laboratories. By providing a workable method for translating a complex quantum state into a succinct classical representation that preserves a rich variety of relevant physical properties, the randomized measurement toolbox strengthens our ability to grasp and control the quantum world.

I. INTRODUCTION AND MOTIVATION

Generic strongly interacting quantum systems cannot be fully and succinctly characterized from classical data, yet increasingly programmable quantum platforms can create and control complex many-body states. Randomized measurements offer a practical route to concise classical data that can be reused to estimate many physically relevant properties without full tomography.

I. INTRODUCTION AND MOTIVATION

  • Powerful programmable platforms already create and control complex states of atoms, spins, photons, and superconducting circuits, making characterization a laboratory imperative.These opportunities support scientific discovery even while large-scale error-corrected quantum computers remain in the future.Source: §I · p. 1
  • Full tomography requires exponentially many experiments and classical processing in qubit number, while many tasks need only a less complete, more efficient state description.The introduction motivates developing practical methods to characterize complex states without reconstructing them completely.Source: §I · p. 1
  • A fixed ensemble of randomly sampled measurements can be tailored later in classical postprocessing; local measurements estimate L local operators in order log(L) repetitions, independent of total qubit count.Subsystem estimates cost exponentially in subsystem size but less than subsystem tomography; global properties can use modest repetitions after information-scrambling unitaries, and randomization simplifies noise mitigation.Source: §I · pp. 1–2
  • Demonstrated applications include estimating state overlap across distant laboratories, probing chaotic dynamics, quantifying entanglement, identifying topological order, and estimating local Hamiltonian expectation and variance.Out-of-time-order correlators can probe chaos without reversing time evolution or introducing ancillas.Source: §I · p. 2
  • Randomized measurements turn repeated preparations of complex quantum states into succinct classical data that retain many physically relevant properties despite discarding much of the state information.The resulting data can be reanalyzed for questions not known when measurements were taken, including data originally collected for another purpose.Source: §I · p. 2

II. EXPERIMENTAL RECIPE AND POSTPROCESSING OF THE MEASUREMENTS · A. Protocol—Data acquisition · B. Protocol—Postprocessing

The section describes how repeated randomized measurements acquire data and how classical postprocessing uses those data to estimate observables and density-matrix functionals.

A. Protocol—Data acquisition

  • For an N-qubit state, randomized measurement applies independently sampled local rotations, measures computational-basis bitstrings, repeats K shots per setting, and repeats across M independent settings (MK runs).The single-qubit rotations are sampled from ensembles that evenly cover each qubit's Bloch sphere.Source: A. Protocol—Data acquisition · pp. 2–3

B. Protocol—Postprocessing

  • With M = 1 and K > 1, repeated measurements estimate the chosen Pauli string and compatible subsystem marginals, but not other Pauli expectations.The same randomized measurement corresponds to measuring one random Pauli string repeatedly.Source: B. Protocol—Postprocessing · p. 3
  • With K = 1, compatible outcomes across random Pauli strings estimate many size-w expectations; for XYZ on three sites, matching probability is 1/27 and M ≥ 3^3/ϵ^2 settings suffice for ϵ accuracy.The sampling requirement depends on subsystem size, not the qubit locations or the particular Pauli strings of interest.Source: B. Protocol—Postprocessing · p. 3, Figure 2 · p. 3
  • For a target observable, empirical averaging over compatible outcomes produces an estimate; combining settings and outcomes yields classical shadows [24] that also handle arbitrary observables and evenly covering unitary ensembles.The method is not limited to Pauli observables or Clifford rotations.Source: B. Protocol—Postprocessing · p. 3
  • Randomized measurements estimate purity by averaging weighted Hamming distances of distinct outcomes from the same setting and then averaging across settings.The passage explains that second-order correlations averaged over local random unitaries yield the purity, tr(ρ^2).Source: B. Protocol—Postprocessing · pp. 3–4
  • Replacing distinct copies with distinct classical shadows also estimates purity, and this procedure extends to arbitrary density-matrix polynomials [23, 35, 39, 40] and reduced-subsystem purities.Reduced density matrices can be selected by restricting the postprocessing to a subsystem.Source: B. Protocol—Postprocessing · p. 4
  • For a fixed MK, the same-setting purity estimator is more accurate with many repetitions of fewer settings, while the shadow estimator performs better with many settings and few repetitions and is expected to better tolerate unitary miscalibration.The same-setting estimator depends on measured bitstrings rather than explicit matrix elements of the applied random unitaries.Source: B. Protocol—Postprocessing · p. 4

C. Rigorous theory and history

Randomized measurements admit rigorous guarantees for estimating many subsystem expectation values, with sample complexity depending on subsystem size and observable count rather than the underlying state. The section situates these guarantees against tomography and earlier shadow-estimation approaches.

C. Rigorous theory and history

  • For evenly distributed single-qubit unitary ensembles, M ∝ log(L)4^w/ϵ^2 measurements suffice to estimate L subsystem-size-w expectation values within ϵ with high success.The guarantee applies to ensembles such as the full unitary or Clifford group.Source: C. Rigorous theory and history · p. 4
  • For Pauli expectations, the improved bound is M ∝ log(L)3^w/ϵ^2; it predates Theorem 1, and repetitions per setting can further improve performance.The result was first proven in Ref. [41, Appendix D], influenced by the quadratically weaker bound in Ref.; the general theorem builds on Ref..Source: C. Rigorous theory and history · p. 4
  • The bound trades exponential dependence on subsystem size for logarithmic dependence on observable count and is independent of the state ρ.For example, M ∝ log(N)/ϵ^2 randomized N-qubit Pauli measurements suffice to estimate all 2-body Pauli expectations.Source: C. Rigorous theory and history · p. 4
  • Estimating selected properties can be easier than full tomography, which generally requires M·K ≥ 4^N/ϵ^2 samples for ϵ-accuracy.Purity estimation still scales exponentially with N, but with a significantly reduced exponent compared with full state tomography.Source: C. Rigorous theory and history · p. 4
  • This near-term shadow-estimation variant predicts subsystem expectations without reconstructing ρ, unlike the original approach requiring quantum memory and entangling computations on multiple copies.Shadow estimation is also connected to Refs., [25], and resource-efficient tomography approaches.Source: C. Rigorous theory and history · p. 4

D. Vignette application: Purity measurements in an ion trap quantum simulator

In the ion-trap experiment, randomized measurements estimated second Rényi entropies across subsystems and evolution times, revealing near-constant total-system purity alongside growing subsystem entanglement.

D. Vignette application: Purity measurements in an ion trap quantum simulator

  • The total-system second Rényi entropy stayed near 0.4 (purity P2 ∼ 0.8), while subsystem entropy grew beyond it, indicating entanglement.The protocol faithfully estimated entropies for different subsystem sizes and evolution times.Source: Figure 3 · p. 5, D. Vignette application: Purity… · p. 5
  • In the 10-qubit ion trap, single-qubit random rotations followed by computational-basis measurements used local z rotations and a global x-axis beam [34].Acquisition used M = 500 measurement settings with K = 150 single-shot repetitions each, processed with the averaged purity formula for K ≫ 1.Source: Figure 3 · p. 5, D. Vignette application: Purity… · p. 5
  • Subsystem purity checks whether a device produces a pure state, while second Rényi entropy S2(ρA) = −log2(P2) quantifies entanglement in quantum many-body systems.The experiment evolved an initial Néel state under an XY Hamiltonian; preparation errors, dephasing, and spontaneous emission yielded a density matrix ρ(t).Source: D. Vignette application: Purity… · p. 5

III. THE MANY APPLICATIONS OF RANDOMIZED MEASUREMENTS

The section surveys randomized measurements across a range of applications in quantum physics, simulation, diagnostics, learning, algorithms, and computation.

III. THE MANY APPLICATIONS OF RANDOMIZED MEASUREMENTS

  • Randomized measurements are applied to many-body physics, quantum simulation, noise diagnostics, machine learning, variational algorithms, and NISQ-device quantum computation.The passage also indicates that this list is not exhaustive.Source: §III · p. 5

A. Characterization of topological order

Randomized measurements (RMs) address the challenge of identifying topological phases, whose global correlations make them undetectable by local measurements, by enabling access to nonlocal topological properties.

A. Characterization of topological order

  • Because topological phases are characterized by global correlations and cannot be detected by local measurements, their identification in quantum simulations is challenging.RMs are proposed as an experimental tool to detect and classify these phases.Source: A. Characterization of topological order · p. 5
  • For SPT phases, reflection-symmetric random unitaries let bitstring statistics infer the reflection invariant ZR, whose purity-based order parameter is quantized at ±1 for trivial versus topological phases.The subsystem must contain 2n spins symmetrically across the central bond, with n large compared with the correlation length.Source: A. Characterization of topological order · pp. 5–6, Figure 4 · p. 6
  • The reflection-invariant protocol distinguishes topological from trivial phases in the extended Su-Schiefer-Heeger model and generalizes to time-reversal and internal-symmetry invariants.RMs have also been developed to access many-body Chern numbers revealing topological order in certain fractional quantum Hall states.Source: A. Characterization of topological order · p. 6, Figure 4 · p. 6
  • RMs detected topological order in a 31-qubit toric-code implementation by measuring partition purities to access topological entanglement entropy Stop, quantized at −1 in the ordered phase.The reported randomized-measurement average corresponds to Stop for a nine-qubit partition.Source: A. Characterization of topological order · p. 6, Figure 4 · p. 6

B. Quantum chaos diagnostics · C. Machine learning for quantum many-body problems

Randomized measurements provide practical diagnostics of quantum scrambling and spectral statistics, while also turning many-body states into classical data for local-property estimation and phase classification.

B. Quantum chaos diagnostics

  • Out-of-time-ordered correlators (OTOCs) diagnose many-body quantum scrambling by tracking how local perturbations spread over time.At infinite temperature, O(t) = tr(ρ∞W(t)VW(t)V), where W and V are local operators and W(t) evolves under the Hamiltonian.Source: B. Quantum chaos diagnostics · p. 6
  • Randomized measurements extract infinite-temperature OTOCs from correlations across two experiments using identical randomized initial states, without ancillas or backward evolution.The protocol also extends to finite-temperature OTOCs by sampling global random states according to their overlaps with the thermal state.Source: B. Quantum chaos diagnostics · p. 7
  • Infinite-temperature OTOCs were measured in trapped-ion and NMR experiments to study scrambling in quantum spin models with tunable long-range interactions [66].Source: B. Quantum chaos diagnostics · p. 7
  • A single-run protocol estimates the spectral form factor K(t) = |tr(exp(−iHt))|^2 from final computational-basis statistics after applying the same local random unitaries before and after evolution.K(t) probes eigenlevel statistics, tests random-matrix predictions and universal thermalization, and accesses dynamical-process properties without an initial state or measured observable.Source: B. Quantum chaos diagnostics · p. 7

C. Machine learning for quantum many-body problems

  • Order log(N)/ϵ^2 randomized measurements suffice to ϵ-approximate all constant-size reduced density matrices, enabling nonlinear functions on those subsystems [57].Source: C. Machine learning for quantum… · p. 7
  • Randomized measurements turn many-body states into bitstring training data for classical ML, whose training size, measurement count, and runtime scale polynomially if a phase-classifying function exists.The model need not know the function explicitly and is guaranteed to find it or something better, potentially yielding new, more compact phase classifiers.Source: C. Machine learning for quantum… · pp. 7–8
  • Topological entanglement entropy identifies topological phases; moderate randomized measurements can estimate it on subsystems larger than the correlation length but independent of system size, or ML can classify phases without known order parameters.Source: C. Machine learning for quantum… · p. 7
  • Using 500 randomized measurements, ML clustered XXZ states by phase and accurately classified toric-code states despite increasing random-circuit depth.The examples cover a 300-spin bond-alternating XXZ model with three phases and a 200-qubit toric code with two phases.Source: C. Machine learning for quantum… · p. 7

D. Fidelity estimation

The randomized measurement toolbox estimates fidelity either between an imperfect state and a known pure target or between states produced in different experiments. These protocols avoid full tomography, though their measurement costs and applicable settings differ.

D. Fidelity estimation

  • Direct fidelity estimation (DFE) importance-samples Pauli observables to estimate pure-target fidelity within F ± ϵ with high probability using O(1/ϵ^2) samples.The target state's known Pauli coefficients define the sampling distribution, while the corresponding observables of the prepared state are estimated empirically.Source: D. Fidelity estimation · p. 8
  • In the worst case, DFE requires O(2^N/ϵ^2) measurements because high-weight Paulis are difficult to estimate and generic fidelities require precision ±1/2^(N/2).This can overstate costs when target-state Pauli weight is concentrated on a small set of low-weight observables, as for local-generator stabilizer states and high-temperature local-Hamiltonian Gibbs states.Source: D. Fidelity estimation · p. 8
  • DFE generalizes to processes and has estimated Toffoli process fidelity, validated a 14-qubit ion-trap preparation, and tested a simpler variant on four-qubit states [75] [76].The variant uses a simpler Pauli measurement scheme and achieves nearly optimal sample complexity.Source: D. Fidelity estimation · p. 8
  • For device benchmarking, shared random unitaries and computational-basis measurements yield tr(ρ1ρ2) from outcome correlations, enabling comparison of mixed-state fidelity.The states may come from distant, asynchronous, or physically different devices, and may be constant-size subsystems of larger many-body systems; measurement cost is exponential in subsystem size but below full tomography.Source: D. Fidelity estimation · p. 9
  • Demonstrations compared states of up to 10 qubits with theory or another preparation in the same experiment, and compared up to 13-qubit states across six trapped-ion or superconducting devices [31].Source: D. Fidelity estimation · p. 9

E. Quantum gate noise characterization · F. Hamiltonian & Liouvillian learning

Randomized dynamics simplify quantum gate noise and enable its characterization, while randomized measurements support learning Hamiltonians and Lindbladians from observable data. The sections cover methods ranging from benchmarking and noise estimation to steady-state, thermal-state, and process-tomography approaches.

E. Quantum gate noise characterization

  • Inserting random Pauli gates projects noise into a Pauli channel, simplifying characterization into Pauli error-rate estimation.Pauli channels include depolarizing, dephasing, bit-flip, and some correlated noise, but not amplitude damping or coherent over-rotation.Source: E. Quantum gate noise characterization · p. 9
  • Randomized benchmarking estimates average gate error rates from the exponential decay of the probability of measuring 0 across random Clifford sequences [90, 91].Under identical, Markovian, time-stationary gate noise, varying sequence lengths reduce state-preparation and measurement noise effects and amplify small gate errors, making RB a standard for one- and two-qubit experiments.Source: E. Quantum gate noise characterization · pp. 9–10
  • Interleaved and simultaneous RB estimate gate-specific and crosstalk errors, while randomized compiling and ACES extend Pauli-noise characterization toward larger circuits [108].Experiments estimated average noise on a 10-qubit Mølmer–Sørensen gate and locally Clifford-averaged Pauli rates in a 14-qubit transmon device; ACES was numerically shown to scale to at least 100 qubits.Source: E. Quantum gate noise characterization · p. 10

F. Hamiltonian & Liouvillian learning

  • For sufficiently generic Hamiltonians, low-weight Pauli expectations in a ground or steady state can reconstruct the Hamiltonian up to scale and an unobservable energy shift.For k-body couplings, the method estimates K_lm = i tr(ρ[W_l, W_m]); the coupling vector lies in its kernel, and a unique kernel vector estimates the couplings up to scale.Source: F. Hamiltonian & Liouvillian learning · p. 10
  • Learning requires a preparable steady state and a sufficiently well-conditioned kernel; randomized measurements estimate the needed observables with weight at most 2k − 1.Time averaging can prepare the steady state, with a Gaussian-quadrature protocol achieving super-exponential convergence in the number of averages; a non-unique approximate kernel or small least nonzero eigenvalue of K^T K makes the problem ill-conditioned.Source: F. Hamiltonian & Liouvillian learning · p. 10
  • Bayesian priors and calibrated controls, Lindbladian fixed points, and quenched dynamics extend Hamiltonian learning [41].Entanglement Hamiltonians were learned from randomized-measurement data for subsystems of up to 7 qubits in a 20-qubit trapped-ion simulator after quenches; the learned Gibbs-state fidelity varies with time and ansatz complexity, with multi-body corrections important at intermediate times.Source: F. Hamiltonian & Liouvillian learning · pp. 10–11, Figure 6 · p. 11
  • At small inverse temperature, few-qubit Pauli expectations can reconstruct Hamiltonians with bounded interaction size and degree when interaction supports are known [116].The approach assumes Gibbs-state preparation and excludes some systems, including two-body Hamiltonians with power-law interactions; randomized measurements can estimate the required expectations.Source: F. Hamiltonian & Liouvillian learning · p. 11
  • Gate set tomography jointly fits gates, measurements, and preparations from varied circuits, while newer variants target faster or scalable learning and Lindbladian error mitigation [123] [125].GST uses error-amplifying germs; fast Bayesian tomography used side information for online processing, and Pauli-Lindbladian learning enabled probabilistic error cancellation despite crosstalk.Source: F. Hamiltonian & Liouvillian learning · pp. 11–12

G. Variational quantum-classical algorithms

Variational quantum-classical algorithms use NISQ devices with classical computers, but repeated measurement of complicated cost functions is a practical obstacle. Randomized measurements can estimate many observables from shared data, while their measurement cost and available refinements depend on observable structure.

G. Variational quantum-classical algorithms

  • These hybrid algorithms pair NISQ quantum co-processors with classical computers for optimization tasks such as many-body ground-state energies and variational circuit compression.In quantum chemistry and quantum field theory, energy cost functions require measurements in many settings for each evaluation.Source: G. Variational quantum-classical… · p. 12
  • Randomized measurements jointly estimate many observables from the same classical-shadow data, with measurement count scaling logarithmically in the number of observables and exponentially improving over one-by-one estimation.This addresses the repeated-measurement burden of variational cost functions involving many Hamiltonian terms.Source: G. Variational quantum-classical… · p. 12
  • ∝log(L) randomized measurements estimate the Schwinger-model Hamiltonian variance involving ∝L2 terms, outperforming an optimized hand-crafted scheme for large L [23, 130].The figure reports an exponential advantage over direct measurement of Hamiltonian terms and hand-crafted schemes for large systems; derandomization can further improve the protocol [131].Source: Figure 7 · p. 12, G. Variational quantum-classical… · p. 12
  • Measurement requirements still scale exponentially with observable weight, which can be problematic for fermionic systems such as quantum chemistry with high-weight Jordan–Wigner Hamiltonian terms.Observable weight includes the locality of the many-body Hamiltonian.Source: G. Variational quantum-classical… · p. 12
  • Importance sampling can use knowledge of target observables or the quantum state, while derandomization gives a deterministic protocol that performs at least as well and sometimes much better [132, 133, 49, 131].The deterministic strategy is tailored to a specific set of target observables.Source: G. Variational quantum-classical… · p. 12

H. Machine learning in quantum-enhanced feature space · I. Higher order polynomial functionals of the density matrix

Randomized measurements support improved quantum-enhanced machine learning and provide access to higher-order density-matrix functionals. These tools enable applications including entanglement detection and polynomial-functional estimation for tensor networks.

H. Machine learning in quantum-enhanced feature space

  • Quantum-enhanced feature-space models transform inputs and train linear predictors by convex optimization, which avoids barren plateaus and allows efficient access to the global optimum.The models predict outputs such as a real number or discrete label.Source: H. Machine learning in… · pp. 12–13
  • Randomized-measurement feature vectors achieve significantly higher prediction accuracy than fidelity-overlap proposals and conventional classical ML, with a quadratic measurement-count speed-up over the original proposals [138].The randomized variant also has a simple proof of quantum speed-up in the fault-tolerant regime.Source: H. Machine learning in… · p. 13
  • The original in-place swap-test proposal can predict worse than classical ML on simple tasks, even when it perfectly fits training data.This limitation motivates the randomized-measurement feature-vector construction.Source: H. Machine learning in… · p. 13

I. Higher order polynomial functionals of the density matrix

  • Classical shadows estimate any n-copy observable tr(Oρ^⊗n) by cross-correlating n distinct shadows, generalizing purity measurement [23].Purity is the n = 2 swap-operator case.Source: §I · p. 13
  • Violating inequalities on subsystem cyclic-permutation moments implies a negative eigenvalue of ρAB^TA and therefore entanglement; this criterion demonstrated mixed-state entanglement in systems up to 7 qubits.The moments involve separate n-copy cyclic permutations acting on subsystems A and B [35, 139].Source: §I · p. 13
  • Randomized measurements also infer symmetry-resolved entropies, quantum Fisher information, and moments of realigned density matrices as multi-copy observables [39, 40, 141].Reference gives closed-form formulas for measurement counts needed to estimate arbitrary n-copy observables via classical shadows.Source: §I · p. 13
  • Randomized-measurement moments can detect multipartite entanglement without a shared reference frame and resist local basis miscalibration; the method was demonstrated with four-photon states [146].It is suited to quantum networks with distantly separated parties.Source: §I · p. 13
  • Randomized measurements have inspired tensor-network sampling methods that estimate polynomial functionals without potentially costly contraction of multiple replicas [148].Source: §I · p. 13

IV. CHALLENGES AND PERSPECTIVES · A. Dealing with experimental imperfections: Robustness and error mitigation

Randomized-measurement estimates can be made robust to experimental noise by simplifying noise through randomization and compensating for characterized errors in classical postprocessing. The approach covers gate-independent unitary errors, known readout or depolarization errors, and—under a noise-strength condition—unknown noise channels.

A. Dealing with experimental imperfections: Robustness and error mitigation

  • NISQ noise and decoherence affect quantum operations, including measurements, so practical property-learning procedures must remain accurate under some level of noise.Source: A. Dealing with experimental… · p. 13
  • Averaging over random unitary rotations can reduce noise in the rotations and projective measurements to an averaged channel that calibration experiments can learn for robust estimation.Source: A. Dealing with experimental… · p. 13
  • Purity and fidelity formulas use only measured bitstrings, not the applied unitaries beyond their 2-design property, making estimates insensitive to gate-independent unitary errors.Examples include random over- and under-rotations.Source: A. Dealing with experimental… · p. 13
  • Calibration experiments can correct estimates for well-characterized depolarization or simple qubit read-out errors, yielding unbiased estimates.Source: A. Dealing with experimental… · p. 13
  • For classical shadows, calibration with an easily prepared state such as |0⟩⊗n and random-unitary twirling converts unknown noise into a characterized stochastic Pauli channel [29] [150].Source: A. Dealing with experimental… · pp. 13–14
  • When that noise is not too strong, classical postprocessing compensates the Pauli channel’s bias, leaving estimated-observable noise at the sampling-error floor of channel characterization [29] [150].Source: A. Dealing with experimental… · p. 14

B. Local vs. global random unitaries · C. Beyond qubits · D. Learning about the quantum world using classical machines

The randomized-measurement toolbox extends from local qubit rotations to global and intermediate random unitaries, non-qubit systems, and classical representations that enable machines to learn about quantum systems.

B. Local vs. global random unitaries

  • Global random unitaries support analytic estimates of some global properties, but general-purpose postprocessing can be costly unless using multi-qubit Clifford circuits, which permit efficient classical processing [25].General-purpose formulas may require explicit knowledge of the random unitaries, whose description becomes challenging as system size grows.Source: B. Local vs. global random unitaries · p. 14
  • Shallow-circuit depth tunes locality: very shallow circuits suit local properties, while runs needed for some global estimates, such as global state fidelity, decrease exponentially with depth [151].Unlike local-random-unitary protocols, this approach lacks simple analytical expressions for classical shadows and succinct property prediction.Source: B. Local vs. global random unitaries · p. 14
  • Hardware choice balances coherence and interactions: local unitaries need system-size-independent coherence but negligible residual couplings, whereas global quenches need size-growing coherence and shallow circuits offer a compromise [154] [151].Global random unitaries can be generated in an interaction background; random quenches approximate a unitary 2-design, while Clifford-circuit size scales quadratically with qubit number.Source: B. Local vs. global random unitaries · p. 14

C. Beyond qubits

  • The toolbox generalizes to qudits, whose single-system random unitaries and state-resolved measurements are implementable on platforms such as trapped ions and Rydberg tweezer arrays [155, 5, 8].Qudits are d-level systems with d ≥ 3 and can represent spins S > 1/2 in quantum simulation.Source: C. Beyond qubits · p. 14
  • For fermionic and bosonic systems, global random unitaries can respect particle-number conservation; local random unitaries are typically unavailable in optical-lattice experiments.Quantum-gas microscopes provide single-site control and site-resolved single-shot readout for studying strongly correlated phases and dynamics.Source: C. Beyond qubits · pp. 14–15

D. Learning about the quantum world using classical machines

  • Randomized measurements convert quantum systems into efficient classical representations, allowing classical learning algorithms to be applied to quantum-mechanical questions.The representation is intended to capture many aspects of the original quantum object.Source: D. Learning about the quantum world… · p. 15

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