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The randomized measurement toolbox
Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, Peter Zoller
TL;DR
Complex many-qubit states are difficult to characterize with measured classical data, while complete tomography is exponentially inefficient. This review presents randomized measurements that convert repeatedly sampled quantum measurements into reusable classical representations, enabling estimates of diverse observables and state properties. The surveyed applications show that these protocols are practical across quantum platforms, while their costs and accuracy depend on the target property and sampling setting.
Problem
Many-qubit quantum systems are difficult to characterize accurately and efficiently, and full state tomography requires exponentially large experimental and classical resources.
Method
The review surveys protocols that repeatedly prepare and measure a state in randomly chosen bases, then use classical postprocessing to estimate desired properties.
Results
Randomized measurements support estimates of observables, purity, state overlap, higher-order functionals, and properties used in Hamiltonian simulation, chaos, entanglement, and cross-platform comparisons.
Takeaways & Limitations
The toolbox provides a succinct classical representation that preserves many physically relevant features and allows measurement data to be reused for later questions.
Takeaways & Limitations
Fidelity estimation can have worst-case measurement complexity O(2^N/ϵ^2) for generic states, despite sampling only O(1/ϵ^2) Pauli observables.
Abstract
from arXiv · showhide
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
Many-qubit quantum states are powerful but difficult to characterize efficiently with classical data. Randomized measurements address this challenge by reusing fixed random measurements and adapting classical postprocessing to estimate diverse properties.
- Motivation: Generic strongly interacting quantum systems cannot be fully and succinctly characterized using classical data.This limits classical understanding while motivating the construction of large-scale quantum systems.
- Motivation: Full state tomography requires exponentially many experiments and exponentially large classical postprocessing in the number of qubits.Many applications instead require only a less complete description of the state.
- Randomized measurement strategy: Randomized measurements sample settings from a fixed ensemble, then adapt classical postprocessing to the property being studied.The measurements can remain simple enough for today’s noisy quantum platforms.
- Advantages: For local operators, the number of experiments need not depend on the total number of qubits.Subsystem properties cost exponentially with subsystem size, while scrambling operations can support global-property estimation with modest repetitions.
- Applications: Randomized measurements simplify noise effects, making measurement imperfections easier to mitigate through modified classical postprocessing.The toolbox has been applied to state overlap, quantum chaos, entanglement, and order-parameter measurements.
- Applications: A single randomized-measurement dataset can be reanalyzed for applications different from its original purpose.This supports the principle of measuring first and asking questions later.
II. EXPERIMENTAL RECIPE AND POSTPROCESSING OF THE MEASUREMENTS
The experimental recipe repeatedly prepares a many-qubit state, applies independently sampled local random unitaries, and measures in the computational basis. Postprocessing can either estimate compatible Pauli observables from repeated settings or combine many random settings to predict broader families of observables.
- Experimental recipe: Each randomized-measurement run prepares ρ, applies a randomly selected unitary, and performs a computational-basis projective measurement.The procedure repeats K times for each of M independently sampled unitaries, totaling M·K runs.
- Experimental recipe: Local random unitaries apply independent single-qubit rotations sampled from ensembles that evenly cover each qubit’s Bloch sphere.Examples include the single-qubit Clifford group and the full unitary group U(2).
- Postprocessing: With M = 1, repeated measurements estimate one randomly selected Pauli string and its compatible subsystem marginals.Other Pauli expectation values are inaccessible in this setting.
- Postprocessing: With K = 1, combining many randomly sampled Pauli strings allows prediction of subsystem expectation values from compatible measurement outcomes.For a three-site Pauli example, compatibility occurs with probability (1/3)^3, requiring M ≥ 3^3/ϵ^2 random settings for the stated approximation scale.
Y X X
Randomized-measurement data can be postprocessed for compatible observables and for nonlinear functionals such as purity. Classical shadows package measurement settings and outcomes into state approximations that support arbitrary target observables and higher-order density-matrix polynomials.
- Pauli compatibility: A Pauli observable is compatible with a measurement string when its non-identity factors match the measured basis at the corresponding qubits.This compatibility rule determines which outcomes contribute to estimating the observable.
- Classical shadows: Classical shadows combine each measurement setting with its associated outcomes to produce an approximation of the underlying N-qubit state ρ.The collection of these approximations is called a classical shadow and supports arbitrary target observables.
- Polynomial functionals: Randomized measurements estimate purity P2 = tr(ρ^2) and other polynomial functionals of the density matrix.Purities of reduced density matrices can be accessed by restricting the postprocessing to a subsystem.
- Polynomial functionals: Purity estimators trade off accuracy according to the number of settings and repetitions, with different estimators favored in different sampling regimes.One estimator is expected to be more robust to miscalibration of random unitaries.
C. Rigorous theory and history
Randomized measurements support rigorous guarantees for estimating many subsystem observables while avoiding full quantum-state tomography. The theory exposes a tradeoff: sample complexity grows exponentially with subsystem size but only logarithmically with the number of observables.
- Rigorous guarantees: M ∝log(L)^4w/ϵ^2 randomized measurements suffice to ϵ-approximate L subsystem-size-w expectation values with high success.The guarantee applies to evenly distributed single-qubit unitary ensembles, including the full unitary and Clifford groups.
- Rigorous guarantees: For Pauli expectation values, the improved scaling is M ∝log(L)^3w/ϵ^2.This bound improves on the general theorem and predates it historically.
- Scaling tradeoffs: Theorem 1 makes subsystem size an exponential cost while the number of observables enters only logarithmically.For N-qubit systems, M ∝log(N)/ϵ^2 measurements suffice for all 2-body Pauli expectation values, independently of the state ρ.
- Relation to tomography: Full quantum state tomography generally requires exponentially many samples and exponentially large classical postprocessing.The motivation for estimating selected properties is that complete reconstruction is inefficient for many-qubit systems.
- Relation to tomography: Shadow estimation predicts selected expectation values directly instead of reconstructing the full state ρ through quantum state tomography.This randomized-measurement procedure is a near-term variant that avoids quantum memory and entangling computations required by the original formulation.
D. Vignette application: Purity measurements in an ion trap quantum simulator
Randomized measurements were used in a trapped-ion simulator to estimate second Rényi entropies and assess purity and entanglement. The experiment used local random rotations, repeated single-shot measurements, and classical postprocessing across subsystems and evolution times.
- Protocol: A 10-qubit trapped-ion experiment estimated second Rényi entropies using single-qubit Haar-random unitaries followed by computational-basis measurements.The unitaries were implemented with rotations decomposed into z- and x-axis operations.
- Protocol: The experiment used M = 500 randomized-measurement settings with K = 150 single-shot repetitions per setting.Postprocessing used an averaged purity formula designed for K ≫1.
- Results: The protocol faithfully estimated second Rényi entropies across different subsystems and evolution times.The results covered partitions of a system with N = 10 ions and times t = 0,...,5 ms.
- Results: The total-system entropy stayed near 0.4, corresponding to purity near 0.8, while subsystem entropy increased with time.The low total-system entropy indicates slight preparation and measurement errors, whereas the subsystem growth signals entanglement.
- Extensions: The same experimental data were later reanalyzed to access additional entanglement properties using the classical-shadow framework.The reanalyses were reported in Refs. [39].
III. THE MANY APPLICATIONS OF RANDOMIZED MEASUREMENTS
Randomized measurements have been applied to detect nonlocal topological order, including symmetry-protected phases and toric-code order. These protocols infer order parameters from randomized-measurement statistics and purity measurements in connected partitions.
- Broader applications: The broader randomized-measurement toolbox spans quantum many-body physics, simulation, noise diagnostics, machine learning, variational algorithms, and NISQ computation.These applications extend beyond topological-order characterization.
- Characterization of topological order: Topological phases are difficult to identify experimentally because their defining global correlations cannot be detected by local measurements.This motivates randomized-measurement protocols for accessing nonlocal order.
- Symmetry-protected phases: Randomized measurements infer the reflection invariant ZR using local random unitaries distributed symmetrically around a central bond.ZR is a nonlocal, nonlinear correlator associated with spatial reflection symmetry.
- Symmetry-protected phases: The quantity [P2(ρI1) + P2(ρI2)]/2 takes quantized values ±1 distinguishing trivial and topological phases.The subsystems I1 and I2 lie on opposite sides of the central bond, with n large compared to the correlation length.
- Toric-code order: Randomized measurements accessed topological entanglement entropy Stop from purities of connected partitions A, B, and C in a 31-qubit toric-code processor.Stop = −1 is the quantized value associated with a topologically ordered phase.
B. Quantum chaos diagnostics
Randomized measurements provide practical routes to diagnose quantum chaos, classify many-body phases, and estimate fidelities without full tomography. These protocols convert repeated randomized measurements into classical estimates of properties that are otherwise difficult to access.
- Quantum chaos diagnostics: Out-of-time-ordered correlators diagnose quantum chaos by revealing how local perturbations spread and scramble quantum information.Randomized-measurement protocols extract infinite-temperature OTOCs from correlations between separate experiments using only forward evolution and no ancillas.
- Quantum chaos diagnostics: Infinite-temperature OTOCs have been experimentally measured in trapped-ion and NMR platforms to study scrambling in quantum spin models.
- Machine learning for quantum many-body problems: Randomized-measurement data can support machine-learning models that predict local properties or consequences of many-body states.The resulting bitstring sequences serve as training data for classical models.
- Machine learning for quantum many-body problems: For the toric-code phase, CNN+Shadow and Shadow models remain accurate as random-circuit depth increases.The figure reports high prediction values despite perturbations of the two phases by random circuits.
- Machine learning for quantum many-body problems: Theoretical analysis guarantees efficient polynomial scaling for phase classification when an underlying phase-classifying function exists, even if the model does not know it explicitly.For the XXZ and toric-code examples, candidate functions include a many-body topological invariant and topological entanglement entropy.
- Fidelity estimation: Direct fidelity estimation rewrites fidelity as an expected value over a known Pauli-based distribution and estimates it using randomized measurements.With O(1/ϵ^2) sampled observables, the estimate is accurate to within F ± ϵ with high probability.
- Fidelity estimation: DFE requires only O(1/ϵ^2) Pauli observables independent of system size, but generic-state measurement complexity can still scale as O(2^N/ϵ^2).The worst case arises from high-weight Pauli strings and the precision needed to resolve generic states.
- Fidelity estimation: Randomized measurements estimate overlaps between states from distant or different quantum devices by correlating outcomes after applying identical random unitaries.The same randomized unitaries are classically communicated to both devices before computational-basis measurements.
E. Quantum gate noise characterization
Randomized dynamics simplify quantum-gate noise into analyzable forms and support practical error-rate characterization. Randomized benchmarking and its extensions improve the separation and precision of gate-error estimates while enabling broader noise diagnostics.
- Noise simplification: Randomized dynamics can transform unwanted interactions into Pauli-channel noise using inserted random Pauli gates or π-pulses.Pauli channels include depolarizing, dephasing, bit-flip, and some correlated noise, but exclude amplitude damping and coherent over-rotation errors.
- Noise simplification: Pauli-channel error rates provide a compact noise metric, support efficient simulation with Clifford circuits, and track progress toward fault tolerance.
- Randomized benchmarking: Randomized benchmarking applies random Clifford sequences of varying lengths, followed by an inverse circuit and computational-basis measurement, to estimate average error rates.
- Randomized benchmarking: Varying sequence lengths decouple preparation-and-measurement noise from gate noise and amplify small gate errors into observable signals.These properties improve gate-error accuracy and estimate precision, making RB standard for one- and two-qubit experiments.
- Randomized benchmarking extensions: Interleaved randomized benchmarking estimates a fixed Clifford gate’s average error by comparing its error rate with a baseline randomized-benchmarking rate.
- Randomized compiling: Randomized compiling reduces circuit depth slightly while projecting noise toward a Pauli-channel form and providing perturbative error analysis.The method has been demonstrated experimentally in superconducting qubits.
- Experimental applications: Noise-characterization methods have been applied to estimate average noise on a 10-qubit Mølmer–Sørensen gate and locally Clifford-averaged Pauli error rates in a 14-qubit transmon device.These approaches are organized within the ACES framework.
F. Hamiltonian & Liouvillian learning
Randomized measurements support learning Hamiltonians and Lindbladians from low-weight observables, including steady-state, dynamical, Bayesian, and entanglement-Hamiltonian settings. These approaches can efficiently reconstruct models but depend on identifiable kernels, suitable state preparation, and structural assumptions.
- Randomized measurements can learn dynamical variables governing quantum evolution, including Hamiltonians and more generally Lindbladians.
- For sufficiently generic Hamiltonians, low-weight Pauli observables measured in a ground or steady state can reconstruct the Hamiltonian up to scale and an energy shift.
- The reconstruction procedure estimates a commutator matrix from low-weight Pauli observables and uses its unique kernel as the Hamiltonian-coupling vector.A separate calibration experiment may be needed to determine the overall scale.
- Bayesian Hamiltonian learning incorporates multiple input states, prior Hamiltonian information, and well-characterized control fields, while related approaches extend learning to Lindbladian fixed points and quenched dynamics.
- Entanglement Hamiltonian tomography learns the Hamiltonian parameterizing a subsystem’s reduced state, whose spectrum contains complete information about bipartite entanglement across the cut.
- Gibbs-state Hamiltonian reconstruction applies to k-local interactions with bounded participation when the support of nonzero interactions is known, excluding some power-law two-body systems.
- Gate set tomography fits models of gates, measurements, and preparations across circuits, while Pauli-Lindbladian learning trades generality for efficient error mitigation.
G. Variational quantum-classical algorithms
Variational quantum-classical algorithms require repeated measurements of costly objective functions. Randomized measurements reuse one data set to estimate many observables, reducing measurement scaling and offering advantages for Hamiltonian-variance estimation, while high-weight observables remain a limitation.
- Variational algorithms repeatedly evaluate measurement-based cost functions on quantum states to optimize tasks such as ground-state energies and circuit compression.
- Randomized measurements jointly estimate many observables from shared classical-shadow data, requiring a number of measurements that scales logarithmically with the observable count.
- This logarithmic scaling is an exponential improvement over direct protocols that estimate observables one by one.
- For Schwinger-model Hamiltonian variance, randomized measurements scale as ∝log(L) for a variance involving ∝L2 terms and outperform optimized hand-crafted schemes at large L.
- Measurement requirements still scale exponentially with observable weight, creating difficulties for high-weight terms produced by Jordan–Wigner encodings in quantum chemistry.
- Derandomization can replace the randomized protocol with a deterministic strategy that performs at least as well, and sometimes better, for specified target observables.
H. Machine learning in quantum-enhanced feature space
Quantum-enhanced feature-space models transform inputs with NISQ devices and fit linear predictors by convex optimization. Their training is efficient, but the original in-place swap-test construction can predict worse than classical machine-learning models despite fitting training data perfectly.
- Quantum machine-learning models transform input feature vectors into higher-dimensional quantum-enhanced feature vectors using NISQ devices.
- The models train a linear function over quantum-enhanced features through convex optimization, whose global optimum can be found efficiently without barren plateaus.
- The original in-place swap-test feature construction can have poor prediction performance, including performance significantly worse than classical models on simple tasks.
- Randomized measurements allow higher-order n-copy observables to be estimated by cross-correlating n different classical shadows.
- Violations of inequalities for cyclic-permutation moments certify entanglement through a negative eigenvalue of the partial transpose, demonstrated for systems of up to 7 qubits.
- Independent local random unitaries enable multipartite-entanglement detection without a common reference frame and with robustness to local measurement-basis miscalibration.
IV. CHALLENGES AND PERSPECTIVES
Randomized-measurement methods must remain accurate on noisy, decohering NISQ devices. Averaging and calibration can simplify measurement noise for classical correction, but effectiveness depends on noise strength and calibration assumptions.
- Noise and decoherence, including in the measurement process, make robustness essential for practical characterization of NISQ quantum systems.
- Randomized measurements can reduce implementation and measurement noise to an averaged noise channel that calibration experiments can learn for robust estimation.
- Purity and fidelity estimators based only on measured bitstrings are insensitive to gate-independent unitary errors when random unitaries form a unitary 2-design.
- Classical shadows can calibrate an unknown noise channel by twirling it into a stochastic Pauli channel and compensating it during classical postprocessing.
- The calibration-based correction is effective only when the noise is not too strong.
B. Local vs. global random unitaries
The randomized-measurement toolbox spans local, global, and intermediate random unitaries, with implementation choices shaped by the target property and available experimental control. Its scope extends beyond qubits to qudits, fermions, and bosons, while hardware constraints can limit local implementations.
- Global random unitaries: Global random unitaries scramble information across the entire system and can be implemented with quantum circuits or approximate random quenches.
- Global random unitaries: Global random unitaries offer analytic expressions for certain global state properties, but their post-processing requirements differ from those of local random unitaries.
- Intermediate random unitaries: Shallow-depth random circuits interpolate between local and global unitaries: shallow circuits favor local properties, whereas greater depth can reduce experimental runs for some global properties.
- Experimental considerations: The local-versus-global choice depends on experimental control and decoherence, while shallow-depth circuits can compromise between interaction requirements and limited coherence time.
- Beyond qubits: The toolbox generalizes beyond qubits to qudits, fermionic systems, and bosonic systems using suitable random unitaries.
- Beyond qubits: Fermionic experiments may lack local random unitaries because physical Hamiltonians obey conservation laws such as atom-number conservation in closed systems.
- Classical representations: Randomized measurements convert quantum systems into efficient classical representations that retain many aspects of the original state.