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Predicting Many Properties of a Quantum System from Very Few Measurements

Hsin-Yuan Huang, Richard Kueng, John Preskill

arXiv:2002.08953v2quant-phcs.ITcs.LG

TL;DR

Predicting many properties of an unknown quantum state requires an efficient measurement strategy with rigorous sample guarantees. Classical shadows construct a reusable classical description from randomized measurements, enabling prediction of many target functions with logarithmic dependence on their number while matching information-theoretic limits.

  • Problem

    The required measurement scaling for predicting observables may reflect a fundamental information-theoretic limitation rather than an artifact of classical-shadow prediction.

  • Method

    Randomized unitary rotations followed by computational-basis measurements produce stored classical snapshots that form an approximate classical description of the quantum state.

  • Results

    Classical shadows predict M arbitrary linear target functions with additive error ϵ using N ≥ (order) log(M) max_i ∥O_i∥2 measurements, and this scaling is generally unavoidable.

  • Takeaways & Limitations

    The constructive guarantee and matching lower bound establish that classical shadows can predict arbitrary collections of target functions with near-optimal measurement complexity.

  • Takeaways & Limitations

    For non-local observables, classical shadows may require exponentially many measurements to predict even one expectation value, whereas direct spin measurements need only order 1/ϵ2 copies.

Abstract

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Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order $\log M$ measurements suffice to accurately predict $M$ different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

PROCEDURE

A classical shadow stores independently generated classical snapshots of an unknown quantum state after randomized measurements, then predicts linear properties using median-of-means estimation.

  • Each snapshot applies a random unitary, measures all qubits computationally, and stores a classical description of the transformed measurement outcome.
  • Applying the inverted measurement channel in classical post-processing produces a single snapshot whose expectation equals the unknown state.
  • Repeating the procedure N times creates the classical shadow, an array of independent classical snapshots.
  • Median-of-means estimation uses equally sized chunks of the shadow to predict linear function values while reducing the effect of outliers.
  • Random Clifford circuits and tensor products of random single-qubit Clifford circuits provide two measurement choices, with stabilizer formalism enabling efficient storage.

RIGOROUS PERFORMANCE GUARANTEES

The performance guarantee predicts many linear target functions with measurement complexity governed by their shadow norm rather than directly by system size.

  • N ≥ (order) log(M) max_i ||O_i||_shadow^2 measurements suffice to predict M arbitrary linear functions within additive error ϵ.
  • The shadow norm depends on the unitary ensemble used to create the classical shadow.
  • When target-function norms remain bounded independently of system size, many properties can be predicted using only a logarithmic number of quantum measurements.
  • Random Clifford measurements relate the shadow norm to the Hilbert-Schmidt norm, supporting efficient prediction of global observables with bounded Hilbert-Schmidt norm.
  • For random Pauli measurements, the norm scales exponentially with observable locality rather than the total number of qubits, enabling efficient prediction of many local observables.

ILLUSTRATIVE EXAMPLE APPLICATIONS

Classical shadows support fidelity, entanglement, and local-observable estimation, while their measurement requirements can become unfavorable for nonlocal global observables.

  • Quantum fidelity estimation: A dimension-independent Clifford shadow can estimate fidelity with any pure n-qubit target state, while a polynomial-size shadow estimates exponentially many pure target fidelities simultaneously.
  • Entanglement verification: Fidelities with pure target states can function as entanglement witnesses by separating an entangled state from all bipartite separable states.
  • Predicting expectation values of local observables: Logarithmic-size random Pauli shadows can efficiently estimate polynomially many expectation values of local observables.
  • Predicting expectation values of global observables (non-example): For nonlocal observables, required shadow size can scale exponentially with the Hilbert-Schmidt norm or observable locality.
  • Predicting expectation values of global observables (non-example): For a spin-chain Pauli expectation with tr(O_i^2)=2^n and locality k=n, an exponential-size shadow may be required, whereas direct spin measurement needs order 1/ϵ^2 copies.

MATCHING INFORMATION-THEORETIC LOWER BOUNDS

The paper shows that classical-shadow measurement scaling is not merely methodological: information-theoretic lower bounds make the dependence unavoidable in general.

  • The required scaling with Hilbert-Schmidt norm or observable locality reflects information-theoretic restrictions rather than an artifact of classical-shadow prediction.
  • Any single-copy measurement procedure predicting M linear functions within additive error ϵ requires at least (order) log(M) max_i ||O_i||_shadow^2 measurements.
  • For tensor products of k single-qubit observables, the relevant scaling can improve to 3k.
  • The lower bound follows by embedding the prediction procedure into a communication protocol constrained by quantum information theory.
  • Together, the two theorems provide a constructive upper bound of order log(M) max_i ||O_i||_shadow^2/ϵ^2 and show that this measurement count is unavoidable in general.

PREDICTING NONLINEAR FUNCTIONS

Classical shadows extend beyond linear observables by estimating nonlinear functions from independent snapshots, with symmetrization and median aggregation reducing prediction error. For second-order Rényi entropy, the measurement cost depends exponentially on subsystem size but not total system size, while nonlinear-function lower bounds remain unresolved.

  • Quadratic functions: Independent classical-shadow snapshots estimate quadratic functions correctly in expectation through tensor-product averaging.The estimator uses tr(Oρ ⊗ρ) and independent snapshots ˆρ_i, ˆρ_j.
  • Error reduction: Symmetrizing over snapshot pairs and taking medians reduces prediction error and the likelihood of outlier corruption.The approach generalizes to higher-order polynomials using U-statistics.
  • Rényi entropy: Second-order Rényi entanglement entropy can be estimated by rewriting tr(ρ_A^2) as an expectation involving the local swap operator.The swap operator acts on two copies of subsystem A.
  • Rényi entropy: For Rényi entropy, required measurements scale exponentially with subsystem size but remain independent of total system size.The method predicts this nonlinear property using classical shadows and is compared with the specialized Brydges et al. protocol.
  • Open limitation: Information-theoretic lower bounds have not been derived for nonlinear functions.The authors note that such bounds might follow by generalizing their methods.

NUMERICAL EXPERIMENTS

The experiments examine classical-shadow scalability and compare it with neural-network tomography across quantum prediction tasks. Classical shadows support efficient storage and prediction, while synthetic-data generation—not feature prediction—is the computational bottleneck.

  • Scalability: Classical shadows are designed for tractable data acquisition, efficient classical storage, and scalable prediction of important quantum features.The implementation uses Pauli measurements for current platforms and Clifford measurements for future quantum computers.
  • Experimental scope: Numerical experiments span systems up to 160 qubits, with synthetic-state and measurement generation forming the computational bottleneck.This bottleneck would not occur in actual experiments.
  • Comparison methods: Neural-network tomography is a promising alternative based on generative models trained on independent local SIC/tetrahedral POVM outcomes.The comparison concerns methods operating in Hilbert spaces whose dimensions can be extremely large.
  • Numerical tasks: Figure 2 evaluates GHZ-state fidelity identification and noisy-state fidelity estimation for classical shadows and NNQST.The left panel targets 0.99 fidelity, while the right uses 6 × 10^4 experiments across Z-error probabilities.

Predicting quantum fidelities (Clifford measurements)

Random Clifford classical shadows efficiently target observables with bounded Hilbert-Schmidt norm, including quantum fidelities. In GHZ-state experiments, their fidelity estimates track the true decline under noise, whereas NNQST overestimates fidelity.

  • Method: Random Clifford measurements are used to predict observables with bounded Hilbert-Schmidt norm, including fidelity with a target state.Efficient stabilizer decompositions can also make median-of-means prediction computationally efficient.
  • Fidelity prediction: Classical-shadow predictions accurately track decreasing GHZ-target fidelity as the Z-error probability p increases.The experiments learn a classical representation of the GHZ source before predicting fidelity.
  • Comparison: NNQST consistently overestimates target fidelity and reports values close to one even when p = 1 makes the state orthogonal to the target.The stated reason is that NNQST efficiently estimates only an upper bound on true fidelity.
  • Computational cost: For stabilizer-state projectors, evaluating the quadratic prediction function takes O(n^2) time under the Gottesman-Knill theorem.The runtime is dominated by evaluating quadratic functions in 2^n dimensions.

Predicting two-point correlation & subsystem entanglement entropy (Pauli measurements)

Local Pauli classical shadows predict correlations and subsystem entanglement entropies using experimentally accessible measurements. They perform comparably or better than NNQST and the Brydges et al. protocol, with favorable classical post-processing and subsystem-size scaling.

  • Pauli measurements: Random local Pauli measurements are easier to implement experimentally than general Clifford circuits and support efficient subsystem-property prediction.Subsystem properties can be predicted by constructing reduced density matrices from the classical shadow.
  • Two-point correlations: Both classical shadows and NNQST predict two-point correlations well for one-dimensional TFIM and two-dimensional Heisenberg ground states.The test systems contain 50 lattice sites and an 8 × 8 lattice, respectively.
  • Subsystem entropies: For second-order Rényi entropy on subsystems of size at most two, 2500 measurements give maximum error 0.052 versus 0.24 for the Brydges et al. protocol.The experiment uses an approximate ground state of a 10-site disordered Heisenberg chain.
  • Two-point correlations: Classical shadows avoid NNQST’s larger Heisenberg-model error and fictitious long-distance oscillations while using the same quantum measurement data.The difference is classical post-processing.
  • Two-point correlations: Roughly 10^4 times faster classical post-processing is achieved with classical shadows than with NNQST for the correlation task.The comparison evaluates processing time against prediction error as measurement count changes.
  • Subsystem entropies: Subsystem entropy measurements scale exponentially with subsystem size but independently of total qubit number, and one shadow predicts many subsystem entropies at once.The comparison includes GHZ states and the left-half subsystem.
  • Subsystem entropies: For small subsystems, classical shadows have smaller prediction error than the Brydges et al. method.Both protocols use random single-qubit rotations and basis measurements; their difference is classical post-processing.

Application to quantum simulation of the lattice Schwinger model (Pauli measurements)

Classical shadows are applied to variational quantum simulation of the lattice Schwinger model to estimate the energy variance from 4-local observables. A deterministic derandomized version is compared with hand-crafted, local-observable, and independent-measurement approaches.

  • Measurement comparison: Figure 5 compares approaches by the number of state copies needed to predict all 4-local Pauli observables in the energy-variance expansion.The comparison includes classical shadows, the hand-crafted scheme from, a recent local-observable method, and independent measurement.
  • Measurement comparison: 100 measurements per local observable define the target error level for estimating the energy variance.The comparison uses an error equivalent to measuring each relevant Pauli observable at least 100 times.
  • Variational simulation: The Schwinger-model example concerns variational states whose energy variance vanishes only for energy eigenstates.Adjusting the variational parameter to minimize the variance therefore prepares an energy eigenstate in the stated setting.
  • Problem setup: The lattice Schwinger model’s 2-local Hamiltonian yields an energy-variance expression that is a sum of expectation values of 4-local observables.The Hamiltonian is not geometrically local in one dimension after eliminating gauge fields.
  • Results: Classical shadows outperform the method from only beyond 50 qubits, and may perform worse for smaller systems.The method from was hand-crafted specifically for the Schwinger-model energy-variance task.
  • Derandomization: The deterministic classical-shadow protocol uses fixed measurements selected to optimize the remaining performance bound instead of randomized Pauli choices.This derandomization simulates the randomized scheme and is fully automated and applicable to any pre-specified set of local observables.

OUTLOOK

The outlook emphasizes classical shadows as succinct classical descriptions extracted from few single-copy measurements, enabling efficient prediction with rigorous guarantees.

  • A classical shadow is a succinct classical description extracted from reasonably simple single-copy measurements on a reasonably small number of state copies.
  • Classical shadows support accurate and efficient prediction of many quantum-state properties with a rigorous performance guarantee.Random Pauli measurements make the method feasible on current quantum platforms.

Competing interests:

The paper reports no competing interests and provides supplementary information.

  • The authors declare no competing interests.
  • Supplementary information is provided.

1. GENERAL FRAMEWORK FOR CONSTRUCTING CLASSICAL SHADOWS

The framework constructs classical shadows from randomized measurements of an unknown multi-qubit state and uses them to predict many properties. The estimator is unbiased in expectation and is obtained by inverting the measurement channel.

  • General framework: The framework considers a fixed but unknown n-qubit state ρ in d = 2^n dimensions and seeks predictions of its properties from measurements.
  • Measurement primitive: The measurement primitive randomly applies a unitary from a tomographically complete ensemble, measures in the computational basis, and records the outcome.Tomographic completeness means distinct states can be distinguished by some allowed unitary and computational-basis outcome.
  • Snapshot construction: Repeated randomized measurements produce classical snapshots by applying the inverse unitary to the measured computational-basis state.
  • Channel inversion: Tomographic completeness makes the measurement channel invertible, allowing the construction of an inverse-channel estimator.
  • Estimator properties: The classical shadow has unit trace but need not be positive semidefinite, while its expectation equals the underlying state exactly.It is the single-shot linear-inversion estimator and is intended to predict many properties with few measurements.

B. Predicting linear functions with classical shadows

Classical shadows provide unbiased estimates of linear functions, with variance controlled by a measurement-dependent norm and accuracy amplified by median-of-means estimation. Their sample complexity grows logarithmically with the number of targets, while measurement choices determine whether global or local observables are efficiently predicted.

  • A single classical shadow gives an unbiased estimate of each linear function tr(O_iρ).The estimator ˆo_i = tr(O_iˆρ) has expectation tr(O_iρ).
  • The shadow norm determines the variance and depends only on the chosen measurement primitive.The variance depends on the traceless component of the observable because classical shadows have unit trace.
  • Median-of-means estimation improves failure-probability scaling by grouping samples into batches and taking the median of their means.The probability of excessive error decreases exponentially with the number of batches K, unlike the 1/δ dependence of sample means.
  • The sample complexity scales logarithmically in M and does not explicitly depend on the problem dimension 2^n.It still depends on the measurement primitive through the shadow norm.
  • Random Clifford measurements: O(log(M) max_i tr(O_i^2)/ϵ^2) random global Clifford measurements suffice to predict M linear functions.For target functions with constant Hilbert-Schmidt norm, the rate is independent of the problem dimension and includes fidelities with pure states and entanglement witnesses.

C. Predicting nonlinear functions with classical shadows

Classical shadows extend to nonlinear target functions by combining independent snapshots with unbiased U-statistics estimators and median aggregation. For quadratic functions, the method provides high-probability simultaneous prediction guarantees, while measurement efficiency depends on locality and the chosen measurement primitive.

  • Quadratic feature prediction: Quadratic functions tr(O_iρ⊗ρ) can be estimated without bias by applying O_i to two independent classical-shadow snapshots.The estimator averages over distinct snapshot pairs, producing a U-statistics estimator whose expectation equals the target function.
  • Quadratic feature prediction: Median aggregation of U-statistics estimators makes quadratic prediction more robust to outliers and exponentially suppresses failure probabilities.The procedure divides samples into groups, constructs separate estimators, and takes their median.
  • Quadratic feature prediction: With probability at least 1 −δ, NK independent classical shadows predict all M quadratic functions within additive error ϵ.The guarantee is stated for estimates satisfying |ô_i(N,K) − tr(O_iρ⊗ρ)| ≤ ϵ simultaneously for every target.
  • Random Pauli measurements: Random single-qubit Pauli measurements support local quadratic features, with variance bounds scaling exponentially in locality rather than total system size.For an observable acting nontrivially on k qubits in each copy, the corresponding 2k-local observable has a variance bound involving 4^k.
  • Random Clifford measurements: Random Clifford measurements require O(log(M) max_i tr(O_i^2)/ϵ^2) measurements for M quadratic functions, but global features can have exponentially large Hilbert–Schmidt norms.The purity function is a non-example because its swap-operator norm scales exponentially with the number of qubits.
  • Applications: Classical shadows reduce simultaneous entanglement-witness estimation from measurement costs scaling linearly in M to log(M)-many measurements.For randomly rotated GHZ states, simulations show the method reaches witness-detection thresholds with exponentially fewer samples than direct measurement.

A. Detailed statement and proof idea

Theorem 5 establishes a lower bound for single-copy measurement procedures that predict many observables, using a communication protocol in which hidden random rotations are revealed only after measurement. The proof connects successful feature prediction to reliable message decoding.

  • Theorem 5: Theorem 5 gives a lower bound on measurements for predicting M observables with bounded Hilbert-Schmidt norm.The result is presented as a detailed restatement of the Hilbert-Schmidt-norm bound.
  • Optimality: The bound matches the order-log(M) upper bound, making classical-shadow feature prediction minimax optimal in the worst case.The paper notes that Theorem 5 identifies cases where the upper-bound measurement count is unavoidable.
  • Communication proof: Alice encodes a uniformly selected message in a quantum state, sends N copies to Bob, and Bob measures each copy before estimating the associated features.Bob uses the predicted feature values to identify the encoded message.
  • Communication proof: Loki applies an unknown random rotation before measurement and reveals it afterward, forcing Bob to measure without knowing which rotated features will be needed.After disclosure, Bob can reinterpret the predicted features as rotated properties.
  • Scope: The lower bound applies to individual-copy measurements and does not apply to protocols using collective measurements on all copies.The paper also states that the bound does not cover features known in advance.

C. Information-theoretic analysis

The information-theoretic analysis bounds how much message information each measurement can reveal under random rotations. It constructs code states and observables that are simultaneously distinguishable by target features yet individually reveal little information.

  • Information bound: Fano’s inequality and data processing imply that successful decoding requires mutual information of order log(M) between Alice’s message and Bob’s measurement outcomes.The argument conditions on Loki’s random unitary, which is independent of the message.
  • Information bound: The measurement outcomes are analyzed one copy at a time, with conditional mutual information decomposed across the N independent outcomes.The proof upper-bounds total information using the entropy of the individual outcomes.
  • Code construction: The construction chooses code states that are distinguishable by linear features while remaining mutually similar enough to reveal little information after random rotation.This tension is achieved with subspace projectors whose pairwise overlaps are controlled.
  • Code construction: A probabilistic projector construction supplies M suitable rank-r subspaces whenever M ≤ exp(rd/32) and d ≥ 4r.The existence claim follows because the probability of failure is shown to be less than one.
  • Local measurements: For local observables, the argument yields an Ω(2^k log(M)/ϵ^2) single-copy lower bound when the number of targets is not extraordinarily large.The local-measurement theorem specializes the construction to Pauli observables acting on k qubits.
  • Local measurements: The local-measurement protocol uses random product unitaries, individual local measurements, and post-measurement disclosure of the rotation.This is the local analogue of the communication protocol used for the general lower bound.
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