Lives or dies on the shadow norm of the target observables under the chosen ensemble. The random-Pauli ensemble is shallow and hardware-ready but costs exponentially in the observable's locality; the global random Clifford ensemble handles dense observables but needs an n-qubit Clifford circuit, which is deep. The guarantee is additive per observable and says nothing about relative error for near-zero expectations. This is what makes the layer worth having: reconstructing $ρ$ itself instead costs $O(rank(ρ)·d/ε²) ≤ O(d²/ε²)$ copies for trace-distance error $ε$ (O'Donnell and Wright, upper bounds only), which at $d = 2^n$ is exponential in the qubit count.

A preparation routine $A$ with $A\lvert 0\rangle = \lvert \psi\rangle$, or repeated copies of $\rho$; a description of $O$; a target additive error $\varepsilon$ and a confidence $1-\delta$. Coherent, controlled access to $A$ and $A^\dagger$ is required by some methods here and by none of the sampling-based ones.

1. State you can prepare → Number with an error barinvert the channel, median-of-means

   Each round rotates the state by a random unitary, $\rho \mapsto U\rho U^{\dagger}$, and measures in the computational basis. Averaged over unitary and outcome that defines a channel, $\mathbb{E}[U^{\dagger}|\hat{b}\rangle\langle\hat{b}|U] = \mathcal{M}(\rho)$. assumption: $\mathcal{M}$ "is invertible if the ensemble of unitary transformations defines a tomographically complete set of measurements" — random $n$-qubit Clifford or single-qubit Clifford circuits are the two the paper uses. Applying the inverse classically to a single outcome gives a snapshot $\hat{\rho} = \mathcal{M}^{-1}(U^{\dagger}|\hat{b}\rangle\langle\hat{b}|U)$ that "exactly reproduces the underlying state in expectation", $\mathbb{E}[\hat{\rho}] = \rho$, and $N$ of them form the shadow. An observable is then read out of the array rather than from a fresh experiment: the shadow is split into $K$ equal batches averaged to $\hat{\rho}_{(k)}$, and $\hat{o}_{i} = \mathrm{median}\{\mathrm{tr}(O_{i}\hat{\rho}_{(1)}), \ldots, \mathrm{tr}(O_{i}\hat{\rho}_{(K)})\}$. approximation: that median of means stands in for a single sample mean, buying robustness to outlier snapshots at some cost in statistical efficiency.

   approximationassumption

A scalar estimate with a stated additive-error guarantee, plus the shot or query budget and the maximum circuit depth actually consumed.

• Prepare an input state runs N = O(log(M) · max_i ||·||²_shadow / ε²) measurements, one preparation each

  Map $|0\ldots0⟩$ to a state whose amplitudes are proportional to a specified vector $b$, to within $\varepsilon$. The cost is set by which description of $b$ you hold, not by the algorithm that consumes it.

  The loop closes through a measurement: every turn ends in a readout and starts from a fresh preparation. The price is a count of runs, not a depth. A measured loop like direct sampling, and the same ε^-2 — what shadows buy is not a shorter loop but a loop whose length no longer grows with the number of observables, since M enters only logarithmically and the observables may be chosen after the data is taken. Formally the turns are grouped: K = 2 log(2M/δ) median-of-means batches of N = 34/ε² · max_i ||·||²_shadow each. The shadow norm is the factor that decides whether that is cheap, and it is exponential in locality under the random-Pauli ensemble.

  Each round's randomness sits in the readout, not the input: the preparation is identical in all $N$ rounds, and the slot supplies one fresh copy of the same unknown $\rho$ per snapshot. The measurement destroys that copy, so none is reused. The protocol consumes $N = O(\log(M)\max_i \|O_i - \mathrm{tr}(O_i)2^{-n}I\|_{\mathrm{shadow}}^{2}/\varepsilon^{2})$ of them to predict $M$ observables to additive error $\varepsilon$, and each returns not a state but one classical bitstring. assumption: the copies are independent and identically prepared — the array the estimator consumes is one of "$N$ independent, classical snapshots of $\rho$" (Eq. 3). Single-copy consumption is the protocol's definition, not a tuning choice, and the paper marks it as the boundary of its own optimality claim: Theorem 5 "assumes that the copies of the state are measured individually" and "does not apply to protocols where collective measurements are applied across many copies."

  assumption

given  a preparation of rho, run once per measurement, a chosen ensemble
       of random unitaries, and a target additive error epsilon -- the M
       observables to be predicted need not be fixed yet

repeat N times, one state preparation per measurement:
    draw a random unitary from the chosen ensemble and apply it
    measure in the computational basis
    keep the pair (unitary, outcome)
    # the draw is the choice the method lives or dies on, through the
    # shadow norm of the target observables under that ensemble:
    #     random Pauli -- shallow and hardware-ready, but costs
    #         exponentially in the observable's locality; shadow norm
    #         bounded by 4^k ||O||^2_infinity for a k-local O, improving
    #         to 3^k for tensor products of single-qubit observables
    #     global random Clifford -- handles dense observables, bounded by
    #         3 tr(O^2), but needs an n-qubit Clifford circuit, deep

# N = O(log(M) * max_i ||O_i - tr(O_i) 2^-n I||^2_shadow / epsilon^2) total
# measurements to predict M linear functions to additive error epsilon;
# formally K = 2 log(2M/delta) median-of-means batches, with
# N = 34/epsilon^2 * max_i ||.||^2_shadow per batch

invert the measurement channel on each kept pair
    # each pair becomes an unbiased single-shot snapshot of rho

choose the observables O_1 ... O_M
    # they may be chosen after the data has been taken, and M enters the
    # count above only logarithmically, so the loop does not lengthen with
    # the number of observables

predict all M at once by median-of-means over the snapshots

return the M predictions

# the guarantee is additive per observable and says nothing about relative
# error for near-zero expectations

# independence of system size holds for observables of bounded shadow norm
# and is not a claim about arbitrary observables: quoting the log(M)
# without the max_i ||.||^2_shadow factor is the standard misreading

# reconstructing rho itself instead costs O(rank(rho) * d / epsilon^2)
# <= O(d^2/epsilon^2) copies for trace-distance error epsilon (O'Donnell
# and Wright, upper bounds only), which at d = 2^n is exponential in the
# qubit count -- which is what makes this layer worth having

# a different construction from Aaronson's shadow tomography, whose
# Otilde(epsilon^-4 * log^4 M * log D) is a copy count for a procedure that
# measures the copies collectively, not a hardware shot count: the two are
# not interchangeable despite the shared word

Huang, Kueng and Preskill: $N = O(\log(M) \cdot \max_i ||O_i − tr(O_i)2^{-n}I||²_shadow / \varepsilon²)$ total measurements to predict $M$ linear functions to additive error $\varepsilon$ — formally $K = 2 \log(2M/\delta)$ median-of-means batches with $N = 34/\varepsilon² \cdot \max_i ||\cdot||²_shadow$ per batch. For random Pauli measurements the shadow norm is bounded by $4^k||O||²_\infty$ for a $k$-local $O$, improving to $3^k$ for tensor products of single-qubit observables; for random $n$-qubit Cliffords it is bounded by $3 tr(O²)$.

"Independent of system size" holds for observables of bounded shadow norm and is not a claim about arbitrary observables; quoting the $\log(M)$ without the max_i ||·||²_shadow factor is the standard misreading of this result. Separately, this is a different construction from Aaronson's shadow tomography, whose Õ(ε^-4 · log⁴M · log D) is a copy count for a procedure that measures the copies collectively, not a hardware shot count. The two results are not interchangeable despite the shared word.

• Clifford-measurement numerics (GHZ, toric code, entanglement witnesses)

  • Predicting Many Properties of a Quantum System from Very Few Measurements

    Hsin-Yuan Huang, Richard Kueng, John Preskill · 2020

  About

  A classical simulation of the shadow protocol under the global random n-qubit Clifford ensemble, run on three targets: GHZ states, toric-code ground states, and witnesses for tripartite entanglement. No quantum device is involved anywhere in this paper's numerics — the target states are generated classically and the measurement outcomes are sampled from them. The authors are explicit that this classical generation, not the shadow prediction, is what bounds the sizes they could reach: "The computational bottleneck is not feature prediction with classical shadows, but generating synthetic data". The baseline differs by target, and the difference matters. For the two fidelity experiments it is neural-network quantum state tomography (NNQST), which the authors ran themselves on the same task. The entanglement-witness experiment is instead compared against directly measuring each witness candidate — neither its main-text section nor its supplement mentions NNQST at all.

  Methods

  For the two fidelity experiments, data acquisition is simulated in the stabilizer formalism, which is what makes the large sizes reachable: "We exploit the Gottesman-Knill theorem for efficient classical computations." Each round samples a Clifford unitary from the Clifford group by a published sampling algorithm, stored as its action on Pauli operators; the unitary is applied to a stabilizer state by updating the stabilizer and destabilizer tableau, and the computational-basis measurement is simulated by the standard stabilizer algorithm. Mixed states are handled by sampling from a pure-state ensemble. The three-qubit witness experiment uses the same random Clifford measurement primitive but not that simulation route: at n = 3 the paper stores and processes all 2^3 = 8 amplitudes directly. Predictions are read out of the stored shadow by the paper's median-of-means estimator (Algorithm 1, "Median of means prediction based on a classical shadow"); a footnote notes that when the target observable is a stabilizer state, Gottesman-Knill evaluates each term in O(n^2) time, which is what keeps the post-processing tractable at these sizes. The NNQST baseline is run from the open-source code released by that method's own authors, with the tetrahedral POVM for GHZ states and the Psi2 (computational-basis) POVM for the toric code — a choice the paper flags as not tomographically complete but empirically better on that target. One qualifier is load-bearing in every fidelity comparison here, and the paper states it in both figure captions: the quantity plotted for NNQST is a classical fidelity, an upper bound on the quantum fidelity, while the quantity plotted for the classical shadow is the quantum fidelity. Any comparison quoted without it is reversed in NNQST's favour.

  Data

  n-qubit GHZ states — Figure 2(a) plots points from about 10 to 120 qubits. A noisy GHZ source in which a phase error occurs with probability p in [0,1], evaluated at n = 2, 5 and 10 under 6 x 10^4 experiments, at p = 0, 0.25, 0.5, 0.75 and 1. Toric-code ground states, taken as the superposition of all closed-loop configurations, at linear sizes L = 2, 3, 4, 5, 7, 9 — the figure legend gives these as n = 8, 18, 32, 50, 98 and 162 qubits. For the entanglement-witness experiment, three-qubit GHZ states rotated by independent random single-qubit unitaries, small enough (n = 3) that the paper simulates them by storing all 2^3 = 8 amplitudes directly. The largest system anywhere in this entry is that n = 162 toric-code point, and the paper describes its own reach inconsistently: the main text puts the whole numerics program at "up to 160 qubits", while the supplement says of this same Clifford experiment "more than 160 qubits".

  Code

  The paper names one repository, in five places: the introduction, the main-text derandomization paragraph, a Code Availability statement — "Source code for an efficient implementation of the proposed procedure is available at https://github.com/momohuang/predicting-quantum-properties" — and twice in the supplement. The arXiv abstract page's comments field advertises it as well, though that is listing metadata rather than text in the paper. Author contributions credit H.H. with conducting the numerical experiments and writing that code. Data availability is narrower than code availability and the paper says so: "Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request."

  Results

  For GHZ target fidelity 0.99, the paper reports that NNQST's required number of measurement repetitions scales linearly in n — confirming the earlier NNQST result it is checking against — while "classical shadows of constant size suffice to accurately estimate GHZ target fidelities, regardless of the actual system size." In Figure 2(a) the NNQST curve reaches about 5 x 10^4 experiments at 100 qubits, while the shadow curve is flat and stays below 10^4 across the full plotted range; the shaded bands are the standard deviation over ten independent runs. On the noisy GHZ source the shadow estimate tracks the true fidelity as p rises, including at p = 1 where the prepared state is orthogonal to the target, whereas NNQST "seems to consistently overestimate this target fidelity" and "reports fidelities close to one" at p = 1. The paper attributes this to the upper-bound nature of the quantity NNQST can estimate efficiently, not to a training failure. On toric-code ground states, NNQST "seems to require a number of samples that scales unfavorably in the system size n", while "fidelity estimation with classical shadows is completely independent of the system size"; in Supplementary Figure 1 the L = 7 and L = 9 NNQST curves stay at zero estimated fidelity across the whole plotted range out to about 10^5 experiments, while every shadow curve has converged near 1 by about 10^4. For entanglement witnesses on the rotated three-qubit GHZ state, where the comparison is against directly measuring each candidate, shadows reach the detection thresholds "with an exponentially smaller number of samples than the naive direct method". None of these are measurements of a physical device.

• Pauli-measurement numerics (correlations, Renyi entropy, Schwinger variance)

  • Predicting Many Properties of a Quantum System from Very Few Measurements

    Hsin-Yuan Huang, Richard Kueng, John Preskill · 2020

  About

  A second body of simulations using the shallow random single-qubit Pauli ensemble rather than global Cliffords, aimed at few-body properties of many-body ground states, plus a measurement-budget study for a variational quantum simulation of the lattice Schwinger model. As with the Clifford numerics, nothing runs on quantum hardware — but the three studies do not share one acquisition route. The two-point-correlation runs sample Pauli outcomes from a DMRG tensor network; the Renyi-entropy runs sample from an approximate ground state built by a strong-disorder renormalization-group recursion, and separately from ideal GHZ states of 4 to 10 qubits; the Schwinger comparison reports no sampled outcomes at all — its figure counts the copies each scheme would need to reach a stated error, and names no target state for the plotted numbers. Two things must not be conflated when reading this half. First, the Schwinger study compares against a measurement scheme designed in a prior trapped-ion experiment; the 20-qubit trapped-ion analog simulator belongs to that prior work, not to any run reported here. Second, the paper constructs a deterministic variant of its own protocol — derandomized classical shadows, which fixes the measurement set in advance instead of sampling it — and plots it beside the randomized protocol; the largest advantages reported in that figure belong to the derandomized variant, whose "full details will appear in upcoming work".

  Methods

  Two-point correlations: the ground state is approximated by a tensor network found with DMRG, and random Pauli measurements on it are simulated from that tensor network. The shadow and the NNQST baseline consume the same measurement data — "the only difference is the classical post-processing" — and the supplement states the data was downloaded from the NNQST authors' repository (https://github.com/carrasqu/POVM_GENMODEL). A local observable is predicted by forming the reduced density matrix on the subsystem it acts on, which the paper does by simply discarding the data for the rest of the system; the inverted snapshot is never materialised as a full 2^n x 2^n matrix. Reported timings are total CPU time, with the machine-learning model trained and evaluated on a multi-core CPU; no specific processor, cluster or core count is named anywhere in the paper. Renyi entropy: tr(rho_A^2) is written as a linear function on two copies via the local swap operator and estimated from the shadow by averaging over all N(N-1) ordered pairs of distinct snapshots, with the whole procedure repeated and the median taken. The shadow here is built from single-shot measurements — a fresh random Pauli basis for every shot — which the paper contrasts with what is easier in a physical experiment. The baseline is the Brydges et al. protocol, which uses the same random single-qubit rotations and basis measurements and "the only difference between the methods is in the classical post-processing"; its two hyperparameters (number of random unitaries, repetitions per unitary) are tuned by the grid search that protocol's own authors advocate. Schwinger model: after a Kogut-Susskind encoding onto a spin-1/2 lattice with an even number of sites and a Jordan-Wigner transform, the energy variance expands into 4-local Pauli observables, and the paper counts the copies each scheme needs to estimate all of them to "an error equivalent to measuring each Pauli observable at least 100 times". The derandomized variant replaces each coin toss with the basis giving the best value of the paper's own performance bound for the remainder of the protocol.

  Data

  One-dimensional critical anti-ferromagnetic transverse-field Ising model, J > 0, a chain of 50 lattice sites at the critical point h = J where correlations decay as a power law rather than exponentially; predictions from 2^9 x 1000 random Pauli measurements, against a true value from MPS. Two-dimensional anti-ferromagnetic Heisenberg model, J > 0, on an 8 x 8 triangular lattice, also from 2^9 x 1000 random Pauli measurements, against DMRG. The cost/accuracy sweep in Figure 3(c) runs over {2^1, ..., 2^9} x 1000 measurements. For Renyi entropies: a disordered Heisenberg spin chain of 10 sites with open boundary conditions, each coupling J_i drawn uniformly and independently from [0,1], whose approximate ground state is a set of singlets built by the strong-disorder renormalization group recursion (repeatedly pair the sites joined by the largest J_i and remove them), with all subsystems of size at most two predicted from 2500 measurements; and GHZ states at n = 4, 6, 8 and 10, where the left-half subsystem of size n/2 has entropy exactly one bit at every n. For the Schwinger model, the plotted axis runs to about 160 lattice sites.

  Code

  The same repository, https://github.com/momohuang/predicting-quantum-properties, named in the Code Availability statement, in the main-text derandomization paragraph — "we emphasize that the derandomization procedure is fully automated (see [the repository] for open source code) and not problem-specific" — and twice more in the supplement for this half of the numerics: once for "the exact details" of how the Renyi-entropy measurement strategy handles repeated bases, and once for "a (roughly linear time) algorithm that derandomizes random Pauli measurements for any collection of target observables with Pauli structure". The two-point-correlation experiment additionally reuses data downloaded from the NNQST authors' repository, https://github.com/carrasqu/POVM_GENMODEL, which the paper cites in the supplement as the source of its measurement data for that comparison.

  Results

  Two-point correlations: both methods predict them well, but "NNQST has a larger error for the 2D Heisenberg model" and at larger separations "produces some fictitious oscillations that are not visible in the results from DMRG and classical shadows". On identical measurement data the paper finds "roughly a 10^4 times speedup in classical processing time using the classical shadow instead of NNQST" — a post-processing time, not a measurement count; Figure 3(c) plots prediction error against an axis labelled "Classical post-processing time (in sec.)", and both methods improve as measurements increase. Renyi entropies on the 10-site disordered chain: from 2500 quantum measurements the shadow's maximum prediction error over all subsystems of size at most two is 0.052, against 0.24 for the Brydges et al. protocol on the same panel — the paper states that 2500-measurement budget for the shadow and does not restate a budget for the baseline. On GHZ states, "to achieve an error of 0.05, classical shadows require several times fewer measurements and the discrepancy increases as we require smaller error" — several times, and the paper gives no single number. Schwinger model: the randomized protocol beats the hand-crafted scheme it is compared against "only for system size larger than 50 qubits, and may actually be worse for small system sizes". The large factors in Figure 5(b) belong to the derandomized variant, not the randomized one: the hand-crafted scheme is annotated as using from 2.1 times as many measurements at the smallest plotted system size up to 14.3 times as many at the largest (about 160 lattice sites), relative to derandomized shadows. The paper frames the comparison as a trade of specialisation for generality — the hand-crafted scheme "was hand-crafted for the particular task of estimating the variance of the energy in the Schwinger model", while shadows "provide a good prediction for any set of local observables".

• Classical-shadow VQE estimation · Qiskit

  From the repository — run, not written up from a paper · unsupported

  About

  Randomized measurements are reused to estimate many observables from a shared data set.

  Methods

  None found yet.

  Data

  None found yet.

  Code

  Qiskit

  Results

  Literature-backed method record; algorithmic scope and evidence boundary reviewed, with no benchmark run claimed.

• Classical-shadow VQE estimation

  Randomized measurements are reused to estimate many observables from a shared data set.