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REVIEW 4 major objections 5 minor 53 references

Hadamard Random Forest: Reconstructing real-valued quantum states with exponential reduction in measurement settings

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that any real-valued pure state of $N_\mathrm{q}$ qubits can be fully reconstructed using only $N_\mathrm{q}+1$ measurement settings rather than the exponential number needed by full tomography.

desk verdict Useful heuristic for dense real quantum states, but the 'any real-valued state' claim is false—W-states break it, and the sample complexity caveat is understated. read the letter →

arxiv 2505.06455 v2 pith:LXJLE2GE submitted 2025-05-09 quant-ph

Zhixin Song , Hang Ren , Melody Lee , Bryan Gard , Nicolas Renaud , Spencer H. Bryngelson This is my paper · ORCID
classification quant-ph
keywords quantumstatetomographyreal-valuedstatesmeasurementsettingsrandomforesthypercubegraphoverlaplinearsolverentanglementandmagicestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Full quantum state tomography needs an exponential number of Pauli settings because it must resolve complex phases, but many useful algorithms prepare states whose amplitudes are real. The paper argues that for this real-valued class the complete state vector is recoverable from just $N_\mathrm{q}+1$ circuits: one computational-basis measurement fixes the magnitudes, and $N_\mathrm{q}$ single-qubit Hadamard-basis measurements fix all relative signs. An exact sign identity converts the extra probabilities into the sign of products $\psi_j\psi_{j+2^k}$, and a random forest over the hypercube graph assigns every amplitude its sign. The cost is shifted to exponential classical post-processing, while the quantum sample collection becomes linear. Experiments on a 10-qubit processor, plus applications to entanglement, magic, overlap, and a quantum linear-solver readout, support the protocol's usefulness.

What carries the argument

The load-bearing object is the relative-sign identity $s_{j,j+2^k}=\operatorname{sgn}\bigl(2|\psi^k_j|^2-|\psi_j|^2-|\psi_{j+2^k}|^2\bigr)$. It is exact because a Hadamard on the $(N_\mathrm{q}-k-1)$-th qubit maps the two amplitudes $\psi_j$ and $\psi_{j+2^k}$ into $(\psi_j\pm\psi_{j+2^k})/\sqrt2$, so the measured probability contains the cross term $\psi_j\psi_{j+2^k}$ whose sign is the relative sign. The second mechanism is the hypercube graph $Q_{N_\mathrm{q}}$: its nodes are bit-string indices, edges join indices differing by a power of two, and each spanning tree gives a consistent set of paths from the root amplitude to every other node. Repeating the sign determination over many random spanning trees and taking a majority vote is the random forest that makes the reconstruction robust.

What would settle it

Run the two-circuit version on the one-qubit state $(|0\rangle+e^{i\varepsilon}|1\rangle)/\sqrt{2}$ with very many shots. For $\varepsilon$ just above $\pi/2$, the sign identity returns $-1$ and HRF outputs $(|0\rangle-|1\rangle)/\sqrt2$, whose fidelity to the true state is about $1/\sqrt2$ even with unlimited samples, locating exactly where the real-amplitude assumption breaks down.

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Extended reading notes

Core claim

The central claim is that for a pure state $|\psi\rangle=\sum_j \psi_j|j\rangle$ with real $\psi_j$, the state vector can be reconstructed to arbitrary precision from $N_\mathrm{q}+1$ sampling circuits. The first circuit samples in the $\sigma_z$ basis and yields the magnitudes $|\psi_j|$; the $k$-th of the remaining circuits applies a Hadamard gate to one qubit before measurement, yielding probabilities $|\psi^k_j|^2$ that involve the superpositions $\psi_j\pm\psi_{j+2^k}$. Comparing these probabilities through the identity $s_{j,j+2^k}=\operatorname{sgn}\bigl(2|\psi^k_j|^2-|\psi_j|^2-|\psi_{j+2^k}|^2\bigr)$ returns the sign of the product $\psi_j\psi_{j+2^k}$. Since every index $j$ is connected to the root $0$ by a path of such pairs along a spanning tree of the hypercube $Q_{N_\mathrm{q}}$, multiplying these relative signs yields every $\operatorname{sgn}(\psi_j)$, and averaging over many random trees suppresses individual sign errors. The paper validates the method by reconstructing 10-qubit hardware states with high fidelity and using the reconstructed vector to estimate logarithmic negativity, stabilizer entropy, state overlap, and a variational linear-solver solution.

Load-bearing premise

The load-bearing assumption is that the prepared state is exactly a pure state with real amplitudes in the computational basis; if any imaginary component enters the wavefunction, the sign identity no longer recovers the true phase, and the algorithm silently outputs a nearby real state instead.

Editorial extensions

If this is right

  • For any application whose output state is real-valued, state-vector readout uses $O(N_\mathrm{q})$ measurement settings instead of $O(3^{N_\mathrm{q}})$, so the sampling bottleneck disappears even though classical post-processing remains exponential.
  • Nonlinear state properties such as logarithmic negativity, stabilizer entropy, and state overlap become computable from the reconstructed vector, and for overlap the method outperforms the standard SWAP test in the paper's noisy simulations.
  • Quantum linear-system solvers can be followed by HRF readout; the paper demonstrates a 7-qubit solution state reconstructed with about 92% fidelity on hardware.
  • The measurement circuits stay shallow, with each extra setting adding only one Hadamard rotation, which makes the protocol compatible with near-term error rates and measurement-error mitigation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same sign identity should carry over to any pure real wavefunction class, such as tensor-network-prepared or Gaussian states, making HRF a general readout for real-amplitude quantum simulation rather than only for linear-system solvers.
  • A dedicated robustness experiment with a one-qubit phase gate could map the imaginary-component threshold at which reconstruction fails; the paper does not report such a boundary.
  • Because sign recovery is a graph traversal, caching subtree results or evaluating trees in parallel could cut a large constant from the $\Omega(2^{N_\mathrm{q}})$ post-processing bill.
  • A natural extension is to real-valued quantum channels: applying the same $N_\mathrm{q}+1$-setting idea to the channel's process matrix would give an exponential reduction in settings for process tomography.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces Hadamard Random Forest (HRF), a tomography protocol aimed at reconstructing the full state vector of an Nq-qubit pure state with real amplitudes in the computational basis. The protocol uses only Nq+1 measurement settings: one Z-basis setting to obtain magnitude probabilities and Nq single-qubit X-basis settings to infer relative signs via the identity in Eq. (4). The relative signs are propagated along paths of random spanning trees of the hypercube Q_Nq, and a majority vote among trees is used to reduce sign errors. The authors provide an error analysis, report experiments on IBM quantum hardware up to 10 qubits, demonstrate extraction of nonlinear properties such as entanglement and magic, and compare against full quantum state tomography (FQST). They also apply the method to read out the solution of a variational quantum linear solver.

Significance. If the central claim held for all real-valued pure states, the protocol would constitute a major practical advance: it would reduce the number of measurement settings for this state class from exponential to linear, at the cost of exponential classical post-processing. The paper is strengthened by reproducible code, hardware experiments with error mitigation, and a transparent derivation of the sign identity for states that satisfy the required support condition. The experimental demonstrations on Ry+CNOT circuits, which produce states with dense support, are concrete and useful. However, the advertised generality of the method is invalidated by simple counterexamples, and the sample-complexity analysis hides an exponential dependence on the state. The manuscript's central correctness claim therefore requires substantial revision.

major comments (4)
  1. [Section 2.1, Eq. (4), Algorithm 1] The claim that any real-valued pure state can be reconstructed from the Z-basis setting plus Nq single-qubit X-basis settings is false. A concrete counterexample is the pair of 3-qubit W states |W+>=(|001>+|010>+|100>)/sqrt(3) and |W->=(|001>+|010>-|100>)/sqrt(3). These are both real pure states, but the nonzero-amplitude basis states have pairwise Hamming distance 2, so every product psi_j psi_{j+2^k} that appears in Eq. (4) vanishes. Consequently, for every relevant edge, the expression 2|psi^k_j|^2 - |psi_j|^2 - |psi_{j+2^k}|^2 is exactly zero, so no relative sign can be inferred. The Z-basis probabilities and all Nq single-qubit X-basis probability distributions are identical for both states, so Algorithm 1 returns the same vector for both and cannot reconstruct either. Furthermore, the algorithm assumes the root amplitude psi_0 > 0, but psi_0 = 0 for these states. Thus infinite samples cannot resolve the sign ambiguity. This directly contradicts the abstract and Section 2.1's statement that HRF reconstructs any real-valued state with Nq+1 circuits.
  2. [Section 2.2, sample bound] The sample bound N_samp >= ln(L/delta)/(2 m^2) depends on m = min_{j,k} psi_j psi_{j+2^k}, but this parameter is not stated as a hypothesis of the central claim. For the W states above, m = 0 and no finite number of samples suffices. For generic real Haar-random states, amplitudes are of order 2^{-Nq/2}, so m is typically of order 2^{-Nq}, making N_samp exponential in Nq. The paper's claim of 'O(Nq) measurement settings' therefore does not, by itself, provide an efficient reconstruction protocol; the sample complexity must be stated explicitly and the class of states for which the protocol is efficient must be restricted to those with m > 0 (or at least with a known nonzero lower bound on m).
  3. [Section 2.2, Eqs. (5)-(6)] The error bounds in Eqs. (5) and (6) rely on unproven independence assumptions. Equation (5) assumes that edge sign errors along a path in a given tree are independent, but adjacent edge decisions use overlapping sets of measured probabilities and are therefore statistically dependent. Equation (6) assumes that the error rates of different random spanning trees are independent, but random spanning trees of the hypercube share many edges, so their errors are correlated. The text labels these as assumptions 'for simplicity,' but then uses them to make quantitative claims, such as that Ntree = 11 trees suffice for a sign error below 3% when p_j = 0.1. Without a proof of the independence approximation, or at least numerical evidence that it is accurate for the tested states, these quantitative guarantees are not justified.
  4. [Section 3.1, comparison to FQST] The experimental comparison in Fig. 4(a) benchmarks HRF against full quantum state tomography of general complex states, which requires 3^{Nq} settings. For the target class of real-valued states, a more appropriate baseline would be a real-state-specific tomography method, such as one that measures only sigma_z and sigma_x settings, or a classical shadow protocol tailored to real states. As presented, the observed runtime and fidelity advantage over FQST does not demonstrate an advantage over the best existing method for the restricted class of states. The authors should include such a baseline or temper the claim that 'HRF outperforms FQST.'
minor comments (5)
  1. [Section 2.1, Eq. (4)] The notation s_{j,j+2^k} is used before it is defined, and the behavior of sgn(0) is not specified; this matters for states with zero edge products such as the W states discussed above.
  2. [Section 2.2] The statement 'assuming (for simplicity) that each edge has the same error probability p_e' is not justified because p_e depends on |psi_j|^2 |psi_{j+2^k}|^2, which varies substantially across edges unless the state is very flat.
  3. [Section 3.1, Fig. 4(b)] The runtime plot appears to use a logarithmic horizontal axis, but the axis label and caption do not state this explicitly, making the scaling claims harder to interpret.
  4. [Section 3.2, Eq. (8)] In the definition of the stabilizer Renyi entropy, the set P should be clarified to be the set of all 4^{Nq} Pauli strings, including the identity term, to match the normalization used in the formula.
  5. [References] The discussion of prior work on real-valued states and classical shadows cites Ref. [31] but does not describe how HRF differs from or improves upon that approach in terms of settings or sample complexity; a few sentences of comparison would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: HRF reconstructs by direct Born-rule inversion, not by fitting a model to the quantities it later reports.

full rationale

The derivation chain is self-contained rather than circular. HRF estimates |ψ_j|^2 from σ_z-basis samples and |ψ^k_j|^2 from single-qubit σ_x-basis samples, and Eq. (4) is an algebraic rearrangement of the Born rule: for real amplitudes, 2|ψ^k_j|^2 − |ψ_j|^2 − |ψ_{j+2^k}|^2 = ±2ψ_jψ_{j+2^k}, so the relative sign is read directly from measured frequencies. No parameter is fitted to the quantities later reported as predictions; entanglement, magic, and overlap are computed from the reconstructed state vector and compared against exact values, FQST, or the SWAP test. The sampling-error analysis (Hoeffding bound, majority-voting bound) only quantifies finite-shot uncertainty and does not inject the target answer. The few self-citations (Refs. [17,18]) are background references on device modeling and state verification, not load-bearing premises. Even if the claimed universality for all real-valued states is questionable—for instance, W-type states with zero hypercube-edge products make some relative signs invisible to Eq. (4)—that is a correctness or assumption gap, not circularity. The paper does not define its conclusion into its inputs or rename a fitted parameter as a prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The method relies on the state being a real pure state and on standard statistical bounds; the only non-standard assumption is the independence of per-tree sign errors, used in the majority-voting guarantee.

free parameters (2)
  • N_tree = 111 (runtime benchmark); ~30 sufficient in simulations
    Chosen by the authors to achieve sign-error suppression; the value trades post-processing cost against fidelity and is not derived from first principles.
  • N_samp = 10^4 for Nq <= 5; 10^5 for Nq > 5
    Set based on the error bound (4) and available device time; impacts the fidelity of the reconstructed state.
assumptions (4)
  • domain assumption The unknown state is a pure state with real amplitudes in the computational basis (Section 2.1, Eq. 2).
    The method reconstructs a state vector, not a density matrix, and the sign identity (4) only holds for real amplitudes.
  • standard math The hypercube graph Q_Nq is connected, allowing sign propagation from a root (Section 2.1).
    Used to justify the spanning tree approach for propagating relative signs from the root amplitude.
  • standard math Hoeffding's inequality applies to the empirical frequency estimates (Section 2.2).
    Used for the single-edge sign error bound, Pr(error) <= exp(-2 N_samp |psi_j|^2 |psi_{j+2k}|^2).
  • ad hoc to paper Sign errors on different spanning trees are mutually independent (Section 2.2, after Eq. 6).
    Assumed in the majority-voting error bound (6) but not proven; trees share the same measured probabilities, so errors are plausibly correlated.

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Pith. "Pith review of Hadamard Random Forest: Reconstructing real-valued quantum states with exponential reduction in measurement settings." pith.science (2026). https://pith.science/paper/LXJLE2GE

@misc{pith2026250506455,
  author       = {Pith},
  title        = {Pith review of: Hadamard Random Forest: Reconstructing real-valued quantum states with exponential reduction in measurement settings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXJLE2GE}},
  note         = {Machine review of arXiv:2505.06455}
}
abstract

Quantum tomography is a crucial tool for characterizing quantum states and devices and estimating nonlinear properties of the systems. Performing full quantum state tomography on an $N_\mathrm{q}$ qubit system requires an exponentially increasing overhead with $O(3^{N_\mathrm{q}})$ distinct Pauli measurement settings to resolve all complex phases and reconstruct the density matrix. However, many potential quantum computing applications, such as linear system solves, require only real-valued amplitudes. We introduce a readout method for real-valued quantum states that reduces measurement settings required for state vector reconstruction to $O(N_\mathrm{q})$; the post-processing cost remains exponential $\Omega(2^{N_\mathrm{q}})$. This approach offers a substantial speedup over conventional tomography. We experimentally validate our method up to 10 qubits on the latest available IBM quantum processor and demonstrate that it accurately extracts key properties such as entanglement and magic. Our method also outperforms the standard SWAP test for state overlap estimation. This calculation resembles a numerical integration in certain cases and can be applied to extract nonlinear properties, which are important in application fields. We further implement the method to readout the solution from a quantum linear solver.

Figures

Figures reproduced from arXiv: 2505.06455 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (e) shows the readout and native two-qubit gate (CZ) error rates. We select a 10-qubit chain from the device to ensure a lower readout error, which is the dominant error source. The real-valued quantum states were generated using the hardware-efficient ansatz [36], but…
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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