{"id":"e3bdce9f-1e7f-480c-9967-fee69dba90b7","arxiv_id":"2505.06455","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A tomography protocol reconstructs real-amplitude quantum states from O(Nq) measurement settings using per-qubit Hadamard measurements plus majority-voted spanning-tree sign recovery.","lead":"Researchers present a new measurement method, the Hadamard Random Forest, that reconstructs real-valued quantum states using only a linear number of distinct measurement circuits instead of an exponential number. The method was tested on IBM's quantum processor up to 10 qubits and used to read out a quantum linear solver's solution, suggesting a practical way to cut readout overhead for a useful class of quantum computations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"HRF's Nq+1 measurement settings do not determine all real-valued states: W-type states with support on one Hamming-weight layer have zero hypercube-edge products, so their relative signs are invisible to Eq. (4).","rationale":"The reader's weakest assumption was exact realness. That is a practical robustness caveat, but it does not identify the more basic failure: even for perfectly real states, the chosen settings do not contain enough information for all real state vectors. The W-state example is a direct, finite-dimensional counterexample to the headline claim. I therefore partially agree with the reader's conditional verdict, but the required condition is different and stronger: the method only works when the support of the real state is connected by nonzero single-bit edges (equivalently m>0 in Sec. 2.2). The experiments with Ry+CNOT ansatz circuits avoid this by generating dense states with probability one, so the failure mode is not visible in the reported data. The paper should state the support assumption in the theorem, restrict all 'any real-valued state' claims, and discuss the exponential shot count implied by the m-bound. With that amendment the core idea remains valid for dense real states; without it the central claim is false. Hence CONDITIONAL rather than UNCHANGED or REJECT.","tokens_in":12818,"tokens_out":14267,"duration_ms":157733,"concrete_test":"Analytical check on Nq=3: compute the exact outcome probabilities for the four HRF circuits (Z; H on qubit 2; H on qubit 1; H on qubit 0) for |W+>=(|001>+|010>+|100>)/sqrt(3) and |W->=(|001>+|010>-|100>)/sqrt(3). Show these four probability distributions coincide pairwise. Then feed the common probabilities into Algorithm 1 (Eq. (4), spanning trees, majority vote) with effectively infinite shots; the reconstructed state is identical for the two inputs, so one must be wrong. This settles whether the central claim holds for 'any real-valued state'; no hardware or simulation sampling noise is involved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2.1 and Algorithm 1 claim that any real-valued state can be reconstructed from the Z-basis setting plus Nq single-qubit X-basis settings. This is false when the nonzero amplitudes do not connect all nodes of the hypercube Q_Nq. Concrete counterexample: the 3-qubit W states |W+>=(|001>+|010>+|100>)/sqrt(3) and |W->=(|001>+|010>-|100>)/sqrt(3) are both real pure states. The supported basis states have pairwise Hamming distance 2, so every edge used in Eq. (3)-(4) has psi_j psi_{j+2^k}=0; 2|psi^new|^2 - |psi_j|^2 - |psi_{j+2^k}|^2 = 0 for every relevant edge, and no spanning-tree path can carry a relative sign. The Z-basis probabilities and the three single-qubit X-basis probability distributions are identical for |W+> and |W->, so Algorithm 1 returns the same vector for both and cannot reconstruct both. The statement is therefore not true for 'any real-valued state'; the paper's own sample bound in Sec. 2.2 assumes m = min_{j,k} psi_j psi_{j+2^k} > 0, a support condition never stated as a hypothesis of the central claim. This failure class is not exotic: particle-number-conserving, Dicke, and symmetric states supported on a fixed Hamming weight are standard. Even for generic dense states, m is typically ~Nq 2^{-Nq} for Haar-random real states, so the Hoeffding bound demands N_samp ~ 4^{Nq}/Nq^2, i.e. exponentially many samples per setting; the 'O(Nq) settings' claim does not by itself provide efficient reconstruction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13233,"tokens_out":6312,"duration_ms":62092,"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":[{"comment":"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.","section":"Section 2.1, Eq. (4), Algorithm 1"},{"comment":"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).","section":"Section 2.2, sample bound"},{"comment":"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.","section":"Section 2.2, Eqs. (5)-(6)"},{"comment":"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.'","section":"Section 3.1, comparison to FQST"}],"minor_comments":[{"comment":"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.","section":"Section 2.1, Eq. (4)"},{"comment":"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.","section":"Section 2.2"},{"comment":"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.","section":"Section 3.1, Fig. 4(b)"},{"comment":"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.","section":"Section 3.2, Eq. (8)"},{"comment":"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.","section":"References"}],"recommendation":"reject","confidential_remarks":"The W-state counterexample in Major Comment 1 is decisive for the paper's central claim as stated. The protocol is correct for states whose support is connected in the hypercube and includes a nonzero root amplitude, but the paper does not state this condition and its title and abstract claim generality that is false. The sample-complexity analysis also reveals an exponential hidden cost for typical states, further weakening the advertised advantage. If the authors wish to resubmit, they would need to substantially reformulate the claim, restrict the state class, and provide a real-state-specific comparison baseline; as it stands, the manuscript's main theorem is not defensible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Couple of things you should know about arXiv:2505.06455. The method—reconstructing real-valued pure states from one Z-basis setting plus Nq single-qubit X-basis settings, with signs resolved via spanning-tree majority voting—works in practice for the dense states they test. The hardware results up to 10 qubits are real, the code is on GitHub, and the VQLS readout is a nice application. That part is solid.\n\nBut the central claim as stated is false. They say 'any real-valued state' can be reconstructed from Nq+1 settings. The W-state counterexample is decisive: |W+> and |W-> have the same Z and single-qubit X marginals, so the protocol returns the same vector for both. The problem is that if the support doesn't connect the hypercube nodes via edges with nonzero products, the sign information vanishes. The error analysis already defines m = min ψ_j ψ_{j+2^k} > 0, but the main theorem never states this as a hypothesis. That's a load-bearing omission.\n\nAlso worth flagging: for generic dense real states, m is exponentially small, so the Hoeffding bound demands exponentially many samples per setting. The paper only says post-processing is exponential; it doesn't emphasize that total sample complexity can be exponential too. So the reduction is in number of distinct circuits, not in total shots. That's a real caveat for scalability, though it may still be a win when circuit execution dominates.\n\nThe error bound for majority voting assumes independent tree errors without proof. That's a heuristic—not a fatal flaw, but it means the formal guarantee is weaker than it looks.\n\nBottom line: the method is a useful heuristic for real states with full or connected support, and the engineering is honest. But the 'any real-valued state' claim needs to be corrected to a support condition, and the sample complexity caveat should be explicit. A good referee could fix this in revision. I'd send it to review.","headline":"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.","tokens_in":13708,"tokens_out":3956,"would_cite":false,"duration_ms":38203,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum state tomography","real-valued quantum states","measurement settings","random forest","hypercube graph","state overlap","quantum linear solver","entanglement and magic estimation"],"falsifier":"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.","tokens_in":12649,"feed_emoji":"⚛️","tokens_out":11086,"duration_ms":106186,"temperature":0.7,"pith_summary":"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.","feed_headline":"Real-valued state tomography uses N+1 settings, not 3^N","feed_subtitle":"Hadamard basis measurements recover every sign; exponential work moves to classical post-processing.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the concentration inequality that converts shot counts into sign-error probabilities.","marker":"[35]"},{"why":"Provides the hypercube graph structure and the fact that $Q_{N_\\mathrm{q}}$ has many spanning trees, which the random forest draws from.","marker":"[34]"},{"why":"Defines full quantum state tomography in Pauli bases, the exponential-settings approach that HRF replaces.","marker":"[19]"},{"why":"Introduces random decision forests, the majority-voting idea used to stabilize sign estimates.","marker":"[32]"},{"why":"Supplies the hardware-efficient ansatz used to prepare the real-valued test states.","marker":"[36]"},{"why":"Defines the variational quantum linear solver whose solution state HRF is used to read out.","marker":"[33]"},{"why":"Defines the SWAP test against which HRF's overlap estimation is compared.","marker":"[46]"},{"why":"Provides the measurement error mitigation used to reach the reported hardware fidelities.","marker":"[37]"}],"fun_headline_variants":["Real-state tomography: N+1 settings instead of 3^N","Hadamard forest cuts tomography from exponential to linear","Measurement settings drop from 3^N to N+1 for real states","IBM 10-qubit test: real states from N+1 measurements"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Real-state tomography: N+1 settings instead of 3^N","Hadamard forest cuts tomography from exponential to linear","Measurement settings drop from 3^N to N+1 for real states","IBM 10-qubit test: real states from N+1 measurements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000663,"raw_usage":{"total_tokens":3077,"prompt_tokens":1041,"completion_tokens":2036,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":1960}},"tokens_in":657,"tokens_out":2036,"duration_ms":15455,"temperature":1.0,"reasoning_tokens":1960,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:42:55.113918+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Harary, J","cited_arxiv_id":null,"evidence_quote":"Supplies the concentration inequality that converts shot counts into sign-error probabilities."},{"cited_title":"Bravo-Prieto, R","cited_arxiv_id":null,"evidence_quote":"Provides the hypercube graph structure and the fact that $Q_{N_\\mathrm{q}}$ has many spanning trees, which the random forest draws from."},{"cited_title":"Shaffer, H","cited_arxiv_id":null,"evidence_quote":"Defines full quantum state tomography in Pauli bases, the exponential-settings approach that HRF replaces."},{"cited_title":"Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American statistical association58(1963) 13–30","cited_arxiv_id":null,"evidence_quote":"Supplies the hardware-efficient ansatz used to prepare the real-valued test states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the variational quantum linear solver whose solution state HRF is used to read out."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the SWAP test against which HRF's overlap estimation is compared."},{"cited_title":"Kandala, A","cited_arxiv_id":null,"evidence_quote":"Provides the measurement error mitigation used to reach the reported hardware fidelities."}],"review_version":1}