{"id":"0677fe49-7862-48cd-8e43-ae3777a0c77b","arxiv_id":"2501.13097","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A distributed, postselection-based quantum phase estimation filter prepares low-variance states faster per iteration than a single-device filter, at the price of extra entanglement and restarts.","lead":"The authors present a distributed quantum filtering algorithm in which two or more quantum devices, connected by entanglement, repeatedly filter a product state toward low energy variance using postselected joint measurements. The protocol is benchmarked on a small Ising model and shown to reduce variance faster per iteration than a single-device filter, at the cost of additional Bell pairs and, for the strong postselection variant, an exponential restart overhead.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Distributed advantage is not isolated from postselection: the single-device baseline rejects no runs, so the faster variance drop may reflect selection bias rather than entanglement.","rationale":"The reader's verdict was already CONDITIONAL, and the reader's rationale explicitly listed the missing single-device postselection baseline as one unresolved issue. However, the reader's stated 'weakest_assumption' focused on the i.i.d.-uniform eigenphase requirement (Proposition 1, Appendix A.2). That assumption is real but is a generic spectral-genericity condition, is explicitly acknowledged by the authors with a counterexample, and can be repaired by shifting the Hamiltonian (H′ = H + εI, as noted in Appendix A.2). The postselection-baseline gap, by contrast, attacks the central comparison itself: the headline claim is about variance reduction relative to single-device implementations, and the only single-device implementation benchmarked is one that never discards outcomes. Since the distributed advantage vanishes when postselection is removed, the burden is on the authors to show that the advantage is not just conditioning on successful runs. A single-device filter with outcome |0⟩ postselection is the natural control and is absent. The other concerns—Conjecture 3's unproven status, the participation-ratio extrapolation, and missing error bars in main figures—are secondary; they affect quantitative details or analytic convenience, whereas the missing baseline affects whether the strongest claim is true at all. The proposed numerical test directly settles this, and until it is run, CONDITIONAL is the appropriate verdict. Thus the reader's verdict should remain unchanged.","tokens_in":32693,"tokens_out":2990,"duration_ms":34319,"concrete_test":"Run the single-device filter (Algorithm 3) with postselection that keeps only ancilla outcome |0⟩ at every iteration, discarding and restarting on |1⟩, using the same Hamiltonian (Eq. 4), initial states |+⟩^{⊗n} and |−⟩^{⊗n}, n = 4, 5, 6, and the same random t_k sampling as in Fig. 2. Compute the average variance of accepted runs versus k and the cumulative success probability. Compare these curves against the distributed weak- and strong-postselection curves in Fig. 2 and Fig. 5(c), both per accepted iteration and per total resource (including restarts and Bell-pair costs). If single-device postselection matches or beats the distributed curves per accepted iteration, the distributed advantage claim collapses to a postselection artifact; if the distributed curves remain strictly below at equal conditioning, the advantage is real.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—'our distributed algorithm reduces the energy variance faster compared to single-device implementations'—is supported numerically by comparing the distributed protocol with weak or strong postselection against a single-device algorithm that accepts every measurement outcome (Algorithm 3, Appendix A). In the distributed case, failed runs are discarded and restarted, and all reported averages are conditioned on surviving runs; in the single-device case, no branch is discarded, so the average includes variance-broadening outcomes. The paper itself notes that the distributed protocol without postselection performs worse than the single-device case (Fig. 2), confirming that the observed advantage is driven by postselection. A single-device Hadamard-test filter can equally postselect, e.g., by keeping only ancilla outcome |0⟩, which applies the filtering operation (I+U)/2 and would lower the conditional variance at the cost of success probability. The paper never benchmarks this single-device postselected baseline. Consequently, the reported speedup may be a selection-bias artifact rather than a genuine benefit of the distributed, entanglement-assisted architecture. This is the most load-bearing gap in the argument: without an equal-conditioning baseline, the abstract's claim is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a distributed quantum filtering algorithm for preparing low-variance eigenstate approximations. Two (or more) devices each hold a copy of the same initial state and run iterative-phase-estimation-style Hadamard-test filters with synchronized random evolution times; a joint measurement on the ancillary qubits, followed by weak or strong postselection, heralds successful runs. The central claim is that this distributed protocol reduces the energy variance faster per iteration than a single-device implementation. Analytical formulas are provided for the success rates, the energy bias, and the eigenstate spread, with a conjectured extension from weak to strong postselection. Numerical simulations on a non-integrable Ising model with n=4,5,6 and up to three devices support the variance reduction, and resource costs (controlled time evolutions and Bell pairs) are analyzed via cumulative success rates and Monte Carlo estimates.","tokens_in":32969,"tokens_out":14614,"duration_ms":143054,"significance":"If properly benchmarked, the protocol would be a useful tool for near-term distributed quantum processors, trading circuit depth for postselection overhead and entanglement links. The paper is transparent about several limitations: the generic-spectrum assumption underlying the random-phase randomization, the exponential overhead of strong postselection, and the conjectural status of the strong-postselection analytic formulas. It provides reproducible numerical procedures, error-bar plots in Appendix E, and a resource-cost analysis. However, the main comparative claim is currently confounded with the effect of postselection because the single-device baseline does not postselect. This is a significant gap that must be addressed before the central claim can be accepted.","major_comments":[{"comment":"The comparison between the distributed and single-device algorithms is not controlled for postselection. The single-device filter in Algorithm 3 accepts every measurement outcome (100% success, as noted in the Fig. 2 caption), whereas the distributed algorithms discard failing runs and report averages conditioned on surviving runs. The paper's own Fig. 2 shows that the distributed algorithm without postselection performs worse than the single-device case, confirming that the observed variance reduction is driven by postselection rather than by the distributed structure. A single-device Hadamard-test filter that postselects on the ancilla outcome |0> (applying (I+U)/2 per round and restarting on failure) is the natural equal-conditioning baseline; such a baseline is not benchmarked anywhere in the manuscript. Without it, the abstract's claim that the distributed algorithm 'reduces the energy variance faster compared to single-device implementations' is not established.","section":"Section 2.2, Fig. 2; Appendix A (Algorithm 3)"},{"comment":"The analytical energy-bias and eigenstate-spread formulas for the strong-postselection case (Eqs. (8), (12), and the associated bounds) rely on the approximation E[N/D] ≈ E[N]/E[D] stated as Conjecture 3. The authors explicitly show in Eq. (84) that the variance of the denominator g(φ) grows faster than the square of its mean for large K, so the approximation is not justified by the same argument as Proposition 2. The main text presents Eq. (12) with 'one can show' without indicating that, for strong postselection, this is a numerically supported conjecture rather than a proven result. This should either be flagged at the point of use, proved, or replaced by a direct asymptotic argument for the K→∞ limit.","section":"Section 2.4, Eq. (12); Appendix B.1.3, Conjecture 3"}],"minor_comments":[{"comment":"The algorithm's filtering mechanism depends on the eigenphases being effectively i.i.d. uniform, which requires (almost) irrational eigenvalue ratios; for Hamiltonians with symmetric spectra (e.g., H = Σ Z_j Z_{j+1}) the protocol fails, as acknowledged in Appendix A.2. The numerical support is for a single non-integrable Ising model, yet the abstract and introduction state the variance-reduction claim without this 'generic spectrum' qualification. The scope should be stated explicitly.","section":"Abstract and Section 2.1 (Proposition 1)"},{"comment":"The caption reads 'The cumulative success rate after postselection at each iteration are shown in (b)'; 'are' should be 'is'.","section":"Figure 2 caption"},{"comment":"The notation for the measurement outcome is inconsistent: the pseudocode writes |m_0 m_A m_B>, while Figure 1 labels the top qubit as 'aux', and the superscript k on the outcomes is not defined. Please align the notation.","section":"Algorithm 1 and Figure 1(b)"},{"comment":"The proposed hybrid protocol that starts with strong postselection and transitions to weak postselection is not numerically demonstrated. Either add a supporting simulation or present it only as a qualitative suggestion.","section":"Section 4, Discussion"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of Quantum and the numerical results are likely reproducible, but the central comparative claim needs a controlled baseline. Adding a single-device postselected filter (or a two-copy local-postselection baseline) and explicitly marking the strong-postselection analytics as conjectural in the main text would address the main concerns. The novelty relative to [16] is the filtering perspective, the two postselection regimes, and the multi-device generalization, which is sufficient for the journal if the presentation is corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new content here is the filtering analysis: the moments calculation, the weak-vs-strong postselection distinction, the energy-bias bound, and the s-device generalization with resource accounting. The paper is clearly written and unusually transparent about its own limitations—it flags the random-phase assumption, gives a concrete counterexample where the algorithm fails, labels Conjecture 3 as unproven, and puts error-bar plots in the appendix. The numerical work is reproducible in principle, and the analytical formulas for weak postselection are derived from the Born rule, not fitted.\n\nThe soft spots are real but not fatal. The stress-test note is on point: the abstract's claim that the distributed algorithm 'reduces the energy variance faster compared to single-device implementations' compares against a single-device algorithm that never discards runs. The distributed protocol with postselection throws away variance-broadening outcomes; the single-device protocol keeps them. So part of the observed speedup is selection bias, not entanglement. The paper even shows the distributed protocol without postselection does worse than the single device—which confirms that postselection is the main driver. A single-device Hadamard-test filter that postselects on the ancilla being |0⟩ would be the natural control, and it is missing. That said, the analytical argument in Appendix B.2 partially addresses this: the two-device strong-postselection protocol has a higher probability of amplifying the dominant eigenstate than the single-device filter, which is a genuine entanglement-assisted effect, not just postselection. But that argument is asymptotic and heuristic, and the numerics do not back it up with the missing baseline.\n\nThe other issues are minor. Conjecture 3 is unproven but clearly labeled and supported by numerics. The claimed scaling of the participation ratio (A exp(B/n)) is stated without a derivation. Main-figure error bars are absent but present in Appendix E. None of these undermine the core contribution.\n\nWho gets value from this: people working on eigenstate preparation, iterative phase estimation, and distributed quantum computing. The paper deserves a serious referee, and I would engage with it, but I would ask for the single-device-with-postselection baseline and a more careful statement of what 'distributed advantage' means once selection effects are controlled.","headline":"A solid analysis of a distributed filtering protocol; the core advantage claim needs one more baseline (single-device with postselection) before I'd take the 'distributed advantage' at face value.","tokens_in":33424,"tokens_out":4033,"would_cite":true,"duration_ms":44630,"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 a distributed filtering algorithm with one auxiliary qubit per device and postselected joint measurement reduces energy variance faster than a single-device filter, shown numerically for the Ising model with up to…","keywords":["distributed quantum computing","eigenstate preparation","quantum filtering","iterative quantum phase estimation","postselection","energy variance","swap test","cyclic permutation test"],"falsifier":"Run the two-device protocol on the symmetric Ising Hamiltonian $H = \\sum_{j=1}^{n-1} Z_j Z_{j+1}$ starting from $|+\\rangle^{\\otimes n}$ and measure the average energy variance versus round number under weak postselection; Proposition 1 predicts no filtering, because the correlated eigenphases of the $\\pm\\lambda$ pairs prevent single-population amplification. A direct numerical check that the variance does not decrease would confirm the claim's scope.","tokens_in":32499,"feed_emoji":"⚛️","tokens_out":8316,"duration_ms":74096,"temperature":0.7,"pith_summary":"This paper claims that a quantum filtering algorithm split across several devices can prepare low-energy-variance states in fewer rounds than running the same filter on a single device. The protocol runs identical iterative phase-estimation filters in parallel on each device, connects the devices' single auxiliary qubits by a swap test, and postselects on the joint measurement outcome. Postselection on the symmetric subspace amplifies the dominant eigenstate population faster than a single-device filter, at the cost of restarting failed runs. The paper proves that the weak-postselection overhead plateaus, analyzes the induced energy bias, and gives numerical evidence that two or three devices beat one device on a non-integrable Ising model for up to six qubits. If correct, the protocol offers a lower-circuit-depth route to eigenstate preparation suited to near-term and early fault-tolerant hardware.","feed_headline":"Distributed filter beats single-device state prep","feed_subtitle":"Postselected joint measurements cut the rounds to low-energy states, with bounded restart overhead.","key_machinery":"The carrying object is the joint controlled measurement of the auxiliary qubits: a swap-test circuit (control-SWAP between each device's single auxiliary qubit) for two devices, generalized to a cyclic permutation test in which an $s$-level qudit controls a permutation operator $D$ on the $s$ auxiliary registers. A single Bell pair per round teleports Bob's auxiliary qubit to Alice so the control-SWAP is applied locally, and the generalization uses $s-1$ Bell pairs. The filtering operation itself is inherited from iterative quantum phase estimation: a round applies $e^{-iH t_k}$ controlled on the auxiliary qubit, and the randomness of large $t_k$ makes the eigenphases nearly independent and uniform, so a postselected successful round multiplies the eigenstate populations by independent factors and sharpens the energy distribution. The analysis tracks the first two moments of the energy distribution, proving the plateauing success rate of weak postselection, the exponential decay of the strong one, and the bounded energy bias of the final state.","core_discovery":"Starting from identical product states on every device, the protocol acts each round $k$ with the same controlled time evolution $e^{-iH t_k}$ on each device for a randomly drawn large time $t_k$, then applies a control-SWAP between the devices' auxiliary qubits, and measures all auxiliary qubits. Postselection keeps runs whose auxiliary outcomes meet the weak criterion (top qubit $0$) or the strong criterion ($000$ or $011$). Because the eigenphases $\\varphi_j^{(k)} = -t_k \\lambda_j \\pmod{2\\pi}$ are effectively i.i.d. uniform whenever the Hamiltonian's eigenvalue ratios are (almost) irrational, each accepted round multiplies eigenstate amplitudes by independent random filtering factors, concentrating the population on the dominant eigenstate faster than the single-device filter. The central analytical results are closed-form expressions for the ensemble energy and variance: weak postselection has a success probability that plateaus at $\\sum_j |c_j^{(0)}|^4$ with overhead linear in the round number, strong postselection has exponentially decaying success but faster variance reduction, and the expected energy converges to $(\\sum_j \\lambda_j |c_j^{(0)}|^4)/(\\sum_j |c_j^{(0)}|^4)$, bounding the postselection bias by at most $0.1716|\\mu|$ for Gaussian spectra. The construction extends to $s \\ge 2$ devices through a cyclic permutation test on an $s$-level qudit, with numerical evidence that three devices reduce variance faster than two.","pith_inferences":["The plateau value $\\sum_j |c_j^{(0)}|^4$ of the weak-postselection success rate is the purity of the initial state in the eigenbasis; for local Hamiltonians the authors show it scales as $A e^{B/n}$, suggesting the protocol remains practical for system sizes beyond the numerics shown.","Because the circuit is a symmetrization onto the symmetric subspace of the two registers, one could test whether replacing the swap with other symmetric projections changes the filtering rate or the final bias.","After many rounds the two devices hold nearly identical eigenstates, so a direct swap-test comparison of the two output registers could certify convergence without measuring the energy.","The bounded bias formula implies that for variational or phase-estimation pipelines the filtered state's energy must be corrected by $\\sum_j \\lambda_j |c_j|^4 / \\sum_j |c_j|^4$, which requires knowing the initial-state populations; estimating those populations is a natural extension."],"forward_implications":["For any Hamiltonian with generic irrational-ratio spectrum, a two-device filter with weak postselection reaches a target energy variance in fewer rounds than the single-device filter, with a success probability that stabilizes instead of decaying.","Strong postselection converges even faster but has exponentially decaying success, so it fits short round budgets; the authors explicitly suggest a hybrid protocol that starts strong and switches to weak postselection.","Each successful run outputs $s$ identical filtered states, so the protocol is a heralded way to prepare multiple copies of an approximate eigenstate in parallel.","The postselection-induced energy bias is bounded and explicitly computable from the initial-state populations, so the final energy can be predicted and corrected.","Distributed filtering trades circuit depth for entanglement links and restart overhead; the paper quantifies this as the per-round Bell-pair cost and the number of controlled time evolutions, with per-state costs shown in the resource-counting figures."],"supporting_citations":[{"why":"Supplies the conditional-dynamics filtering mechanism (controlled $e^{-iHt}$ with phase kickback) that the distributed protocol iterates.","marker":"[6]"},{"why":"Introduced the two-device controlled-swap circuit for eigenstate broadcasting that this work extends to symmetric filtering with postselection.","marker":"[16]"},{"why":"Established that iterative single-device spectral projection decreases the average energy variance, the baseline the distributed filter is compared against.","marker":"[15]"},{"why":"Provides the Gaussian density-of-states result for local Hamiltonians used in the continuous approximations for the energy bias and eigenstate spread.","marker":"[5]"},{"why":"Provides the Gaussian eigenstate-population result for product states used to analyze the initial-state dependence and the $A e^{B/n}$ success-rate scaling.","marker":"[11]"},{"why":"The symmetrization circuit that realizes the swap test acting as the entanglement link between the two devices.","marker":"[24]"},{"why":"The generalized parallelized permutation-test construction used for the $s$-device extension with a qudit-controlled derangement.","marker":"[29]"},{"why":"Weyl's equidistribution criterion, used in Proposition 1 to justify treating the eigenphases as i.i.d. uniform for large time intervals.","marker":"[30]"}],"fun_headline_variants":["Distributed postselection cuts state-prep rounds","Joint measurement speeds low-variance state prep","Faster energy filtering via multiple quantum devices","Distributed quantum filter reduces variance quicker","Multi-device postselection beats single-device filtering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The protocol filters only when the Hamiltonian's eigenvalue ratios are (almost) irrational, so that random evolution times scramble the eigenphases into effectively independent uniform variables; for a Hamiltonian with a symmetric spectrum such as $H = \\sum_j Z_j Z_{j+1}$ the algorithm fails to converge.","fun_headline_variants_meta":{"raw":{"variants":["Distributed postselection cuts state-prep rounds","Joint measurement speeds low-variance state prep","Faster energy filtering via multiple quantum devices","Distributed quantum filter reduces variance quicker","Multi-device postselection beats single-device filtering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1399,"prompt_tokens":996,"completion_tokens":403,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":336}},"tokens_in":612,"tokens_out":403,"duration_ms":4332,"temperature":1.0,"reasoning_tokens":336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:28:10.225385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the two-device protocol on the symmetric Ising Hamiltonian $H = \\sum_{j=1}^{n-1} Z_j Z_{j+1}$ starting from $|+\\rangle^{\\otimes n}$ and measure the average energy variance versus round number under weak postselection; Proposition 1 predicts no filtering, because the correlated eigenphases of the $\\pm\\lambda$ pairs prevent single-population amplification. A direct numerical check that the variance does not decrease would confirm the claim's scope.","supporting_citations":[{"cited_title":"Quan- tumAlgorithmProvidingExponentialSpeed Increase for Finding Eigenvalues and Eigen- vectors","cited_arxiv_id":null,"evidence_quote":"Supplies the conditional-dynamics filtering mechanism (controlled $e^{-iHt}$ with phase kickback) that the distributed protocol iterates."},{"cited_title":"Quan- tum eigenstate preparation assisted by a coherent link","cited_arxiv_id":null,"evidence_quote":"Introduced the two-device controlled-swap circuit for eigenstate broadcasting that this work extends to symmetric filtering with postselection."},{"cited_title":"Quantum algorithm for spectral projection by measur- ing an ancilla iteratively","cited_arxiv_id":null,"evidence_quote":"Established that iterative single-device spectral projection decreases the average energy variance, the baseline the distributed filter is compared against."},{"cited_title":"Spectral Densities and Partition Functions of Modular Quantum Systems as Derived from a Central Limit Theorem","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian density-of-states result for local Hamiltonians used in the continuous approximations for the energy bias and eigenstate spread."},{"cited_title":"Matrix product state approximations to quantum states of low energy variance","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian eigenstate-population result for product states used to analyze the initial-state dependence and the $A e^{B/n}$ success-rate scaling."},{"cited_title":"Generalized concen- tratable entanglement via parallelized per- mutation tests","cited_arxiv_id":null,"evidence_quote":"The generalized parallelized permutation-test construction used for the $s$-device extension with a qudit-controlled derangement."},{"cited_title":"Über die Gleichverteilung von Zahlen mod. Eins","cited_arxiv_id":null,"evidence_quote":"Weyl's equidistribution criterion, used in Proposition 1 to justify treating the eigenphases as i.i.d. uniform for large time intervals."}],"review_version":1}