{"id":"cd4473f0-24df-4941-b3ae-49a11afb6d77","arxiv_id":"2505.24626","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A dynamic-circuit linear solver that measures and classically reconstructs the quantum state after every discrete adiabatic step claims to make circuit depth independent of the number of steps, with numerical fidelities above 80%.","lead":"This paper proposes breaking a quantum adiabatic linear-system calculation into tiny steps, measuring the quantum state after each step, and rebuilding it classically before the next step. It claims this cuts circuit depth from depending on the number of steps to depending only on one step, which would matter for noisy near-term quantum computers, but the scheme only works by knowing the whole state on a classical computer.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed O(depth(U)) depth reduction hides an exponentially costly full-state measurement and classical re-preparation loop, so the central resource claim is unsupported.","rationale":"The reader's verdict is REJECT, and my analysis reinforces that rejection. The single most load-bearing weakness is the full-state measurement/reconstruction/re-preparation loop underpinning the claimed depth reduction. If this loop works perfectly with negligible overhead, the depth reduction is real; if it requires many shots per component or fails under noise, the central claim collapses. The paper provides neither a sample-complexity analysis of the amplitude estimation nor a circuit-level simulation that would account for measurement shot noise and classical reconstruction errors. Section 3.1 explicitly says 'thorough measurement immediately following each discrete operation step' yields the new quantum state, but no mechanism is given for obtaining 2N amplitudes and signs at scale. The unresolved '[cite]' placeholder in Section 3.1 is additional evidence that this key step is not fully specified. Therefore the advertised O(depth(U)) scaling is not established, and the paper should remain rejected until the missing resource analysis and a realistic sampling-based benchmark are provided.","tokens_in":12260,"tokens_out":2295,"duration_ms":34321,"concrete_test":"Simulate the proposed measurement-and-reprepare loop with finite shot noise: for N = 2, 4, 8, 16 and L = 200, 500, 1000, 2000, estimate all 2N amplitudes each step from a fixed shot budget (e.g., 10^3, 10^4, 10^6 shots per component), assign signs by Eq. (19), re-prepare the reconstructed state, and compare final fidelity to the ideal noiseless state-vector evolution of Eq. (16). If matching the claimed >80% fidelity requires a shot count that grows faster than any fixed polynomial in N, or if fidelities collapse for N = 16 at realistic budgets, the O(depth(U)) depth claim is not a genuine speedup.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim (abstract, Section 1, Fig. 1) is that circuit depth scales as O(depth(U)) instead of O(steps × depth(U)). The mechanism, however, is not a shallower circuit but a stop-measure-reconstruct-reprepare loop: after every discrete step the full 2N-dimensional state is measured, its amplitudes are obtained as |x_i|^2, signs are assigned by the heuristic Eq. (19), and the state is re-encoded for the next segment. This makes the depth claim a redefinition of the resource metric. The full state of 2N amplitudes cannot be extracted with constant or few shots; amplitude estimation to relative precision η requires Ω(2N/η^2) repetitions per segment, so total executions are Ω(L · 2N/η^2 · depth(U)). No shot-count, sign-error, or classical reconstruction cost is analyzed. In fact, reconstructing an arbitrary 2N-dimensional state classically is exponentially expensive in N unless the state has special structure, and the paper does not establish such structure beyond the real/imaginary split. The sign rule Eq. (19) is particularly fragile: when |x_i^k| < δ, a wrong sign flips the reconstructed component, and no analysis bounds how often this occurs under realistic noise. The unresolved '[cite]' in Section 3.1 and the absence of a circuit-level or sampling-based numerical simulation further weaken the claim. Thus the advertised depth reduction does not translate into a resource reduction for solving a linear system; it converts the algorithm into a quantum subroutine embedded in a classical state-vector simulation loop.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a co-designed quantum discrete adiabatic linear system solver that uses dynamic circuits and real-time classical processing. The idea is to split the adiabatic evolution into discrete segments and, after each segment, measure the evolved state, reconstruct it classically, and re-encode it for the next segment. The authors claim this reduces circuit depth from O(steps × depth(U)) to O(depth(U)). They benchmark the method on 2×2, 4×4, 8×8, and 16×16 linear systems with condition numbers 10–50, reporting over 80% fidelity under a noise model. The central claim, however, is that the depth reduction is achieved without accounting for the cost of measurement, classical reconstruction, and repeated state preparation; the paper also leaves the non-unitary nature of the first-order evolution operator unanalyzed.","tokens_in":12516,"tokens_out":3545,"duration_ms":44887,"significance":"If the central claim were sound, this would be a significant practical advance for near-term quantum linear solvers, because it would decouple circuit depth from the number of adiabatic evolution steps. The paper correctly identifies a structural feature of the block Hamiltonians used (real/imaginary component separation), and it provides numerical benchmarks for small systems. However, the advertised O(depth(U)) scaling is not established as a resource bound: it is obtained by measuring and classically reconstructing the full state after every step, and the costs of that reconstruction are never counted. The real-imaginary separation alone does not remove the need for tomography-like measurement overhead. Since the central complexity claim is load-bearing and unsupported, the contribution as stated cannot be accepted.","major_comments":[{"comment":"The claimed reduction from O(steps × depth(U)) to O(depth(U)) is not a resource reduction. Equations (2) and (16) still describe L sequential applications of the evolution operator, and the proposed protocol inserts a full measurement, classical reconstruction, and re-encoding after every step. The paper counts only the depth of a single segment, but the total execution includes L rounds of state preparation, measurement, and classical processing. No shot count, total runtime, or combined circuit-plus-classical depth is provided, so the O(depth(U)) statement is a restatement of 'each segment is shallow' rather than a bound on the full algorithm.","section":"Section 3.1, 'Dynamic Circuit Design Scheme'; Fig. 1"},{"comment":"The sign-prediction rule is load-bearing and unanalyzed. Reconstructing the 2N-dimensional intermediate state requires the sign of every component, but Eq. (19) can assign the wrong sign whenever |x_i^k| < δ, and the paper gives no bound on how often this occurs, on the resulting fidelity loss, or on the number of measurement shots needed to estimate the amplitudes |x_i|^2 to the required precision. For a general state, estimating all 2N amplitudes to fixed relative precision requires Ω(2N/η^2) shots per segment, so the total cost is Ω(L · 2N/η^2) executions of the step circuit. The paper does not analyze any of these costs, and without them the claimed O(depth(U)) scaling collapses.","section":"Section 3.1, Eq. (19)"},{"comment":"The first-order operator I − iH dt is non-unitary, yet the paper applies it as though it were a standard quantum operation. Appendix A invokes block encoding for non-unitary matrices, but block encoding is a probabilistic embedding: the desired operation succeeds only with a certain probability, and the success probability and any amplitude-amplification overhead are not analyzed. The numerical simulations in Section 4 appear to apply the non-unitary operator exactly to a state vector, so they do not demonstrate that the per-segment circuit can be implemented at the claimed depth or success rate.","section":"Section 3.1, Eqs. (9)–(10) and Appendix A"},{"comment":"The numerical experiments are not circuit-level or sampling-based simulations. They do not include mid-circuit measurement shot noise, classical reconstruction, the sign-prediction heuristic, or re-encoding costs. The 'decoherence noise' model is not specified (the text only says '0.001 decoherence noise'), and the reported fidelity curves therefore cannot validate the dynamic-circuit protocol described in Section 3. A statevector simulation that directly applies the non-unitary operator is not evidence that the proposed stop-measure-reconstruct-reprepare loop works under realistic conditions.","section":"Section 4, Figs. 3 and 4"}],"minor_comments":[{"comment":"The heading 'Quanutm Linear System Problem' contains a typo and should read 'Quantum Linear System Problem'.","section":"Section 2.2 heading"},{"comment":"There is an unresolved citation '[cite]' in the sentence introducing the initial condition (b1, b2, 0, 0)^T; this should be replaced with a proper reference or removed.","section":"Section 3.1, paragraph before Eq. (15)"},{"comment":"The term 'circuit multiplexing' is used repeatedly but never defined; Fig. 1(b) shows dynamic circuits, but the multiplexing mechanism is not described.","section":"General / Fig. 1"},{"comment":"The linear extrapolation x'_i = 2 x_i^k − x_i^{k−1} is used without specifying how x_i^{k−1} is handled at k = 1, and without stating conditions under which this approximation is valid.","section":"Section 3.1, Eq. (19)"},{"comment":"Figure 2's axes and curves are not fully explained; the caption should state what quantity is plotted on each axis and what the blue and red curves represent quantitatively.","section":"Fig. 2"},{"comment":"Appendix A contains an equation numbered '(1)' that conflicts with the numbering of Eq. (1) in Section 2.1; renumbering is needed.","section":"Appendix A"}],"recommendation":"reject","confidential_remarks":"The manuscript's central complexity claim is not supported by the presented analysis: the depth reduction is obtained by hiding the cost of measurement and classical reconstruction. The unresolved '[cite]' and the lack of circuit-level simulations suggest the manuscript was not ready for review. If the authors can provide a full resource count including measurement shots, classical reconstruction cost, and block-encoding success probability, a revised version might be reconsidered, but as written the central claim is not fixable within the current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The depth-reduction claim doesn't survive contact with the paper's own mechanism. The authors split the adiabatic evolution into L segments, and after each segment they measure the full state, reconstruct it classically with a sign heuristic, and re-prepare it for the next segment. The O(depth(U)) scaling is then obtained by counting only one segment. That is a redefinition of the metric, not a resource reduction: extracting the 2N amplitudes requires many shots, the sign prediction fails near zero crossings, and the classical reconstruction cost is exponential in the worst case. The paper never accounts for any of this, so the abstract's promise of depth independent of steps is unsupported.\n\nTo give credit where it's due: the observation that the state retains a real/imaginary split under the block-structured Hamiltonian is a small genuine insight, and the idea of co-designing dynamic circuits with a discrete adiabatic solver is worth exploring. The numerical experiments show that the underlying adiabatic algorithm works for small matrices, with fidelity improving as steps increase, which is consistent with existing theory. But those numerics are state-vector calculations, not circuit-level or hardware simulations, and the noise model is barely described.\n\nOther soft spots: the first-order Taylor step I - iHdt is non-unitary, so its block encoding requires postselection; the success probability and overhead are never analyzed. The sign-prediction rule in Eq. (19) has a threshold delta that is not chosen or justified, and its failure rate under noise is not bounded. There's also an unresolved '[cite]' placeholder in Section 3.1, which alone tells you the manuscript isn't ready.\n\nThis is not a case where the central argument holds up. It's a case where the advertised improvement is an artifact of ignoring measurement and classical post-processing costs. The paper has a kernel of an idea for future work, but as written it would mislead a reader who trusts the complexity claim. I would not send it to peer review in this state; it needs a complete resource analysis and real circuit-level benchmarking before it deserves referee time.\n\nRecommendation: reject, and tell the authors to either establish the state-reconstruction cost or drop the O(depth(U)) claim.","headline":"The advertised O(depth(U)) depth reduction is a resource-accounting artifact: it ignores the cost of measuring, classically reconstructing, and re-preparing the full state after every segment.","tokens_in":13124,"tokens_out":2625,"would_cite":false,"duration_ms":29957,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A discrete adiabatic quantum linear system solver can be run at constant circuit depth by measuring, classically reconstructing, and re-encoding the state after each evolution step.","keywords":["quantum linear systems","discrete adiabatic algorithm","dynamic quantum circuits","mid-circuit measurement","circuit depth","block encoding","noisy intermediate-scale quantum","adiabatic evolution"],"falsifier":"A direct test is to simulate the exact state after each segment alongside the state reconstructed by Eq. (19): if any segment has a sign or amplitude mismatch, the final fidelity will fail to converge as the number of steps increases. Counting the number of measurement shots required per segment also settles the point: if the total reconstruction overhead scales with the number of steps or the Hilbert space dimension, the wall-clock advantage over the serial circuit disappears.","tokens_in":11988,"feed_emoji":"⚛️","tokens_out":5907,"duration_ms":68014,"temperature":0.7,"pith_summary":"Existing discrete adiabatic quantum linear system solvers pay a circuit depth that grows linearly with the number of evolution steps, quickly exceeding the coherence time of current hardware. This paper proposes a co-designed alternative that stops the evolution after each short segment, measures the resulting state, reconstructs it classically, and re-encodes it for the next segment. Because the Hamiltonian block structure keeps the first half of the state vector real and the second half imaginary, only the sign of each component is unknown, and a continuity-based sign predictor supplies it. The authors claim this reduces the depth scaling from $O(\\text{steps}\\times\\text{depth}(U))$ to $O(\\text{depth}(U))$ and, in simulations, keeps solution fidelity above 80% under decoherence noise for systems up to $16\\times 16$ with condition numbers up to 50.","feed_headline":"Mid-circuit measurements cut adiabatic linear-solver depth","feed_subtitle":"Stop, measure, and re-encode each segment: depth scaling drops from steps × depth(U) to depth(U).","key_machinery":"The load-bearing object is the real-imaginary component separation identity: for $H_s=\\begin{pmatrix}0&B\\\\C&0\\end{pmatrix}$, applying $I - iH_s\\,dt$ to a vector whose first $N$ entries are real and last $N$ are pure imaginary yields a vector of the same form. This makes the phase of every measured basis amplitude known modulo a sign, reducing full state tomography to $2N$ sign bits. The sign bits are supplied by the continuity rule Eq. (19), which compares each component with a linear extrapolation of the two previous segments, switching sign only when the amplitude is below a noise threshold. With the full state known, a classical controller schedules the next segment unitary on the fly and a dynamic-circuit re-encoding step restarts the evolution, so the total circuit depth is bounded by one segment rather than by the number of segments.","core_discovery":"The paper's central claim is that a discrete adiabatic linear system solver can be made depth-efficient by hardwiring the algorithm to the structure of the state it produces. Under the first-order evolution operator $I - iH\\,dt$ with block off-diagonal $H$, the state after every step retains the form $(x_1,\\ldots,x_N,\\,i x_{N+1},\\ldots,i x_{2N})^T$, so amplitude measurements determine every component up to a sign. The sign is predicted from continuity across steps with Eq. (19). Once the full intermediate state is known, it can be re-encoded classically, so the deep product of step unitaries never needs to be run as one coherent circuit. Circuit multiplexing then brings the circuit depth to $O(\\text{depth}(U))$, and a post-processing step that truncates the imaginary components raises the fidelity of the final solution. The benchmarks are offered as evidence that this segmentation does not destroy adiabatic accuracy: systems up to $W=16$ and $\\kappa=50$ stay above 80% fidelity under 0.001 decoherence noise, and above 90% fidelity noiseless for $W=16$.","pith_inferences":["The measure-reconstruct-re-encode pattern is not restricted to linear systems: any time-dependent simulation whose evolution operator preserves a known phase structure could be segmented in the same way, provided the classical reconstruction cost stays smaller than the saved circuit depth.","The $O(\\text{depth}(U))$ claim counts quantum circuit depth; the end-to-end wall-clock cost also includes the classical sign-prediction and re-encoding computation plus the number of measurement shots, so the true quantum-classical tradeoff will depend on those constants.","A hardware implementation with feed-forward mid-circuit measurement could encode predicted signs directly as classical bit conditions instead of post-processing, turning the sign predictor into an online controller and removing one round-trip to classical storage."],"forward_implications":["Increasing the number of adiabatic steps no longer deepens the quantum circuit; the same $O(\\text{depth}(U))$ circuit is reused regardless of precision.","On simulated decoherence noise of level 0.001, the solver keeps solution fidelity above 80% after 2000 steps for matrix dimensions 2, 4, 8, 16 and condition numbers 10 to 50.","Precision requirements are decoupled from decoherence time, so higher-accuracy runs can be attempted on the same hardware without lengthening coherent execution.","The real-imaginary separation plus truncation of imaginary components gives a noise-resilient post-processing step that works for any $N\\times N$ system of this block form."],"supporting_citations":[{"why":"Supplies the discrete adiabatic QLSP formulation and the $O(\\text{steps}\\times\\text{depth}(U))$ baseline depth that this paper sets out to eliminate.","marker":"[3]"},{"why":"Provides the discrete adiabatic linear-system solver and complexity analysis that the new solver inherits its Hamiltonian construction from.","marker":"[7]"},{"why":"Gives the block-encoding technique used to implement the non-unitary first-order operator $I - iH\\,dt$.","marker":"[10]"},{"why":"Demonstrates mid-circuit measurement with real-time classical communication, the hardware capability that makes stop-measure-re-encode segments possible.","marker":"[21]"},{"why":"Provides fast approximate block-encoding circuits (FABLE) used to reduce the cost of the re-encoding step.","marker":"[28]"},{"why":"Supplies the time-optimal adiabatic QLSP construction and the scheduling of evolution steps that the present solver adapts.","marker":"[5]"}],"fun_headline_variants":["Co-designed adiabatic solver cuts depth to single step","Mid-circuit tricks slash adiabatic solver depth scaling","Adiabatic linear solver depth drops via dynamic circuits","Re-encode states to shrink adiabatic solver depth","Quantum adiabatic solver beats depth limits with resets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scheme assumes that after every short segment the $2N$-dimensional state can be fully recovered from amplitude measurements plus the continuity-based sign prediction, and that re-preparing that state costs no more than one segment's unitary; if prediction errors or measurement shot counts grow, the claimed $O(\\text{depth}(U))$ depth reduction no longer holds.","fun_headline_variants_meta":{"raw":{"variants":["Co-designed adiabatic solver cuts depth to single step","Mid-circuit tricks slash adiabatic solver depth scaling","Adiabatic linear solver depth drops via dynamic circuits","Re-encode states to shrink adiabatic solver depth","Quantum adiabatic solver beats depth limits with resets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1521,"prompt_tokens":1006,"completion_tokens":515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":440}},"tokens_in":622,"tokens_out":515,"duration_ms":6482,"temperature":1.0,"reasoning_tokens":440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:17:59.507976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test is to simulate the exact state after each segment alongside the state reconstructed by Eq. (19): if any segment has a sign or amplitude mismatch, the final fidelity will fail to converge as the number of steps increases. Counting the number of measurement shots required per segment also settles the point: if the total reconstruction overhead scales with the number of steps or the Hilbert space dimension, the wall-clock advantage over the serial circuit disappears.","supporting_citations":[{"cited_title":"Optimal scaling quantum linear- systems solver via discrete adiabatic theo- rem.PRX quantum, 3(4):040303, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete adiabatic QLSP formulation and the $O(\\text{steps}\\times\\text{depth}(U))$ baseline depth that this paper sets out to eliminate."},{"cited_title":"The discrete adiabatic quan- tum linear system solver has lower con- stant factors than the randomized adiabatic solver","cited_arxiv_id":null,"evidence_quote":"Provides the discrete adiabatic linear-system solver and complexity analysis that the new solver inherits its Hamiltonian construction from."},{"cited_title":"Quantum algorithm for systems of linear equations with expo- nentially improved dependence on precision","cited_arxiv_id":null,"evidence_quote":"Gives the block-encoding technique used to implement the non-unitary first-order operator $I - iH\\,dt$."},{"cited_title":"Com- bining quantum processors with real-time classical communication.Nature, pages 1–5, 2024","cited_arxiv_id":null,"evidence_quote":"Demonstrates mid-circuit measurement with real-time classical communication, the hardware capability that makes stop-measure-re-encode segments possible."},{"cited_title":"Fa- ble: Fast approximate quantum circuits for block-encodings","cited_arxiv_id":null,"evidence_quote":"Provides fast approximate block-encoding circuits (FABLE) used to reduce the cost of the re-encoding step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the time-optimal adiabatic QLSP construction and the scheduling of evolution steps that the present solver adapts."}],"review_version":1}