{"id":"c7f11b74-119c-4e3e-b936-2f712c791925","arxiv_id":"2607.06040","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"A classically-enriched quantum floating-point variable stores offset and scaling classically, reducing approximation error by up to 89% after repeated additions compared to prior mono-quantum coding.","lead":"The paper introduces a hybrid quantum-classical floating-point encoding (CEFV) that stores global offset and scaling parameters classically while keeping only values in the quantum register, reducing rounding error accumulation during repeated arithmetic. A generalist might read it to understand how near-term quantum computers can handle real-valued arithmetic with fewer qubits and less depth.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The 89% error reduction is an artifact of an engineered example where the offset advantage is maximally amplified; generalization is untested and the advantage likely shrinks with larger registers or wider input distributions.","rationale":"The reader's verdict of CONDITIONAL with MODERATE confidence is appropriate. The correctness proofs are sound — the tolerance propagation in Props S2.9 and S2.17 is straightforward and I find no logical gap. The CEFV encoding is a legitimate contribution over mono-quantum coding, adding useful degrees of freedom (offset, arbitrary scaling factors). However, the headline quantitative claim (89% error reduction) rests on a single engineered 3-qubit example that maximally amplifies the offset's advantage. The paper itself acknowledges this is an 'ad hoc example' (§2.5), and the extended results in Fig S10 merely repeat the same example with more visualization. No systematic benchmarking across register sizes, input distributions, or operation sequences is provided. The code is not publicly available. These limitations justify the CONDITIONAL verdict: the method is promising and the theory is correct, but the empirical evidence is too narrow to support the broad claims about significant error reduction in general use. My concern aligns precisely with the reader's weakest_assessment, so I recommend no change to the verdict. The concrete test I propose would settle whether the 89% figure is representative or best-case-only, which is the key open question.","tokens_in":25601,"tokens_out":2007,"duration_ms":149095,"concrete_test":"Run the repeated-addition benchmark across a grid of configurations: register sizes n∈{3,4,6,8,12}, input distributions including (i) the current {6,7} example, (ii) uniform on {0,...,2^n-1}, (iii) a distribution concentrated near 2^{n-1}, and (iv) a wide-tail distribution; for each, perform 1 through 10 self-additions. Report ε±_out for CEFV-with-offset vs. CEFV-without-offset (equivalent to MFV) at each configuration. If the mean error reduction across all configurations falls below ~30%, or if the advantage vanishes for n≥6, the headline 89% claim should be qualified as best-case-specific rather than general.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the core concern. The headline 89% reduction comes from a single example (§2.5, Fig. 3): a 3-qubit register encoding values {6,7}, summed with itself six times. This example is specifically constructed to maximize the offset's benefit. Without offset, the 3-qubit register represents {0,...,7}; after six additions the sum ranges over [36,42], forcing a large rescaling factor that makes the grid extremely sparse (ε+=28). With offset a=36, the same 3 qubits represent {36,...,43}, which is dense in the relevant range (ε+=3). The advantage is real but is an extreme case: it requires (a) values concentrated far from zero, (b) a narrow support relative to the register range, and (c) a very small register where grid sparsity is most painful. In scenarios where the input distribution already spans the register range (e.g., uniform on {0,...,7}), or where the register is large enough that the grid is already dense, the offset provides little to no benefit. The paper provides no systematic evidence that the advantage persists outside this engineered regime. The correctness proofs (Props S2.9, S2.17) are sound and I find no gap in the tolerance propagation arguments. The concern is purely empirical: the breadth of the headline claim is unsupported by the breadth of the evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript introduces Classically-Enriched Floating Point Variables (CEFVs), a hybrid quantum-classical encoding for floating-point numbers on quantum registers. The CEFV augments a quantum register storing integer values with classical side information: an offset $a$, a scaling factor $b$, and upper/lower tolerance parameters $ε^±$. The authors define addition (Algorithm S1) and multiplication (Algorithm S2) operations on CEFVs, prove that both operations are overflow-free and correctly propagate tolerances (Propositions S2.9 and S2.17), and benchmark the approach against mono-quantum floating variables (MFV) from Ref. [17]. The headline empirical result is an 89% reduction in approximation error after six repeated additions on a 3-qubit register, attributed to the classical offset enabling a denser representable grid in the relevant range.","tokens_in":25861,"tokens_out":1087,"duration_ms":224772,"significance":"The CEFV representation is a natural and well-motivated extension of mono-quantum coding (Ref. [17]). The correctness proofs for addition and multiplication (Props. S2.9, S2.17) are the core technical contribution and appear sound: the tolerance propagation is derived from the encoding structure, not fitted to data, and the no-overflow guarantee is properly established via the weight-bounding condition in Algorithm S1 (Step 5). The observation that classical offsets and scaling factors can be adjusted without quantum circuit manipulation — enabling zero-overhead ClassicalAdd and ClassicalProd operations — is a genuine practical advantage. The framework is falsifiable: the tolerance bounds are constructive and can be checked against any concrete input. However, the empirical claim of ~90% error reduction rests on a single engineered example, which limits the demonstrated significance.","major_comments":[{"comment":"§2.5, Fig. 3: The headline 89% error reduction is demonstrated on a single ad hoc example — a 3-qubit register encoding values {6,7} summed with itself six times. This example is specifically constructed to maximize the offset's benefit: without offset, the 3-qubit register represents {0,...,7}, and after six additions the sum ranges over [36,42], forcing $b_{out}$ to be large and making the grid extremely sparse ($ε^+=28$); with offset $a=36$, the same 3 qubits densely cover the relevant range ($ε^+=3$). The paper provides no systematic evidence that this advantage persists outside this engineered regime. For instance, when the input distribution already spans the register range (e.g., uniform on {0,...,7}), or when the register is large enough that grid sparsity is not painful, the offset provides little benefit. The heat maps in Fig. 2 study $b_{lead}$ selection but do not address the","section":null}],"minor_comments":[{"comment":"§2.1: 'fomr' should be 'from' in the definition of quantum variable.","section":null},{"comment":"§2.6: 'Converesely' should be 'Conversely'.","section":null},{"comment":"Fig. S10 caption: panel (b) lists 'j = 3' twice instead of 'j = 3, j = 4' or similar sequential labeling.","section":null},{"comment":"Formula box S2: The notation 'max v' for the maximum of a five-entry vector is introduced in a footnote-like remark marked (⋆), but is used in the formula above its definition. Reordering would improve readability.","section":null},{"comment":"§S2.2, Remark S2.10: The claim that Step 3 initialization guarantees the lower bound on weight sums and that Step 5 requires few iterations is stated with 'The proof of these facts is left to the reader.' For a journal publication, these should be proven or explicitly marked as straightforward consequences with a brief argument.","section":null},{"comment":"Fig. 2 and Fig. S9: The relationship between the main-text Fig. 2 and the extended Fig. S9 could be stated more explicitly (e.g., which panels of S9 correspond to Fig. 2).","section":null},{"comment":"References: The Qrisp package [22] is mentioned as now including SBPEval; a brief note on whether the authors' implementation differs would help reproducibility.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core theoretical contribution (CEFV definition + correctness proofs) is sound and publishable. The issue is purely the gap between the breadth of the headline empirical claim ('up to 90% reduction') and the narrowness of the evidence (one engineered 3-qubit example). This is fixable within the manuscript's scope by adding a modest benchmarking section and qualifying the claim. The authors should not be asked for exhaustive benchmarks — even 3-4 additional examples spanning different regimes would suffice. The paper is not a candidate for rejection; the theoretical framework is self-contained and the proofs check out."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the constructive assessment. The referee correctly identifies the core technical contributions (correctness proofs for addition and multiplication, the no-overflow guarantee, the zero-overhead ClassicalAdd/ClassicalProd operations) and acknowledges the framework as a natural, well-motivated, and falsifiable extension of mono-quantum coding. The sole major concern is that the headline 89% error-reduction claim rests on a single engineered example, with no systematic evidence that the advantage persists outside that regime. We agree that the current manuscript over-claims the breadth of the empirical demonstration, and we will revise accordingly: we will add systematic benchmarks across multiple input distributions and register sizes, reframe the 89% figure as an upper bound achieved in a specific regime rather than a general result, and explicitly characterize the regimes where the offset provides minimal benefit. The theoretical contributions (Propositions S2.9 and S2.17, the tolerance propagation formulas, the no-overflow condition) are independent of this empirical point and are unaffected.","responses":[{"response":"The referee is correct that the example in Figure 3 is engineered to showcase the offset's benefit in a regime where it is maximally effective, and that the manuscript does not currently provide systematic evidence across input distributions and register sizes. We accept this criticism and will revise the manuscript in three concrete ways. First, we will add a systematic numerical study varying (i) the input distribution (including the uniform-on-{0,...,7} case the referee specifically suggests, as well as distributions concentrated on subranges of varying widths), (ii) the register size n (from 3 to 8 qubits), and (iii) the number of repeated additions. This will show explicitly how the offset's benefit scales with these parameters. Second, we will reframe the 89% figure throughout the manuscript (abstract, Section 2.5, Discussion) as an upper bound on the achievable error reduction in the regime where the target distribution is concentrated in a narrow range far from zero — precisely the regime where mono-quantum coding (MFV) is forced to allocate a large exponent, making the grid sparse. We will state plainly that when the input distribution already spans the full register range, the offset provides little to no benefit, as the referee anticipates. Third, we will add a discussion characterizing the regimes where CEFV's advantage over MFV is expected to be significant (narrow-range distributions, repeated operations causing range drift) versus negligible (wide distributions, large registers). We note that the theoretical contributions — the correctness proofs (Props. S2.9, S2.17), the tolerance propagation formulas, the no-overflow guarantee, and the zero-overhead ClassicalAdd/ClassicalProd operations — are independent of the empirical breadth concern and remain valid","revision_made":"yes","referee_comment":"The headline 89% error reduction is demonstrated on a single ad hoc example (3-qubit register, values {6,7} summed six times), specifically constructed to maximize the offset's benefit. No systematic evidence that the advantage persists outside this engineered regime. When the input distribution already spans the register range, or when the register is large enough that grid sparsity is not painful, the offset provides little benefit. The heat maps in Fig. 2 study b_lead selection but do not address the advantage of offsets across input distributions."}],"tokens_in":25380,"tokens_out":711,"duration_ms":133108,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The core contribution is the CEFV representation: a quantum register paired with classically stored offset, scaling factor, and tolerance bounds. This extends Seidel et al.'s mono-quantum coding (which stores only the exponent classically) by adding the offset and arbitrary scaling factors, plus tracked tolerances that propagate through arithmetic. The addition and multiplication algorithms are cleanly defined, and the correctness proofs (Props S2.9, S2.17) are sound — the no-overflow guarantee and tolerance propagation are properly derived from the encoding structure, not fitted to data. The framework is self-contained and the math checks out. The b_lead heat maps (Fig. 2) are a reasonable parameter study for the scaling factor choice, and the circuit depth improvements in Remark S2.12 are concrete and well-demonstrated (Fig. S11 shows a 99% depth reduction in a pathological case). The observation that classical add/multiply on CEFVs requires no quantum gates is a genuine practical advantage over MFV. The soft spot is empirical and the reader and stress-test both land on it correctly: the headline 89% error reduction comes from a single engineered example. The setup is a 3-qubit register encoding {6,7}, summed with itself six times. This is specifically constructed to maximize the offset's benefit — without offset, the 3-qubit grid is forced to span [0,42] making it extremely sparse, while with offset a=36 the same qubits densely cover [36,43]. The advantage is real in this regime but requires values concentrated far from zero, narrow support relative to register range, and a very small register. The paper provides no systematic benchmarking across input distributions, register sizes, or operation sequences. The claim that this generalizes is unsupported by the breadth of evidence presented. That said, the stress-test overstates things slightly when it calls the advantage likely to shrink — the paper does show (Fig. S6) that wider registers delay approximation, and the offset mechanism is structurally sound, so the advantage should persist in some form for concentrated distributions. The issue is that we don't know how broadly. No code is publicly available, which limits reproducibility. This paper is for researchers working on quantum arithmetic circuit design, particularly for applications like quantum finance where values are concentrated in specific ranges. The encoding framework and correctness proofs are the real contribution; the empirical claims need more support. It deserves a serious referee. The referee should push hard for broader benchmarks — varied input distributions, multiple register sizes, and comparison against MFV in regimes where the offset advantage is not maximized — and for public code. The formal framework is solid enough to warrant engagement; the empirical validation is what needs rounding out.","headline":"CEFV encoding is a legitimate extension of mono-quantum coding; the 89% error reduction is real but narrowly demonstrated.","tokens_in":26368,"tokens_out":625,"would_cite":false,"duration_ms":108975,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Classical offset slashes quantum arithmetic error by up to 89%","keywords":["quantum arithmetic","floating-point","hybrid quantum-classical","error propagation","quantum computing","random variable encoding"],"falsifier":"If one constructs a repeated-addition scenario where the target values are uniformly or near-uniformly distributed across a wide range centered at zero (so that the offset provides no grid-densification benefit), the CEFV tolerance should be no better than the MFV tolerance, and the 89% reduction claim would not hold.","tokens_in":25875,"feed_emoji":"🔢","tokens_out":1226,"duration_ms":224722,"temperature":0.7,"pith_summary":"This paper introduces the Classically-Enriched Floating Point Variable (CEFV), a hybrid quantum-classical encoding for real-valued random variables on a quantum register. A CEFV stores a quantum register of qubit values alongside a classical register holding four parameters: an offset (a), a scaling factor (b), and two tolerance bounds (epsilon-minus, epsilon-plus). The offset shifts the representable grid away from zero, allowing a fixed qubit budget to cover a narrower, more relevant range with higher density. The scaling factor stretches or compresses that grid. The tolerances track how far the true target random variable can stray from the nearest representable value. The paper defines addition and multiplication operations on CEFVs that guarantee no overflow: the output scaling factor is automatically adjusted by a power of two to use all available output qubits while keeping the result in range. After each operation, the tolerances are propagated forward so the output CEFV carries an honest error bar. The central claim is that the classical offset, which costs no quantum gates for addition because it is summed classically, densifies the representable grid in the region where the data actually lives, and this dramatically reduces the accumulated rounding error after repeated additions compared to mono-quantum floating variables (MFV), which lack an offset.","feed_headline":"Classical offset cuts quantum arithmetic error by 89%","feed_subtitle":"Storing a range-shifting offset in classical memory densifies the representable grid for quantum floating-point numbers, sharply reducing 误差","key_machinery":"CEFV (Classically-Enriched Floating Point Variable): a quantum register of n qubits paired with classical parameters (offset a, scaling factor b, tolerances epsilon-minus and epsilon-plus). The encoding maps basis state |z> to value x = a + b*z. Addition and multiplication are implemented via semi-boolean polynomial evaluation (SBPEval), where input qubit weights are rounded to integers and the output scaling factor is auto-tuned to a power-of-two multiple of a reference value b_lead to prevent overflow and minimize rounding.","core_discovery":"The paper's central object is the CEFV encoding, and its core result is that introducing a classically stored offset into a quantum floating-point representation narrows the tolerance window around arithmetic results. In the paper's benchmark, a 3-qubit register encoding a random variable taking values 6 or 7 with equal probability is summed with itself six times. Without an offset, the tolerance grows to epsilon-plus = 28; with an offset, it stays at epsilon-plus = 3, an 89% reduction. The correctness of the addition (Proposition S2.9) and multiplication (Proposition S2.17) is proven: for any inputs compatible with the input CEFVs, the output state is guaranteed to be compatible with the ar","pith_inferences":["If the offset advantage is real and general, then any quantum algorithm that chains many additions (e.g., iterative solvers, Monte Carlo path averaging) would benefit from switching to CEFV encoding, because the error compounding rate per operation would be lower. The paper demonstrates this for six additions on one distribution; the implicit claim is that the mechanism (denser grid near the data)","The auto-tuning of the output scaling factor to a power-of-two multiple of b_lead suggests a natural extension: a compiler-level pass that, given a sequence of CEFV operations, optimizes the choice of b_lead at each step to minimize cumulative tolerance growth across the entire circuit, not just per-operation.","The fact that offsets add no quantum gate overhead for addition but only modest overhead for multiplication (linear terms only, not the quadratic cross-term) implies that CEFVs are most cost-effective in addition-heavy circuits; the break-even point where multiplication overhead cancels the precision gain would depend on the ratio of additions to multiplications in a given algorithm."],"forward_implications":["Quantum algorithms requiring repeated arithmetic on real-valued distributions, such as option pricing or quantum amplitude estimation for integration, could achieve materially higher output precision on near-term hardware with limited qubit counts.","The principle of classically storing a range-shifting offset to densify the representable grid is not specific to floating-point arithmetic; it could be applied to any quantum encoding where the data occupies a narrow band far from zero, including fixed-point and amplitude-encoded representations.","The tolerance-propagation framework provides a built-in error accounting system: any quantum algorithm using CEFVs can report a guaranteed error bound on its output without needing separate error analysis, which is useful for certification of quantum computation results."],"fun_headline_variants":["Classical offset narrows quantum floating-point tolerance by 89%","Hybrid offset register halves quantum arithmetic depth, cuts error 89%","Storing range classically reduces quantum float error by 89%","Classical-global-register method cuts quantum addition error 89%","Offset-encoded quantum floats keep tolerance at 3 after six additions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The 89% error reduction is demonstrated on a single example: a 3-qubit register encoding a variable taking values 6 or 7, summed with itself six times. The claim that this advantage generalizes to other input distributions, register sizes, and operation sequences rests on the assumption that the offset's grid-densification benefit persists broadly, which is not systematically benchmarked in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Classical offset narrows quantum floating-point tolerance by 89%","Hybrid offset register halves quantum arithmetic depth, cuts error 89%","Storing range classically reduces quantum float error by 89%","Classical-global-register method cuts quantum addition error 89%","Offset-encoded quantum floats keep tolerance at 3 after six additions","Classical range info cuts quantum float rounding error by 89%","Hybrid quantum-classical floats reduce repeated-addition error 89%","Classically offset quantum floats: 89% less error after six sums","Quantum floats with classical offset avoid overflow, cut error 89%","Global classical register densifies quantum float grid, error down 89%"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":772,"prompt_tokens":565,"completion_tokens":207,"prompt_tokens_details":null},"tokens_in":565,"tokens_out":207,"duration_ms":19478,"temperature":1.0,"reasoning_tokens":52,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T18:19:44.315633+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one constructs a repeated-addition scenario where the target values are uniformly or near-uniformly distributed across a wide range centered at zero (so that the offset provides no grid-densification benefit), the CEFV tolerance should be no better than the MFV tolerance, and the 89% reduction claim would not hold.","supporting_citations":[],"review_version":1}