{"id":"e89c9afc-a475-4f62-a1f7-1d19841de7e3","arxiv_id":"2501.13469","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Penta-O sets QAOA parameters level by level using five energy measurements per level, cutting parameter-setting cost to O(p^2) circuit executions for depth p.","lead":"The paper introduces Penta-O, a level-wise recipe for choosing QAOA circuit angles using a few quantum measurements at each depth instead of a classical optimizer. If it works, it cuts the classical overhead of running QAOA on optimization problems, but the method depends on a hand-picked constant.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The trigonometric theorem and five-trial reconstruction are sound, but the fixed cost angle γ0 is a free hyperparameter; Penta-O's reported performance depends on problem-class-specific tuning that the paper leaves unresolved.","rationale":"I read the paper in good faith and checked the central analytical claim. The proof in Section S1 is essentially correct: for a fixed level-p cost angle γ_p and fixed preceding parameters, the conjugation of H_C by the final mixer unitary e^{-iθ_p H_M} yields terms in cos^2(2θ_p), sin^2(2θ_p), and sin(4θ_p), which combine into the trigonometric form of Eq. (4) with coefficients independent of θ_p. The proposed five-point reconstruction is also mathematically sound; the probe set kπ/6 (k=1..5) gives a full-rank linear system for the five coefficient degrees of freedom, and the field-free three-point version is nonsingular as well. The non-decreasing property follows because setting θ_p=0 leaves the previous-level energy unchanged, since the additional cost evolution e^{-iγ_p H_C} commutes with H_C. Thus I do not see an internal inconsistency or a false theorem. The most load-bearing weakness is the role of γ0. The paper fixes γ0 by hand per problem class and acknowledges that its optimal choice is open. The coefficients in Eq. (S8) depend on γ_p through off-diagonal expectations, so the per-level improvement and the depth needed to reach a given approximation ratio vary with γ0. The benchmarks use different γ0 values for different problem families and give no rule to select γ0 from the instance, so the strong claim of a stand-alone, outer-loop-free, parameter-free strategy is not fully supported. This matches the reader's identified weakest assumption. The concern is a limitation on the practical, parameter-free claim rather than a refutation of the core trigonometric result, so the reader's CONDITIONAL verdict is appropriate and should remain unchanged. A systematic γ0 sensitivity study would settle whether the reported performance is an artifact of the chosen values or robust across a wide range.","tokens_in":14674,"tokens_out":21438,"duration_ms":191980,"concrete_test":"Run the paper's noiseless u3r MaxCut benchmark (all nonisomorphic graphs with N≤14 plus the random replicas for N=16,18,20 used in Fig. 2(c)) with Penta-O for γ0 swept over {0.01, 0.025, 0.05, 0.075, 0.1, 0.2, 0.5}. For each instance record the converged approximation ratio and the first level p at which r exceeds the u3r classical bound r_u3r≈0.9326. If the paper's fixed γ0=0.075 is not within 5% of the best γ0 for a substantial fraction of instances, or if the optimal γ0 varies widely across instances, then the reported performance depends on an unstated prior and the method needs a γ0-selection rule before it can be called outer-loop-free. If performance is flat across the sweep, the concern is resolved and the conditionality can be relaxed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (4) and its proof in S1 are internally sound: conjugating H_C by the final mixer unitary produces the claimed 4θ and 2θ harmonics, and the five probe angles (kπ/6, or kπ/8 for field-free cases) give a nonsingular linear system for the five coefficients. The load-bearing weakness is the fixed cost angle γ0. The paper sets γ0=0.2 for the hardware experiment, 0.075 for unweighted 3-regular MaxCut, and 0.05 for the SK model, and explicitly states that determining its optimal value remains an open question. The coefficients in Eq. (S8) depend on γ_p through the off-diagonal expectations O_YY and O_ZY, so the achievable decrease per level and the depth p needed to reach a target approximation ratio are functions of γ0. Without a rule relating γ0 to the instance (e.g., to max|w_ij| or to spectral properties of H_C), Penta-O is not a parameter-free recipe: the classical outer loop over θ is removed, but an equally consequential hyperparameter remains. The non-decreasing guarantee holds for any fixed γ0, so this does not refute the theorem, but it does limit the practical claim that the method stands alone. The benchmarks use different γ0 values per problem class without a selection procedure, and no error bars or code are provided, so the near-optimality claims are conditional on tuning choices not derived in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a p-level QAOA on an Ising/QUBO Hamiltonian, the energy expectation J_p is a trigonometric function of the final-level mixer angle, specifically J_p = A_p sin(4θ_p + φ_p) + A'_p sin(2θ_p + φ'_p) + C_p, with coefficients independent of θ_p. On the basis of this identity, the authors propose Penta-O, a level-wise parameter-setting method that determines each level's final mixer angle from a small number of probe measurements (five in general, three in the field-free case), removing the classical outer loop over variational parameters. They claim an O(p^2) time complexity and 5p+1 trial sampling overhead, and demonstrate the method on a superconducting processor for a small MaxCut instance and in noiseless simulations for MaxCut and Sherrington-Kirkpatrick models, reporting near-optimal approximation ratios and low-energy sampling probabilities. The authors also discuss the role of the fixed cost angle γ0 and potential quantum-classical crossover. The central mathematical derivation is presented in Supplemental Material S1, and numerical details are in S2 and S3.","tokens_in":14910,"tokens_out":12741,"duration_ms":78603,"significance":"If the trigonometric identity and the reconstruction protocol are correct, this is a valuable contribution: it replaces a variational optimization over a continuous parameter with a fixed small number of quantum measurements per level, provides a non-decreasing performance guarantee (via θ_p = 0 recovering the previous level), and yields a clean O(p^2) circuit-preparation cost. The proof in S1 is explicit and, once the notation is fixed, appears mathematically sound; the coefficient formulas in Eq. (S8) follow from a straightforward trigonometric expansion. The empirical demonstrations support the plausibility of the method on small instances, though they are conditional on a hand-picked γ0. The strengths of the paper include a first-principles derivation rather than a black-box fit, a simple and falsifiable prediction about the functional form of J_p, and a clear protocol for determining the five coefficients from measurements. The main weaknesses are the internal notation reversal and the presence of a free hyperparameter γ0 that is tuned per benchmark class without a selection rule.","major_comments":[{"comment":"The roles of γ and θ are exchanged between the main text and the proof. In Eq. (2) of the main text, θ_l multiplies H_C and γ_l multiplies H_M, so θ_p is the final cost angle. In S1, however, U_C(l) = e^{-iγ_l H_C} and U_M(l) = e^{-iθ_l H_M}, and the identity (S2) is derived for U_M† Z_i U_M, so θ_p is the final mixer angle. The state written at the start of S1 (∏ e^{-iγ_l H_M} e^{-iθ_l H_C}) contradicts the definitions of U_C and U_M immediately following it. As written, Eq. (4) in the main text therefore asserts that J_p is a trigonometric function of the final cost angle, which is not what the proof establishes; the proof establishes the dependence on the final mixer angle. This ambiguity affects the core algorithm, since the probe angles θ_x in the Penta-O protocol must be applied to the final mixer unitary. The authors should adopt a single convention throughout, restate Eq. (2) and Eq. (4) consistently, and ensure the S1 proof uses the same pairing of parameters and unitaries as the main text.","section":"§S1 and Eq. (2)"},{"comment":"The fixed cost angle γ0 is a free hyperparameter whose values are chosen per benchmark class: γ0 = 0.2 for the hardware experiment, γ0 = 0.075 for unweighted 3-regular MaxCut, and γ0 = 0.05 for the SK model (Fig. 2 captions and main text). The coefficients in Eq. (S8) depend on γ_p through the state U_C(p)|ψ_{p-1}⟩, so the achievable per-level energy decrease and the depth p needed to reach a target approximation ratio are functions of γ0. The paper acknowledges that determining the optimal γ0 remains an open question. This means Penta-O does not fully eliminate the classical outer loop; it replaces optimization over θ_p with a per-problem-class choice of γ0. The reported performance claims are therefore conditional on tuning choices that the paper does not derive or justify. To make the method a standalone recipe, the authors should either provide a rule for setting γ0 from the instance (e.g., based on max_i,j |w_ij| or other spectral properties), or demonstrate through a sensitivity analysis that the results are robust over a wide range of γ0 values.","section":"Discussion and outlook (γ0 choice) and Fig. 2"},{"comment":"The numerical evidence is presented without error bars for the hardware experiment and without a public code or data repository. The hardware results in Fig. 2(a) are based on single runs with M = 3000 repetitions per trial; quantum measurement statistical fluctuations and gate errors are not quantified, so it is difficult to assess whether the observed monotonic decrease in energy is significant. For the simulations, the use of Qiskit with exact statevector evolution is not explicitly stated, nor is a reproducibility package provided. Given that the empirical demonstrations are a central part of the paper's practical claims, the authors should provide at least confidence intervals for the hardware data and make the simulation scripts and instance lists available.","section":"§S2 and Fig. 2(a)"}],"minor_comments":[{"comment":"The product notation ∏_{l=1}^p e^{-iγ_l H_M} e^{-iθ_l H_C} is ambiguous for quantum circuits; specify explicitly whether the l = 1 factor is on the right (applied first) or left, as the ordering affects the subsequent derivations.","section":"Eq. (2)"},{"comment":"The word 'unprecedented' oversells the result; other outer-loop-free methods such as FALQON already remove the variational outer loop, so the novelty is the fixed number of trials and quadratic cost, not the general concept. Suggest rephrasing to avoid a misleading uniqueness claim.","section":"Abstract and main text"},{"comment":"The phrase 'sampling overhead proportional to 5p+1' should be more precise: the total number of shots is (5p+1)M, where M is the number of repetitions per trial. This distinction matters for practical resource estimation.","section":"Main text, sampling overhead"},{"comment":"The comparison of the ground-state probability 0.2733 to 'the classical algorithm' is vague; specify which classical algorithm (e.g., Goemans-Williamson) and what probability is being compared, and note that the QAOA probability is for the specific 2×3 grid instance rather than a worst-case bound.","section":"Fig. 2(b)"},{"comment":"The quantum-classical crossover estimate depends on speculative p-N scaling laws (p ∝ N^2, p ∝ N, p ∝ log N) and on an assumed t0 and M; these assumptions should be clearly labeled as speculative and not as a proven advantage result.","section":"S3"},{"comment":"There is a typo: 'proportion to' should be 'proportional to', and the statement about the discretization error should refer to the operator norm of the commutator or a more precise bound rather than an informal proportionality.","section":"Discussion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a sound core identity but is currently impaired by an internal notation reversal between the main text and the Supplemental proof, and by a per-benchmark tuned hyperparameter γ0 that limits the strength of the outer-loop-free claim. The authors should be asked to fix these issues and provide the missing numerical reproducibility details. I see no grounds for rejection, as the central derivation appears correct and the empirical results, while conditional, are promising."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing: the central claim is true. S1 gives a clean conjugation proof that J_p is a trig polynomial in the final mixer angle with coefficients independent of that angle, and the five-probe reconstruction is nonsingular. The p=1 version was known, but extending it to general p and turning it into level-wise Penta-O — inherit previous parameters, fix gamma0, fit the curve with five energy measurements, pick the minimizer — is genuinely new. It removes the classical outer loop over theta and gives non-decreasing depth performance. That is a useful practical result, and the relevant p=1 derivations are cited fairly.\n\nWhere the soft spots are. The gamma/theta notation is swapped between Eq. (2) and the S1 proof. It matters because the non-decreasing argument only works if theta_p is the mixer angle, as in S1. The main text's Eq. (2) has them reversed. Fixing this is not a pedantic point; a reader following Eq. (2) won't get the guarantee.\n\nThe bigger issue is gamma0. It is set by hand (0.2 hardware, 0.075 u3r, 0.05 SK), and the paper admits the optimal value is open. Since the coefficients in Eq. (S8) depend on gamma_p, the convergence speed and final approximation ratio are conditional on this choice. Non-decreasing holds for any fixed gamma0, so the theorem is fine, but Penta-O is not a parameter-free recipe. At minimum the paper needs a sensitivity analysis or a plausible selection rule. The benchmarks also lack error bars and code, which matters for the hardware and near-optimality claims.\n\nThe complexity claim is O(p^2) in level count, with t0 proportional to N per level, so total gate cost is O(p^2 N). The abstract's \"quadratic time complexity\" is defensible but could mislead. The N approximately 500 crossover estimate is explicitly speculative because p-N scaling is unknown; I would not treat it as a result.\n\nThis is a solid methods paper with one clear free parameter. The theorem is correct, the algorithm is simple and testable, and the comparison with FALQON is reasonably fair. It deserves proper peer review. I would send it, but with referees asked to pin down the gamma0 problem and demand code and data.","headline":"Correct trig theorem and a genuinely useful level-wise parameter-setting scheme, but the hand-picked gamma0 and the notation swap keep it from being the stand-alone recipe it claims.","tokens_in":15536,"tokens_out":4134,"would_cite":true,"duration_ms":36075,"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":"Penta-O proves that the energy expectation of a p-level QAOA on any QUBO/Ising Hamiltonian is a two-sine trigonometric function of the final mixer angle, so that angle can be set from five measurements and the classical outer loop…","keywords":["quantum approximate optimization algorithm","QAOA parameter setting","QUBO","Ising model","outer-loop-free variational optimization","MaxCut","Sherrington-Kirkpatrick model","trigonometric energy landscape"],"falsifier":"Take a small weighted MaxCut instance with a longitudinal field, set $\\gamma_0$ to a fixed value, and measure $J_2$ at many $\\theta_2$ values; if a two-sine fit from five probe angles does not reproduce the measured curve within sampling error, Eq. (4) is wrong. In the practical direction, run Penta-O on one graph family with $\\gamma_0$ spanning 0.01 to 1; if performance depends sharply on a narrow $\\gamma_0$ range and no heuristic predicts that range, the method is not yet a standalone recipe.","tokens_in":14446,"feed_emoji":"⚛️","tokens_out":10570,"duration_ms":88805,"temperature":0.7,"pith_summary":"This paper seeks to remove the most expensive step in the Quantum Approximate Optimization Algorithm (QAOA): the classical outer loop that repeatedly adjusts circuit angles. It proves that for any quadratic unconstrained binary optimization (QUBO) problem written as an Ising Hamiltonian, the energy expectation of a $p$-level QAOA, seen as a function of the final mixer angle $\\theta_p$, is exactly a two-sine trigonometric function whose coefficients are independent of $\\theta_p$. Five energy measurements at fixed probe angles therefore recover the whole curve and let the minimizing angle be chosen directly; repeating this level by level builds a $p$-level circuit in $O(p^2)$ time with sampling overhead proportional to $5p+1$. Because the previous level is recovered at $\\theta_p=0$, the construction guarantees non-decreasing performance. Experiments on MaxCut and simulations on weighted graphs and the Sherrington-Kirkpatrick model with a longitudinal field show near-optimal ratios at substantially lower depth than prior outer-loop-free methods.","feed_headline":"Five energy probes set QAOA angles, no optimizer needed","feed_subtitle":"A level-by-level fit from five probe angles cuts quantum optimization time to quadratic, with non-decreasing performance.","key_machinery":"The machinery is the trigonometric identity for the terminal mixer angle, Eq. (4), derived from the single-qubit rotation formula $U_M^\\dagger(\\theta_p) Z_i U_M(\\theta_p) = \\cos(2\\theta_p) Z_i + \\sin(2\\theta_p) Y_i$. This identity separates $\\theta_p$ from the Hamiltonian and expresses the five coefficients as $\\theta_p$-independent expectations, for example sums of $\\langle Z_i Z_j \\rangle$, $\\langle Y_i Y_j \\rangle$, $\\langle Z_i Y_j + Y_i Z_j \\rangle$, $\\langle Z_i \\rangle$, and $\\langle Y_i \\rangle$, evaluated on the state $U_C(p)|\\psi_{p-1}\\rangle$. Penta-O is the level-wise fitting procedure that uses five (or three) energy samples to recover these coefficients and then selects the minimizing $\\theta_p$ for each new level.","core_discovery":"The paper's central discovery is that at any level $p$, once the earlier levels are fixed, the energy expectation $J_p = \\langle \\psi_p | H_C | \\psi_p \\rangle$ is exactly $$J_p = A_p \\sin(4\\theta_p + \\phi_p) + A'_p \\sin(2\\theta_p + \\phi'_p) + C_p,$$ with $A_p, A'_p, \\phi_p, \\phi'_p, C_p$ all independent of $\\theta_p$. The proof pulls the last mixer rotation through the Pauli operators via $U_M^\\dagger Z_i U_M = \\cos(2\\theta_p)Z_i + \\sin(2\\theta_p)Y_i$, leaving $\\theta_p$ only in scalar sine and cosine factors. Measuring at five probe angles $\\theta_x = k\\pi/6$ for $k=1,\\dots,5$ therefore determines $J_p(\\theta_p)$ completely, and the minimizer of that fitted curve is the chosen $\\theta_p$. When the Ising field $w_{ii}$ is absent, the period shortens to $\\pi/2$ and three probes at $k\\pi/8$ suffice. Because $\\theta_p=0$ reproduces the previous level, the fitted minimizer always yields $J_p \\le J_{p-1}$, giving a non-decreasing performance guarantee.","pith_inferences":["Beyond the paper: the rotation-identity proof does not use the fact that the problem is quadratic in a deep way, so a similar finite-harmonic trigonometric form should hold for Hamiltonians whose terms are products of $Z$ operators, with more harmonics appearing at higher interaction order.","Beyond the paper: five probe angles are a minimal interpolation set, so using redundant probe angles with a least-squares fit would give a noise-robust variant whose sampling overhead is still $O(p)$; this is a cheap experimental upgrade.","Beyond the paper: the open choice of $\\gamma_0$ is the main obstacle to a fully automatic method, and a per-instance heuristic for $\\gamma_0$ based on graph degree, weight statistics, or a short classical prescreen would complete the recipe.","Beyond the paper: the benchmark design suggests a falsifiable scaling test — on random 3-regular MaxCut graphs with $N$ growing beyond 20, one should check whether the level count needed to cross the classical bound stays near 40 or grows with $N$."],"forward_implications":["At each level the minimizing $\\theta_p$ is obtained from a closed-form fit, so no gradient-based or black-box classical optimizer is ever invoked.","The full parameter search costs $O(p^2)$ circuit-preparation time and at most $(5p+1)M$ samples, with only $3p+1$ for field-free problems, giving the first quadratic-time level-wise setting with a non-decreasing guarantee.","QAOA performance cannot decrease as $p$ grows, because $\\theta_p=0$ exactly reproduces the previous level's circuit and the fitted minimizer is at least as good.","On unweighted 3-regular MaxCut, the average approximation ratio crosses the classical lower bound around $p\\approx 30$ and in the worst case by $p\\approx 40$, roughly an order of magnitude shallower than the leading feedback-based outer-loop-free method.","For spin-glass instances with longitudinal fields, more than half of the tested replicas sample low-energy states with probability above 0.8, and the paper's time-to-solution extrapolation locates a practical quantum-classical crossover near $N\\approx 500$ if the required QAOA level grows only logarithmically with $N$."],"supporting_citations":[{"why":"Defines the QAOA ansatz (cost and mixer unitaries) whose final-level parameter dependence is the object of study.","marker":"[24]"},{"why":"One of the p=1 analytical expectation-value results this proof extends, and a source of the Sherrington-Kirkpatrick infinite-size background.","marker":"[26]"},{"why":"Gives the single-layer fermionic-view derivation for MaxCut that the trigonometric proof generalizes.","marker":"[50]"},{"why":"Provides the single-layer Ising expectation-value decomposition whose observable pattern (ZZ, YY, ZY, Z, Y) the proof reuses for level p.","marker":"[53]"},{"why":"Introduces the feedback-based method FALQON, the main outer-loop-free baseline against which Penta-O's depth is compared.","marker":"[39]"},{"why":"The Lyapunov-control companion of FALQON; together with [39] it defines the previous state of the art for outer-loop-free QAOA parameter setting.","marker":"[40]"},{"why":"Supplies the inapproximability thresholds 16/17 and 331/332 used to frame how close the achieved approximation ratios are to optimal.","marker":"[57]"},{"why":"Classical semidefinite-programming bound and optimal-sampling probability used as the reference line for the hardware MaxCut experiment.","marker":"[58]"},{"why":"Gives the improved classical lower bound for unweighted 3-regular MaxCut used in the p-versus-ratio convergence comparison.","marker":"[63]"},{"why":"Defines the Sherrington-Kirkpatrick spin-glass Hamiltonian used as the longitudinal-field benchmark test.","marker":"[65]"}],"fun_headline_variants":["Five probe angles fix QAOA parameters without optimizer","Quadratic time QAOA: fit energy from five measurements","Penta-O sets QAOA angles level-wise, no classical loop","Trigonometric fit yields non-decreasing QAOA performance","QAOA parameter setting reduced to five probe runs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method depends on a fixed cost-evolution angle $\\gamma_0$ that is chosen by hand for each problem family, and the paper explicitly leaves the optimal choice of this angle as an open question; without a reliable way to set it, the reported performance is not a fully parameter-free recipe.","fun_headline_variants_meta":{"raw":{"variants":["Five probe angles fix QAOA parameters without optimizer","Quadratic time QAOA: fit energy from five measurements","Penta-O sets QAOA angles level-wise, no classical loop","Trigonometric fit yields non-decreasing QAOA performance","QAOA parameter setting reduced to five probe runs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1856,"prompt_tokens":993,"completion_tokens":863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":795}},"tokens_in":609,"tokens_out":863,"duration_ms":7850,"temperature":1.0,"reasoning_tokens":795,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:56:12.446073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small weighted MaxCut instance with a longitudinal field, set $\\gamma_0$ to a fixed value, and measure $J_2$ at many $\\theta_2$ values; if a two-sine fit from five probe angles does not reproduce the measured curve within sampling error, Eq. (4) is wrong. In the practical direction, run Penta-O on one graph family with $\\gamma_0$ spanning 0.01 to 1; if performance depends sharply on a narrow $\\gamma_0$ range and no heuristic predicts that range, the method is not yet a standalone recipe.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the single-layer fermionic-view derivation for MaxCut that the trigonometric proof generalizes."},{"cited_title":"Ozaeta, W","cited_arxiv_id":null,"evidence_quote":"Provides the single-layer Ising expectation-value decomposition whose observable pattern (ZZ, YY, ZY, Z, Y) the proof reuses for level p."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Lyapunov-control companion of FALQON; together with [39] it defines the previous state of the art for outer-loop-free QAOA parameter setting."},{"cited_title":"Berman and M","cited_arxiv_id":null,"evidence_quote":"Supplies the inapproximability thresholds 16/17 and 331/332 used to frame how close the achieved approximation ratios are to optimal."},{"cited_title":"Halperin, D","cited_arxiv_id":null,"evidence_quote":"Gives the improved classical lower bound for unweighted 3-regular MaxCut used in the p-versus-ratio convergence comparison."},{"cited_title":"Panchenko, The Sherrington-Kirkpatrick model (Springer New York, 2013)","cited_arxiv_id":null,"evidence_quote":"Defines the Sherrington-Kirkpatrick spin-glass Hamiltonian used as the longitudinal-field benchmark test."}],"review_version":1}