{"id":"fc6431ef-73f3-4a99-83e6-6f7c5bda4693","arxiv_id":"2412.15081","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A dissertation presenting adiabatic evolution with optimal control, the Rodeo Algorithm, and a new Variational Rodeo Algorithm that uses Rodeo success probability as a variational cost function.","lead":"This thesis studies three quantum algorithms for preparing eigenstates of Hamiltonians: adiabatic evolution with optimal control, the Rodeo Algorithm, and a new Variational Rodeo Algorithm. The new method is tested only in noiseless simulations, while the Rodeo Algorithm is demonstrated on an IBM quantum computer for a one-qubit system.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"VRA's claimed advantage is not yet supported: all Section 5.3 evidence uses the noiseless expectation P_N of Eq. (5.13) with E=E0 supplied exactly, and no noise-model or hardware test exists; a practical benchmark with inexact E and noise is needed.","rationale":"The reader correctly identified the weakest point: VRA's headline results are obtained from noiseless exact-state calculations using Eq. (5.13) with the exact ground-state energy supplied to the cost function. This is load-bearing because the method's entire advantage over energy minimization is supposed to be practical scalability on near-term devices, and no evidence is given that the advantage survives gate noise, finite shot statistics, or the realistic case where only an approximate eigenvalue is known. I read the derivation of Eq. (5.13) in good faith and find it internally consistent; the concern is not a mathematical error but an unsupported extrapolation from idealized simulation to hardware. A concrete noise-model and inexact-energy test would settle whether the central claim holds. I also note a separate, non-central typographical issue: the probability of an ancilla being in |1> is described as the success probability in Section 5.1, whereas the RA derivation identifies success with the |0> outcome; this does not affect the numerical results but should be corrected.","tokens_in":55249,"tokens_out":10838,"duration_ms":102173,"concrete_test":"Rerun the Section 5.3.2 6-qubit VRA-vs-QAOA comparison under two realistic modifications: (1) use E = E0 + δ for δ ∈ {0.25Δ, 0.5Δ}, where Δ is the gap to the first excited state, and (2) replace the exact-state evaluation of Eq. (5.13) with a statevector simulation of the full QAOA+RA circuit (controlled time evolutions and mid-circuit measurements) under a depolarizing noise model with per-gate error around 10^-3 and finite shot counts (e.g., 8192 shots per cost call). If VRA's final-state fidelity or success probability no longer exceeds the energy-minimization baseline, the noiseless exact-E0 results do not support the scalability claim; if it still wins, the concern is mitigated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that VRA prepares eigenstates with higher fidelity than energy minimization and 'allows for better scalability'—rests on the assumption that the noiseless, infinite-statistics Rodeo success probability P_N in Eq. (5.13) is a faithful proxy for near-term device performance, and that a usable approximation to the target energy E is available. Section 5.3 explicitly computes the exact QAOA final state classically ('no circuit executions are necessary') and evaluates Eq. (5.13) directly; no noisy circuit simulation, finite-shot sampling, or hardware run of VRA is presented. Section 5.3.2 further sets E = E0 exactly, the very quantity the algorithm is supposed to find; this idealization is unavailable for the nuclear-physics targets motivating the thesis. Because VRA appends the RA controlled-evolution circuit to the QAOA ansatz, its circuit depth is at least double that of QAOA alone, so the noise-free advantage reported in Figures 5.3–5.4 could vanish or reverse under gate errors, decoherence, or the post-selection cost of conditioning on all Rodeo ancilla successes. The excited-state results in Section 5.3.1 likewise place E near the target level rather than testing robustness to energy uncertainty. The derivation of Eq. (5.13) is internally consistent, but the numerical evidence for the headline scalability claim stops short of the conditions under which the algorithm would actually be used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This thesis (arXiv:2412.15081) develops and tests three quantum algorithms for eigenstate preparation: adiabatic evolution with optimal control (Chapter 3), the Rodeo Algorithm (RA, Chapter 4), and the novel Variational Rodeo Algorithm (VRA, Chapter 5). The RA section derives the success probability for N cycles with Gaussian-sampled time parameters, demonstrates eigenvalue scans and eigenstate preparation on a single-qubit Hamiltonian on IBM Casablanca hardware (achieving 0.08% relative energy error with measurement-error mitigation), and argues for an exponential advantage over phase estimation and adiabatic evolution. The VRA section proposes maximizing the RA success probability as a variational cost function for QAOA parameters, derives the gradient structure of three cost functions, and reports noiseless classical simulations on random 6- and 10-qubit Hamiltonians showing that VRA prepares ground and excited states with higher fidelity than energy minimization for low-depth QAOA circuits.","tokens_in":55574,"tokens_out":5426,"duration_ms":46900,"significance":"The strengths of the manuscript are the clean analytic derivations: the RA success probability (Eqs. 4.5–4.10) and the VRA cost-function gradients (Section 5.2) are carefully derived from the initial-state overlap without assuming knowledge of the target eigenstate. The hardware demonstration of RA (Section 4.2.2) is a concrete, reproducible experiment with measurement-error mitigation, and the 0.08% eigenvalue error is a useful benchmark. The VRA idea is a plausible and interesting extension that directly addresses RA's dependence on initial-state overlap, and the comparison of energy minimization versus overlap maximization is informative. However, the central claim that VRA 'allows for better scalability' is currently supported only by noiseless simulations with the exact ground-state energy injected into the cost function; the numerical evidence does not yet cover the operational conditions of a near-term device.","major_comments":[{"comment":"The VRA ground-state comparison sets the target energy E in Eq. (5.13) to the exact ground-state energy E0, a quantity that is not known in practice for the systems that motivate the algorithm. Because the abstract claims better scalability under realistic conditions, this oracle-like input is load-bearing; the manuscript does not test how VRA performs when E is only known approximately, which would require energy scanning or a broader energy filter and could substantially reduce the reported success probabilities.","section":"Section 5.3.2, Eq. (5.13)"},{"comment":"All VRA results are obtained by classically computing exact final states and evaluating the noiseless expectation value P_N from Eq. (5.13); no noisy circuit simulation, finite-shot sampling, or hardware experiment is presented for VRA. Since the VRA circuit appends the RA controlled time evolutions and an ancilla register to the QAOA ansatz, the circuit depth is at least double that of QAOA alone, so the noiseless advantage shown in Figures 5.3 and 5.4 could vanish or reverse under gate errors, decoherence, and the post-selection cost of conditioning on all ancilla successes. A benchmark with a standard noise model and finite shots is needed to support the claim that VRA allows better scalability on near-term devices.","section":"Section 5.3 (including 5.3.1 and 5.3.2)"},{"comment":"The claimed exponential speedup of RA over phase estimation and adiabatic evolution for eigenstate preparation is based purely on scaling in the residual error Δ, omitting the total coherent evolution time per preparation, which grows with the number of cycles N and the standard deviation σ of the time parameters. The complexity comparison should include these resources, along with the cost of implementing the controlled time evolution (e.g., Trotterization), before concluding that RA is exponentially more efficient for large systems; this is a load-bearing point for the motivation of VRA.","section":"Section 4.1.1, Eqs. (4.9)–(4.11)"}],"minor_comments":[{"comment":"The notation \"||c|^2_exp>\" in the confusion-matrix equation is malformed; it should read \"|c^2_exp>\" or similar, with the superscript applied to the probability vector.","section":"Section 3.3.1.1, Eq. (3.14)"},{"comment":"The definition of P_ψ uses M for the number of cycles, while the surrounding text and figures (e.g., Figure 5.3) use N; unify the notation to avoid confusion.","section":"Section 5.2, Eq. (5.13)"},{"comment":"The phrase \"ascanwith\" appears to be a typo; it should read \"a scan with\".","section":"Section 4.2.2.3"},{"comment":"The captions state that the figures are reused from [31] but do not specify the source or permission; for an arXiv posting, this is acceptable but should be clarified if the manuscript is submitted elsewhere.","section":"Chapter 3, Figures 3.2 and 3.3"}],"recommendation":"major_revision","confidential_remarks":"This is a PhD dissertation with substantial introductory material (Chapters 1–2) that goes beyond a typical research article; for journal publication, it would benefit from being condensed to the novel contributions. The central VRA claim is promising but currently supported only under idealized conditions; the authors should either add noise-robustness benchmarks or moderate the scalability claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Joey,\n\nRead the thesis. The genuinely new piece is the Variational Rodeo Algorithm (Chapter 5), and the math around it is the strongest part. The derivation of the RA success probability (Eq. 5.13) and the cost-function gradient analysis are clean and internally consistent; the argument that maximizing RA success probability reduces to maximizing ground-state overlap in the right parameter limit is correct as far as it goes. The hardware demonstrations in Chapters 3 and 4 are real: two-qubit adiabatic evolution with optimal control, and a single-qubit RA eigenvalue scan on IBM Casablanca with measurement error mitigation, achieving 0.08% eigenvalue error. Figure reuse is explicitly marked, which is honest, and the self-citations to prior RA work are appropriate.\n\nThe soft spot is where the abstract makes its strongest claim. VRA's \"better scalability\" is supported only by noiseless classical simulation: Section 5.3 explicitly computes exact final states with no circuit executions, and the main ground-state comparison feeds E=E0 into Eq. (5.13) — exactly the quantity the algorithm is supposed to find. There is no noise model, no finite-shot sampling, no hardware run for VRA. Since VRA appends RA controlled evolutions to QAOA, the circuit depth is approximately double, so the reported fidelity advantage could plausibly reverse under gate errors or decoherence. The text hedges with \"suggest\" and \"should be effective\" in places, but the abstract's \"allowing for better scalability\" goes beyond what the evidence shows. No code or data artifacts are provided, which makes independent replication harder.\n\nThese gaps are addressable. A noisy simulation with finite shots and an inexact energy guess, or a small hardware demonstration, would substantially close the gap. I don't see a load-bearing technical error; the missing piece is evidence under realistic conditions.\n\nWho gets value: anyone working on variational eigenstate preparation or NISQ algorithms will find the VRA idea worth knowing, and the RA analysis is a useful reference. But as it stands, it is a proposal with promising but incomplete validation. I'd send it to peer review rather than desk-reject: it deserves referee time, but I'd expect a serious referee to ask for the noisy benchmark before accepting. If I were refereeing, I'd recommend major revision.","headline":"Genuinely new VRA algorithm with clean math, but its central scalability claim currently rests on noiseless simulations with an idealized energy target; the hardware demos are real.","tokens_in":56122,"tokens_out":2809,"would_cite":false,"duration_ms":21241,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Lx","03.67.-a"],"model":"deepseek-v4-flash","headline":"This thesis argues that maximizing the Rodeo algorithm's success probability, rather than minimizing energy, prepares eigenstates with higher fidelity on shallow quantum circuits and also reaches excited states.","keywords":["eigenstate preparation","Rodeo algorithm","variational quantum eigensolver","QAOA","adiabatic evolution","optimal control","quantum simulation","NISQ"],"falsifier":"Run VRA on a real noisy device for a random 6-qubit Hamiltonian, using only an estimated energy (for example from a prior VQE calculation) as the Rodeo target, and compare the measured ground-state overlap and success probability against QAOA energy minimization with equal circuit depth. If the VRA advantage disappears under device noise, or if the optimized parameters fail to increase the measured Rodeo success probability relative to the simulated value predicted by Equation 5.13, the central claim is not borne out on hardware.","tokens_in":55012,"feed_emoji":"⚛️","tokens_out":9914,"duration_ms":81967,"temperature":0.7,"pith_summary":"Solving eigenvalue problems—finding the eigenstates and energies of a Hamiltonian—is the bottleneck for quantum simulation of nuclei and molecules. This thesis proposes a practical route through three quantum algorithms: adiabatic evolution with optimal control, the Rodeo algorithm, and the Variational Rodeo Algorithm (VRA). The central claim is that VRA, which uses the Rodeo success probability as its cost function, prepares eigenstates more faithfully than energy minimization when the variational circuit is too shallow to reach the exact ground state, and that it can target excited states by shifting the Rodeo energy window. If correct, this gives near-term quantum computers a concrete way to turn low-depth circuits into high-fidelity eigenstates, a prerequisite for simulating nuclear structure and reactions.","feed_headline":"Variational Rodeo prepares eigenstates better than energy minimization","feed_subtitle":"A cost built from Rodeo success probability targets any eigenstate, including excited states, on shallow quantum circuits.","key_machinery":"The central object is the Rodeo success probability used as a variational cost function (Equation 5.13). The circuit is: prepare a parameterized ansatz, then for each of $M$ Rodeo cycles apply a Hadamard to an ancilla, perform a controlled time evolution $e^{-iH_{\\text{obj}}t}$, apply a phase rotation $e^{iEt}$, apply another Hadamard, and measure the ancilla. The probability that all ancilla measurements give $|0\\rangle$ is $$\\sum_n |c_n|^2\\left[\\frac{1+$e^{{-(E_n-E)^2\\sigma^2/2}}$}{2}\\right]^M,$$ which exponentially suppresses eigenstates whose energies lie outside a window around $E$. Maximizing this probability is the mechanism that steers the parameterized circuit toward a chosen eigenstate, and it is what distinguishes VRA from energy-minimization approaches.","core_discovery":"The paper's central proposal is the Variational Rodeo Algorithm: append a Rodeo circuit to a parameterized state-preparation ansatz (here QAOA) and classically optimize the ansatz parameters to maximize the probability that every Rodeo ancilla measurement returns $|0\\rangle$, i.e. to minimize $1-P_N$. For a state $|\\psi\\rangle=\\sum_n c_n|E_n\\rangle$ of the object Hamiltonian and a Rodeo energy parameter $E$, the success probability is $$P_\\psi=\\sum_n |c_n|^2\\left[\\frac{1+$e^{{-(E_n-E)^2\\sigma^2/2}}$}{2}\\right]^M.$$ Because each eigenstate's contribution is damped by a Gaussian factor centered at $E$, maximizing this probability concentrates the state on the eigenstate with eigenvalue closest to $E$. In noiseless simulations on random 6- and 10-qubit Hamiltonians, the paper finds that this cost function drives the QAOA output toward the targeted eigenstate—including excited states—and that for shallow circuits (12–20 parameters) it reaches higher ground-state overlap than energy minimization. The same simulations show that using VRA as a fine-tuning step after energy minimization improves overlap in cases where the variational principle is not in effect. The paper also demonstrates the two building blocks: adiabatic evolution with optimal-control custom gates reaches roughly 95% fidelity in emulations against about 60–85% on cloud processors, and the Rodeo algorithm recovers single-qubit eigenvalues to 0.08% relative error on hardware.","pith_inferences":["Editorial inference: the same cost function could be applied to any parameterized ansatz beyond QAOA, such as hardware-efficient circuits, as long as controlled time evolution under $H_{\\text{obj}}$ is available; the paper only tests QAOA.","Editorial inference: because the Rodeo success probability is a binary-measurement count rather than a signed energy expectation, it may be more robust to certain systematic errors; this is testable by comparing gradient noise under a depolarizing noise model.","Editorial inference: VRA's ability to target excited states, combined with the Hellmann-Feynman technique demonstrated for the Rodeo algorithm, could enable spectrum and observable calculations without orthogonality-constrained excited-state variational methods.","Editorial inference: the noiseless-proxy assumption will be the first thing to break as system size grows: the exact energy input $E_0$ used in the tests must in practice be replaced by an estimate, and error in that estimate should degrade the overlap advantage."],"forward_implications":["VRA turns an arbitrary parameterized ansatz into an eigenstate filter: by choosing the Rodeo energy $E$, the same circuit can be steered to the ground state or to a specific excited state.","For shallow QAOA circuits, optimizing the Rodeo success probability yields higher ground-state overlap than energy minimization, so VRA can replace or follow VQE/QAOA optimization.","The success probability is estimated by counting ancilla measurement outcomes, avoiding the need for full Hamiltonian expectation-value estimation in the optimization loop.","Rodeo's exponential suppression of off-target eigenstates means the added cost of the Rodeo tail grows only logarithmically with the desired eigenstate purity, making the hybrid approach compatible with NISQ-era circuit depths."],"supporting_citations":[{"why":"Supplies the Rodeo algorithm itself: the controlled-evolution filter, the success-probability formula, and the exponential suppression that VRA uses as its cost function.","marker":"[30]"},{"why":"Provides the hardware demonstration of the Rodeo algorithm and the energy-scanning procedure whose success-probability estimates VRA optimizes.","marker":"[105]"},{"why":"Defines QAOA, the parameterized circuit family whose parameters VRA optimizes in all simulations.","marker":"[38]"},{"why":"Defines the VQE energy-minimization paradigm that serves as the comparison baseline in the ground-state tests.","marker":"[84]"},{"why":"Supplies the variational principle underlying energy minimization, the contrast to the overlap-based Rodeo cost.","marker":"[49]"},{"why":"Supplies the adiabatic theorem behind the thesis's first eigenstate-preparation route and motivates the QAOA ansatz.","marker":"[21]"},{"why":"Supplies the optimal-control method used to design the custom two-qubit gates in the adiabatic evolution demonstration.","marker":"[89]"}],"fun_headline_variants":["Variational Rodeo targets any eigenstate, not just ground","Variational Rodeo finds excited states on shallow circuits","Rodeo plus optimization beats energy minimization for eigenstates","New variational method prepares arbitrary eigenstates with Rodeo","Variational Rodeo improves eigenstate overlap on quantum hardware"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that VRA is an effective eigenstate-preparation method rests on the assumption that the noiseless success probability of the Rodeo circuit is a faithful stand-in for real-device performance, since all VRA results come from exact classical simulations and the ground-state tests supply the exact ground-state energy as input.","fun_headline_variants_meta":{"raw":{"variants":["Variational Rodeo targets any eigenstate, not just ground","Variational Rodeo finds excited states on shallow circuits","Rodeo plus optimization beats energy minimization for eigenstates","New variational method prepares arbitrary eigenstates with Rodeo","Variational Rodeo improves eigenstate overlap on quantum hardware"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000788,"raw_usage":{"total_tokens":3585,"prompt_tokens":1166,"completion_tokens":2419,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":782,"completion_tokens_details":{"reasoning_tokens":2335}},"tokens_in":782,"tokens_out":2419,"duration_ms":14594,"temperature":1.0,"reasoning_tokens":2335,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:38:56.334803+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run VRA on a real noisy device for a random 6-qubit Hamiltonian, using only an estimated energy (for example from a prior VQE calculation) as the Rodeo target, and compare the measured ground-state overlap and success probability against QAOA energy minimization with equal circuit depth. If the VRA advantage disappears under device noise, or if the optimized parameters fail to increase the measured Rodeo success probability relative to the simulated value predicted by Equation 5.13, the central claim is not borne out on hardware.","supporting_citations":[],"review_version":1}