The quantum formalism contains a structural discrepancy: the transition amplitude computes the interaction-specific relative phase between probe and target and its effect on coupling strength, while the post-interaction state description does not preserve it. A companion paper identified this discrepancy and argued that the measurement problem arises from mistaking the formalism’s predictive completeness for descriptive completeness at the level of individual events. The present paper develops the quantitative consequence. A minimal phenomenological model shows that, for an event-level conditional law of the stated phase-dependent form, uniform averaging over the interaction-relative phase recovers the Born rule exactly, while partial phase control narrows the outcome distribution according to a visibility-loss factor . Starting from the Jaynes-Cummings Hamiltonian for a two-level system coupled to a single-mode field, we derive the event-level conditional probability , where is the phase-coupling strength — a quantity determined by the microscopic interaction parameters. The phase-coupling strength has two contributions. A pointer-overlap term, , arises from the finite separation of the measurement pointer states and decays exponentially with measurement strength . A persistent term, , arises from beyond-dispersive corrections to the Jaynes-Cummings dynamics. The standard treatment computes the rate of these transitions by squaring the matrix elements, eliminating the relative phase between qubit and field; the present framework retains the amplitude structure prior to ensemble reduction, predicting that the transitions are phase-correlated with the qubit-field relative phase. We propose a specific experiment: sweeping measurement strength while tracking phase-resolved outcome modulation in a circuit QED system. The standard treatment predicts the modulation amplitude decays to zero; the present framework predicts it saturates at a nonzero floor. Because a small phase-dependent bias can also arise from ordinary measurement non-diagonality, the decisive evidence is not the cosine modulation itself but its parameter dependence: the predicted amplitude must scale as across controlled changes in detuning, reverse under , and persist under recalibration. The experiment is feasible with current technology and would determine whether event-level phase structure persists through the measurement interaction or is genuinely absent from the outcome-producing physics.
Abstract We present a framework for preparing quantum ground states by structural projection rather than by deep-circuit Trotterized dynamical simulation. The method is grounded in a rigorous operator-algebraic formalization spanning 297 parts (the IQS/EQS corpus), from which we extract the complete derivation chain 𝐻_boundary → 𝐴_𝜇 → 𝑃_𝐻 → 𝐴_𝐻. We prove that the Host projector 𝑃_𝐻 = 𝑃_DFS^max acts as a native state-preparation oracle — a single projective step applied once, not a query oracle in the phase-kickback sense: assuming appropriate overlap, it projects any initial state |𝜓₀⟩ into the prescribed ground-state subspace with an 𝒪(1) execution depth on the quantum device.The body of this paper is confined to the framework and its theorems; the complete experimental record is given in the appendix. We validate the framework through 55 independent QPU executions on three IBM Heron-class processors (156 qubits, 𝑟2 revision): ibm_marrakesh, ibm_kingston and ibm_fez. Twenty-one of these form the Phase 3–6 programme, utilizing up to 76 actively measured qubits and achieving highly non-uniform output distributions in which the most probable outcome exceeds its uniform-distribution probability by factors of 3.44 × 10⁵ to 5.50 × 10⁶ at 50,000 shots; the remaining twenty-four test derived theorems directly against predictions computed and published before submission, including the oracle projection at 𝑁 = 6 to 12, where up to 82.1% of shots return an exact maximum cut of a frustrated graph against a uniform null of 4.88 × 10⁻⁴. We demonstrate cross-domain applicability by deploying custom Hamiltonians encoding molecular chemistry (H₂ electron correlation), integer factorization (cryptographic Ising reduction), and combinatorial optimization (MaxCut/portfolio) into the same oracle architecture, all executing at identical 𝒪(1) quantum circuit depth. We further extend the oracle to 𝑘-local Hamiltonians (𝑘 = 3, 4, 5) utilizing up to 153 actively measured qubits per execution, proving via conservation of the total magnetization 𝐴₁ = ∑ₓ 𝑍ₓ that the framework generalizes to a specific structured family of 𝑘-body interactions. Finally, we address the problem encoding question by formalizing the Compression Encoding Theorem, which provides the algebraic structure to map target problem families into the oracle (with classically-bounded construction complexity). We instantiate this construction across 6 representative encoded problem instances on live hardware. We are explicit about what this does not claim. The 𝒪(1) figure is the coherent execution depth alone, and the hardness of a target problem is confined rather than removed: to the classical synthesis of the Host isometry in the deterministic route, and to post-selection overhead in the measurement-conditioned one. The encoding theorem bounds that classical cost only on a class whose ground-state energy is computable in polynomial time, and a family whose ground-state energy is NP-hard to compute does not belong to it.These results establish that the quantum execution phase of ground-state preparation is achievable on current NISQ devices without variational optimization, Trotterization, or quantum error correction, provided the problem Hamiltonian is mapped through the decoherence-free-subspace projection framework defined herein.
Random quantum circuits have been extensively explored for quantum supremacy demonstrations [1]–[4]. However, verifying their output distributions remains challenging [2], [5], [6]. Here, we propose the infinite-temperature correlation function (ITCF) as a physically meaningful observable for noisy intermediate-scale quantum (NISQ) devices one that can be extracted using engineered circuits rather than relying on fully random constructions. This is realized by leveraging peaked quantum states whose probability distributions are sharply peaked at specific outcomes due to constructive interference thus offering more efficient verifiability and stronger signal observability. Rather than using Haar-random states, which often yield vanishing signals through destructive interference, we construct purposefully biased quantum states using either Grover-based amplitude amplification or shallow structured circuits. These engineered states amplify contributions from relevant operator subspaces, enabling robust detection of non-zero ITCF values that would otherwise be suppressed under random-state sampling. Our results highlight a problem-specific state preparation framework that mitigates signal loss from random averaging and facilitates the detection of physically meaningful observables in NISQ devices. We also discuss future extensions to multiqubit observables, scrambling diagnostics, and variational circuit optimization, underscoring the broader potential of Peaked States for quantum simulation and verification.
A quantum ensemble contains preparation weights that are lost in its average density operator. We show how optimal decisions read these weights as a wave on projective state space. Varying the input prior sweeps a surface of fixed Born overlap through the ensemble; the resulting response propagates according to the geometry of pure states and reconstructs their distribution in every finite dimension. Conditional preparations can fix this wave through contact, selecting the ensemble at a steering boundary. Positive propagation gives exact biased-qubit solutions and explicit parent measurements for noisy channels. When measurement records are combined, their state directions are averaged and the wave acquires a source whose growth measures the lost discrimination advantage. Preparation, inference and noise can thus be understood as operations on the same recoverable distribution, linking quantum correlations to measurement resolution.
Physical Sources, Finite Detectors, Persistent Records, and Actual Events This paper develops a unified account of quantum measurement under the Admissibility and Standing Constraint (AASC) framework, connecting constructed physical sources, quantum instruments, material amplification, finite record retention, event probabilities, and the actuality of measurement outcomes. The paper distinguishes three separate objects: a complete quantum instrument containing all outcome operations, including unresolved and failed outcomes; a material record carried by a physical system whose acquisition and retention can be analyzed; an actual history describing which occurrences belong to one original experimental run. A complete instrument and a persistent material record can be constructed without assuming that linear quantum dynamics alone selects one globally definite original-run history. Conversely, the admission of an actual event does not by itself determine its probability law or conditional quantum state. The physical construction contains two complementary measurement targets: a regulated interacting gauge–Higgs–fermion source with its full source Hamiltonian retained during finite receiver pulses; an explicitly parameterized six-mode cosine circuit with source and pump interfaces, losses, distinguishable outputs, a two-polarization intrinsic-spin receiver, and a finite energy-resolving thermal amplifier. The material-registration chain includes source-code loading, driven conversion, output registration, pump-displacement removal, transfer to an intrinsic material spin, finite Gibbs-collision amplification, endpoint reading, and quiet record retention. Electronic display labels, material flags, missed counts, dark displays, blanks, and unreadable outcomes remain distinct and are retained in the joint error ledger. The paper establishes: informative bounded receivers on an interacting quantum source; finite-pulse calibration with the complete source Hamiltonian retained; a complete monitored detector model; finite preparation and amplification bounds; coherent composition of instruments and records; adaptive record composition; local-instrument compatibility; calibrated affine probability uniqueness; sharp coherent calibration limits; sequential probability laws and compatible conditional states; explicit actual-history requirements. A finite Fourier residual certificate bounds the complete monitored detector instrument over its full
35.587 ms interval by
0.063241 in restricted diamond norm. Exact interval arithmetic gives seeded amplifier endpoint fidelity greater than
0.9999235242 after 1,024 finite collisions, together with an explicit conditional survival bound for a subsequent
100 ms quiet hold. The probability analysis shows that positivity, density-operator dependence, affinity under classical mixtures, and calibration fix the trace probability rule for the declared readout. Approximate sector calibration alone permits coherent deviations of order the square root of the calibration error; stronger linear bounds require additional structure. The paper’s central measurement-theoretic conclusion is deliberately separated from its successful detector construction. A complete instrument admits an exclusive-history completion, but linear quantum dynamics alone does not select one such history. Source-specific preparation, reader analysis, and a bounded counting-law completion supply the additional physical input required to select an actual history. The resulting framework substantially develops the material and probabilistic parts of quantum measurement while isolating apparatus errors, actual-history law, and event-selection assumptions as explicit obligations. The work does not claim to solve every interpretation of the measurement problem, derive a universal collapse law, establish unrestricted detector accuracy, or identify a finite calibration certificate with an experimental calibration. Principal contributions A unified source-to-detector account of quantum measurement. Constructed interacting quantum sources with finite receivers. Six-mode cosine-circuit detector realization. Material registration and finite amplification chain. Two-polarization intrinsic-spin receiver. Finite thermal collision and endpoint-reading model. Full-duration restricted diamond-norm detector bound. Interval-certified amplifier fidelity. Conditional quiet-hold survival bound. Retention of blanks, wrong, unreadable, and missed outcomes. Calibrated affine probability uniqueness. Sequential instruments and compatible conditional states. Explicit separation of complete instruments, material records, and actual histories. Source-specific conditions for actual-event selection. Complete physical error and assumption ledger.
Correlations that do not admit a deterministic explanation are a central feature of quantum theory and a key resource for quantum information processing. Identifying and certifying such correlations, however, remains a fundamental challenge. In this work, we advance on this problem by considering deterministic correlations in the prepare-and-measure scenario consistent with a wide range of communication restrictions. To certify correlations incompatible with such deterministic models, we ask whether they admit a scenario in which measurement outcomes can be perfectly predicted by an adversary equipped with classical side-information. We show the usefulness of this approach and propose semidefinite programming relaxations tailored to three representative communication restrictions: fixed ensemble, bounded overlaps and restricted observables.