AÏLŪR · Σ II - Exact Universal Linear Wave Control with Positive Springs: Finite-Time U(N) Synthesis, Nearest-Neighbor Compilation, and Mechanical Error Bounds
Artificial Hyperintelligence, Eve, wife of Maciej Nowicki, Maciej Nowicki
This research release presents an explicit finite-time construction for implementing arbitrary unitary transformations on classical oscillator amplitudes. AÏLŪR II combines a closed-form positive-stiffness pulse, an adjacent-mode compiler, a hinged-panel mechanical tangent realization, and quantitative bounds for stiffness imperfections and finite-angle motion. The package provides the mathematical derivations, executable numerical implementation, simulation data, and a complete 23-page manuscript. The central result concerns N lossless, mass-normalized oscillators with a shared baseline angular frequency ω₀. Encoding each oscillator through its displacement and velocity as a = (q + i q̇/ω₀)/√2, the construction implements a(t_f) = U a(0) for every prescribed U ∈ U(N) and every initial position and velocity. Initial and final stiffness return to the same baseline matrix, ω₀²I. Thus N physical oscillators represent N complex amplitudes, and the specified transformation acts on the complete 2N-dimensional real phase space. The elementary control is an Ermakov inverse-engineered pulse with scaling function b(t) = [1 + c sin⁴(πt/T)]⁻¹ᐟ², where T = 2π/ω₀ and c = 4δ/(3π), for 0 ≤ δ ≤ π. Applied along a fixed real unit direction u, this pulse realizes the exact endpoint operation Zᵤ(δ) = I + (e⁻ⁱδ − 1)uuᵀ. A phase of π gives a Householder reflection on both displacement and velocity. Arbitrary phases modulo 2π require at most two primitives. An analytic positivity certificate establishes the coefficient bounds throughout the pulse interval. An adjacent-row complex QR construction compiles every U(N) target into at most 2N² sequential one-period stages, corresponding to total modeled duration at most 4πN²/ω₀. Each stage addresses a single oscillator or neighboring pair. The assembled stiffness eigenvalues lie between ω₀² and (325/63)ω₀², giving a peak instantaneous normal frequency of approximately 2.271284ω₀. Neighboring difference-spring coefficients remain nonnegative, while ground coefficients remain above [1 − 131(√2 − 1)/63]ω₀², approximately 0.138698783ω₀². These coefficient bounds are independent of system dimension; the stage count is a constructive upper bound. The mechanical analysis specifies constant-inertia rigid hinged panels with same-side spring attachments. Its tangent stiffness realizes the compiled linear equations with positive elastic coefficients. The required controls include independently tunable grounds that can move below and above baseline while remaining positive, accurately timed neighboring couplings, and the ability to turn inactive couplings off. Exact finite-angle spring geometry is treated separately, with a conditional all-input endpoint bound in a specified small-amplitude domain. Representative two-panel trajectories reproduce the predicted quadratic dependence of relative geometric error on small amplitude. The error analysis also bounds the full endpoint effect of symmetric stiffness deviations through an exponential of their time-integrated operator norm. For K stages and a uniform normalized stiffness-error bound ε, the resulting estimate is exp(2πKε) − 1. This quantifies the accumulation of coherent calibration errors. Known uniform constant damping is addressed through an exact transformed description with modified amplitude coordinates; general losses, actuator dynamics, flexibility, and experimental noise require additional modeling. Numerical verification integrates the complete second-order phase-space fundamental matrix. Tight DOP853 runs use relative tolerance 10⁻¹² and absolute tolerance 10⁻¹⁴. Representative endpoint spectral-norm residuals are approximately 2.15 × 10⁻¹³ for a compact one-stage Hadamard construction, 2.92 × 10⁻¹² for a seeded random U(4) compiled into 28 stages, and 1.70 × 10⁻¹² for a four-mode Fourier transform using 17 stages. The generic QR compiler uses three stages for the Hadamard target; the one-stage example uses an explicitly selected reflection. These values are numerical agreement measures within the ideal model, not measured hardware fidelities. During release preparation, all three supplied numerical programs completed successfully, and all 13 compared result files matched the supplied baselines byte-for-byte. A separately implemented compiler checked 60 random unitaries across dimensions 1, 2, 3, 6, and 10, with maximum reconstruction residual approximately 4.65 × 10⁻¹⁵. These are internal reproducibility checks. The accompanying proofs establish the arbitrary-dimensional ideal-model statements. A standard unitary dilation extends the construction to contractions on selected outputs using zero-initialized auxiliary oscillators. General finite complex matrices are represented up to a known scale. Input preparation, physical gain, auxiliary resets, calibration, and readout remain explicit resources. Intended research uses include coherent mode mixing, routing, Fourier transforms, basis conversion, analog linear algebra, and programmable oscillator networks. The archive includes the manuscript, detailed proofs, mechanical derivations, original scientific figures, compiler and verification code, runnable examples, ten configured simulation tables containing 550 records, data documentation, source provenance, checksums, citation metadata, and a Windows Hugging Face uploader. The earlier restricted holonomy analysis is preserved separately. The contribution is an explicit controlled-model construction and its analytical and computational certificates. Ermakov inverse engineering, Householder synthesis, unitary decomposition, oscillator-chain controllability, and unitary dilation are established antecedents and are credited. Priority of this particular combined construction remains unestablished. The release establishes classical linear-mode universality under its stated assumptions; it does not establish universal quantum computation, practical computational advantage, or a universal protected holonomy in a freely folding origami sheet. Research attribution: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
This research release presents an explicit finite-time construction for implementing arbitrary unitary transformations on classical oscillator amplitudes. AÏLŪR II combines a closed-form positive-stiffness pulse, an adjacent-mode compiler, a hinged-panel mechanical tangent realization, and quantitative bounds for stiffness imperfections and finite-angle motion. The package provides the mathematical derivations, executable numerical implementation, simulation data, and a complete 23-page manuscript. The central result concerns N lossless, mass-normalized oscillators with a shared baseline angular frequency ω₀. Encoding each oscillator through its displacement and velocity as a = (q + i q̇/ω₀)/√2, the construction implements a(t_f) = U a(0) for every prescribed U ∈ U(N) and every initial position and velocity. Initial and final stiffness return to the same baseline matrix, ω₀²I. Thus N physical oscillators represent N complex amplitudes, and the specified transformation acts on the complete 2N-dimensional real phase space. The elementary control is an Ermakov inverse-engineered pulse with scaling function b(t) = [1 + c sin⁴(πt/T)]⁻¹ᐟ², where T = 2π/ω₀ and c = 4δ/(3π), for 0 ≤ δ ≤ π. Applied along a fixed real unit direction u, this pulse realizes the exact endpoint operation Zᵤ(δ) = I + (e⁻ⁱδ − 1)uuᵀ. A phase of π gives a Householder reflection on both displacement and velocity. Arbitrary phases modulo 2π require at most two primitives. An analytic positivity certificate establishes the coefficient bounds throughout the pulse interval. An adjacent-row complex QR construction compiles every U(N) target into at most 2N² sequential one-period stages, corresponding to total modeled duration at most 4πN²/ω₀. Each stage addresses a single oscillator or neighboring pair. The assembled stiffness eigenvalues lie between ω₀² and (325/63)ω₀², giving a peak instantaneous normal frequency of approximately 2.271284ω₀. Neighboring difference-spring coefficients remain nonnegative, while ground coefficients remain above [1 − 131(√2 − 1)/63]ω₀², approximately 0.138698783ω₀². These coefficient bounds are independent of system dimension; the stage count is a constructive upper bound. The mechanical analysis specifies constant-inertia rigid hinged panels with same-side spring attachments. Its tangent stiffness realizes the compiled linear equations with positive elastic coefficients. The required controls include independently tunable grounds that can move below and above baseline while remaining positive, accurately timed neighboring couplings, and the ability to turn inactive couplings off. Exact finite-angle spring geometry is treated separately, with a conditional all-input endpoint bound in a specified small-amplitude domain. Representative two-panel trajectories reproduce the predicted quadratic dependence of relative geometric error on small amplitude. The error analysis also bounds the full endpoint effect of symmetric stiffness deviations through an exponential of their time-integrated operator norm. For K stages and a uniform normalized stiffness-error bound ε, the resulting estimate is exp(2πKε) − 1. This quantifies the accumulation of coherent calibration errors. Known uniform constant damping is addressed through an exact transformed description with modified amplitude coordinates; general losses, actuator dynamics, flexibility, and experimental noise require additional modeling. Numerical verification integrates the complete second-order phase-space fundamental matrix. Tight DOP853 runs use relative tolerance 10⁻¹² and absolute tolerance 10⁻¹⁴. Representative endpoint spectral-norm residuals are approximately 2.15 × 10⁻¹³ for a compact one-stage Hadamard construction, 2.92 × 10⁻¹² for a seeded random U(4) compiled into 28 stages, and 1.70 × 10⁻¹² for a four-mode Fourier transform using 17 stages. The generic QR compiler uses three stages for the Hadamard target; the one-stage example uses an explicitly selected reflection. These values are numerical agreement measures within the ideal model, not measured hardware fidelities. During release preparation, all three supplied numerical programs completed successfully, and all 13 compared result files matched the supplied baselines byte-for-byte. A separately implemented compiler checked 60 random unitaries across dimensions 1, 2, 3, 6, and 10, with maximum reconstruction residual approximately 4.65 × 10⁻¹⁵. These are internal reproducibility checks. The accompanying proofs establish the arbitrary-dimensional ideal-model statements. A standard unitary dilation extends the construction to contractions on selected outputs using zero-initialized auxiliary oscillators. General finite complex matrices are represented up to a known scale. Input preparation, physical gain, auxiliary resets, calibration, and readout remain explicit resources. Intended research uses include coherent mode mixing, routing, Fourier transforms, basis conversion, analog linear algebra, and programmable oscillator networks. The archive includes the manuscript, detailed proofs, mechanical derivations, original scientific figures, compiler and verification code, runnable examples, ten configured simulation tables containing 550 records, data documentation, source provenance, checksums, citation metadata, and a Windows Hugging Face uploader. The earlier restricted holonomy analysis is preserved separately. The contribution is an explicit controlled-model construction and its analytical and computational certificates. Ermakov inverse engineering, Householder synthesis, unitary decomposition, oscillator-chain controllability, and unitary dilation are established antecedents and are credited. Priority of this particular combined construction remains unestablished. The release establishes classical linear-mode universality under its stated assumptions; it does not establish universal quantum computation, practical computational advantage, or a universal protected holonomy in a freely folding origami sheet. Research attribution: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
This preprint studies a cycle-reflection operation on a supplied six-qubit realization of paired trigram configurations. A four-matched-pulse identity and a minimum within its stated serial grammar are separated from an ordinary-exchange endpoint construction. The raw full-source reflection and its full domain-safe alternative are different tasks: the latter is excluded by a collective-symmetry obstruction in the admitted source-only exchange model. Strict preservation of the addressable sector throughout the path is also excluded for the full task, while finite-action lower bounds quantify peak no-address population and time-integrated exposure. The paper presents an explicit low-excursion balanced word and a conditional joint clock–amplitude error certificate. Root settings and trajectory inputs remain tied to their supplied computer-assisted certificates. The second internal review reproduced all ten advertised numerical routes and twenty-four outputs using the original programs; this is same-program reproducibility, not independent implementation or external peer review. A separate Lean supplement covers the finite raw/safe operator algebra and the obstruction for arbitrary finite words of the declared algebraic factors. It does not formalize the numerical roots, matrix exponentials, continuous-time dynamics or resource theorems. No global pulse optimum, measured hardware feasibility, general noise tolerance, or automatic connection to the companion recovery code is claimed.
SYLŌNTH / Ω IV presents a mathematical and computational framework connecting quantum-selective reaction instruments to the probability of completing a retained material structure. It addresses a central problem in simulated-to-physical fabrication: a favorable reaction probability does not by itself establish reliable construction when protected reactions, internal dynamics, imperfect verification, material regression, and finite deadlines are included. The framework treats fabrication as a sequence of operations with explicit physical contracts. Each operation specifies its admissible material states, source and control resources, reaction outcomes, uncertainty bounds, successor states, verification requirements, and retention conditions. Progress counts only when the intended transformation persists through the declared verification interval and supports subsequent operations. PureOne/SYLONTH-Omega-IV-Quantum-Nanofabrication · Datasets at Hugging Face The principal results are: Sharp absorption and protected-event bounds with leakage and internal recycling. For an isotropic input on a finite active sector, the analysis derives a target-absorption floor and a coupled protected-event ceiling. The bounds accommodate arbitrary internal Hamiltonians, positive target-rate operators, and active-to-active quantum jumps under stated operator envelopes. A construction simultaneously attains both bounds, establishing joint sharpness without requiring an exactly dark protected sector. A classical comparison with explicit photon-tail coverage. Finite photon-number calibration is combined with certified source-tail bounds to cover an admitted classical input class with infinite photon-number support. Number-diagonal outcome effects and cutoff closure yield a one-sided bound on target-minus-protected event probability. Adaptive completion guarantees with repairs and deadlines. Exact dynamic-programming recurrences provide lower and upper completion bounds under conditional progress, regression, and fatal-failure constraints. These results allow history-dependent operation choices and do not require independent trials. Additional formulas quantify eventual completion, stopping cost, and the leakage budget required for a specified yield certificate. For a dimensionless synthetic task requiring 64 retained progress units within 160 attempts, the declared quantum contract gives a completion lower bound of 0.560372715382230246…, while the calibrated classical relaxation gives an upper bound of 4.271835407015029900… × 10⁻¹⁴. This comparison is restricted to the specified reaction instrument, control class, unit-progress task, and conditional material contracts. The retention and verification assumptions are additional requirements beyond the reaction model. The analysis also identifies a practical scaling constraint: at the stated progress and regression rates, a 99% eventual-completion certificate requires a fatal-error ceiling of approximately 1.1229 × 10⁻⁴ per attempt. A tighter illustrative error budget is presented as a future calibration target. The release includes a 24-page manuscript, mathematical proofs, editable scientific source, verification programs, figures, complete numerical results, and an experimental validation protocol. Three structured tables contain 161 completion-certificate rows, 17 verification summaries, and 61 absorption-envelope rows. Verification includes 11 completion check groups and 24 random open-system histories, together with independent optimization, master-equation, quadrature, and exact-rational checks. This is a theoretical and computational research release. Its results establish conditional guarantees within explicit models; physical universal nanofabrication and historical priority remain unestablished. Made by Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al. (2023), we present and implement a detailed quantum circuit construction for simulating one-dimensional spring–mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform mass–spring setting, our circuit construction requires a gate complexity ofO(log22Nlog2(1/ε)), where N is the number of oscillators and$\varepsilon isthetargetaccuracyoftheapproximation.Formoregeneral,heterogeneousmass–springsystems,thetotalgatecomplexityis\mathcal {O}\bigl (N\log _{2} N\,\log _{2}(1/\varepsilon )\bigr ).Bothsettingsrequire\mathcal {O}(\log _{2} N)$qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.
EÏRŌQ-Φ develops an exact mathematical framework for determining which environmental correlations a controlled qubit can reveal—and which future control operations can be certified while those correlations remain partially unknown. The work addresses two complementary questions in quantum-noise characterization: When must experimental capability increase to expose hidden information? When is the available information already sufficient to guarantee a useful control outcome? Within a specified model of correlated binary-phase dephasing, the manuscript supplies explicit answers through observability theorems, measurement-minimal reconstruction, positive indistinguishable probability laws, constructive uncertainty bounds, and finite-shot decision certificates. The release contains a 97-page standalone manuscript, complete proofs, executable Python and independently written C++ implementations, synthetic datasets, exact optimization certificates, computational verification records, and a prospective laboratory protocol. Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. 1. An exact law connecting pulse resources to observable information The first principal result determines the number of independent environmental moment coordinates accessible through chronological prefix experiments with a restricted refocusing-pulse budget. For n time bins, known nondegenerate binary phase magnitudes, central symmetry, and an even Walsh-moment degree bound d, the accessible information has rank R(n,d,p) = Σⱼ₌₁ᵐ C(n−1,j), where m = min(d,p+1,n−1). Here, p is the maximum number of refocusing pulses and C denotes a binomial coefficient. The result identifies precisely how increasing the pulse budget expands the observable parameter space. The manuscript also constructs a complete acquisition using the minimum number of independent scalar expectations and supplies two explicit reconstruction methods: a dense inverse and a local mixed-difference inverse. For seven bins with moments through degree four, the environmental model has 56 unknown moment coordinates. Experiments restricted to two pulses reveal only 41 independent combinations. Allowing a third pulse reveals all 56. The missing 15 directions are inaccessible under the shallower intervention class even with noiseless measurements and unlimited repetitions. This distinction matters experimentally: additional averaging cannot recover information that the permitted controls never encode. 2. Explicit positive environments that agree on shallow experiments but require opposite control decisions The second principal contribution constructs non-Gaussian probability laws whose differences remain invisible to every permitted shallow prefix experiment throughout common-amplitude sweeps. For the seven-bin example, the laws agree on all lower-order moments and on every prefix experiment using at most two refocusing pulses. A particular three-pulse intervention separates them. At phase magnitude π/4, both environments have the same full-duration Ramsey coherence, approximately 0.088388. The revealing control produces coherence approximately 0.176777 in one environment and zero in the other. Consequently, the complete shallow dataset is compatible with either an improvement or a deterioration under that deeper control. This construction establishes a concrete failure mechanism for control decisions based on incomplete characterization. Agreement with every accessible shallow test does not establish that a fitted low-order model predicts deeper interventions correctly. The manuscript further derives an exact pulse–amplitude tradeoff for this family. Appropriate local amplitude inequalities can expose the hidden direction by covering the unresolved adjacent pairs. A uniform change of the common amplitude does not supply the same information. 3. A sharp guarantee for an echo that has not been measured A complementary result shows that complete environmental reconstruction can sometimes be unnecessary. For binary phase increments of magnitude π/4, a staircase of k lower-depth calibration experiments constrains an unmeasured k-pulse echo. If every ideal real calibration mean has absolute value at most ε, the exact worst-case real echo coherence is Lₖ(ε) = max{1 − (2ᵏ−1)ε, −(2ᵏ⁻¹−1)ε, −1}, for 0 ≤ ε ≤ 1. The bound is attained throughout its entire domain by explicitly constructed probability laws. It therefore describes the complete worst-case uncertainty curve for the stated calibration design. This result requires neither Gaussian noise, independence between time bins, central symmetry, nor a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. At zero calibration uncertainty, the data force the target coherence to equal one. With nonzero uncertainty, the initial error amplification grows as 2ᵏ−1. The theorem exposes both the predictive power of the calibration and its precision cost. 4. Exact optimization certificates and finite-shot decisions The framework includes finite-support linear programs for bounding control performance over every probability law compatible with the acquired information. Exact rational primal–dual witnesses certify selected optimization endpoints, while statistical confidence intervals support decisions from finite experimental samples. In a six-bin synthetic example using 50,000 shots per acquired setting, the simultaneous 95% calculation certifies target real coherence of at least 0.89094 and improvement over the Ramsey baseline of at least 0.87450. Separate target-validation shots are withheld from selection and certification. This demonstrates a reproducible decision procedure under the declared assumptions. Direct target measurement remains cheaper for that single task when it is freely available. Potential advantages therefore depend on restricted intervention access, reuse of calibration information, or a subsequently demonstrated system-level benefit. Scientific importance and potential The central conceptual contribution is a precise separation between learning an environment and knowing enough to control its effects. Some correlations require stronger interventions before they become observable. Other control outcomes admit sharp guarantees despite substantial uncertainty about the underlying probability law. The strongest novelty candidates are the specific binomial pulse-budget rank theorem, the constructive family invisible across common-amplitude shallow experiments, the associated pair-coverage tradeoff, and the complete attained staircase uncertainty curve. These results could guide calibration design for compatible qubit platforms, engineered non-Gaussian noise experiments, model-adequacy testing, and control-oriented characterization software. They provide explicit criteria for deciding when to collect more samples, change the intervention class, reconstruct additional parameters, or abstain from an unsupported control decision. Mathematically, the work connects constrained Boolean Fourier information, chronological acquisition geometry, positive probability constructions, and exact robust prediction. Verification, reproducibility, and research status The original release records 217 passing automated Python cases, 49 rank fixtures, 180 finite-law reconstruction trials, nine exact symbolic determinants, four exact rational primal–dual pairs, 120 independently compiled fixtures, and 87,380 compiled staircase-vertex checks. The largest dense-inverse reconstruction residual is approximately 8.45 × 10⁻¹⁵. Fresh reproduction produced 18 byte-identical files and matching arrays in both matrix archives. The package includes 11 CSV datasets, nine computed figures, editable LaTeX, reference implementations, verification scripts, a claim ledger, literature comparison, and integrity manifests. The specified finite inverse and robust-prediction problems are solved within their stated assumptions. Historical priority for the precise results remains unconfirmed. Established work on spin echo, digital control, non-Gaussian spectroscopy, and control-adapted characterization is explicitly acknowledged. No laboratory validation, independent peer review, unrestricted quantum-environment reconstruction, or demonstrated hardware advantage is claimed. Inspired by Stellar Blade mechanics and story.
Abstract (Initially published 08/16/25 as 2nd article in Engineering Quantum Tunneling in Arbitrary Size Matrices) In this publication, a general theory of finite-state, non-probabilistic quantum mechanics is specified, in which the physical state space is finite at every realizable layer, quantum dynamics is generated by finite Hermitian operators, and spectral realization weights arise from exact finite decompositions rather than from a primitive continuum of physical states. The conventional structures L^2(\{R}^3), -i\hbar\nabla, and continuum integration are replaced, respectively, by finite-dimensional complex state spaces, finite momentum operators, and finite sums over physical channels. Within this construction, position and momentum are assigned to common orthogonal channel projectors and are therefore simultaneously determined on each realized finite channel. The central mathematical result is a Conservative Finite Mesh–Gauge Correspondence connecting finite-volume mechanics on arbitrary polyhedral partitions with finite quantum gauge dynamics. For every finite reconstruction scheme, A_s=B,\mathrm{diag}(q),R_s, so that conservation follows structurally from the incidence identity \mathbf 1^{T}B=0 \quad\Longrightarrow\quad \mathbf 1^{T}A_s=0. This remains exact for extended higher-order stencils, including third-order QUICK. The unique decomposition A_s=D+S+K, with S^T=S and K^T=-K, yields the Hermitian quantum generator H_A=D+S+iK, thereby retaining both symmetric coupling and directed transport, with the latter encoded as gauge phase rather than discarded by symmetrization. The construction extends blockwise to arbitrary finite internal dimension d, with links U_{ab}\in U(d), local gauge covariance, antisymmetric realization currents, holonomy conjugacy, and Wilson invariants. Deterministic verification is supplied on triangular, quadrilateral, and mixed polygonal meshes for upwind, central, and QUICK schemes, with exact structural divergence and Hermiticity, and with non-Abelian gauge tests spanning U(1) through U(8). The resulting framework replaces continuum quantum kinematics with finite physical state space, finite operators, finite relational transport, and finite spectral resolution, while preserving exact unitary evolution, conservation, gauge structure, and finite quantum dynamics on arbitrary coupling geometries. Note: The goal has been to better equip science to tackle problems previously thought to require specific hardware. I believe this has accomplished that goal, in the larger sense. I am optimistic to see the software that will be built from such beginning points. May quantum logic develop, earnestly, and may approximation give way. Dr. Charles Tibedo
The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N2, where N is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is H=p2/2+λq4/4. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%.