We study finite-size adiabatic state preparation on Boolean hypercubes using graph-local drivers built from sector/path coordinates related to monotone Gray-code representatives. The construction is not presented as a new all-$n$ Gray-code existence theorem; rather, it provides finite representatives, explicitly checked through the cases used in the numerical experiments, for testing problem-dependent graph-local drivers. For ordinary diagonal-cost transverse-field annealing, the ordering does not yield a robust advantage, and we include this negative result as a baseline. For non-diagonal target Hamiltonians whose geometry is expressed in the same sector/path coordinates, hybrid drivers combining sector, path-window, and small transverse-field components can substantially improve the final ground-state fidelity in centered barrier instances. Reproduction runs from the accompanying code confirm a representative centered original-window barrier value of approximately \(0.9799\) for the fixed-control hybrid parameters \((w,α,ε)=(8,0.50,0.15)\), while also showing that the improvement is target-class dependent. Randomized and ablation controls indicate that the dominant contribution is the sector-preserving skeleton, with strict one-bit completion acting as a secondary refinement. We provide code, finite certificates, CSV files, validation logs, and reproduction scripts to make the finite-size claims traceable.
We study finite-size adiabatic state preparation on Boolean hypercubes using graph-local drivers built from sector/path coordinates related to monotone Gray-code representatives. The construction is not presented as a new all-n Gray-code existence theorem; rather, it provides finite representatives, explicitly checked through the cases used in the numerical experiments, for testing problem-dependent graph-local drivers. For ordinary diagonal-cost transverse-field annealing, the ordering does not yield a robust advantage, and we include this negative result as a baseline. For non-diagonal target Hamiltonians whose geometry is expressed in the same sector/path coordinates, hybrid drivers combining sector, path-window, and small transverse-field components can substantially improve the final ground-state fidelity in centered barrier instances. Reproduction runs from the accompanying code confirm a representative centered original-window barrier value of approximately (0.9799) for the fixed-control hybrid parameters ((w,α,ε)=(8,0.50,0.15)), while also showing that the improvement is target-class dependent. Randomized and ablation controls indicate that the dominant contribution is the sector-preserving skeleton, with strict one-bit completion acting as a secondary refinement. We provide code, finite certificates, CSV files, validation logs, and reproduction scripts to make the finite-size claims traceable.
We explicitly carry through to the Hubbard endpoint the product-state (Hartree-Fock-type) case that Waite et al. state is available within their framework. The closest route is their Theorem 7 with a subset of cardinality one. We give explicit constants, track each guiding spinor through the reduction, and provide a finite description of the resulting site-local Slater determinant. The target is a repulsive Fermi-Hubbard Hamiltonian on a full open rectangular square grid, with spatially uniform positive hopping and repulsion, local vector fields, and the entire fixed-number sector Nf = M. With inverse-polynomial squared overlap and normalized decision precision, the resulting promise problem is PromiseBQP-complete; hardness holds on the subclass with a unique sector ground state and an inverse-polynomial spectral gap. The guiding spinors are retained from their creation-time penalty ground states and rounded to normalized dyadic-pair descriptions. The proof includes explicit global perturbation bounds and preparation and energy-estimation costs. MSC2020: Primary 68Q12; Secondary 81P68, 81V70. The archive contains the complete LaTeX source, vector figures, numerical test data, build instructions, a manifest, and SHA-256 digests. AI-use disclosure: The natural-language proofs, diagnostic code, partial formal proofs, and manuscript were generated using GPT-6 Astra and Claude Fable 5.1 under the author's direction on the research question and acceptance criteria. AI review comprised four independent audits and one heterogeneous adversarial audit across the two model families; completed human mathematical proof reviews number zero. Formal checking is peripheral: five source-side and five finite-endpoint declarations have Lean kernel proofs, Schrieffer-Wolff composition is zero of eight original targets, and one overgeneral transfer signature has a singleton counterexample. The full uniform completeness theorem is not formally proved. The author supplied the presentation revisions and remains responsible for final acceptance of the work. Appendix C distinguishes the earlier high-precision numerical example from the new certified Arb interval calculation.
This record freezes the analytical and computer-assisted successor protocol to the HLV Golden Gauge Forge finite-family study. Previous work established an exact golden 6→3 cut-and-project carrier fingerprint, a nested bounded-locality exhaustion, maximum vertex degree 12, maximum edge–face incidence 10, volume-independent positive HLV coefficient bounds, and a strictly positive physical-sector gap for every fixed finite connected complex satisfying the declared hypotheses. A preregistered exact Z3 finite-family study successfully constructed the controlled carrier sequence but did not support the frozen positive volume-uniform-gap candidate. This negative result motivates a locality-based analytical programme rather than further unrestricted enlargement of finite matrices. The primary system is a fixed-lattice Z3 gauge Hamiltonian on a declared class of finite oriented two-complexes with uniformly bounded vertex degree, face incidence, interaction range, and positive electric and magnetic coefficients. The protocol freezes a sequential theorem programme consisting of: an explicit volume-uniform gap for the electric product-state backbone; derivation of system-size-independent locality and interaction constants; a strong-coupling perturbative stability theorem with a volume-independent spectral gap; exponential clustering of spatially separated gauge-invariant observables; a van-Hove/Følner theorem for the canonical HLV origin-connected exhaustion; and, only if these gates are closed, construction of a fixed-cutoff thermodynamic-limit state. The computer-assisted component freezes an eight-radius geometry audit, graph-boundary layers of thickness one to three, radius-two rooted patch-frequency diagnostics, exact local norm and interaction-overlap calculations, a global exact-diagonalization cap at cycle rank 10, independent numerical verification, and complete deterministic replay. HLV carrier specificity is tested against matched controls consisting of uniform weights on the same complex, 50 exact histogram-preserving type-stratified weight permutations, 12 deterministic internal-window phason shifts, and three rational Fibonacci-slope periodic approximants. The protocol explicitly forbids replacing a local perturbation estimate by the global magnetic operator norm, which grows with system size and cannot establish a bulk spectral gap. It also keeps the thermodynamic limit, representation-cutoff removal, and continuum lattice-spacing limit strictly separate. A positive result under this protocol would establish, at most, a fixed-Z3, fixed-lattice strong-coupling theorem for a class of aperiodic bounded-locality gauge complexes and a separate finite test of whether the HLV geometric assignment produces effects not reproduced by matched controls. It would not establish full SU(3) dynamics, representation-cutoff removal, a continuum Yang–Mills theory, or the Yang–Mills existence and mass-gap problem.
This preprint presents a finite-dimensional quantum-simulation framework for selected TEBAC 9D/9D+ topological phases, with emphasis on synthetic-dimensional encoding, spectral-gap diagnostics, portal-channel observables, and Qiskit-ready Hamiltonian construction. The manuscript formulates a finite-lattice simulation target for the internal TEBAC sector and encodes the model through Pauli-sum Hamiltonians suitable for NumPy exact diagonalization, Qiskit Aer simulation, and future IBM Runtime execution. The central simulation Hamiltonian is organized schematically asHsim=HDirac(N)+Hgauge(N)+Htop(N)+Hchannel(N). The paper develops a concrete two-qubit benchmark v0.1 and a four-qubit finite-lattice sector v0.2. For the four-qubit sector, the manuscript gives an explicit Pauli decomposition, exact spectrum, spectral-gap diagnostic, topological proxy, link-sector markers, fidelity diagnostic, and determinant diagnostic. A key contribution of this version is the transition from a hardware-ready protocol to an executed local emulator report. The v0.2 four-qubit sector is evaluated through NumPy exact computation and local Qiskit Aer execution, including ideal sampling, readout-noisy sampling, and readout-mitigation recovery. For the topological proxyQ4=⟨ZZZZ⟩,the reported values are:Q4exact=0.986000,Q4Aer,ideal=0.986572±0.001805,Q4Aer,noisy=0.775391±0.006977,andQ4mitigated=0.993137±0.008936. The manuscript also includes a Qiskit Aer / IBM Runtime execution skeleton, an error-mitigation roadmap, and a validation ledger. The included code package provides the Python scripts used for the NumPy and local Qiskit Aer runs. This preprint should be read as a companion algorithmic module to the TEBAC portal-channel preprint. The portal-channel preprint formulates the geometric and quantum-informational target model; the present manuscript formulates and partially executes a finite-dimensional quantum-simulation protocol for selected spectral, topological, and channel-level diagnostics. The RH and BSD modules of the broader TEBAC 9D/9D+ programme are discussed only methodologically. They are not used as physical inputs for quantum simulation. Their role is to motivate the common operator--spectrum--trace/determinant--comparison architecture used across the TEBAC programme. This version does not claim a laboratory realization of a physical TEBAC portal, a physical 9D/13D geometry on quantum hardware, or a remote IBM/Google hardware backend result. It reports a local NumPy and Qiskit Aer execution and provides a hardware-ready path toward future cloud-backend experiments.
Quantum error-correcting codes with translation symmetry and local checks have been studied extensively, leading to a wide variety of fracton codes in three or more dimensions which lack a complete unifying picture. Recently, the study of translation-invariant codes with long-range checks has revealed impressive performance for small fixed-size instances in two dimensions. Here, we provide a unifying picture for a large family of translation-invariant codes, both local and long-range, that captures many fracton codes and all Abelian Two-Block Group Algebra (A2BGA) codes, including the Bivariate Bicycle (BB) codes. The balanced product structure of A2BGA codes leads to a local parent code that is a hypergraph product fracton model in a higher dimension. Different compactifications of a parent code produce a wide variety of descendant codes which provides a unifying picture for their properties. In particular, all BB codes with the same check weight are derived from a single parent hypergraph product fracton model. This construction allows us to extend Wang and Pryadko's code-parameter bounds for Generalized Bicycle codes to A2BGA codes. We conjecture that the transversal gates and energy barriers of the translation-invariant descendant codes are limited by those of their parent fracton models.
We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians H 0 defined by ⟦ N , K , d ⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance d ≥ c log ( N ) , implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by H 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the O ( 1 ) smallest eigenvalues of H 0 . The band originating from the smallest eigenvalue has 2 K states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth δ = C N e − Θ ( d ) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.
One of the distinct features of quantum mechanics is that the probability amplitude can have both positive and negative signs, which has no classical counterpart as the classical probability must be positive. Consequently, one possible way to achieve quantum speedup is to explicitly harness this feature. Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes (with respect to the computational basis), a non-stoquastic Hamiltonian can be eventually stoquastic or properly non-stoquastic when its ground state has both positive and negative amplitudes. In this paper, we describe that, for some hard instances which are characterized by the presence of an anti-crossing (AC) in a transverse-field quantum annealing (QA) algorithm, how to design an appropriate XX-driver graph (without knowing the prior problem structure) with an appropriate XX-coupler strength such that the resulting non-stoquastic QA algorithm is proper-non-stoquastic with two bridged anti-crossings (a double-AC) where the spectral gap between the first and second level is large enough such that the system can be operated diabatically in polynomial time. The speedup is exponential in the original AC-distance, which can be sub-exponential or exponential in the system size, over the stoquastic QA algorithm, and possibly the same order of speedup over the state-of-the-art classical algorithms in optimization. This work is developed based on the novel characterizations of a modified and generalized parametrization definition of an anti-crossing in the context of quantum optimization annealing introduced in [4].