Higher Categorical Coherence Breakdown as a Quantum Process: Measurement without Classicality and Its Dual Role in Quantum Computing Limitations and Algorithmic Advantages

Higher Categorical Coherence Breakdown as a Quantum Process: Measurement without Classicality and Its Dual Role in Quantum Computing Limitations and Algorithmic Advantages Andrei T. Patrascu1 1FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We introduce a framework in which higher categorical coherence breakdown (HCCB) is formulated as a fully quantum process, distinct from conventional decoherence models that appeal to a quantum-to-classical transition. In this approach, standard quantum mechanics emerges from the breakdown in coherence of triangle diagrams in a monoidal category of physical processes, while breakdown of higher coherence conditions—pentagon and hexagon diagrams—induces controlled departures from linearity, unitarity, and invertibility. These departures are captured by state-dependent Hamiltonian shifts, sector-selective non-invertible maps, and curvature terms on the purification bundle, all of which preserve quantum probabilistic structure within each context yet reshape the mapping between contexts. This structure resolves the measurement problem without invoking classicality and naturally produces correlations between sectors of higher categorical curvature. We analyse how such correlations both limit and enhance quantum computation. In gate-based architectures, HCCB can generate correlated, state-dependent error processes that challenge standard fault-tolerance thresholds, while offering potential advantages in dissipative state preparation and nonlinear gate shortcuts. In annealing-based systems, HCCB modifies the adiabatic path near small gaps, introducing directed irreversibility that can degrade reversible protocols but accelerate convergence in rugged landscapes. We present algorithmic templates— a gate-based constraint solver and an annealing-based biased sampler— that harness HCCB-induced probability flow as a computational resource. Numerical simulations illustrate both performance limits and attainable gains. Our results motivate the treatment of HCCB not solely as a noise source but as a programmable quantum control mechanism with implications for error modelling, compilation strategies, and hybrid quantum–classical algorithm design. INTRODUCTION Historical and Scientific Context Since the inception of quantum mechanics in the early 20th century [1–3], the nature of quantum measurement and the emergence of classical outcomes from quantum theory have been subjects of sustained debate [4–6]. The so-called measurement problem — the question of how and when definite outcomes arise from quantum superpositions — has motivated a wide range of interpretations and technical frameworks, including decoherence theory [7–9], hidden variable theories such as Bohmian mechanics [10, 11], objective collapse models [12, 13], and many-worlds approaches [14, 15]. While decoherence successfully explains the suppression of interference terms in reduced density matrices [16, 17], it does so by appealing to an effectively classical environment, leaving unresolved questions about the ontological status of the wavefunction and the physical realisation of a “collapse”. Hidden variable theories, in contrast, reproduce the Born rule via additional classical degrees of freedom, but at the price of introducing nonlocality and deviating from the fully probabilistic character of standard quantum mechanics. In parallel, category theory and higher category theory have become increasingly prominent in the mathematical formulation of quantum theory [18–20], offering diagrammatic calculi for processes, compositional structures for systems, and higher-dimensional generalisations that encode intricate algebraic and topological constraints [21, 22]. In particular, the study of coherence laws in monoidal and braided monoidal categories — embodied in commutative diagrams such as the triangle, pentagon, and hexagon — provides a structural backbone for how subsystems combine and evolve. Higher Categorical Coherence and Standard Quantum Mechanics In the monoidal category of finite-dimensional Hilbert spaces with the standard tensor product ⊗, the triangle coherence diagram expresses the natural isomorphism between (A⊗I)⊗Band A⊗B, where Iis the monoidal 2 unit. This coherence condition ensures the consistency of associators and unitors, and, when interpreted physically, underpins the linear and unitary structure of standard quantum mechanics. The breakdown of coherence of the triangle diagram leads directly to the canonical commutation relations of quantum mechanics. In the operator algebra setting, the preservation of coherence implies that the product of exponentials eXeYsatisfies the Baker–Campbell–Hausdorff formula without additional defect terms beyond the standard nested commutators [23]: eXeY= exp X+Y+1 2[X, Y ] + 1 12[X, [X, Y ]] −1 12[Y, [X, Y ]] + ···.(1) Here, [X, Y ] is the usual commutator, and higher nested commutators arise from the purely algebraic structure enforced by coherent associativity and unitarity. Higher Coherence Diagrams and Breakdown Beyond the triangle diagram, higher categorical structures impose more complex coherence laws: •The pentagon diagram encodes the associativity coherence for fourfold tensor products. Its commutativity ensures that all ways of rebracketing a fourfold tensor yield the same morphism. •The hexagon diagram arises in braided monoidal categories, enforcing compatibility between the braiding and associativity. Higher Categorical Coherence Breakdown (HCCB) occurs when these higher diagrams fail to commute exactly. We quantify this failure via: 1. A coherence defect Φ (e.g., an operator-valued 3-cocycle) representing the phase and amplitude mismatch along different paths in the diagram. 2. A curvature Fon the bundle of purifications (Uhlmann connection), capturing geometric obstruction. 3. A possible attenuation factor η > 0 rendering certain morphisms non-invertible. Physically, such breakdown leads to: Φ⇒state-dependent Hamiltonian shifts (nonlinearity),(2) η⇒sector-selective irreversibility (non-unitarity),(3) F⇒gauge curvature affecting parallel transport in state space.(4) In the BCH expansion, these defects appear as extra terms modifying the commutators and nested commutators, with coefficients depending on the state and context. Measurement Problem without Classicality In our framework, the measurement problem is addressed without invoking a quantum-to-classical transition or hidden variables. The key idea is: •Within a fixed categorical context (fixed bracketing, fixed morphism set), the Born rule and unitary evolution hold exactly. •When the physical process changes context — for example, when different subsystems become the “active” interacting pair — the mapping between contexts is governed by an associator that may carry the defects (Φ, F, η). •These defects induce quantum probability reweighting and sector correlations, but do not collapse the state into a classical mixture. Thus, outcomes remain quantum-probabilistic, with structured correlations between sectors defined by the higher categorical curvature. This approach is fundamentally distinct from Bohmian mechanics or other hidden variable models: it neither supplements the wavefunction with classical degrees of freedom nor imposes deterministic trajectories. 3 Implications and Scope Because HCCB is itself a quantum process, it can be: 1. A limiting factor for certain quantum computational architectures, introducing correlated, state-dependent errors not captured by standard noise models. 2. A computational resource when harnessed deliberately, enabling sector-selective dissipation, biasing of probability flows, and nonlinear state steering. In the remainder of this work, we formalise these ideas, quantify the onset of HCCB effects in both gate-based and annealing-based quantum computing, and present algorithmic templates that exploit the unique structure of HCCB. FROM TRIANGLE COHERENCE TO COMMUTATORS, AND THE EFFECT OF HIGHER COHERENCE BREAKDOWN Triangle Coherence in Monoidal Categories Let Cbe a (strict) monoidal category modelling physical processes, with tensor product ⊗, unit object I, associator αA,B,C : (A⊗B)⊗C→A⊗(B⊗C), and left/right unitors λA:I⊗A→A,ρA:A⊗I→A. The triangle coherence law is the requirement that, for all A, B, αA,I,B ◦(ρA⊗idB) = idA⊗λB.(5) Diagrammatically, (5) is expressed as: (A⊗I)⊗B A ⊗(I⊗B) A⊗B αA,I,B ρA⊗idBidA⊗λB Physically, the triangle law asserts that appending an “idle” system Ito either side of a composite and rebracketing does not change the composite’s physical effect. This structural requirement has direct implications for the algebra of observables. Triangle Coherence and Quantum Commutators In the category of finite-dimensional Hilbert spaces (Hilb,⊗, I) with I=C, the triangle diagram holds strictly: αA,I,B = idA⊗B, λA=ρA= idA. Dynamics are given by unitary morphisms Ut=e−itH for self-adjoint generators H(observables). Composition of time evolutions UX=eX,UY=eYis associative and unitary, and the triangle law ensures there is no defect term in the rebracketing of tensor products involving the unit object. However, quantum mechanics itself is the first level breakdown of categorical coherence at the triangle level. We therefore allow for a defect at the level of the triangle diagram Ω∆but constrain it such that •The defect is pure phase (an element of U(1)), acting multiplicatively, and •It depends only on the order/composition of the morphisms, not on the state Mathematically, this is equivalent to having a 2-cocycle in the monoidal structure valued in U(1). The restricted phase defect changes the Baker-Campbell-Hausdorff expansion in exactly the way that produces the canonical commutator [X, Y ] and the standard Heisenberg algebra but does not introduce any amplitude damping or norm change. 4 This structural form guarantees that the product eXeYof exponentials of generators satisfies the Baker–Campbell–Hausdorff (BCH) expansion with the standard commutator [X, Y ]: eXeY= expX+Y+1 2[X, Y ] + 1 12[X, [X, Y ]] −1 12[Y, [X, Y ]] + ···,(6) where [X, Y ] = XY −Y X is bilinear, antisymmetric, and satisfies the Jacobi identity. The triangle diagram ensures that the associator αacts trivially on products of exponentials, so that (6) contains no stateor context-dependent modifications. Higher Coherence Laws: Pentagon and Hexagon For tensor products of more than three objects, higher coherence laws are needed: •The pentagon coherence law (Mac Lane’s pentagon) asserts that, for any A, B, C, D, αA,B,C⊗D◦αA⊗B,C,D = (idA⊗αB,C,D)◦αA,B⊗C,D ◦(αA,B,C ⊗idD).(7) This ensures the unambiguous reassociation of fourfold tensor products. •In braided monoidal categories, the hexagon coherence laws ensure compatibility between braiding βA,B and associativity, with two hexagon diagrams commuting for all A, B, C. When (7) or the hexagon laws fail to hold exactly, we have a higher categorical coherence breakdown (HCCB). Coherence Defects and Modified Commutators Let ΩA,B,C,D be the pentagon defect: ΩA,B,C,D =αA,B,C⊗D◦αA⊗B,C,D ◦[(idA⊗αB,C,D)◦αA,B⊗C,D ◦(αA,B,C ⊗idD)]−1.(8) In the coherent case, Ω = id. In the presence of HCCB, Ω = expiΦ−η 2, with: •Φ: operator-valued phase (coherence defect, e.g., a 3-cocycle); •η≥0: attenuation factor rendering certain morphisms non-invertible. Physically, inserting Ω into the rebracketing path for exponentials eXeYeZproduces extra terms in the BCH expansion: eXeYeZ= expX+Y+Z+1 2[X, Y ] + 1 2[X+Y, Z] + ····Ω = expX+Y+Z+1 2[X, Y ] + 1 2[X+Y, Z] + ···+δΦ,η(X, Y, Z),(9) where δΦ,η(X, Y, Z) contains state-dependent contributions from Φ and non-unitary logarithmic terms from η. 5 Physical Interpretation •In the coherent (triangle breakdown + pentagon coherence + hexagon commute) case, the associator acts trivially and the algebra of generators closes under the usual commutator [ ·,·]. •In the HCCB case, higher associator defects act like additional generators that depend on the state and context, yielding nonlinear Hamiltonian terms K(ρ) and dissipators that act only on certain sectors. •The Born rule still holds within each fixed context, but mapping between contexts is altered by Ω 6= id, producing quantum correlations between sectors of nonzero curvature. We can model the effect on dynamics with a GKSL-type generator: ˙ρ=−i[H+KΦ,F (ρ), ρ] + X j γj(ρ)Lj(ρ)ρL† j(ρ)−1 2{L† j(ρ)Lj(ρ), ρ},(10) where KΦ,F (ρ) encodes the Φ-induced nonlinearity and Ljproject onto sectors where ηinduces non-unitarity. Well-Posedness and Non-Signalling A natural concern when introducing scope-local corrections of the form ˙ρ=LGKSL(ρ) + KΦ,F (ρ),(11) is whether such modifications compromise operational consistency, in particular the non-signalling principle. We now establish sufficient conditions ensuring that Higher Categorical Coherence Breakdown (HCCB) does not enable superluminal signalling or external detectability outside the affected scope. Lemma 1 (Non-Signalling under Scope-Local HCCB).Let ρAB be a bipartite density matrix, with subsystem A subject to HCCB-corrected dynamics and subsystem Buntouched. Suppose that the HCCB-correction KΦ,F satisfies: 1. Complete positivity: KΦ,F extends to a CP map on A⊗HEfor any environment E. 2. Trace preservation: Tr[KΦ,F (ρA)] = 0 for all density matrices ρA. 3. Scope locality: KΦ,F acts trivially on B, i.e. KΦ,F ⊗IB. Then the reduced state of Bevolves trivially: d dt TrA[ρAB] = TrA[ ˙ρAB] = 0.(12) Hence, no choice of Φ, F, η, κ within the scope of Aenables signalling to B. Sketch. By assumption (iii), the dynamics is generated by LA GKSL +KA Φ,F on A, tensored with the identity on B. Taking a partial trace over Acommutes with the identity action on B, and trace preservation (ii) ensures no leakage of norm. Complete positivity (i) guarantees that the extension to any AB system is physically valid. Combining these conditions, the derivative of the reduced state ρBvanishes identically, establishing non-signalling. Remark. This result emphasizes that while HCCB introduces scope-dependent non-unitarity and effective nonlinearity on A, these effects cannot be used to transmit information outside A. Operationally, the breakdown remains local: external observers accessing only Bcannot distinguish whether HCCB has occurred. Having established consistency, it is natural to ask how HCCB relates to other established frameworks. To pre-empt confusion, we include here a focused comparison before turning to algorithmic consequences. 6 Φ (coherence defect) F (purification curvature) η (attenuation) κ (scope tuning) HCCB-augmented dynamics ˙ρ=LGKSL(ρ) + KΦ,F (ρ) (scope-local, CP & TP; see Non-Signalling Lemma) Gate-Based QC under HCCB Annealing under HCCB CNOT backreaction decay signatures Depth-sweep drift / clause-violation descent Outputs: performance trade-offs, drift bounds, stability windows Gap spectroscopy & window localisation Reverse-anneal asymmetry ⇒η(s) Biased sampling tilt ⇒κ(s) Interferometric loops ⇒Θ(s) Outputs: spectral protection, calibrated windows, feasibility FIG. 1: Overview of HCCB structure and diagnostics. (Φ, F, η, κ) feed the HCCB dynamics ˙ρ=LGKSL(ρ) + KΦ,F (ρ), which branch into Gate-Based QC under HCCB and Annealing under HCCB. Diagnostics used later: gates—CNOT backreaction decay and depth-sweep drift; annealing—gap spectroscopy/window localisation, reverse-anneal asymmetry (estimate η(s)), biased sampling tilt (estimate κ(s)), interferometric loops (estimate Θ(s)). RELATION TO EXISTING APPROACHES To situate Higher Categorical Coherence Breakdown (HCCB) within current theory, we contrast it with four neighbouring lines of work. Throughout we keep the notation introduced earlier: the defect data (Φ, F, η, κ), the curvature-corrected generator ˙ρ=LGKSL(ρ) + KΦ,F (ρ), and the scope locality and non-signalling properties established in the previous subsection. Categorical Quantum Mechanics (CQM) CQM models quantum processes in dagger-compact (or related) symmetric monoidal categories, with coherence ensured by strict (or coherent) associators and unitors. In that setting, string diagrams commute by design, and dynamics is typically represented abstractly via morphisms satisfying those coherence laws. How HCCB differs. •Coherence is not assumed but parameterised. HCCB treats departures from higher coherence as structured defects captured by (Φ, F, η). The associator “failure” is encoded as an operator-valued 3-cocycle (the coherence 7 defect Φ) together with a purification curvature F(Uhlmann geometry) and an attenuation ηthat governs sector-selective non-unitarity. •From diagrammatics to dynamics. Rather than remaining purely equational, HCCB transports the categorical data into state evolution via the curvature term KΦ,F (ρ), yielding testable corrections in a GKSL-type evolution. This provides quantitative, scope-local predictions (drift bounds, decay signatures) that can be calibrated from observables. •Safety contract. While CQM presumes exact functoriality, HCCB weakens it in a controlled, scope-local manner and proves non-signalling under the conditions stated previously, maintaining operational consistency. Non-Hermitian and Non-Unitary Circuits Non-Hermitian models replace Hby Heff or allow non-unitary gates to model loss, postselection, or PT-symmetric effects. In many such formulations, non-unitarity is global and, without care, may threaten complete positivity, trace preservation, or even no-signalling in composite settings. How HCCB differs. •Locality of the modification. HCCB imposes its correction as KΦ,F acting on a specified scope S(tensored with the identity outside S), with trace preservation and complete positivity enforced as part of the construction. This is precisely the hypothesis of the non-signalling lemma proved earlier. •Geometric origin of non-unitarity. The non-unitary contribution is not inserted ad hoc but arises from curvature/defect data (Φ, F, η) tied to higher coherence. This yields structured dependencies (e.g. on interferometric loops for Θ, sector projectors for η) and diagnostic predictions rather than free phenomenology. •Compatibility with GKSL. The correction KΦ,F integrates with a GKSL generator so that, at fixed scope, the evolution remains a valid quantum channel semigroup (under stated assumptions), distinguishing HCCB from generic non-Hermitian ans¨atze. Measurement-Induced Transitions (MIT) MIT studies random circuit ensembles where intermittent projective measurements drive entanglement phase transitions. The hallmark is a stochastic, measurement rate–controlled transition between volume-law and area-law entanglement phases. How HCCB differs. •Deterministic vs. stochastic mechanism. MIT relies on randomness in measurements; HCCB encodes a deterministic breakdown via curvature and associator defects. The parameters (Φ, F, η, κ) are continuous control/data objects, not measurement rates in an ensemble. •Scope and calibration. HCCB’s effects are confined to the active scope and are calibrated by concrete diagnostics (gap spectroscopy, reverse-anneal asymmetry for η(s), biased tilt for κ(s), loops for Θ(s)), whereas MIT characterises phases of random-circuit statistics. •Operational contract. The non-signalling and CP/TP conditions are explicit in HCCB; in MIT, signalling is not the central question (the model remains standard QM with stochastic projections). Resource Theories of Coherence (RTC) RTC quantifies coherence relative to a preferred basis via monotones and studies state transformations under incoherent (or related) operations. It treats coherence as a consumable resource, with performance limits captured by monotone bounds. How HCCB differs. 8 •Structural vs. quantitative viewpoint. RTC assigns numerical monotones to states under allowed maps; HCCB modifies the structure of the process theory itself through higher-coherence defects. The defect data (Φ, F, η) enter dynamics to create scope-local, controllable non-unitarity compatible with CP/TP. •Process-level semantics. While RTC is typically basis-dependent, HCCB’s curvature/associator language is basis-agnostic and compositional, mapping the categorical layer to master-equation corrections. •Predictive diagnostics. HCCB yields concrete, falsifiable signatures (e.g. CNOT backreaction decay laws, drift bounds, spectral window protection) beyond monotone inequalities. Summary. CQM assumes exact coherence; HCCB models its breakdown as geometric/categorical data and propagates it into valid open-system dynamics. Non-Hermitian circuits insert global non-unitarity; HCCB enforces scope locality with CP/TP and non-signalling. MIT describes stochastic entanglement phases; HCCB gives deterministic, calibratable corrections at fixed scope. RTC quantifies state resources; HCCB alters process structure in a controlled, testable way. Taken together, these differences justify treating HCCB as a distinct framework rather than a reformulation of existing approaches. GATE-BASED QUANTUM COMPUTING UNDER HCCB Gates, Contexts, and Associator Defects In the categorical formulation, an n-qubit register is an object Q1⊗Q2⊗···⊗Qn in the monoidal category C. A gate gacting on a subset (scope) S⊂ {1, . . . , n}is a morphism Ug:O i∈S Qi→O i∈S Qi tensored with identities on qubits outside S. When two consecutive gates act on overlapping but distinct scopes, e.g. S1={a, b}, S2={b, c}, the change of active scope requires a rebracketing of the total tensor product: (Qa⊗Qb)⊗Qc−→ Qa⊗(Qb⊗Qc). If the associator αa,b,c for this rebracketing carries a higher coherence defect, we write αa,b,c = expiΘabc −ηabc/2 2Pab 11 ⊗Ic, where: •Θabc is the phase defect from Φ, •ηabc ≥0 is the attenuation parameter, •Pab 11 =|11iabh11|projects onto the doubly-excited state of the (a, b) pair. Modified Gate Composition Law Let Ug1act on S1and Ug2on S2. In the presence of HCCB, the composition is modified: Ug2◦Ug17→ ΩS1→S2(ρ)Ug2Ug1,(13) where ΩS1→S2(ρ) is the defect map induced by the associator αand may depend on the current state ρ. For the simple αabove, we have: ΩS1→S2(ρ) = expiΘabc −ηabc/2 2Pab 11 ,(14) so that the action of Ug2is preceded by a conditional Z-rotation and damping on the |11iab sector. 9 GKSL Representation of the Defect Action The infinitesimal form of the defect action is given by a GKSL-type generator: ˙ρ=−i[Hg+KΦ,F (ρ), ρ] +γ(ρ)Pab 11 ρPab 11 −1 2{Pab 11 , ρ},(15) where: •Hgis the intended gate Hamiltonian on scope S, •KΦ,F (ρ) = κ 2hPab 11 iρPab 11 is the curvature-induced nonlinear term, •γ(ρ) = ηabc/(2τg) is the sector-selective damping rate, with τgthe gate duration. Leading-Order Expansion of Gate Error Let δgate be the allowable per-gate error budget (e.g. 10−3). Expanding (13) for small Θabc,ηabc, and κτg: EHCCB(ρ)≈UgρU† g+iΘabc 2[Pab 11 , ρ] +ηabc 2Pab 11 ρPab 11 −1 2{Pab 11 , ρ}−iτg[KΦ,F (ρ), ρ].(16) The average gate infidelity to leading order is bounded by: 1−Favg .|Θabc| 2+ηabc 2+κτg.(17) Onset condition: HCCB effects become non-negligible when the right-hand side of (17) approaches δgate. This provides the per-gate budget: |Θabc| 2≤δgate,ηabc 2≤δgate, κτg≤δgate.(18) Noise Model Characteristics The map EHCCB has several features: 1. Non-Pauli: the defect acts conditionally on multi-qubit projectors Pab 11 , not on single-qubit Pauli operators. 2. State-dependent: the nonlinear term KΦ,F (ρ) depends on hPab 11 iρ. 3. Non-Markovian: when ΩS1→S2depends on the history of context changes, the error channel has memory. These properties violate the i.i.d. error assumption used in standard threshold theorems, potentially lowering faulttolerance thresholds. Implications for Fault Tolerance Consider a code with distance dand threshold pth under i.i.d. Pauli noise. If HCCB induces correlated two-qubit errors with effective rate pHCCB, then the logical error rate scales as: pL∼C(d)pdd/2e HCCB, with pHCCB computed from (17). Since pHCCB here aggregates both coherent (unitary) and incoherent (dissipative) defects, its effect is generally more damaging than uncorrelated Pauli errors of the same marginal strength. 16 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 context-switch time (a.u.) 0.55 0.56 0.57 0.58 0.59 0.60 Population in |11 Sector-selective decay of |11 under HCCB (fixed initial p 11 = 0.6) moderate HCCB strong HCCB FIG. 2: CNOT backreaction: |11ipopulation under moderate vs. strong HCCB. (Generated from our simulation; see accompanying CSVs.) Annealing-Based HCCB Biased Sampler (HCCB–BQA) Setup and generators Consider the standard annealing path H(s) and let s∗denote a (possibly multiple) small-gap region. We shape the HCCB parameters to peak near s∗: η(s) = X c Γcgc(s), κ(s) = κ0g(s), with narrow windows gc(s) supported near s∗and R1 0g(s)ds = const. For each clause scope Sc, choose local isometries Lc,k(s) = g(c) k(s)EDv(c) k(s)analogous to the gate-based case. The annealing generator is ˙ρ=−iH(s) + Ks(ρ), ρ+X c,k γc(s)DLc,k(s)[ρ], Ks(ρ) = κ(s) 2X c wchPciρPc. Population master equation and tilted distribution In the instantaneous eigenbasis {|ψm(s)i}, under adiabatic separation and weak dissipation, the populations follow d dtpm(s) = X nWmn(s, ρ)pn−Wnm(s, ρ)pm, with Wmn(s, ρ)≈Pc,k γc(s)|hψm|Lc,k(s)|ψn|i|2. Under mild conditions (local detailed-balance-like relations inherited from Lc,k design and slow Ks), the late-time sampling distribution over bit strings xobeys πHCCB(x)∝exp−βE(x) + κeff G(x), G(x) = X c wc1{xviolates c},(37) where κeff ∝Rs≈s∗κ(s)ds. 17 Physical intuition. Near s∗the unitary adiabatic flow is fragile; HCCB adds a directed wind that drains population from locally “bad” sectors and tilts the exploration toward lower violation—without globally suppressing coherence. Onset & budgets Away from s∗we enforce the local adiabatic safety margin kKs(ρ)k ∆(s)≤ε, X c γc(s)∆(s), and globally constrain irreversibility via Z1 0 η(s)ds dt ds ≤2δanneal. Numerical diagnostics: ground-state probability Figures 3–5 show the effect of HCCB on a two-qubit Ising anneal: (1) moderate HCCB approximately tracks unitary behaviour; (2) strong, gap-aligned HCCB produces a measurable improvement in the instantaneous and final ground-state probabilities. 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 time (a.u.) 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 Instantaneous ground-state probability Annealing: HCCB-assisted (moderate) vs unitary HCCB (moderate) Unitary FIG. 3: QA: instantaneous ground-state probability vs. time for unitary vs. moderate HCCB. Risk–benefit summary •Benefit: improved success on rugged landscapes via targeted dissipation at bottlenecks and adaptive curvature tilt. •Risk: excessive or misaligned dissipation traps in wrong basins and degrades reverse-anneal capabilities; hence the need for spectral shaping and a global ηbudget. 18 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 time (a.u.) 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 Instantaneous ground-state probability Annealing: HCCB-assisted (strong) vs unitary HCCB (strong) Unitary FIG. 4: QA: instantaneous ground-state probability vs. time for unitary vs. strong HCCB. Unitary HCCB (moderate) HCCB (strong) 0.00 0.05 0.10 0.15 0.20 0.25 Final ground-state probability Annealing outcomes FIG. 5: Final ground-state probability (unitary vs. moderate/strong HCCB). Resource Trade-offs and Practical Notes No-signalling compliance. Both designs use K(ρ) and Lc,k that depend only on reduced states on clause scopes Sc, keeping the dynamics local in the categorical sense and respecting no-signalling. Fault-tolerance implications. GBQC: the effective per-gate defect budget is |Θ| 2+η 2+κτg≤δgate. QA: safe shaping obeys kKsk/∆(s)≤εaway from small gaps and Rη(s)ds ≤2δanneal. Calibration strategy. Begin with diagnostic two-qubit experiments (as in Fig. 2) to extract (Θ, η, κ), then scale to clause scopes and annealing windows using spectroscopy of ∆(s) to place dissipation only where it helps. 19 EXPERIMENTS & CALIBRATION: IDENTIFYING (Θ, η, κ)ON HARDWARE This section specifies experimental protocols to measure and calibrate the HCCB defect parameters: •Θ (phase defect from the coherence 3-cocycle Φ), dimensionless, •η(sector-selective attenuation; non-invertibility strength), dimensionless, •κ(curvature-induced nonlinearity strength), in s−1. We give circuit-level (gate-based) and annealing-level (continuous) procedures together with estimators and uncertainty analysis. All protocols respect no-signalling by operating on reduced scopes in which the defect acts. Gate-Based Calibration: Two-Scope Context Switch We consider a three-qubit register with scopes S1={A, B}and S2={B, C}to probe the defect that arises when rebracketing from (A⊗B)⊗Cto A⊗(B⊗C). Estimating the phase defect Θ(context-switch interferometry) Protocol. 1. Prepare A, B in a controllable superposition that loads the |11iAB sector with known probability p∈[0,1]: |ψAB(p)i=p1−p|00i+√p|11i, ρAB =|ψihψ|. Prepare Cin |+i= (|0i+|1i)/√2. 2. Apply a gate on S1(e.g. CNOTA→B), then switch context and apply a Ramsey sequence on S2with a short free-evolution window τgand a phase sweep ϕon C: R(C) xπ 2free τg −−−−→ R(C) ϕπ 2. 3. Measure hXCi(ϕ) to obtain Ramsey fringes for different p. Estimator. Under the HCCB model, the context switch applies a conditional phase Uassoc = exp+iΘ 2PAB 11 , PAB 11 =|11ih11|. Hence the Ramsey fringe is shifted by ∆φ(p) = Θ 2p. A linear fit of the phase shift vs. pyields b Θ = 2 d∆φ dp p∈[0,1]. Uncertainty: obtain σΘfrom the standard error of the linear fit; shot noise gives σ∆φ∼1/√Nshots. Estimating the attenuation η(sector-selective T1extraction) Protocol. 1. Prepare AB in |11i(or any state with p(0) ≡Tr[PAB 11 ρ(0)] ≈1), Cin |0i. 2. Perform the same context switch S1→S2and hold for a variable dwell time τ∈[0, τmax] with all other drives off. 3. Measure p(τ) = Tr PAB 11 ρ(τ)by local projective readout or weak measurement tomography. 20 Estimator. The HCCB dissipator on the violating sector produces d dtp(t) = −Γp(t)⇒p(τ) = e−Γτp(0), with Γ = η/(2τg) if the dissipation arises during a gate window of duration τg, or Γ = η/Thold for a static hold of duration Thold. A log-linear fit gives bη= 2 Γ τg,Γ = −d dτ ln p(τ). Control: verify that dephasing-only channels (jump PAB 11 ) do not change p(τ); a nonzero Γ indicates genuine sectordraining non-invertibility. Estimating the nonlinearity κ(population-dependent frequency pull) Protocol. 1. Prepare AB in |ψAB(p)ias above and Cas a spectrometer qubit in |+i. 2. Insert a context switch and a short free evolution τg; optionally weakly drive Cto perform spectroscopy around its transition frequency. 3. Sweep p∈ {0,0.25,0.5,0.75,1}and extract the effective Z-rotation rate ωC(p) from Ramsey fringes on C. Estimator. The nonlinear Hamiltonian shift K(ρ) = κ 2hPAB 11 iρPAB 11 produces an additional conditional phase δφ(p) = 1 2κ p τg(to leading order). A linear fit of δφ(p) vs. pyields bκ= 2 d δφ dp 1 τg . Cross-check: repeat with AB in classical mixture having the same pto distinguish genuine state dependence from classical post-selection effects. Hardware notes (superconducting vs. trapped ions). In superconducting devices, use calibrated DRAG pulses for the Ramsey probes and measure ωC(p) shifts at fixed detuning; in trapped ions, use local addressing to vary pand interrogate spectator modes. In both, isolate crosstalk by interleaving randomized compiling on idle qubits. Annealing Calibration: Spectral Shaping and Gap Windows For annealing systems, the parameters become functions of the path parameter s: Θ(s), η(s), κ(s). Calibration focuses on small-gap windows s∗where HCCB is most impactful. Gap spectroscopy and window localisation Estimate the instantaneous gap ∆(s) by: •Local spectroscopy: short-time quench-and-probe around fixed svalues; fit Rabi frequencies to infer ∆(s). •Landau-Zener scans: rapid passages with varying speed ˙s; fit excitation probability Pex ∼exp{−π∆2/2~v}to estimate ∆. Identify s∗as the minima of ∆(s). 21 Estimating η(s)from reverse-anneal asymmetry Protocol. 1. Perform forward anneals with a narrow engineered dissipation burst g(s) centred at s∗(small amplitude). 2. Perform matched reverse anneals through the same window. 3. Measure asymmetry in success probability ∆P=Pfwd −Prev as a function of burst area Aη=Rη(s)ds. Estimator. To leading order, irreversibility grows linearly with Aη: ∆P≈CηAη+O(A2 η), yielding b Aη= ∆P/Cηand thus an estimate of the local η(s) amplitude given the known burst shape. Estimating κ(s)from biased sampling tilt Protocol. 1. Implement anneals with weak curvature drive g(s) at s∗and collect bitstring samples xfrom the final distribution for several drive amplitudes. 2. Compute the empirical tilt statistic hG(x)iwith G(x) = Pcwc1{xviolates c}. Estimator. From the tilted Boltzmann model πHCCB(x)∝e−βE(x)+κeff G(x), κeff ∝Zs≈s∗ κ(s)ds, we obtain a linear response for small κeff : d dκeff hGi= Var0(G), where Var0is the variance under the un-tilted distribution. Thus bκeff =hGiHCCB −hGi0 Var0(G). Given the burst shape, deconvolve to obtain the local amplitude κ(s∗). Estimating Θ(s)via interferometric loops Execute a closed loop in sthat encircles the small-gap region and compare the geometric phase accumulated with and without the engineered associator drift. The phase difference ∆φloop scales with the line integral of ˙ Θ(s) along the loop: ∆φloop =1 2Zloop ˙ Θ(s)ds dt dt =1 2∆Θ. Use Ramsey interferometry at the loop end to read out ∆Θ. Global Budgets, Model Fit, and Uncertainties Budget checks. Verify that the fitted parameters obey the safe-operation budgets established earlier: GBQC: |Θ| 2+η 2+κτg≤δgate. QA: kKsk ∆(s)≤ε(away from s∗),Z1 0 η(s)ds dt ds ≤2δanneal. 22 Joint parameter inference. Many observables depend on combinations of (Θ, η, κ). Perform a joint maximumlikelihood (or Bayesian) fit to: •Ramsey phase shifts vs. prepared sector population p(sensitive to Θ and κ), •sector population decay curves p(τ) (sensitive to η), •reverse/forward asymmetry ∆Pin anneals (sensitive to η(s)), •sampling tilt metrics hGi(sensitive to κ(s)). Include cross-terms only if required by residual analysis. Uncertainty decomposition. Report σΘ, ση, σκwith: •shot noise (binomial or multinomial), •SPAM systematics (mitigate with randomized compiling / readout calibration), •drift (interleave calibration shots; use Allan variance). Hardware-specific notes. Superconducting platforms: guard against residual ZZ by echoing spectator qubits during probes; use tunable couplers to isolate scopes Sc. Trapped-ion platforms: exploit mode spectroscopy to localise s∗-like bottlenecks in effective spin models; use local addressing to vary pcleanly. CNOT Backreaction Decay Signatures As a concrete illustration of the sector-selective non-invertibility parameter ηand its state-dependence via β(cf. Sec. ), we simulate the decay of the |11isector during a context-switch window, for both moderate and strong HCCB parameter sets. The dissipator is applied on both qubits, draining |11iinto |10iand |01i, with a rate γ=γ0+β p11(t), where p11(t) = Tr PAB 11 ρ(t)is the instantaneous |11ipopulation. We initialise with a controllable load p11(0) ∈ {0.2,0.4,0.6,0.8}and track p11(t) during the context switch. Figures 6 and 7 show the population decay curves for moderate and strong HCCB settings. The strong setting yields visibly faster decay for all initial loads due to both a higher baseline γ0and a stronger state-dependent term β. From each decay curve, we fit an effective early-time decay rate Γeff by linear regression on ln p11(t) over the initial half of the window. The resulting rates are plotted in Fig. 8 as a function of p11(0). The approximately linear dependence matches the model γ=γ0+β p11 and provides a direct calibration of (γ0, β). Calibration use. In an experiment, this protocol directly yields: bγ0≈Γeff (p11 = 0), b β≈dΓeff dp11 , thereby separating the baseline attenuation from the state-dependent part. These parameters feed into the global budgets in Sec. and the safe-operation conditions in Eqs. (32)–(35). LIMITATIONS AND OPPORTUNITIES UNDER HCCB We collect sharp statements about how higher categorical coherence breakdown (HCCB) constrains quantum computation and, conversely, how its induced correlations/entanglement can be used as a resource. 23 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 context-switch time (a.u.) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Population in |11 Sector-selective decay of |11 (moderate HCCB) p 11(0) = 0.2 p 11(0) = 0.4 p 11(0) = 0.6 p 11(0) = 0.8 FIG. 6: Sector-selective decay of |11iduring a context switch for initial loads p11(0) ∈ {0.2,0.4,0.6,0.8}under moderate HCCB parameters (γ0, β, κ) = (0.02,0.35,0.6). Higher initial occupation drains faster due to the p11-dependent term in the dissipator. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 context-switch time (a.u.) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Population in |11 Sector-selective decay of |11 (strong HCCB) p 11(0) = 0.2 p 11(0) = 0.4 p 11(0) = 0.6 p 11(0) = 0.8 FIG. 7: Same as Fig. 6, but for strong HCCB parameters (γ0, β, κ) = (0.08,0.90,1.2). Both the baseline and the state-dependent term are larger, producing visibly faster decay for all initial loads. Notation and Sign Conventions Let (Θ, η, κ) denote, respectively, the phase defect (from the coherence 3-cocycle Φ), the sector-selective attenuation (non-invertibility), and the curvature-induced nonlinearity strength. For clause-structured problems we write Pcfor 24 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Initial p 11 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Effective decay rate eff (a.u.) Extracted eff vs initial p 11 moderate strong FIG. 8: Extracted early-time decay rate Γeff vs. initial |11iload p11(0) for moderate and strong HCCB. The slopes provide an experimental estimate of the state-dependent coefficient β, and the intercepts estimate γ0. the projector onto the violating sector of clause cand G(x) = Pcwc1{xviolates c}for the weighted violation count. For annealing we adopt the penalty convention: πHCCB(x)∝exp−βE(x)−κeff G(x), κeff ≥0,(38) so that larger κeff penalises violations. (This absorbs earlier sign choices into the definition of κeff .) Gate-Based QC: Limitations State-dependent, non-Pauli, and non-Markovian error. Let a gate Ugon scope Sbe preceded by a context-change map ΩS1→Sgenerated by (Θ, η, κ) with active projector P(S1) bad . To first order in small parameters, EHCCB(ρ) = UgρU† g+iΘ 2[P(S1) bad , ρ] + η 2P(S1) bad ρP(S1) bad −1 2{P(S1) bad , ρ}−iτg[K(ρ), ρ],(39) with K(ρ) = κ 2hP(S1) bad iρP(S1) bad . This channel is: (i) non-Pauli (jump is a multi-qubit isometry/projector), (ii) statedependent (nonlinear K), (iii) history-dependent if scopes cycle, hence non-Markovian. Average gate fidelity lower bound. Let d= 2|S|and Favg be the average gate fidelity of EHCCB to Ug. Then, for two-qubit scopes with active P11, 1−Favg &1 2|Θ| 2+η 2pΦ(11) + cκκτg+O(Θ2, η2,(κτg)2),(40) where pΦ(11) = Tr[(I⊗Φ−1 d)(PT 11 ⊗P11)]/d2is the Choi weight of the active sector (pΦ(11) = 1 4for two qubits) and cκ∼O(1) depends on scope alignment. Thus the per-gate budget |Θ| 2+η 2+κτg.δgate (41) is necessary to preserve threshold assumptions. Impact on entanglement generation. Let ENbe the logarithmic negativity across a bipartition A|Bafter applying EHCCB ◦Ugto a pure input ψ. Then for active sector supported within A, Eout N≤Eideal N−cηηΠA(ψ) + O(Θ2, κτg),ΠA(ψ) = hP(S1) bad iψ,(42) showing entangling power is reduced in proportion to the weight of the active (drained) sector. 25 Gate-Based QC: Opportunities We can program HCCB to be helpful by (a) choosing sector-draining isometries Lc,k =g(c) kEDv(c) kon clause scopes (cf. Sec. ??), and (b) keeping K(ρ) weak and aligned. Exponential drift out of violations. Within a clause block of duration τc, d dtTr(Pcρ) = −γcTr(Pcρ)⇒Tr(Pcρ)7→ e−γcτcTr(Pcρ),(43) so after Tsweeps (with mixer between blocks) EV(ρT)≤max ce−γcτcT EV(ρ0)+OX c κcτc.(44) Hence to reach E[V]≤εone needs T=O(Pcγcτc)−1log(1/ε), provided PcκcτcPcγcτc. Nonlinear amplitude steering. During each block, Kc(ρ) = κc 2hPciρPcgenerates a state-dependent phase/energy penalty. Linear response gives, for any observable Osupported on the scope, d dκchOiκc=0 =−iτc 2h[Pc, O]i ⇒ constructive/destructive steering depending on [Pc, O].(45) By compiling clause orderings so that [Pc, O] has the desired sign, one can bias amplitude toward low-violation sectors without destroying intra-context coherence. Annealing: Limitations Adiabatic fragility and irreversibility. With H(s) and a shaped HCCB profile, the local safety condition away from the minimum-gap window(s) s∗is kKs(ρ)k ∆(s)≤ε, X c γc(s)∆(s),(46) and the global reversibility budget is Z1 0 η(s)ds dt ds ≤2δanneal.(47) Violation of either reduces reverse-anneal fidelity and can trap dynamics in undesired basins. Misaligned bias. Under the penalty convention (38), small-signal response at κeff = 0 yields d dκeff hEi0=−Cov0(E, G),d dκeff hGi0=−Var0(G),(48) where h·i0denotes expectation under the un-tilted distribution. Thus if Gis poorly correlated with E, Cov0(E, G)≈0 and the penalty does not help (wasting irreversibility budget); if Cov0(E, G)<0 (perverse alignment), the penalty increases energy on average. Annealing: Opportunities Gap-aligned relaxation. Concentrate dissipation near s∗via a narrow shape g(s) with area Aη=Rs≈s∗η(s)ds, and set Ks(ρ) = κ(s) 2PcwchPciρPc. The instantaneous population master equation in the eigenbasis, ˙pm=X nWmnpn−Wnmpm, Wmn ≈X c γc(s)hψm|Lc|ψn|i2, promotes downward flow if the matrix elements of Lcconnect excited to lower-energy eigenstates in the violating sector. As Aηgrows from zero, the final ground probability Pgs(1) initially increases provided (48) yields Cov0(E, G)>0. 32 [20] J. C. Baez and M. Stay, “Physics, topology, logic and computation: a Rosetta Stone,” in New Structures for Physics, B. Coecke, ed. (Springer, 2011), pp. 95–172. [21] T. Leinster, Higher Operads, Higher Categories (Cambridge University Press, 2004). [22] J. C. Baez and J. Dolan, “Higher-dimensional algebra and topological quantum field theory,” J. Math. Phys. 36, 6073 (1995). [23] B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed. (Springer, 2015).