Quantum programmes can agree on the operation performed within each branch whilst differing in the interference produced when those branches are recombined. I propose a Quantum Riemann Machine as a finite architecture for checking whether specified implementations admit a common quantum realisation and determining its environmental cost. Its geometric input records the continuation of Kraus frames around loops; its physical consistency condition is the positive Gram condition of channel gluing. For three branches, the eigenphases of a single loop give an exact coherence threshold and pure-environment spectrum. I extend this calculation to one reference branch and an arbitrary number of mutually calibrated branches, allowing their loop transports to be noncommuting. The singular values of the sum of these transports determine the complete spectrum through explicit cubic polynomials, and hence determine feasibility and the minimum environment dimension. A four-branch example has feasible restrictions to every three branches but no joint realisation. Reorienting one transport while preserving its eigenvalues restores feasibility, demonstrating information that separate loop spectra do not retain. Leaving one comparison unspecified instead permits an environment of dimension six, whereas prescribing it can require dimension eight or make the programme impossible. Spectral error bounds and constructive compression retain every local branch channel exactly. These certificates distinguish incompatible interference specifications from insufficient environmental resources and quantify the consequences of reducing an implementation, connecting its transport description to the design and verification of coherently controlled quantum programmes.
Quantum programmes can agree on the operation performed within each branch whilst differing in the interference produced when those branches are recombined. I propose a Quantum Riemann Machine as a finite architecture for checking whether specified implementations admit a common quantum realisation and determining its environmental cost. Its geometric input records the continuation of Kraus frames around loops; its physical consistency condition is the positive Gram condition of channel gluing. For three branches, the eigenphases of a single loop give an exact coherence threshold and pure-environment spectrum. I extend this calculation to one reference branch and an arbitrary number of mutually calibrated branches, allowing their loop transports to be noncommuting. The singular values of the sum of these transports determine the complete spectrum through explicit cubic polynomials, and hence determine feasibility and the minimum environment dimension. A four-branch example has feasible restrictions to every three branches but no joint realisation. Reorienting one transport while preserving its eigenvalues restores feasibility, demonstrating information that separate loop spectra do not retain. Leaving one comparison unspecified instead permits an environment of dimension six, whereas prescribing it can require dimension eight or make the programme impossible. Spectral error bounds and constructive compression retain every local branch channel exactly. These certificates distinguish incompatible interference specifications from insufficient environmental resources and quantify the consequences of reducing an implementation, connecting its transport description to the design and verification of coherently controlled quantum programmes.
When a quantum operation is specified exactly, irrational amplitudes in its unitary description can disappear from its action on density matrices, leaving an operation with rational coefficients whose implementation still requires additional resources. I develop a Quantum Galois Machine that determines whether an algebraically specified quantum channel can be executed using rational matrices and a finite environment initially prepared in a computational basis state. For every unitary channel with rational coefficients, the minimum dimension of an environment that may be discarded is exactly the least number of rational squares needed to represent a positive rational parameter extracted from the channel. This gives a complete classification with possible dimensions one, two, three and four, each attained by an explicit example. The associated square class multiplies under parallel and sequential composition, so two identical operations admit a joint implementation without an environment even when one operation requires four dimensions. Returning the environment to its initial basis state or retaining a specified coherent control imposes different arithmetic obstructions. Although unrestricted approximation removes this exact environment requirement, a bound on rational denominators restores a quantitative obstruction: a nonsquare output probability with reduced denominator S forces diamond-norm error at least 2/(SB^2) when every entry denominator of an environment-free implementation is at most B. A constructive Householder approximation supplies a complementary upper bound, giving logarithmic denominator bit cost in inverse precision for each fixed channel with nonsquare parameter. These results provide finite certificates that account for the coefficient field, phase convention, auxiliary systems and numerical description size.
A quantum von Neumann machine that retains its stored programme in superposition must execute the selected operations without destroying coherence between programme branches, creating a resource requirement that cannot be assessed by examining each programme separately. I establish this distinction for programmes that select controlled-Z operations on the edges of a graph. Every classically selected programme uses only Clifford operations, yet coherent execution requires a number of injected non-Clifford T states proportional to the programme length for every fixed normalised diamond-distance tolerance below one half. For a graph with m edges and n active data vertices, the lower bound for exact execution is m + n states, whilst an explicit implementation uses at most 4m. The lower bound permits arbitrary programme-dependent phases, stabilizer workspace, measurements and adaptive control. Its extension to imperfect execution bounds both the probability of substantial resource consumption and the expected consumption, without imposing a separate limit on workspace or measurement count. The proof derives a uniform bound on Pauli expectations of the joint programme-data output and uses its stabilizer subspaces to constrain every low-resource trajectory. A supporting criterion identifies precisely when a Clifford instruction library admits an exact Clifford interpreter under the chosen binary programme encoding. These results separate the cost of interpreting coherent instructions from the cost of the operations selected by a classical programme, providing a quantitative constraint on compilation and resource estimation for stored-programme quantum computation.
Quantum programmes described through successive approximations need a consistent way to retain their limiting behaviour when individual approximating settings are discarded. I propose a Quantum Cantor–Bendixson Machine that certifies and repairs quantum operations at boundaries of a classical control space. For continuous matrix-valued observables over a compact zero-dimensional metric space and a finite retained boundary, the smallest completely bounded repair cost is exactly twice the largest boundary leakage, and a continuous unital completely positive map attains it. I extend this result to simultaneous finite boundary constraints, identifying when their costs combine without an additional penalty. For independent operations on any finite product of convergent-sequence spaces, a product repair preserves every Cantor–Bendixson derivative, including the infinite intermediate boundaries, at the exact cost already forced by the final boundary. A coherent qubit example establishes why the corresponding cost for a quantum subspace can instead scale as the square root of leakage. Computable operations can have noncomputable optimal repair costs: their possible leakages are exactly the lower semicomputable reals in the unit interval. An explicit tail certificate permits finite bounds and a computable repair arbitrarily close to the optimum. These results specify when one operation can respect several topological reductions, what this consistency costs, and which additional information makes its certification effective.
Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{d-compatibility} for measurements, channels, and instruments whose parent passes a d-dimensional quantum system and unlimited classical information to the post-processing. We formulate d-compatibility as a common linear factorization of operator spaces and operator systems and show that d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a d-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are d-incompatible in certain regimes of d. We also link d-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.
How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.
Everettian quantum mechanics describes alternative histories within one evolving quantum state, while quantum Darwinism relates their accessibility to information retained in environmental records. I examine how finite record memory constrains the number of distinguishable histories and the organisation of their conditional ancestry. When d + 1 histories must each be decoded from a d-dimensional fragment with success d/(d + 1), the established discrimination optimum forces a regular-simplex geometry. Its conditional mixtures then determine exact ancestral edge costs and a joint criterion for the permissible topology, depth, total variation and largest step. In the three-history binary example, equally weighted labels decoded with success 2/3 require 61.6% greater minimum conditional variation in a qubit than in an optimally designed three-dimensional record. For six histories, minimising total variation and minimising the largest step select different genealogies. Quantitative stability estimates show that the additional constraint persists under sufficiently small departures from ideal decoding and weights. At this threshold, trace-distance contraction prevents a fixed channel acting only on an unrestricted optimum from compressing it into the smaller record at unchanged accuracy, although an explicit controlled preparation realises the alternative encoding. A von Neumann-inspired application distinguishes hereditary function, present configurations and retained mutation histories, and a comparison of finite and infinite counts separates these quantities from cosmological multiplicity. The resulting theory makes the limitations of environmental memory consequential for readable branching while keeping conditional variation distinct from physical rates of change. Its bounds apply to specified records and histories without assigning an absolute number of worlds since the universe’s beginning.