A finite reversible model can approximate individual quantum gates while retaining freedom in its transition tables and preparation contexts. We formulate the simultaneous certification problem in which one stationary controller and one preparation law must reproduce an entire collection of circuit probabilities. For a specified frame catalogue, accuracy graph and calibration constraints, we give an exact shared-table feasibility formulation and distinguish certificates for a fixed table from certificates for the complete model class. A four-frame qubit example exhibits a strict separation between fitting two circuits individually and fitting them jointly. A larger exact benchmark uses 464 frames and a phase symmetry that removes preparation-prior dependence. We independently reconstruct its admissible graph and certify a sharp repeated-circuit horizon: one controller stays within probability error 1/3 through 192 blocks, whereas every allowed controller violates that threshold by block 193. A separate controller reproduces all 8,190 nonempty Hadamard-phase words of length at most twelve within the same tolerance. We also derive a finite-window spectral obstruction and finite-shot confidence tests, illustrated by a reproducible sampling experiment. The conclusions concern the stated finite reversible model classes; they do not establish a universal memory bound or a discrepancy for physical quantum hardware.
Comments
15 pages, 3 figures, includes exact computational certificates and a reproducible finite-shot simulation study
A finite reversible model can approximate individual quantum gates while retaining freedom in its transition tables and preparation contexts. We formulate the simultaneous certification problem in which one stationary controller and one preparation law must reproduce an entire collection of circuit probabilities. For a specified frame catalogue, accuracy graph and calibration constraints, we give an exact shared-table feasibility formulation and distinguish certificates for a fixed table from certificates for the complete model class. A four-frame qubit example exhibits a strict separation between fitting two circuits individually and fitting them jointly. A larger exact benchmark uses 464 frames and a phase symmetry that removes preparation-prior dependence. We independently reconstruct its admissible graph and certify a sharp repeated-circuit horizon: one controller stays within probability error 1/3 through 192 blocks, whereas every allowed controller violates that threshold by block 193. A separate controller reproduces all 8,190 nonempty Hadamard-phase words of length at most twelve within the same tolerance. We also derive a finite-window spectral obstruction and finite-shot confidence tests, illustrated by a reproducible sampling experiment. The conclusions concern the stated finite reversible model classes; they do not establish a universal memory bound or a discrepancy for physical quantum hardware.
Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a 40Ca+ trapped-ion processor and certify the X-gate rotation eigenvalue using 22000 circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of 99.94(3)% with 99% confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.
Quantum executions sharing a provenance record can produce different output distributions. We ask which execution details to retain, and at what precision, to guarantee agreement within a declared Hellinger tolerance. Under a specified channel model, we combine independent stochastic mechanisms and coherent rotation differences to certify recording precision without output simulation or explicit dependence on register dimension. For four-qubit quantum approximate optimization at depth one, a 13-bit comparison record suffices at tolerance 0.05 over a declared continuous noise range, uniformly in programmed angles for each fixed task. Pairwise separation constraints characterize sufficient field sets on finite context classes, exposing omissions that pass individually but fail jointly. Separately, six-qubit hardware measurements over six tested contexts certify minimum label counts of two for raw outputs and one after a fixed decoder, at tolerance 0.15 with 97\% simultaneous confidence under stationary independent shots. Execution records thereby carry checkable agreement guarantees.
Two labelled rank-one quantum measurement contexts can have identical static transition probabilities while differing in their relative complex realization. This preprint studies reconstruction in a specified one-system access model with state preparation, context readout, and independently calibrated projector-phase controls. It gives a finite-pulse extraction of fourth-order Bargmann invariants, constructive reconstruction of the relative unitary matrix modulo row and column phases, a common-unitary completion condition, stability bounds, and a fit-free four-setting composition identity. Combined with simultaneous finite-sample confidence regions, the latter yields a conservative level-alpha rejection rule for the stated stable-control model class. Numerical evidence is synthetic and limitations, blind spots, and unequal hardware access in comparisons are reported explicitly.
We prove that a quantum circuit together with measurement apparatuses and EPR sources can be fully verified without any reference to some other trusted set of quantum devices. Our main assumption is that the physical system we are working with consists of several identifiable sub-systems, on which we can apply some given gates locally. To achieve our goal we define the notions of simulation and equivalence. The concept of simulation refers to producing the correct probabilities when measuring physical systems. To enable the efficient testing of the composition of quantum operations, we introduce the notion of equivalence. Unlike simulation, which refers to measured quantities (i.e., probabilities of outcomes), equivalence relates mathematical objects like states, subspaces or gates. Using these two concepts, we prove that if a system satisfies some simulation conditions, then it is equivalent to the one it is purposed to implement. In addition, with our formalism, we can show that these statements are robust, and the degree of robustness can be made explicit (unlike the robustness results of [DMMS00]). In particular, we also prove the robustness of the EPR Test [MY98]. Finally, we design a test for any quantum circuit whose complexity is linear in the number of gates and qubits, and polynomial in the required precision.
Reversible Finite-Frame Dynamics for Rational Quantum Grids: Preparation Laws and Matching-Dependent Statistical Predictions This paper develops a finite, reversible model of quantum control motivated by Tim Palmer’s rational quantum mechanics (RaQM). Rational restrictions on quantum states do not by themselves determine how nominal quantum gates should act when their exact outputs lie outside the admissible state set. The paper addresses this problem by constructing finite catalogues of quantum frames on which commands act as permutations, together with explicit preparation laws that assign statistical predictions to complete control sequences. The analysis distinguishes a conditional rational grid from its occupation-compatible subset, identifies their common projective monomial gate group, and proves an obstruction to uniformly accurate fixed-grid repair by local gates. Equal-measure partitions and perfect matchings are then used to construct reversible approximations of local and entangling controls on enlarged finite ray sets. A compensated preparation law yields a common random-unitary channel for each control word, while a finite consistency criterion determines whether a proposed preparation law works for all words. For fixed circuit depth, the resulting statistics converge uniformly to the nominal quantum predictions as the catalogue is refined. At sufficiently large depths, finite recurrence forces disagreement for a repeated entangling circuit. An explicit rematching of the CNOT control removes a seven-block discrepancy without increasing the catalogue size or the stated error bounds, showing that those coarse conditions alone do not determine the earliest observable deviation. The construction is therefore a mathematically explicit candidate completion of finite-state quantum kinematics, while leaving the physical selection of a particular matching, preparation law and reduction dynamics as additional assumptions. Exact Python and Wolfram Language checks accompany the finite examples.