We prove early preparation and entropy bounds for an explicit evaporating random-circuit model. Independent Haar two-qubit gates scramble an active register before each emission, and emitted qubits never interact again. Preparation complexity allows arbitrary one- and two-qubit gates and any finite number of initially pure ancillas, at fixed trace-distance error. We specialize the established two-copy purity method to permanent deletion and bound its evolution by a moving hyperbolic-cosine comparison profile. The first scrambling round gives a uniform relative bound on expected subsystem purity; positivity preserves this bound through every subsequent emission, even if later scrambling rounds are shortened or omitted. For the stipulated schedule, radiation before a logarithmic window around half emission has a linear preparation upper bound with an explicit failure bound. The entropy of every radiation prefix differs from its dimension ceiling by at most a logarithmic term, simultaneously with high probability. Numerical recurrence calculations, exact subset identities and small-system state-vector simulations provide reproducible checks. These results settle the early-preparation and entropy clauses of the benchmark. They do not establish its post-Page complexity lower bound at the accumulated gate-count scale. We identify the additional distributional information that such a lower bound requires and explain why the entropy Page law alone is insufficient. This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized this deposit and is responsible for it.
We prove early preparation and entropy bounds for an explicit evaporating random-circuit model. Independent Haar two-qubit gates scramble an active register before each emission, and emitted qubits never interact again. Preparation complexity allows arbitrary one- and two-qubit gates and any finite number of initially pure ancillas, at fixed trace-distance error. We specialize the established two-copy purity method to permanent deletion and bound its evolution by a moving hyperbolic-cosine comparison profile. The first scrambling round gives a uniform relative bound on expected subsystem purity; positivity preserves this bound through every subsequent emission, even if later scrambling rounds are shortened or omitted. For the stipulated schedule, radiation before a logarithmic window around half emission has a linear preparation upper bound with an explicit failure bound. The entropy of every radiation prefix differs from its dimension ceiling by at most a logarithmic term, simultaneously with high probability. Numerical recurrence calculations, exact subset identities and small-system state-vector simulations provide reproducible checks. These results settle the early-preparation and entropy clauses of the benchmark. They do not establish its post-Page complexity lower bound at the accumulated gate-count scale. We identify the additional distributional information that such a lower bound requires and explain why the entropy Page law alone is insufficient. This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized this deposit and is responsible for it.
We introduce a method for studying state preparation complexity in dense quantum p-spin Hamiltonians on n qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly p qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of n for sufficiently large fixed p. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds.
Applying this framework, we show that attaining near-ground-state energy requires Ω(n2/logn) one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width O(n). In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+T circuits with o(n)T-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.
We determine the minimum complex-Haar variance of the average purity over exactly n distinct balanced bipartitions of an n-party pure state, identifying complementary subsystems. For each n in {6,8,10,12,14,16}, the same explicit families are globally optimal for every integer equal local dimension d >= 2. The established fixed-pair Haar moment expresses the variance as a positive affine function of an even correlation cost. A parity floor settles n congruent to 2 modulo 4. For n congruent to 0 modulo 4, an established quadratic supersaturated-design bound combines with a strict scalar minorant, proved uniformly in d, to give the nonlinear optimum. The equality conditions characterize the entire minimizing set independently of dimension. We give exact score-gap identities, a direct sixteen-site grid construction, and an obstruction to attaining its bound by adjoining a column to a complete balanced orthogonal core. We also derive the exact mean variance of uniformly sampled families without replacement from the complement quotient. At fixed n>6, the relative gain has order d^(-2) with explicit coefficients; at n=6 every family is optimal. Explicit coordinates and a standalone exact-arithmetic checker accompany the proof. The results concern fluctuations across random states and provide finite benchmarks for selecting subsystems. Correspondence: Ryosuke.Shiroshita@exactory.ai This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, reviewed the full content and is responsible for it.
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum p-spin glasses, random Hamiltonians whose interactions act on p-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement.
We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with n, we prove that, for all sufficiently large fixed p, any circuit preparing an n-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth Ωp(logn). In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed D, a sufficiently large degree prefactor rules out depth-D preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits.
Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.
We study prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after independently measuring the observed copies with the one-copy Haar POVM. For the resulting fixed separable observation, the Bayes predictive state under quantum relative-entropy loss is the full-rank posterior mean. We evaluate it exactly through permanents of minors of the outcome Gram matrix, prove positivity and normalization of the permanent representation, and establish minimaxity among decision rules based on these fixed outcomes. Using Hilbert--Schmidt projection of the posterior mean and an exactly analyzed linear-inversion competitor, we prove a finite-sample posterior-purity bound and an explicit relative-entropy guarantee with sample count O((d2/ε)log(d/ε)). Using the exact collective benchmarks proved in a companion paper, we also obtain entropy and posterior-purity comparisons. To connect the fine product data to a regular ray-valued experiment without assuming deterministic finite-sample uniqueness, we draw one random orientation for a finite projective mesh, retain it for the entire sample, and apply a selected finite-alphabet MLE in the resulting reference experiment. A nested recovering mesh uniformly recovers the fine Fisher information. Consequently every sufficiently fine fixed mesh gives a regular finite-alphabet model. Its selected MLE has Haar-averaged inverse-Fisher coefficient ak, and refining the mesh after the fixed-mesh large-sample limit makes ak approach d−1. This yields epsilon-optimal leading relative-entropy and posterior-overlap bounds for the fine posterior and the sharper fixed-dimensional asymptotic scale (d/ε)log(d/ε). The latter is not asserted as a uniform finite-sample guarantee. No deterministic finite-sample MLE uniqueness or triangular mesh schedule is used.