Quantum circuits, information, channels, cryptography and computation, including entangled games and quantum complexity.
For every finite two-player entangled game G with entangled value below one, the value of winning all n independent repetitions decays exponentially in n; the Chapter 6 bound depends on the gap and answer alphabet sizes.
Gu, Kemeny, Yelin and Lukin report lower Toffoli and peak-qubit counts in seven specified circuit benchmarks, plus a new prime-field modular-inversion architecture. For fixed-point Gaussian evaluation, their stated benchmark falls from about 24,000 Toffolis and 362 qubits to 1,583 Toffolis and 139 qubits; the reported worst-case error is 9.99×10⁻⁸.
For any finite-Kraus 1→2 qubit channel whose mean fidelity over the six Pauli eigenstates is at least 5/6−ε, the Kraus operators have total squared Frobenius distance at most 6ε from the span of the two Bužek–Hillery Kraus operators. For every input matrix ρ, the channel output is within 200√ε‖ρ‖F of the Bužek–Hillery channel in Frobenius norm.
For every n, one quantum oracle circuit generated in polynomial time from n using H, T, T† and CNOT approximates the channel of every n-qubit unitary to full diamond-norm error at most 1/2 after a suitable Boolean oracle is chosen; circuit size, qubits, queries and query length are polynomial in n.
There are total Boolean functions whose bounded-error randomized query complexity cannot obey any universal bound of the form O((1+Q(f))^α) with α<4, where Q is bounded-error quantum query complexity.
A polynomial-time uniform family of quantum circuits over one fixed finite bounded-arity gate set outputs the complete prime factorization of every integer N≥2 with probability exactly one, using polynomially many gates and qubits in the input bit length.
For every finite two-player game with finite-dimensional entangled value v<1 and arbitrary correlated questions, the maximum probability of winning at least a fraction v+δ of k repetitions decays at least as exp(−κδ¹³k/(1+log(|A||B|))) for one universal κ>0 and 0<δ<1−v.
Approximating the electronic Coulomb spectral infimum in three-dimensional continuum space is QMA-hard when positive integer nuclear charges use binary encoding, nuclei have distinct rational positions and electron count is unary; a deterministic polynomial-time reduction produces a promise gap of at least one.
For every fixed depth, polynomial qubit bound and ε>0, sufficiently large measured-output quantum circuits using arbitrary one-qubit and unbounded-arity Toffoli gates have some input on which parity succeeds with probability below 1/2+ε.
OpenAI reports that the maximum number of mutually unbiased orthonormal bases in complex dimension six is three, using an exhaustive computer-assisted exclusion of four bases under specified arithmetic and compiler conditions.
The electronic Coulomb ground-energy promise problem is QMA-hard for distinct rational positions of unit-charge nuclei, unary electron count and rational thresholds separated by at least an inverse polynomial, with minimization over the full three-dimensional spinful fermionic continuum.
Independent random one-qubit complex Cliffords, or exact one-qubit 2-designs, form a depth-one n-qubit tensor-product ensemble with entangled anticoncentration error (2/3)^n and negligible distinctness for polynomially many queries.
Bostanci and coauthors construct a unitary quantum oracle U for which QMA(2)^U differs from QMA^U. Their oracle problem has a one-query QMA(2) protocol with linear-size unentangled proofs, while QMA needs exponential resources in query count or proof size.
For the two-qubit state ρ=(|Φ+⟩⟨Φ+|+|01⟩⟨01|)/2, where |Φ+⟩=(|00⟩+|11⟩)/√2, Holevo and Shirokov show that entanglement of formation has no global supporting affine functional. Along a perturbation toward |10⟩, it decreases on a square-root scale.
Stempin, Llorens and Huber refute the proposed universal bound χ_f(B_ε(ρ)) = O(ε⁻²). Their Theorem A constructs states and Pauli sets from lexicographic powers of the complement of a seven-cycle for which χ_f(B_ε(ρ)) grows faster. Theorem B gives a broader counterexample criterion using anticommutation graphs with α(G) < β(G).
At canonical point C of the DiVincenzo family, an explicit Schmidt-rank-two tight-frame witness proves one-copy undistillability but two-copy distillability for every local dimension d≥3.
Three explicit Hermitian self-dual MDS codes with parameters [12,6,7] over F25 and [18,9,10] over F121 and F169 establish AME(12,5), AME(18,11), and AME(18,13); one-party projection establishes AME(17,11) and AME(17,13). The two length-18 matrices come from nine-entry group-circulant kernels; the theorem asserts existence, not classification.
Lalonde gives a bipartite inner-product game with four and three inputs and six outputs. It has a perfect strategy using a nonmaximally entangled state, but no perfect finite-dimensional strategy using a maximally entangled state.
In the Boolean-output quantum random oracle model, every finite-round perfectly complete key-agreement protocol with quantum local operations, independent initial private states and classical communication admits a passive eavesdropper that recovers the key with certainty using O((q_A+q_B)^5) classical oracle queries. The eavesdropper may use unbounded computation.
For an unknown d-dimensional state and m adaptively chosen observables 0≤A≤I, Chen, O’Donnell, Pelecanos and Wright give online expectation estimates within ε with success probability at least 9/10 using O(log(m)√log(d)/ε³) or O(√m/ε²) copies, respectively.
For a thermal attenuator with environment photon number ν=1, a rank-two input on six Fock levels has certified coherent information at least 4.7×10⁻⁴ qubits per channel use at transmissivity η=0.8, while all single-mode Gaussian inputs have at most zero. A larger witness is certified positive at η=0.7841.
For finite-dimensional quantum states at trace distance at most δ and d=dim A, the optimal dimension-only conditional-entropy continuity modulus is h₂(δ)+δ log(d²−1) up to δ=1−d⁻², then 2 log d. Equality is attainable when dim B≥d.
GPT-5.5 Pro’s Proposition 2 expresses the rank-two partial-trace quadratic form q⁽²⁾₋₁/₂(C) as projection terms. Song and Chen then use a separate human-developed lemma to prove nonnegativity and two-copy nondistillability at the Werner threshold.
Ananth, Ragavan and Sahai give efficient, perfectly correct, one-time private-key encryption of one bit using a random nonidentity phase-free n-qubit Pauli and a random eigenstate. After key disclosure, both separated receivers recover the bit with probability at most 1/2+O(2^(−n/2)) against information-theoretic attacks.
The selected contribution is GPT-5-Thinking’s original rational-trace strategy for the completeness barrier; the current published proof uses Phillip Harris’s determinant instead.
Quantum circuits, information, channels, cryptography and computation, including entangled games and quantum complexity.
Publication dates and editorial changes.
Claude-found two-qubit state lacks an entanglement supporting functional
Codex-found Pauli scheme gives one-bit unclonable encryption
ChatGPT-led protocols match classical online shadow-tomography rates
ChatGPT-found inputs improve a bosonic quantum-capacity bound
GPT-5.5 Pro contributes a Werner-state projection decomposition
AI-discovered attack on perfectly complete quantum key agreement
ChatGPT-origin Clifford construction gives depth-one distinctness
Codex-found game needs nonmaximal entanglement for a perfect strategy
AI-derived counterexamples to a Pauli shadow-tomography conjecture
Claude-assisted proof gives quantitative stability for optimal qubit cloning
GPT-5's rational-function step in an early QMA amplification proof
OpenAI reports polynomial unitary synthesis with a Boolean oracle
OpenAI reports exact quantum factoring with fixed finite gates
LLM-discovered quantum circuits reduce Toffoli and qubit costs
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