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The timeline by which quantum computers can solve breakthrough problems in science depends on improved quantum algorithms as much as on better hardware. Such algorithmic advances are typically achieved through many years of expert human collaboration. In mathematics and computer science, large language models have accelerated progress through automated discovery loops. However, quantum algorithms can only be evaluated by classical computers on restricted or small instances. Here we demonstrate automated discovery of quantum programs beyond these regimes including order-of-magnitude improvements on scientific applications. To achieve this, we build a quantum program library in the Lean theorem prover in which Gemini 4 agents synthesize novel programs against formal problem specifications and verify their correctness and cost. Agent teams collaborate, sharing results and knowledge through wikis and discussing ideas at organized conferences. Our results improve on prior baselines for parallel search (up to 2.55x), quantum arithmetic (290x), and simulations relevant to fusion, photochemistry and drug-metabolism by 11x, 8x and 2.5x respectively. Combining human and artificial intelligence at scale, under the rigor of mathematical proof, lets researchers design algorithms that today's classical and quantum computers cannot test, shortening the path to practical quantum computation.

We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian $H$ and a projected unitary encoding of the stacked jump operator $B=\sum_{k=1}^m \lvert k\rangle\otimes L_k$, with normalization factors $α_H$ and $α_B$, respectively. For evolution time $t$, set $τ=(α_H+α_B^2)t$. The algorithm approximates the evolution channel to diamond-norm error $\varepsilon$ using $O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)$ oracle queries, matching the query lower bound for Hamiltonian simulation. The number of additional one- and two-qubit gates is linear in the query complexity up to polylogarithmic factors. The query- and gate-complexity bounds extend to Lipschitz-continuous time-dependent Lindbladians under coherent time-indexed oracle access. Our construction uses a one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst. We bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels. The gate implementation combines a compressed Kraus-label representation, which stores only the positions and values of the nonzero labels, with the rotation factorization of Chen et al.

We give a short proof of the Aaronson-Ambainis conjecture. If a partial Boolean function $f$ is defined on a $δ$-fraction of all $n$-bit strings and have quantum query complexity $Q(f)$, then, for every $0<ε<1$, a deterministic algorithm computes $f$ correctly on at least a $1-ε$ fraction of its domain using $O( Q(f)^6 / (εδ)^2)$ queries. Our proof uses the completion framework and establishes a more general bound in terms of the classical adversary measure. The only result we use from Turturean's recent proof is a special case of his certificate inequality, for which we provide a self-contained proof. We also derive the stronger variant of the Aaronson-Ambainis Conjecture, the bounded-polynomial influence conjecture, from our main result in the appendix.

Fundamental tasks in linear systems analysis and control require evaluating infinite-horizon output energies for autonomous systems and optimal value functions for controlled systems. These quantities are characterized by Lyapunov and algebraic Riccati equations, respectively. In this work, we study promise problems for their evaluation when the system matrices are sparse and represented succinctly by Boolean circuits specifying their nonzero entries, with input length polynomial in the logarithm of the system dimension. For dissipative autonomous systems with coordinate-vector initial states, we define the succinct Lyapunov value problem (SLV). Its extension to linear-quadratic optimal control is the succinct LQR value problem (SLQRV). We prove that SLV is $\BQP$-hard: every problem in bounded-error quantum polynomial time ($\BQP$) reduces to it in polynomial time. We then establish that the more general SLQRV problem admits a quantum algorithm running in time polynomial in the succinct input length. Since SLV is a special case of SLQRV, both problems are $\BQP$-complete. Consequently, unless bounded-error probabilistic polynomial time coincides with $\BQP$, no classical randomized algorithm can solve either problem on all promised instances in time polynomial in the succinct input length.

We construct a recursive oracle $O$ and a unary language $L\in\mathsf{EQP}^O\setminus(\mathsf{IP}^O\cup\mathsf{PH}^O)$, strengthening the $\mathsf{BQP}$ versus $\mathsf{IP}$ oracle separation of Bouland et al. to exact quantum computation. The separation is based on a variant of the Forrelation problem in which the normalized correlation of one Boolean function with the Fourier transform of another is promised to be exactly $+1/2$ or $-1/2$. The main technical contribution is a repair method that constructs distributions supported on these exact levels while preserving the indistinguishability needed for the $\mathsf{IP}$ and $\mathsf{PH}$ lower bounds. A uniform polynomial-time quantum algorithm distinguishes the two cases with certainty using only three oracle queries.

The Deep Tensor Network is a family of sequence operators built from contractions of queries, keys, and values. Tensor Attention mixes tokens through the Gram operators $Q(K^\top K)Q^\top$ and $K(Q^\top Q)K^\top$; Tensor Interaction mixes channels through operators derived from $C=Q^\top K$. Associativity lets us evaluate these products exactly without forming $n\times n$ token matrices. For query/key width $d$ and value width $d_v$, the Gram constructions use $O(d^2+dd_v)$ work and summary state per causal token, or $O(d^2)$ when $d_v=O(d)$. We derive prefix recurrences and distinguish trace, diagonal, and row normalization, including the conditions under which the weights are nonnegative and row-stochastic. The unmasked token operators are positive semidefinite with rank at most $d$, but their entries can be negative. We also give exact evaluations and costs for Hadamard and matrix-exponential variants.