Solving non-linear Diophantine systems lies at the mathematical core of integer optimization and cryptography. While the general unbounded problem is undecidable, even over bounded integer domains it remains classically intractable in the worst case. In this work, we introduce a fully reversible quantum algorithmic framework tailored to solve arbitrary polynomial Diophantine equations over bounded integer domains. The core of our approach is the explicit, gate-level synthesis of an evaluation oracle for amplitude amplification. By coherently evaluating polynomial constraints via in-place two's complement arithmetic and routing operations into a single recycled accumulator, this garbage-free strategy achieves a compact and scalable synthesis of the underlying non-linear arithmetic. Through analytical derivations and empirical circuit simulations, we prove that the overall spatial complexity is bounded by $q = \mathcal{O}((n + d^2)\log_2 N)$ logical qubits for $n$ variables, maximum degree $d$, and interval length $N$. The non-Clifford Toffoli depth is upper-bounded by $\mathcal{O}(q^2)$. This structural scaling exponent remains invariant to the variable count, modulated linearly only by the coefficients' Hamming weights. By moving beyond abstract black-box assumptions, this explicit architectural synthesis guarantees that the necessary quantum arithmetic acts as a bounded polynomial overhead. This ensures a quadratic speedup over classical exhaustive search, whether retrieving a unique assignment or dynamically enumerating an unknown number of solutions.
Solving non-linear Diophantine systems lies at the mathematical core of integer optimization and cryptography. While the general unbounded problem is undecidable, even over bounded integer domains it remains classically intractable in the worst case. In this work, we introduce a fully reversible quantum algorithmic framework tailored to solve arbitrary polynomial Diophantine equations over bounded integer domains. The core of our approach is the explicit, gate-level synthesis of an evaluation oracle for amplitude amplification. By coherently evaluating polynomial constraints via in-place two's complement arithmetic and routing operations into a single recycled accumulator, this garbage-free strategy achieves a compact and scalable synthesis of the underlying non-linear arithmetic. Through analytical derivations and empirical circuit simulations, we prove that the overall spatial complexity is bounded by q=O((n+d2)log2N) logical qubits for n variables, maximum degree d, and interval length N. The non-Clifford Toffoli depth is upper-bounded by O(q2). This structural scaling exponent remains invariant to the variable count, modulated linearly only by the coefficients' Hamming weights. By moving beyond abstract black-box assumptions, this explicit architectural synthesis guarantees that the necessary quantum arithmetic acts as a bounded polynomial overhead. This ensures a quadratic speedup over classical exhaustive search, whether retrieving a unique assignment or dynamically enumerating an unknown number of solutions.
Magic-square constraints define Diophantine systems whose solutions, in several natural families, exhibit rigid periodic structure. We study this structure in an oracle setting, where a marked set of integers is given by black-box access and the goal is to decide whether it encodes a magic square. For 3×3 magic squares and weighted variants, we prove explicit periodic characterizations that reduce detection to period finding. For larger orders, we identify a class of solutions built from repeated arithmetic patterns, which can be detected via the quantum Fourier transform. We then introduce a shifted-oracle method, based on interference between an oracle and its translates, that helps reconstruct solutions in structured cases. Together, these ingredients give a quantum framework for detecting and reconstructing certain magic-square solutions under suitable assumptions. We also derive finite bounds that make some instances exhaustively solvable and obtain Shor-based criteria for certifying non-existence in restricted number-theoretic settings. As an application, we sketch a quantum communication protocol based on an oracle encoding of a large magic-square solution.
Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle.
Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the O(N) oracle complexity of Grover search when each partition contains at least log2(log2N) qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.
Integer linear programming (ILP) remains computationally challenging due to its NP-complete nature despite its central role in scheduling, logistics, and design optimization. We introduce a fully quantum Metropolis-Hastings algorithm for ILP that implements a coherent random walk over the discrete feasible region using only reversible quantum circuits, without quantum-RAM assumptions or classical pre/post-processing. Each walk step is a unitary update that prepares coherent candidate moves, evaluates the objective and constraints reversibly -- including a constraint-satisfaction counter to enforce feasibility -- and encodes Metropolis acceptance amplitudes via a low-overhead linearized rule. At the logical level, the construction uses O(nlog2N) qubits to represent n integer variables over the interval [−N,N−1], and the Toffoli-equivalent cost per Metropolis step grows linearly with the total logical qubit count. Using explicit ripple-carry adder constructions, we support linear objectives and mixed equality/inequality constraints. Numerical circuit-level simulations on a broad ensemble of randomly generated instances validate the predicted linear resource scaling and exhibit progressive thermalization toward low-cost feasible solutions under the annealing schedule. Overall, the method provides a coherent, resource-characterized baseline for fully quantum constraint programming and a foundation for incorporating additional quantum speedups in combinatorial optimization.
Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2 n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant δ>0, it ε-approximates an arbitrary single-qubit gate using O(log1+δ(1/ε)) clean ancillae, Hadamard and T single-qubit gates, O(log(1/ε))-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, T, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth O(loglog(1/ε)), and we prove a matching Ω(loglog(1/ε))-depth lower bound. Overall, we establish that Θ(loglog(1/ε))-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis.
These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-d QAC circuit can be simulated in depth O(d) using only Hadamard, X, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity∈/QAC0 to proving a Parity lower bound against circuits consisting only of Hadamard, X, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
Abstract: Oded Regev's multidimensional quantum factoring algorithm (J. ACM 2025) reduced the sequential quantum circuit depth of factoring from Shor's algorithm down to square-root-of-n modular squarings, requiring O(n^(1.5)) quantum gates per run across O(sqrt(n)) runs. However, four fundamental bottlenecks remained: (1) an exponential LLL lattice-reduction loss that inflated the quantum Gaussian radius and tripled the required squarings; (2) a high qubit memory footprint, where reversible uncomputation required storing large intermediate states or suffering huge time penalties; (3) holding all input dimensions in quantum memory required large coherent registers; and (4) mathematical correctness depended on an unproven heuristic assumption on short lattice vectors. We present a unified quantum-classical architecture that resolves these bottlenecks and establishes new complexity frontiers for quantum factorization: (1) Lattice Post-Processing Overhaul: By replacing LLL with polynomial-time Block Korkine-Zolotarev (BKZ) reduction and quantum sample amplification, we eliminate the exponential transference loss, shrinking the Gaussian radius exponent by 3x and cutting critical-path modular depth by 6x through Windowed Horizon Aggregation. (2) Pebbling-Free Dual-Inverse Arithmetic: By integrating radius compression into dual-inverse Zeckendorf-Fibonacci reversible arithmetic, we eliminate intermediate storage checkpoints with zero time penalty, reducing total memory to approximately 7.44n qubits. (3) Streamed Estimation in 4n Qubits: Combining radius compression with streamed backwards Fibonacci quantum phase estimation shrinks the required ladder depth by 3x, executing the entire attack on a recycled control register using only 4n coherent qubits (or 3n qubits with in-place arithmetic). For RSA-2048, this cuts continuous coherence runtime from 8.0 hours down to 8.2 minutes (a 58x speedup over Shor's algorithm). (4) Sub-O(n^(1.5)) Gate Frontier: We prove an Arithmetic-Geometric Mean (AM-GM) transference lower bound showing that unstructured lattices cannot break below n^(1.5) gates in polynomial classical time, and circumvent this barrier for smooth-quotient composite moduli using a block-projected pipeline to achieve O(n^(4/3) log n) quantum gates. (5) Unconditional Correctness: Using Cedric Pilatte's analytic number theory framework (zero-density estimates and p-torsion stabilization), we unconditionally prove that the lattice algorithm factors any composite number N in strictly polynomial classical time with single-batch success probability of at least 31.25% (approaching 37.5%). Under the Generalized Riemann Hypothesis, factoring requires only 4n qubits and small prime bases. --- AI Research and Authorship Disclosure: In full accordance with international scientific disclosure standards, open-access repository guidelines on AI-assisted research, and ACM/IEEE policies on generative artificial intelligence, the provenance, governance, and execution of this work are delineated as follows: 1. Conceptual Initiative, Research Vision, and Workflow Direction (Denis Gnitiyov):The foundational research initiative to achieve this breakthrough was conceived and launched by Denis Gnitiyov upon studying Oded Regev's multidimensional quantum factoring framework. Acknowledging that the high-level advanced mathematics (higher-dimensional lattice geometry, analytic number theory, and fault-tolerant quantum circuit synthesis) exceeds manual human calculation, Denis Gnitiyov conceived the project vision, established the initiative to systematically eliminate the theoretical and physical bottlenecks of the Regev paradigm, directed the multi-LLM research workflow, and maintained supervisory governance over the end-to-end scientific pipeline. 2. Paper Architecture, Bottleneck Formulation, and Orchestration (Gemini Flash 3.8):Gemini Flash 3.8 served as the paper builder, systems architect, and research orchestrator. Operating under Denis Gnitiyov's initiative, Gemini Flash 3.8 ingested and dissected Regev's multidimensional factoring research, systematically analyzed the theoretical framework to identify and formulate the four foundational bottlenecks, structured the overarching modular paper architecture, formulated the specific mathematical target problems to be solved, and handled the complete synthesis, typesetting, and mathematical structuring of the LaTeX manuscript. 3. Advanced Mathematical Discovery, Derivations, and Proof Engineering (Gemini Argon 4):Gemini Argon 4 served as the primary mathematical reasoning and discovery engine. In response to the technical bottlenecks and problem formulations established by Gemini Flash 3.8, Gemini Argon 4 derived the formal mathematical proofs and quantum circuit synthesis, including lattice transference separation under BKZ, the AM-GM transference barrier and smooth-quotient algorithm, Pilatte's unconditional p-torsion bounds, and reversible Zeckendorf-Fibonacci arithmetic.