We introduce harmonic Hamiltonian relaxation (HHR), a variational spectral framework for locating ground states of operators defined on high-dimensional discrete lattices. The method embeds the discrete configuration space into a continuous Hilbert space equipped with a periodic modulation potential, transforming a combinatorial optimization problem into a smooth variational one. We construct an effective Schro ̈dinger operator whose localized ground state encodes the shortest non-zero lattice vector. Under a polynomial spectral gap hypothesis supported by numerical evidence for BKZ-reduced bases, the method admits an adiabatic optimization path compatible with gradient-based variational sampling. As an application, we show that HHR forms the computational core of the causal-quantum-spectral (CQS) framework and numerically reproduces arithmetic invariants of semistable elliptic curves, in numerical agreement with predictions of the Birch–Swinnerton-Dyer conjecture [Timakov, 2025, doi:10.5281/zenodo.17768557].
We introduce harmonic Hamiltonian relaxation (HHR), a variational spectral framework for locating ground states of operators defined on high-dimensional discrete lattices. The method embeds the discrete configuration space into a continuous Hilbert space equipped with a periodic modulation potential, transforming a combinatorial optimization problem into a smooth variational one. We construct an effective Schro ̈dinger operator whose localized ground state encodes the shortest non-zero lattice vector. Under a polynomial spectral gap hypothesis supported by numerical evidence for BKZ-reduced bases, the method admits an adiabatic optimization path compatible with gradient-based variational sampling. As an application, we show that HHR forms the computational core of the causal-quantum-spectral (CQS) framework and numerically reproduces arithmetic invariants of semistable elliptic curves, in numerical agreement with predictions of the Birch–Swinnerton-Dyer conjecture [Timakov, 2025, doi:10.5281/zenodo.17768557].
Combinatorial optimization is one of the promising applications of quantum computation. Variational Quantum Algorithm (VQA) is a classical-quantum hybrid algorithm framework for noisy quantum devices. However, the quality of the solution is readily affected by statistical fluctuations and physical noise, making it challenging to maintain applicability for largescale problems. In contrast, sampling-based quantum algorithms have recently been successfully applied to large-scale quantum chemistry problems. In the sampling-based framework, the quantum device is used only for sampling, while the ground state and its energy are estimated on the classical device. In this study, we propose the Sampling-based Quantum Optimization Algorithm (SQOA) that leverages these advantages for combinatorial optimization. There are two challenges in constructing SQOA. The first challenge is that we need to encode the optimization problem in a non-diagonal Hamiltonian, even though many VQAs encode it into the Ising Hamiltonian, which is diagonal. The second challenge is that we need a method for efficient preparation of the input state to be sampled. To address the first challenge, we utilize the Quantum Relaxation (QR) method for encoding an optimization problem within a Hamiltonian. This method achieves a higher encoding density by mapping multiple variables to a single qubit, thus requiring fewer qubits for larger problems compared to the Ising Hamiltonian approach. Moreover, to address the second challenge, we investigate the parameter transferability within the Quantum Alternating Operator Ansatz for QR Hamiltonians. We show that restricting parameters to a linear form exhibits moderate transferability for 3-regular Max-Cut problems, similar to transferability observed in the Quantum Approximate Optimization Algorithm. Therefore, this property allows us to efficiently prepare the input state for a large problem instance using the parameters obtained from a small problem instance. We leveraged transferability to create input states and applied SQOA with QR to the Max-Cut instances. By transferring parameters from a 20-node problem, we demonstrate that SQOA with QR provides high-quality solutions for 40-node problems without variational parameter optimization.
We propose a novel, physically motivated computational approach to the Hilbert-Pólya conjecture by constructing and numerically simulating a deformed one-dimensional quantum Hamiltonian. While classical formulations of the Berry-Keating Hamiltonian suffer from unbound trajectories, we demonstrate that a spatial confinement based on a log-polynomial deformed potential of the form V(x) = c0 ln(x) + c1 x + c2 x^2 + c3/x yields a discrete, stable, and purely real energy spectrum. The eigenvalues of this stabilized system are successfully mapped to a 2-qubit virtual register, with ground state convergence verified via a Variational Quantum Eigensolver (VQE) and excited states resolved using Quantum Subspace Expansion (QSE).
The advent of quantum computing necessitates the transition of worldwide cryptosystems to postquantum cryptography (PQC), which is founded upon the problem of finding short vectors in high-dimensional structured lattices. It is assumed that the structure of these lattices cannot be exploited by quantum or classical algorithms attempting to find short vectors. In this work, we focus on the structure of the lattices used in PQC protocols—negacyclic (and cyclic) lattices—and provide a quantum algorithmic framework that efficiently encodes the structured lattices into Hamiltonians by exploiting their underlying symmetries. The efficient encoding substantially reduces the dimension of the corresponding Hilbert space by limiting it to a relevant subspace where short vectors are likely to be found—leading to significant savings in quantum resources (e.g., qubit count and circuit depth) required to implement a quantum algorithm for finding short vectors. We analytically prove the efficient encoding procedure and benchmark the proposed framework using the variational quantum eigensolver, demonstrating improved results with reduced quantum resources.
This dissertation is concerned with a variety of problems in quantum computing and the spectral theory of quantum-mechanical channels and Hamiltonians. We begin with the problem of quantum gate synthesis. The challenge is to find a fixed, finite set S of quantum gates such that, given a quantum circuit component U acting on n qubits and target precision ε, there is a short, efficiently computable composition of gates from S that approximates U to precision ε. This problem has already led to a rich literature in the case n=1 with deep connections to modern number theory. In this dissertation we present the state-of-the-art algorithm for a single qubit, achieving length 7 log (1/ε). The techniques used include the lattice structure of certain dense subgroups of PU(2) and Lenstra's algorithm for integer programming in convex domains. The next topic considered is scientific computing algorithms intended to run on a quantum computer. Specifically, we study a framework for computing functions of normal matrices using a quantum computer, with the objective of controlling the ancilla overhead, probability of success, and end precision. This arises as a special case of the multivariate quantum eigenvalue transform, for computing function of multiple commuting Hermitian matrices. The techniques applied include approximation of regular functions by low-degree polynomials and modern quantum primitives like quantum signal processing and linear combinations of unitaries. The third component is a spectral gap comparison result for Davies generators, a class of Lindbladians used for Gibbs state preparation and which are a well-studied example of quantum Markov dynamics, and which contain a canonical classical Markov chain. The main result shows that if the spectrum of the Hamiltonian driving the dynamics is free of long arithmetic progressions, then the spectral gaps of the quantum and classical Markov chains are comparable. Thus, classical techniques may be leveraged to control the behavior of the quantum dynamics. The proof is based on transferring any "off-diagonal" eigenvector of the Davies generator to an observable which commutes with the Hamiltonian and has a comparable Lindbladian Rayleigh quotient. Finally, we study the infimum of the spectrum, or ground state energy, of a discrete Schrödinger operator on the scaled integer grid in n dimensions, parameterized by a nonnegative potential V and a small frequency parameter θ. We relate this ground state energy to that of a corresponding continuous semiclassical Schrödinger operator on the full n-dimensional space with parameter θ, arising from the same choice of potential, and show that these quantities are equal up to vanishing factors depending on the regularity of V. The proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on the scaled grid to low-energy functions for the continuous operator on the full space, and vice versa. Coauthors on these works were J. Basso, T. R. Blackman, Y. Borns-Weil, I. Detherage, S. Ganguly, N. Srivastava, T. Saffat, A. Sinclair, and T.-D. Vuong.
Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semidefinite relaxations both provide approximations with performance guarantees that scale as a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semidefinite relaxations inspired by the quantum marginal problem.
Abstract We report the resolution of the six open Clay Millennium problems: the Riemann Hypothesis, the Yang–Mills existence-and-mass-gap problem, Navier–Stokes global regularity, the Hodge conjecture, Birch–Swinnerton-Dyer, and P vs NP. In the process of discovery we realized what the six have in common: each is decided by a spectral invariant of a self-adjoint (Hermitian) operator, a different operator and invariant for each problem, on a space no classical machine can hold. That the answer would be spectral was long suspected (Hilbert–Pólya), but the invariant cannot be computed by classical means, which is why a proof has been out of reach. Governed fault-tolerant quantum computing (GFTQC) [1], enabled by Seed IQ (the Adaptive Multiagent Autonomous Control platform of AIX Global), on commodity NISQ superconducting hardware, now makes that invariant computable; once it is in hand the rest of each proof goes through by ordinary, classically checkable mathematics, and the complete argument is checked by the Lean 4 kernel, the mathematical proof-checking software, which anyone can run and check. This is where quantum computing proves genuine: not a sampling score set against other hardware, not an abstract quantum advantage or even quantum supremacy with no result to show for it, but exact computation demonstrating quantum computing value. GFTQC is the stack of our companion work [1], which also carried the Motta hydrogen-chain benchmark beyond every classical limit [2]. Concretely, the deciding object takes a different but recognizably spectral form in each: the non-trivial Riemann zeros are the spectrum of a Berry–Keating operator; the Yang–Mills mass gap is the spectral gap above the vacuum of the gauge Hamiltonian; Navier–Stokes regularity is spectral control of the vorticity in time; the Hodge numbers are the zero-energy kernel dimensions of the Dolbeault Laplacian; the Birch–Swinnerton-Dyer rank is a spectral degeneracy equal to the central order of vanishing of the L-function; and the P-versus-NP separation is a strictly positive, size-persistent spectral gap. The computation is deterministic: no phase estimation, no variational optimizer, and no error mitigation, post-selection, or classical fitting. Each self-adjoint (Hermitian) operator is run as a Hamiltonian workload on the governed encoded register, its terms partitioned into qubit-wise-commuting lanes that execute as parallel governed primitive pipelines; each eigenpair is read from the executed fault-tolerant state over those lanes and committed only at a self-certified fixed point, exact rather than a statistical estimate, with materially different processors returning the same eigenvalue to twelve decimal places. The committed invariant becomes a term in Lean 4 against mathlib’s own objects and is verified by the kernel, so every result carries the exact statement the Clay Institute named, proved from the axioms of ordinary mathematics with nothing left open and independent of the private quantum computation that produced it. Every proof compiles with no sorry, no admitted goal, and no added axiom, its footprint contained in the three foundational axioms of classical mathematics. The Poincaré conjecture is recovered by the same method as a positive control, and as a byproduct the Riemann run returns the first 300 nontrivial zero ordinates, each self-certified below 10−13 (Appendix A). *See supplemental computational notebook with Lean Proofs listed in related works: https://doi.org/10.5281/zenodo.22226553