Quantum-Native-Solvers

A compilation of quantum-native solver techniques that can be mapped and ran on a quantum computer. Compiled by Onri Jay Benally.

Primary URL for the repository: https://github.com/OJB-Quantum/Quantum-Native-Solvers

The purpose of this repository is to monitor computational techniques (over time) that can be used to determine whether a working mathematical or computational framework of interest may be eligible for quantum simulation.

Interestingly, some models such as the Landau-Lifshitz-Gilbert (LLG) equation (used in micromagnetism studies) can be systematically derived from Lindbladian dynamics, which are based on the general form of Markovian master equations used to describe open quantum systems. Another quantum analog of the LLG also exists based on quantum correlation dynamics, which acknowledges a difference from open quantum systems in the previous example. LLG solvers are typically implemented on classical computing resources, especially that of general-purpose graphics processing units (GPUs). However, through careful derivation, supported by literature, such a method can transcend the classical description and becomes eligible for quantum simulation algorithms, as well as numerically exact and analytical verifications. Although this is not always the case for other classical or semi-classical frameworks and models, examples like these that are physics-informed or physics-supported should encourage one to explore the limits.

Quantum-native (adj.): designed for, and inherently dependent upon, quantum-mechanical resources, so that its core function, scaling, or correctness requires nonclassical phenomena (for example, superposition, interference, entanglement), rather than merely imitating them on classical hardware.
quantum: “Borrowed from Latin quantum,” historically ‘how much; as much as’ (neuter of quantus), later specialized to a discrete physical amount. - Oxford English Dictionary; see also Oxford’s gloss tying quantum to quantus.
native: “From Latin nativus (‘inborn; produced by birth’), via Middle English/French,” yielding senses such as ‘innate, natural; belonging by birth.’ - Oxford English Dictionary.

Classification Tree for Quantum-Native Solvers/ True Quantum Simulation

(A Quantum Simulation Taxonomy)

Quantum‑Native Simulation Techniques 
├─ I. Lattice‑kinetic/ Automata family
│   ├─ A. Quantum Lattice Boltzmann (QLBM)
│   │   ├─ A1. “Quantum-walk‑equivalent” formulations (QLBM ≈ quantum walk)
│   │   ├─ A2. Carleman‑linearized LB (LB-Carleman, fault‑tolerant oriented)
│   │   ├─ A3. Multi‑circuit QLBM for noisy devices
│   │   └─ A4. Early hardware demos (advection-diffusion on QPU)
│   ├─ B. Quantum Lattice Gas Automata (QLGA)
│   │   ├─ B1. Measurement‑based QLG for Navier-Stokes
│   │   ├─ B2. Efficient QLGA single‑step schemes
│   │   └─ B3. Fully‑quantum LGA building blocks (MHD/NS‑ready)
│   └─ C. Quantum Walks/ Quantum Cellular Automata (QCA)
│       ├─ C1. Dirac‑type dynamics (transport, diffusion, barriers & tunneling)
│       ├─ C2. Schrödinger‑type dynamics (continuous‑time quantum walk ↔ free‑particle propagation,
│       │      discrete‑time quantum walk approximating Laplacian)
│       └─ C3. Gauge‑field‑coupled quantum walk/QCA (U(1), SU(2) links for both Dirac & Schrödinger)
│
├─ II. Quantum PDE & Linear‑Differential Equation solvers
│   ├─ D. Quantum Linear‑System Algorithms (HHL & block‑encoding, Poisson)
│   ├─ E. Linear Differential Equations (LDE) oracles (Berry-Childs-Kothari)
│   ├─ F. Carleman linearization for nonlinear ODE/PDE (Navier-Stokes regimes)
│   ├─ G. Quantum Finite Elements (Qu‑FEM/ Q‑FEM) + q‑Multigrid
│   ├─ H. Dirac‑PDE solvers
│   │   ├─ H1. Quantum-walk‑based Dirac first‑order hyperbolic solvers
│   │   ├─ H2. QLGA spinor kernels for relativistic fluids
│   │   └─ H3. LCU/ qubitization of the Dirac operator (mass + kinetic terms)
│   └─ I. Schrödinger‑PDE solvers
│       ├─ I1. Spectral (Fourier) discretization + Trotter/LCU propagation
│       ├─ I2. Chebyshev‑polynomial and interaction‑picture schemes
│       └─ I3. Variational linear‑solver (VQLS) for stationary Schrödinger equation
│
├─ III. Variational & Hybrid Dynamics
│   ├─ J. Variational Quantum Simulation (VQS: real & imaginary time)
│   ├─ K. Quantum Imaginary‑Time Evolution (QITE) + accelerators
│   ├─ L. Variational Quantum Linear Solver (VQLS) for scattering/PDE
│   ├─ M. Quantum Car‑Parrinello Molecular Dynamics (QCPMD, NISQ)
│   ├─ N. Variational Dirac solvers
│   │   ├─ N1. Real‑time VQS for Dirac spinor dynamics
│   │   └─ N2. QITE for massive Dirac ground‑state preparation
│   └─ O. Variational Schrödinger solvers
│       ├─ O1. Real‑time VQS with hardware‑efficient ansätze for non‑relativistic dynamics
│       └─ O2. Imaginary‑time VQS/ QITE for ground‑state Schrödinger problems
│
├─ IV. Hamiltonian Simulation & Scattering
│   ├─ P. Real‑space electron‑dynamics (LCU/ qubitization/ Trotter)
│   ├─ Q. Scattering & cross‑section algorithms (Green’s functions, S‑matrix)
│   ├─ R. Quantum‑walk tunneling/ barrier transport
│   ├─ S. Dirac‑Hamiltonian simulation
│   │   ├─ S1. Product‑formula (Trotter‑Suzuki) for γ·p + m terms
│   │   ├─ S2. LCU/ qubitization of relativistic kinetic operator
│   │   └─ S3. Interaction‑picture and Dyson‑series methods for time‑dependent Dirac fields
│   └─ T. Schrödinger‑Hamiltonian simulation
│       ├─ T1. Finite‑difference/plane‑wave discretization + Trotter
│       ├─ T2. LCU/ qubitization of kinetic + potential operators
│       └─ T3. Interaction‑picture schemes for driven potentials
│
├─ V. Stochastic & Monte Carlo family
│   ├─ O. Quantum Monte Carlo (classically executed, physics‑native)
│   │   ├─ O1. Variational Monte Carlo (VMC; |Ψ|² sampling)
│   │   ├─ O2. Diffusion Monte Carlo (DMC; imaginary‑time projection, fixed‑node)
│   │   ├─ O3. Path‑Integral QMC (PIMC; finite‑T & ground‑state PIGS)
│   │   ├─ O4. Auxiliary‑Field QMC (AFQMC; Hubbard-Stratonovich)
│   │   └─ O5. Stochastic Series Expansion (SSE; loop/cluster updates)
│   ├─ P. Quantum‑accelerated Monte Carlo (hardware‑native)
│   │   ├─ P1. Quantum Amplitude Estimation (QAE; quadratic sampling speedups)
│   │   ├─ P2. Quantum Metropolis sampling (thermal states without sign problem)
│   │   └─ P3. Hybrid QMC w/ QC‑generated trials (e.g., QSCI → AFQMC)
│   └─ Q. Quantum Trajectories (Monte Carlo wave function, stochastic Schrödinger)
│       ├─ Q1. Quantum jumps/ MCWF unraveling of Lindblad
│       └─ Q2. Adiabatic‑master‑equation trajectories; restricted‑MCWF diagnostics
│   ├─ R. Schrödinger‑QMC variants
│   │   ├─ R1. Diffusion Monte Carlo with quantum‑enhanced trial‑wave‑function optimization
│   │   └─ R2. Path‑Integral QMC for real‑time (Keldysh) Schrödinger evolution
│   └─ S. Dirac‑QMC variants
│       ├─ S1. Stochastic Dirac equation (stochastic quantization of relativistic fields)
│       └─ S2. Quantum‑walk based Monte Carlo for relativistic barrier tunneling
│
├─ VI. Open‑System (Lindblad) solvers & Magnetization links
│   ├─ R. Lindblad simulators on quantum hardware
│   │   ├─ R1. LCU/ Stinespring‑dilation & qubitization‑style algorithms
│   │   ├─ R2. Randomized product‑formula & sampling compilers for GKSL
│   │   └─ R3. Quantum‑trajectory simulators with additive O(T + log 1/ε) scaling (class‑dependent)
│   ├─ S. Classical‑HPC Lindblad solvers with quantum trajectories (large‑N pure‑state sampling)
│   └─ T. Lindblad → LL/LLG mappings for magnetization dynamics
│       ├─ T1. LL from Lindbladian dynamics (scale‑separated local mean‑field regimes)
│       ├─ T2. Microscopic LLG coefficients via NEGF/scattering formulations
│       └─ T3. Quantum‑LLG analogs and constraints from entanglement/ non‑Markovianity
│
├─ VII. Emerging quantum‑native simulation families
│   ├─ A. Quantum Tensor‑Network Simulators (MPS, PEPS, TTN) for PDEs & many‑body states
│   ├─ B. Quantum Signal Processing/ QSVT‑based PDE solvers
│   │        (Chebyshev and polynomial transformations for evolution and linear systems)
│   ├─ C. Fractional & Non‑local Operator Solvers
│   │        (fractional Laplacian, Riesz derivatives, etc.)
│   ├─ D. Quantum Stochastic Differential Equations (QSDE) &
│   │        non‑Markovian open‑system dynamics
│   ├─ E. Quantum Lattice‑Gauge‑Theory Engines
│   │        (U(1), SU(2), SU(3) gauge fields, plaquette unitaries)
│   ├─ F. Quantum‑Accelerated Adaptive‑Mesh‑Refinement (AMR)
│   │        (amplitude‑estimation‑driven error indicators)
│   ├─ G. Quantum‑Enhanced Real‑Time Time‑Dependent Density‑Functional Theory (RT‑TDDFT)
│   ├─ H. Quantum Wigner‑Function/ Phase‑Space Solvers
│   │        (discrete Wigner representation + FFT‑based evolution)
│   ├─ I. Quantum Uncertainty Quantification (UQ) for PDEs
│   │        (parameter‑sampling via QAE)
│   ├─ J. Quantum Spectral‑Element/ hp‑Finite‑Element Methods
│   │        (high‑order basis encoded in qubit registers)
│   ├─ K. Quantum Metropolis‑Hastings for Thermodynamic Ensembles of PDE Solutions
│   ├─ L. Hybrid Quantum‑Classical Domain Decomposition (Schwarz) for large‑scale PDEs
│   ├─ M. Quantum‑Accelerated Sparse‑Matrix Preconditioners
│   │        (quantum BiCGSTAB, quantum multigrid)
│   └─ N. Quantum Embedding & Multi‑Scale Methods
│           (DMFT, QM/MM, impurity solvers, quantum‑accelerated self‑energy evaluation)
│
├─ VIII. Dirac Equation Solvers (relativistic quantum dynamics)
│   ├─ A. Quantum‑Walk‑based Dirac solvers
│   │   ├─ A1. 1‑D split‑step Dirac quantum walk (continuous limit → Dirac equation)
│   │   ├─ A2. 2‑D/ 3‑D multi‑step Dirac quantum walks (including spin‑orbit coupling)
│   │   ├─ A3. Gauge‑field‑coupled Dirac quantum walks (U(1) electromagnetic, SU(2) weak)
│   │   └─ A4. Multi‑circuit Dirac quantum walks for NISQ devices (error‑mitigated implementations)
│   ├─ B. Quantum Lattice‑Gas Automata for Dirac dynamics
│   │   ├─ B1. Spinor QLGA kernels (γ‑matrix encoding on qubits)
│   │   ├─ B2. Dirac-Maxwell QLGA (self‑consistent coupling of spinor and EM fields)
│   │   └─ B3. Relativistic fluid‑LGA (Dirac‑MHD, relativistic Navier-Stokes)
│   ├─ C. Hamiltonian‑simulation of the Dirac operator
│   │   ├─ C1. Trotter‑Suzuki product formulas for γ·p + m terms
│   │   ├─ C2. LCU/ qubitization of the Dirac Hamiltonian (including mass‑gap)
│   │   ├─ C3. Interaction‑picture/ Dyson‑series Dirac simulators for time‑dependent fields
│   │   └─ C4. Spectral‑method (Fourier) Dirac simulation (momentum‑space discretization)
│   ├─ D. Variational & Imaginary‑time Dirac solvers
│   │   ├─ D1. Real‑time VQS for Dirac spinors (hardware‑efficient ansätze)
│   │   ├─ D2. QITE for massive Dirac ground‑state preparation
│   │   └─ D3. Adaptive‑VQE style Dirac dynamics (ADAPT‑VQD)
│   ├─ E. Carleman‑linearized Dirac‑type nonlinear models
│   │   ├─ E1. Dirac-Klein‑Gordon coupling (scalar‑field interactions)
│   │   ├─ E2. Non‑linear Dirac (Thirring, Gross‑Neveu) via Carleman embedding
│   │   └─ E3. Dirac-Maxwell-Higgs hybrid systems
│   ├─ F. Quantum Finite‑Element/ Multigrid for Dirac
│   │   ├─ F1. QFEM‑Dirac (mesh‑based spinor discretization)
│   │   └─ F2. Dirac‑multigrid preconditioners realized with quantum subroutines
│   ├─ G. Scattering & S‑matrix for relativistic particles
│   │   ├─ G1. Green‑function based Dirac cross‑section algorithms
│   │   ├─ G2. Relativistic quantum‑walk tunneling analysis (Klein paradox)
│   │   └─ G3. Dirac‑time‑dependent scattering (time‑varying potentials)
│   └─ H. Hybrid classical‑quantum Dirac workflows
│       ├─ H1. Classical spectral Dirac solver + quantum subroutines for high‑frequency modes
│       └─ H2. Domain‑decomposition: QC solves interior relativistic region, classical for exterior
│
└─ IX. Schrödinger Equation Solvers (non‑relativistic quantum dynamics)
    ├─ A. Direct Hamiltonian‑simulation of Schrödinger dynamics
    │   ├─ A1. Spatial discretization (finite‑difference, plane‑wave) + Trotter‑Suzuki
    │   ├─ A2. LCU/ qubitization of kinetic (∇²) + potential V(x) operators
    │   ├─ A3. Interaction‑picture/ Dyson‑series approaches for time‑dependent potentials
    │   └─ A4. High‑order product formulas (e.g., Forest‑Ruth, Suzuki‑23) for improved accuracy
    ├─ B. Quantum‑PDE solvers specialized for Schrödinger
    │   ├─ B1. Linear‑Differential‑Equation oracle tailored to time‑dependent Schrödinger equation
    │   ├─ B2. Spectral (Fourier) method via Quantum Fourier Transform (QFT) - plane‑wave basis
    │   ├─ B3. Chebyshev‑polynomial expansion of the propagator e^{-iHt}
    │   └─ B4. Pseudo‑spectral split‑operator schemes (FFT‑based) on quantum hardware
    ├─ C. Variational & Hybrid Schrödinger solvers
    │   ├─ C1. Real‑time VQS with hardware‑efficient ansätze (e.g., hardware‑efficient, ADAPT‑VQE)
    │   ├─ C2. Imaginary‑time VQS/ QITE for ground‑state preparation of Schrödinger Hamiltonians
    │   ├─ C3. Variational Quantum Linear Solver (VQLS) for stationary Schrödinger equation (Ax = b with A = H−E)
    │   └─ C4. Adaptive‑VQD for excited‑state dynamics
    ├─ D. Quantum‑Walk implementations of Schrödinger dynamics
    │   ├─ D1. Continuous‑time quantum walk mapping to free‑particle Schrödinger (H = −Δ)
    │   ├─ D2. Discrete‑time quantum walk reproducing Laplacian kinetic term via coin‑shift construction
    │   └─ D3. Quantum‑walk based tunneling simulations for arbitrary barrier profiles
    ├─ E. Quantum Cellular Automata for Schrödinger equation
    │   ├─ E1. Local unitary update rules approximating the Laplacian operator
    │   └─ E2. Multi‑step QCA with on‑site potential encoding
    ├─ F. Quantum Monte Carlo methods for Schrödinger equation
    │   ├─ F1. Diffusion Monte Carlo (imaginary‑time projection) with QC‑accelerated trial‑wave‑function refinement
    │   ├─ F2. Path‑Integral QMC for real‑time evolution (Keldysh contour) using quantum‑enhanced sampling
    │   ├─ F3. Variational Monte Carlo with quantum‑generated ansätze (QANSAT)
    │   └─ F4. Reptation QMC with quantum‑assisted move proposals
    ├─ G. Many‑body Schrödinger solvers (second‑quantized)
    │   ├─ G1. Electronic‑structure Hamiltonian simulation (quantum chemistry) via LCU/ qubitization
    │   ├─ G2. Quantum Dynamical Mean‑Field Theory (QDMFT) for lattice models
    │   ├─ G3. Real‑time Green‑function methods (Kadanoff‑Baym) on quantum processors
    │   └─ G4. Tensor‑network‑inspired hybrid algorithms (e.g., VQE‑DMRG hybrids)
    ├─ H. Hybrid Classical‑Quantum Schrödinger solvers
    │   ├─ H1. Classical spectral (Fourier) solver + quantum subroutine for high‑frequency phase propagation
    │   ├─ H2. Domain‑decomposition: QC handles sub‑grid potentials; classical handles bulk
    │   └─ H3. Multi‑scale coupling of quantum‑accelerated short‑range dynamics with classical long‑range Poisson solver
    ├─ I. Scattering & S‑matrix for non‑relativistic particles
    │   ├─ I1. Green‑function based cross‑section computation (Lippmann‑Schwinger equation)
    │   ├─ I2. Quantum‑walk tunneling analysis for arbitrary 1‑D/ 2‑D barrier profiles
    │   └─ I3. Real‑time S‑matrix extraction via phase‑estimation on scattered wave‑packets
    └─ J. Quantum Finite‑Element/ Multigrid for Schrödinger
        ├─ J1. Mesh‑based Schrödinger solvers with quantum enforcement of continuity/ boundary conditions
        ├─ J2. Quantum‑accelerated multigrid preconditioners for the Laplacian‑plus‑potential operator
        └─ J3. Adaptive mesh refinement driven by quantum‑measured error estimates

Quantum‑Native Solver/ Simulation Techniques - Ordered by Hardware‑Readiness

Compact Version

Tier	Typical depth/ qubits	Representative methods
I. NISQ-ready	Shallow circuits, few qubits (today’s IBMQ Rigetti, IonQ)	Quantum walks, QLBM demos, VQS/QITE, tiny HHL demos, basic amplitude-estimation, minimal Lindblad simulators
II. Near-term/ hybrid	Moderate error-correction, multi-circuit layouts	Multi-circuit QLBM, LDE oracles, tiny Q-FEM, prototype QMetropolis, quantum-accelerated Monte Carlo, photonic-co-processor variants
III. Fault-tolerant eligible	Full error-corrected logical qubits, deep circuits	Full-scale HHL/ block-encoding, qubitized Dirac/Schrödinger Hamiltonians, QSVT-based PDE solvers, quantum-enhanced AMR, large-scale tensor-network simulation

Extended Version

I. NISQ‑ready (shallow‑depth, few‑qubit) techniques

These algorithms can be executed today on IBM Q, Rigetti, IonQ, or any other gate‑model device with error‑mitigation. All required primitives are already available in Qiskit ≥ 2.2 (e.g. QuantumWalkCircuit, VQSDynamics, QITE, AmplitudeEstimation, LindbladEvolution).

Category	Techniques
A. Quantum walks & quantum cellular automata	1. Dirac‑type dynamics (split‑step quantum walk) 2. Schrödinger‑type dynamics (continuous‑time quantum walk + discrete‑time Laplacian quantum walk) 3. Simple U(1) gauge‑field coupling (single‑link)
B. Quantum lattice‑Boltzmann/ lattice‑gas (toy lattices)	1. “Quantum-walk‑equivalent’’ QLBM 2. Early hardware demos (advection-diffusion on a few‑site lattice) 3. Efficient single‑step QLGA for 1‑D/2‑D Navier-Stokes (few sites)
C. Variational & hybrid dynamics	1. Real‑time Variational Quantum Simulation (VQS) 2. Quantum Imaginary‑Time Evolution (QITE) 3. Variational Quantum Linear Solver (VQLS) for small linear systems 4. Real‑time VQS for Dirac or Schrödinger spinors with hardware‑efficient ansätze 5. Imaginary‑time VQS/ QITE for Dirac or Schrödinger ground‑state preparation
D. Direct Hamiltonian‑simulation (small grids)	1. Spectral (Fourier) discretization + Trotter for 1‑D Schrödinger 2. quantum-walk‑based Dirac first‑order hyperbolic solver (few lattice sites) 3. Quantum‑walk tunnelling/ barrier transport (single‑step quantum walk)
E. Simple linear‑system/ HHL demos	1. HHL for a 2 × 2 or 4 × 4 Poisson‑type matrix (demonstration only)
F. Quantum‑accelerated Monte Carlo (basic)	1. Iterative Amplitude Estimation (IAE) for expectation‑value speed‑up 2. Quantum jump/ Monte‑Carlo wave‑function (MCWF) for a single Lindblad trajectory (few qubits)
G. Minimal Lindblad simulators	1. Randomised product‑formula simulation of short‑time GKSL dynamics (≤ 5 qubits)
H. Other NISQ‑ready families	1. Quantum tensor‑network simulators (MPS/PEPS with shallow circuits) 2. QSP/QSVT for low‑degree polynomial approximations (e.g. low‑order Chebyshev filters) 3. Quantum algorithms for fractional Laplacian on very small lattices (Fourier‑space LCU) 4. Stochastic Schrödinger‑equation discretizations for a few modes (QSDE prototype)

II. Near‑term/ hybrid (requires modest error‑correction, multi‑circuit layouts, or classical‑quantum co‑processing)

These approaches are still limited on today’s noisy devices but become practical with modest error‑mitigation, circuit parallelism, or when used as sub‑routines inside a classical workflow (e.g. hybrid QMC, domain‑decomposition).

Category	Techniques
A. Multi‑circuit lattice‑kinetic methods	1. Multi‑circuit QLBM for noisy devices (parallel streams) 2. Measurement‑based QLGA for Navier-Stokes (classical feedback after each step)
B. Linear‑differential‑equation & Carleman‑linearization prototypes	1. LDE oracles (Berry-Childs-Kothari) as proof‑of‑concept 2. Carleman linearization for low‑order Navier-Stokes regimes (small lifted dimension)
C. Quantum finite‑element/ multigrid (tiny meshes)	1. Qu‑FEM/ Q‑FEM for 2‑D Poisson on ≤ 4 × 4 meshes 2. QFEM‑Dirac for a 2‑D spinor mesh (≤ 8 qubits)
D. LCU/ qubitization for modest problem sizes	1. LCU/ qubitization of the Dirac operator on a 2 × 2 lattice 2. VQLS for stationary Schrödinger on ≤ 8‑dimensional basis
E. Hamiltonian simulation (intermediate depth)	1. Trotter‑Suzuki product formula for Dirac kinetic + mass on a 3‑site lattice 2. Finite‑difference Schrödinger Trotter on a 4‑site grid
F. Hybrid quantum‑accelerated Monte Carlo	1. Prototype Quantum Metropolis (single‑spin Ising) 2. Hybrid QMC where the quantum computer supplies trial wavefunctions for AFQMC 3. QAE‑based variance reduction inside Diffusion Monte Carlo 4. Stochastic Dirac‑equation sampling on a few momentum modes
G. Quantum trajectories with modest scaling	1. Additive O(T + log 1/ε) trajectory algorithm for a 2‑qubit Lindbladian (research prototype)
H. Quantum Car‑Parrinello Molecular Dynamics (NISQ prototype)	1. QCPMD for a single‑atom, few‑electron system (small basis)
I. Other near‑term families	1. Quantum lattice‑gauge‑theory engines for 1‑D U(1) link models (few links) 2. Quantum‑accelerated adaptive‑mesh‑refinement on a 2‑D grid (error estimate via IAE) 3. Quantum‑enhanced real‑time TDDFT for a 2‑electron molecule (VQE‑derived Kohn‑Sham potential) 4. Wigner‑function evolution on a 2‑point phase‑space (QFT‑based) 5. Quantum‑accelerated uncertainty quantification for a 1‑D diffusion PDE (QAE sampling of input parameters)

III. Fault‑tolerant eligible (requires error‑corrected hardware, large ancilla registers, deep circuits)

These algorithms deliver asymptotic speed‑ups or polynomial‑time reductions that become advantageous only when a full‑scale fault‑tolerant quantum computer is available. Qiskit provides the low‑level building blocks (LinearCombinationOfUnitaries, SelectOracle, PhaseEstimation, QSVT), but high‑level wrappers are still under development.

Category	Techniques
A. Large‑scale linear‑system solvers	1. Full‑scale HHL/ block‑encoding for Poisson matrices of size 2ⁿ × 2ⁿ 2. Fault‑tolerant VQLS for high‑dimensional Schrödinger eigenproblems
B. Dirac & Schrödinger Hamiltonian simulation (qubitization)	1. Qubitization of the relativistic kinetic operator for 3‑D lattices 2. Qubitization of kinetic + potential for high‑resolution Schrödinger grids 3. Full LCU implementation of the Dirac Hamiltonian (mass + gauge coupling) 4. LCU/ qubitization of Schrödinger kinetic + potential
C. Scattering & S‑matrix algorithms	1. Green’s‑function based cross‑section computation using a quantum resolvent (QPE or QSVT) 2. Dirac scattering (Klein‑paradox) with QSVT‑based resolvent 3. Non‑relativistic Lippmann‑Schwinger solver via QSVT
D. Quantum signal‑processing/ QSVT frameworks	1. High‑degree Chebyshev and polynomial approximations for time‑evolution and matrix inversion (generic PDEs) 2. QSP‑based fractional‑operator solvers for non‑local dynamics
E. Quantum lattice‑gauge‑theory (full models)	1. Digital simulation of 2‑D/3‑D U(1) and SU(2) gauge fields with plaquette operators (requires many qubits and T‑gate depth)
F. Quantum stochastic differential equations & non‑Markovian open systems	1. QSDE simulation using Stinespring dilation and phase‑estimation (large ancilla) 2. LCU/ Stinespring dilation of general GKSL generators
G. Quantum‑accelerated multigrid & preconditioners	1. Quantum multigrid V‑cycle as a sub‑routine for linear‑system solvers (requires fault‑tolerant VQLS) 2. Quantum‑accelerated sparse‑matrix preconditioners (quantum BiCGSTAB, quantum multigrid)
H. Quantum‑enhanced adaptive‑mesh‑refinement	1. AMR driven by quantum‑estimated error norms (QAE required at each refinement step)
I. High‑order spectral‑element/ hp‑FEM	1. Quantum spectral‑element method with hp‑refinement (mesh encoded in qubits, block‑encoded stiffness matrices)
J. Quantum tensor‑network simulation at scale	1. Large‑scale MPS/PEPS evolution using quantum circuits (requires fault‑tolerant resources for bond‑dimension growth)
K. Quantum Metropolis‑Hastings for generic distributions	1. Full‑scale Quantum Metropolis algorithm (detailed‑balance via phase estimation)
L. Hybrid quantum‑classical domain decomposition	1. Schwarz‑type decomposition where each subdomain is solved with a fault‑tolerant quantum linear‑solver and the interface is handled classically
M. Quantum‑enhanced real‑time TDDFT & many‑body dynamics	1. Fully fault‑tolerant RT‑TDDFT where the Kohn‑Sham potential and the electron density are updated via quantum sub‑routines (qubitization of the KS Hamiltonian)
N. Wigner‑function/ phase‑space quantum dynamics	1. Exact quantum‑Fourier‑transform‑based evolution of high‑dimensional Wigner functions (large QFT registers required)
O. Quantum uncertainty quantification for large‑scale PDEs	1. Propagation of input‑parameter distributions through quantum‑accelerated forward models (QAE + QSVT)
P. Quantum‑accelerated scattering with many‑body final states	1. Dirac time‑dependent scattering with multi‑particle production (requires fault‑tolerant many‑body state preparation)
Q. Quantum‑enhanced Linear‑Landau‑Lifshitz‑Gilbert (LLG) mapping	1. Extraction of microscopic LLG coefficients from entanglement‑aware quantum simulations (requires full open‑system simulation + phase estimation)

An Example Architecture of What a Fault-Tolerant Heterogenous/ Hybrid Quantum System Could Look Like - by Onri

Heterogeneous Quantum Computer (Architected for Fault-Tolerant Compatibility)
├─ Main QPU: Neutral-Atom (Rydberg/ tweezer)
│   ├─ Fast parallel CZ gates (~99.5%); long coherence (≈12.6 s, hyperfine)
│   ├─ Reconfigurable layouts; mid-circuit measurement & erasure
│   └─ On-chip nanophotonics for coupling/ imaging
├─ Co-Processor: Quantum Photonic IC (QPIC)
│   ├─ Time-bin/ cluster-state generation; fusion operations
│   ├─ Waveguide arrays, thin-film LiNbO₃ modulators, PNR detectors
│   └─ Fiber network to other racks/ fridges
├─ Translator(s): μw ↔ optical
│   ├─ Electro-optic (LiNbO₃), opto-mechanical, Rydberg ensembles
│   ├─ Targets: internal η ≥ 0.1-0.5, added noise ≲ 1 photon
│   └─ With JPA/ JTWPA pre-amps (paramps), pump-noise filtering
├─ Quantum Memory: Superconducting Cat (bosonic)
│   ├─ Passive bit-flip suppression (noise bias)
│   ├─ Repetition-cat outer code; bias-preserving gates (SNAP-enabled)
│   └─ Logical store/ refresh; interface to ancilla
├─ Ancilla Layer: Transmons (tight to cat memories & readout)
│   ├─ Microwave-tunable Transmon (all-microwave, fixed-freq)
│   │   ├─ Effective ZZ/ CZ tuning via microwave dressing (MAP/ CR/ MATC-style)
│   │   └─ Flux-noise immunity; no DC-flux lines; compatible with fixed-freq layouts
│   └─ Voltage-tunable Transmon (advanced gatemon)
│       ├─ Frequency agility via electrostatic gating (semiconductor JJ)
│       └─ Syndrome extraction, parity checks, and cavity SNAP orchestration
└─ Classical Control & RAM
    ├─ Cryo-CMOS SRAM/ FBRAM/ MRAM/ GC-eDRAM near 4-12 K; microcode/ waveform cache
    ├─ Single-Flux-Quantum Digital (4 K stage; higher-integration families)
    │   ├─ RSFQ (DC-biased; ultrafast legacy baseline; static-power overhead); reference-grade timing/ waveform source or local clocks
    │   ├─ eSFQ/ ERSFQ (DC-biased, zero static power; preferred for scalable SFQ logic & SFQ-DACs/ JAWS)
    │   └─ RQL (AC-powered; no on-chip static power; multi-phase AC clock/ power; AC/ DC converters as needed)
    │       • SFQ-based AWG/ DAC for qubit drive (JAWS); isolate to mitigate quasiparticles; near-deterministic timing
    └─ Compiler for biased/ erasure noise and transduction-aware routing

Decision Tree to Determine Which Quantum-Native Solver/ Simulation Techniques Could Run on an Proposed Fault-Tolerant Heterogenous Quantum Computer

Fault‑tolerant heterogeneous/hybrid quantum system
├─ Main QPU: Neutral‑Atom (Rydberg/ tweezer)
│   ├─ NISQ‑ready techniques (shallow circuits, few qubits)
│   │   ├─ I.C.1  Dirac‑type quantum‑walk/ QCA
│   │   ├─ I.C.2  Schrödinger‑type quantum‑walk/ QCA
│   │   ├─ I.C.3  Simple gauge‑field‑coupled quantum walk/QCA (U(1) link)
│   │   ├─ I.A.1  “Quantum-walk‑equivalent’’ Quantum Lattice‑Boltzmann (QLBM)
│   │   ├─ I.B.3  Fully‑quantum Lattice‑Gas Automata (MHD/NS‑ready)
│   │   ├─ III.J  Variational Quantum Simulation (VQS, real‑time)
│   │   ├─ III.K  Quantum Imaginary‑Time Evolution (QITE)
│   │   ├─ III.L  Variational Quantum Linear Solver (VQLS, small systems)
│   │   ├─ III.N.1 Real‑time VQS for Dirac spinors
│   │   ├─ III.O.1 Real‑time VQS for Schrödinger spinors
│   │   ├─ IV.P   Real‑space electron‑dynamics (short‑depth Trotter)
│   │   ├─ IV.R   Quantum‑walk tunnelling/ barrier transport
│   │   ├─ V.P1   Iterative Amplitude Estimation (IAE)
│   │   └─ VI.R2  Randomised product‑formula GKSL simulation (≤ 5 qubits)
│   │
│   ├─ Near‑term/ hybrid techniques (moderate error‑correction, multi‑circuit)
│   │   ├─ I.A.3  Multi‑circuit QLBM for noisy devices (parallel streams)
│   │   ├─ I.B.1  Measurement‑based QLGA for Navier-Stokes (mid‑circuit readout)
│   │   ├─ II.E   Linear‑Differential‑Equation (LDE) oracles (Berry‑Childs‑Kothari)
│   │   ├─ II.F   Carleman‑linearized ODE/PDE prototypes (low‑order Navier-Stokes)
│   │   ├─ II.G   Quantum Finite‑Element/ Q‑FEM on tiny meshes (≤ 4×4)
│   │   ├─ II.H.1 Quantum walk‑based Dirac first‑order hyperbolic solver (few sites)
│   │   ├─ II.I.1 Spectral Schrödinger solver (small Fourier grid)
│   │   ├─ III.M   Quantum Car‑Parrinello MD (prototype, few electrons)
│   │   ├─ IV.S.1 Trotter‑Suzuki Dirac kinetic + mass (3‑site lattice)
│   │   ├─ IV.T.1 Finite‑difference Schrödinger Trotter (4‑site grid)
│   │   ├─ V.P2   Prototype Quantum Metropolis sampler (single‑spin Ising)
│   │   ├─ V.P3   Hybrid QMC - QC‑generated trial wavefunctions for AFQMC
│   │   └─ VI.R3   Additive O(T + log 1/ε) quantum‑trajectory algorithm (research demo)
│   │
│   └─ Fault‑tolerant‑eligible techniques (require logical qubits, bias‑preserving gates)
│       ├─ II.D    Full‑scale HHL/ block‑encoding linear‑system solver
│       ├─ II.E    Large‑scale LDE oracle (high‑precision phase estimation)
│       ├─ II.F    Complete Carleman‑linearization for nonlinear PDEs
│       ├─ II.G    Full Qu‑FEM + quantum multigrid preconditioner
│       ├─ IV.S.2  Qubitization of Dirac kinetic operator (3‑D lattice)
│       ├─ IV.T.2  Qubitization of Schrödinger kinetic + potential
│       ├─ IV.Q    Scattering & S‑matrix via Green’s‑function resolvent (QSVT/QSP)
│       ├─ IV.S.3  High‑order Dirac product‑formula (Forest‑Ruth, Suzuki‑23)
│       ├─ IV.T.3  Interaction‑picture Schrödinger simulation (time‑dependent V)
│       ├─ V.S.1   Stochastic Dirac‑equation (QSDE) Monte Carlo
│       ├─ V.R.1   Diffusion Monte Carlo with QC‑enhanced trial wavefunctions
│       ├─ VI.R1   LCU/ Stinespring‑dilation implementation of GKSL Lindbladians
│       ├─ VI.T1‑T3  Lindblad → LL/ LLG mappings (microscopic coefficient extraction)
│       ├─ VII.A.*  All Dirac‑walk families (A1-A4)
│       ├─ VII.B.*  Dirac QLGA kernels (B1-B3)
│       ├─ VII.C.*  Dirac Hamiltonian simulation (C1-C4)
│       ├─ VII.D.*  Variational & imaginary‑time Dirac solvers (D1-D3)
│       ├─ VII.E.*  Carleman‑linearized Dirac nonlinear models (E1-E3)
│       ├─ VII.F.*  Dirac FEM/ multigrid (F1-F2)
│       ├─ VII.G.*  Dirac scattering & S‑matrix (G1-G3)
│       ├─ VII.H.*  Hybrid classical‑quantum Dirac workflows (H1-H2)
│       ├─ VIII.A.*  Schrödinger Hamiltonian simulation (A1-A4)
│       ├─ VIII.B.*  Schrödinger‑specific QPDE solvers (B1-B4)
│       ├─ VIII.C.*  Variational Schrödinger techniques (C1-C4)
│       ├─ VIII.D.*  Quantum‑walk implementations of Schrödinger dynamics (D1-D3)
│       ├─ VIII.E.*  Quantum Cellular Automata for Schrödinger (E1-E2)
│       ├─ VIII.F.*  Schrödinger Quantum Monte Carlo (F1-F4)
│       ├─ VIII.G.*  Many‑body Schrödinger solvers (G1-G4)
│       ├─ VIII.H.*  Hybrid classical‑quantum Schrödinger workflows (H1-H3)
│       ├─ VIII.I.*  Scattering & S‑matrix for non‑relativistic particles (I1-I3)
│       └─ VIII.J.*  Schrödinger FEM/ multigrid (J1-J3)
│
├─ Co‑Processor: Quantum Photonic IC (QPIC)
│   ├─ NISQ‑ready photonic techniques
│   │   ├─ I.C.1   Dirac‑type quantum walk in time‑bin encoding
│   │   ├─ I.C.2   Schrödinger‑type continuous‑time quantum walk
│   │   ├─ I.B.1   Measurement‑based QLGA (cluster‑state generation)
│   │   └─ V.P1    Iterative Amplitude Estimation with single‑photon detectors
│   │
│   ├─ Near‑term/ hybrid photonic techniques
│   │   ├─ V.P2    Quantum Metropolis sampler realised with linear‑optical interferometers
│   │   ├─ V.P3    Hybrid QMC - photonic circuit produces trial wavefunctions for AFQMC
│   │   └─ II.E    Prototype photonic LDE oracle (cluster‑state linear ODE)
│   │
│   └─ Fault‑tolerant‑eligible photonic techniques
│       ├─ VI.R1   LCU/ Stinespring‑dilation of GKSL Lindbladians using bosonic cat encodings
│       └─ VIII.F.1 Bosonic Diffusion Monte Carlo (photon‑number‑encoded walkers)
│
├─ Translator(s)  μw ↔ optical
│   └─ Enables cross‑platform operations:
│       ├─ Swap neutral‑atom‑prepared states into the photonic QPIC for measurement‑based QLGA
│       ├─ Transfer photonic trial wavefunctions into the neutral‑atom QPU for VQS/ QITE loops
│       └─ Carry Lindblad jump outcomes between the two layers for hybrid trajectory simulations
│
├─ Quantum Memory: Superconducting Cat (bosonic)
│   ├─ Near‑term/ hybrid use
│   │   └─ Store intermediate probability distributions for multi‑circuit QLBM (I.A.3) between Trotter steps
│   └─ Fault‑tolerant‑eligible use
│       ├─ Logical storage for large‑scale HHL/VQLS registers (II.D, III.L)
│       ├─ Buffer for Dirac/ Schrödinger qubitization registers (IV.S.2, IV.T.2)
│       ├─ Hold error‑estimate registers for quantum‑driven AMR (VII.F.2, VIII.J.3)
│       └─ Preserve many‑body wavefunctions for quantum‑dynamical mean‑field theory (VIII.G.2)
│
├─ Ancilla Layer: Transmons (bias‑preserving, syndrome extraction)
│   └─ Provide fault‑tolerant CZ/ZZ and SNAP‑enabled gates used by all LCU/ qubitization kernels in the FT‑eligible branch
│
└─ Classical Control & RAM
    ├─ Cryogenic‑CMOS/ SFQ waveform engines compile bias‑aware, erasure‑aware circuits for every technique
    ├─ Run the outer optimization loops for VQS/ QITE/ VQLS (III.J-III.O)
    ├─ Dispatch hybrid QMC/ Metropolis proposals to the photonic co‑processor
    └─ Coordinate domain‑decomposition (VIII.H, VII.H) and adaptive‑mesh‑refinement decisions (VII.F.2, VIII.J.3)

Estimated Metrics for Qubit Count Per Example Technique

d = 7 → ~98 physical/logical
d = 13 → ~338 physical/logical
Occupancy mapping: for 2D D2Q5, 5M² qubits for an M×M lattice (one per discrete velocity population)

A) Tier I - NISQ‑ready (physical qubits, shallow circuits)

Description: Estimated qubit counts for methods that can be run on current NISQ hardware without error correction.

Method	Qubit estimate (today’s NISQ)	How the count was derived	Representative instance
Quantum walks	6–14 qubits typical; formula ≈ ⌈log₂ N⌉ + 2	Position register + coin + one small ancilla	N = 1,024 nodes → 10 (pos) + 1 (coin) + 1 (anc) = 12
QLBM demos – occupancy encoding	5 M² qubits (scales with lattice size)	D2Q5 mapping: one qubit per discrete‑velocity population	M = 8 → 5 · 64 = 320
QLBM demos – amplitude‑encoded	~10 qubits (≈9 data + 1 ancilla)	Amplitude/state‑encoding keeps width constant	Demonstrations use 9 + 1 qubits
VQS/ QITE	4–20 qubits	VQE chemistry demos: H₂ (4 q), LiH (12 q); QITE demos up to ~20 q	H₂ minimal basis (4 q), LiH frozen‑core (12 q)
Tiny HHL demos	4–8 qubits	Small 2×2–4×4 linear systems using QPE + ancillas	Early NMR/photonic/IBM‑style proofs of concept
Basic amplitude estimation	0–1 ancilla beyond problem registers	Low‑depth variants avoid large phase‑estimation registers	ML‑AE and IQAE use 0–1 extra qubit
Minimal Lindblad simulators	2–4 qubits for one system qubit + environment ancilla(s)	Stinespring/Kraus dilation for a single‑qubit channel	Amplitude damping on one qubit → 2 total; more channels add ancillas

B) Tier II - Near‑term/Hybrid (logical qubits, modest error correction)

Description: Logical‑qubit estimates for methods that employ modest error correction (e.g., surface‑code distance d = 7 or 13) and may involve multiple circuit stages.

Method	Algorithmic logical qubits (typical)	Physical qubits @ d = 7	Physical qubits @ d = 13	Reasoning
Multi‑circuit QLBM (amplitude‑encoded, with mid‑circuit measure/reset)	~10–20 logical	~980–1,960	~3,380–6,760	Uses ~9 data + 1 ancilla kernel; multi‑circuit orchestration reuses the small kernel
Linear‑differential‑equation (LDE) oracles in a QLSS/QSVT pipeline	n + 3–10 (n = log₂ grid DOFs) → e.g. ~32–64 logical	~3,100–6,300	~10,800–21,600	QLSS uses n algorithmic qubits plus ancillas for block‑encoding and QSP
Tiny Q‑FEM prototypes (hybrid FEM + VQLS)	~8–24 logical	~784–2,352	~2,704–8,112	Near‑term VQLS FEM and hybrid CFD/FEM demos show few‑to‑few‑dozen‑qubit runs
Prototype QMetropolis/ QMS	n + r + 2–5 → ~24–64 logical	~2,400–6,300	~8,100–21,600	System register n, r‑bit energy precision, plus small ancillas for accept–reject
Quantum‑accelerated Monte Carlo (QAE‑based risk/option demos, shallow variants)	10–40 logical	~980–3,920	~3,380–13,520	Demos typically use 5–10 qubits for distributions, a payoff register, and 0–1 ancilla
Photonic co‑processor variants	Not qubit‑encoded (modes/photons)	N/A	N/A	Photonic sampling/counting uses modes/photons rather than qubits; surface‑code overhead not applicable

C) Tier III - Fault‑tolerant eligible (full error correction)

Description: Logical‑qubit counts for fully fault‑tolerant algorithms, together with physical‑qubit totals for surface‑code distances d = 7 and d = 13.

Method	Representative problem & logical qubits	Physical @ d = 7	Physical @ d = 13	Notes
Full‑scale HHL/ block‑encoding linear solver	Sparse N = 2²⁰ (n = 20) with QSVT/QLSS → ~30–60 logical (20 data + 1–3 QSP ancillas + small oracles)	~2,900–5,900	~10,100–20,300	Low‑ancilla block‑encoding/QSP; depth/T‑count dominate cost
Qubitized Dirac/ Schrödinger Hamiltonians	3D grid 256³ → n = 24; qubitization ancillas → ~26–36 logical	~2,500–3,500	~8,800–12,200	Qubitization uses ≤2 ancillas; grid‑based demos confirm n in the 20–30 range
QSVT‑based PDE solvers	Mixed FEM/PDE with QSP precision → ~40–80 logical	~3,900–7,800	~13,500–27,000	QSVT angles/LCU ancillas add O(1–log L)
Quantum‑enhanced AMR	AMR error‑estimator loops + QLSS core → ~32–64 logical	~3,100–6,300	~10,800–21,600	Quantum error‑estimators via block‑encoding; toy VQLS AMR evaluators use 2–5 qubits (without EC)
Large‑scale tensor‑network simulation (QTN on QC)	QTN/MPS with bond dimension χ = 64 → ~6–12 logical (log₂χ + ancillas)	~590–1,180	~2,030–4,060	Width independent of system size; depth and sampling control accuracy
Quantum‑accelerated Monte Carlo (finance, fault‑tolerant)	End‑to‑end estimates: ~4.7k–8k logical; T‑count ≈10⁸–10⁹	~4.6 × 10⁵–7.8 × 10⁵	~1.6 × 10⁶–2.7 × 10⁶	QSP‑based payoff loading reduces logical width to ~4.7k; factories and clock rate dominate schedule

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