std-quantum-deep-dive

std.quantumquantum standard libraryquantum deep dive
trident cyber article

std.quantum: A Quantum Standard Library for trident

Provable Quantum Computing Through Prime Field Arithmetic

The Idea

Trident's std.nn provides neural network primitives — linear layers, activations, normalization — all as native field arithmetic over $\mathbb{F}_p$. The same principle applies to quantum computing. Quantum operations are unitary transformations on Hilbert spaces. When the dimension is prime, these transformations are arithmetic over $\mathbb{F}_p$. A quantum standard library for Trident — std.quantum — would provide quantum computing primitives that are simultaneously:

  • Classically simulatable on Triton VM with STARK proofs
  • Quantum-executable on qudit hardware via compilation to Cirq/QuForge
  • Composable with std.nn for verifiable quantum machine learning

The library does not simulate quantum mechanics as an afterthought. It expresses quantum mechanics in its native algebra — the same Goldilocks field algebra (GFP) that Trident already uses for everything else.

Part I: The Primitives

1. Quantum State: Qstate

A quantum state of $n$ qudits in dimension $p$ is a vector of $p^n$ complex amplitudes. In $\mathbb{F}_p$ arithmetic, we represent amplitudes using pairs of field elements (real and imaginary parts), or more precisely, elements of the quadratic extension field $\mathbb{F}_{p^2}$.

Triton VM already supports extension field arithmetic — it uses $\mathbb{F}_{p^3}$ (cubic extension) for STARK proof soundness amplification. The quadratic extension $\mathbb{F}_{p^2}$ is simpler.

/// A complex amplitude: a + bi where a, b ∈ F_p
struct Complex {
    re: Field,  // real part
    im: Field,  // imaginary part
}

/// Quantum state of n qudits, each of dimension d
/// Total amplitudes: d^n
/// For Trident: d = p (Goldilocks prime) or d = 3 (qutrit) etc.
struct Qstate<const N: u32, const D: u32> {
    amplitudes: [Complex; D.pow(N)],
}

On classical Triton VM: Qstate is a concrete vector of field elements. Operations on it produce execution traces. STARK proofs verify correctness.

On quantum hardware: Qstate maps to an actual quantum register. The amplitudes are the physical state. Operations become quantum gates.

The same source code, two compilation targets, both verifiable.

2. Quantum Gates: Unitary Matrices over $\mathbb{F}_{p^2}$

A quantum gate on a $d$-dimensional qudit is a $d \times d$ unitary matrix with entries in $\mathbb{C}$ — or in our representation, entries in $\mathbb{F}_{p^2}$.

/// A single-qudit gate: d×d unitary matrix
struct Gate<const D: u32> {
    matrix: [Complex; D * D],
}

/// Apply gate to qudit at position `target` in the state
fn apply_gate<const N: u32, const D: u32>(
    state: &mut Qstate<N, D>,
    gate: &Gate<D>,
    target: u32,
) {
    // For each basis state of the other qudits,
    // apply the gate matrix to the target qudit's subspace
    let stride = D.pow(N - 1 - target);
    for outer in 0..D.pow(N) / (D * stride) {
        for inner in 0..stride {
            let base = outer * D * stride + inner;
            // Gather the D amplitudes for this target qudit
            let mut slice: [Complex; D] = gather(state, base, stride, D);
            // Matrix-vector multiply: gate.matrix × slice
            slice = matvec(gate.matrix, slice);
            // Scatter back
            scatter(state, base, stride, D, slice);
        }
    }
}

The matvec operation is multiply-accumulate over $\mathbb{F}_{p^2}$ — the same operation as a linear layer in std.nn, just over the extension field instead of the base field. The STARK proof mechanism is identical.

3. Standard Gate Set

The generalized Pauli gates for prime dimension $p$ are:

Shift gate (generalized X): $X|j\rangle = |j+1 \bmod p\rangle$

This is field addition on the basis label. In Trident: a permutation of the amplitude vector, which is a reindexing — zero arithmetic cost.

fn shift_gate<const D: u32>() -> Gate<D> {
    // X_{jk} = δ_{j, k+1 mod D}
    let mut m = [Complex::zero(); D * D];
    for k in 0..D {
        m[((k + 1) % D) * D + k] = Complex::one();
    }
    Gate { matrix: m }
}

Clock gate (generalized Z): $Z|j\rangle = \omega^j |j\rangle$ where $\omega = e^{2\pi i/p}$

This is a diagonal phase gate. The phases are $p$-th roots of unity. In $\mathbb{F}_{p^2}$, we represent $\omega$ using the quadratic extension (or, for simulation purposes, as a lookup table of precomputed values).

fn clock_gate<const D: u32>(omega: Complex) -> Gate<D> {
    // Z_{jk} = ω^j · δ_{jk}
    let mut m = [Complex::zero(); D * D];
    let mut phase = Complex::one();
    for j in 0..D {
        m[j * D + j] = phase;
        phase = complex_mul(phase, omega);
    }
    Gate { matrix: m }
}

Hadamard gate (generalized): $H|j\rangle = \frac{1}{\sqrt{p}} \sum_k \omega^{jk} |k\rangle$

This is the Quantum Fourier Transform on a single qudit — a dense matrix of roots of unity. In $\mathbb{F}_{p^2}$: a matrix-vector multiply with $p^2$ multiply-accumulate operations. Expensive classically, but this is exactly the operation that quantum hardware performs in one step.

fn hadamard_gate<const D: u32>(omega: Complex) -> Gate<D> {
    // H_{jk} = ω^{jk} / √D
    let norm = field_inv_sqrt(D);  // 1/√D in F_{p^2}
    let mut m = [Complex::zero(); D * D];
    for j in 0..D {
        for k in 0..D {
            m[j * D + k] = complex_scale(
                complex_pow(omega, j * k),
                norm
            );
        }
    }
    Gate { matrix: m }
}

Controlled gates: A controlled-U gate on two qudits applies U to the target qudit conditional on the control qudit's state.

fn controlled_gate<const D: u32>(
    gate: &Gate<D>,
    control_value: u32,  // which basis state activates the gate
) -> Gate2<D> {
    // (D^2 × D^2) matrix
    // Acts as identity unless control qudit is in |control_value⟩
    let mut m = identity_2qudit::<D>();
    for j in 0..D {
        for k in 0..D {
            let row = control_value * D + j;
            let col = control_value * D + k;
            m[row * D * D + col] = gate.matrix[j * D + k];
        }
    }
    Gate2 { matrix: m }
}

4. Measurement

Quantum measurement collapses superposition to a definite outcome with probabilities given by the squared amplitudes.

On classical Triton VM: measurement is deterministic — the prover uses divine() to inject the measurement outcome, and the constraints verify that the probability is nonzero (the amplitude for that outcome is nonzero).

On quantum hardware: measurement is physical — the qudit register collapses and returns a basis state.

fn measure<const N: u32, const D: u32>(
    state: &Qstate<N, D>,
    qudit: u32,
) -> (u32, Qstate<N, D>) {
    // Compute probabilities for each outcome
    let probs = measurement_probabilities(state, qudit);
    
    // Classical: prover injects outcome via divine()
    let outcome: u32 = divine();
    
    // Verify: probability of outcome is nonzero
    assert(probs[outcome] != Complex::zero());
    
    // Collapse: project state onto outcome, renormalize
    let collapsed = project_and_normalize(state, qudit, outcome);
    
    (outcome, collapsed)
}

The divine() call is the bridge between classical simulation and quantum execution. Classically, the prover chooses the outcome (and proves it's consistent). Quantumly, the hardware chooses the outcome (and the proof covers the post-measurement computation). In both cases, the STARK proof verifies that the subsequent computation was correct given the measurement result.

5. The Quantum Fourier Transform

The QFT over $\mathbb{Z}/p\mathbb{Z}$ is the workhorse of quantum algorithms — Shor's factoring, quantum phase estimation, and the quantum speedup for NTT in STARK proving all depend on it.

For a single qudit, QFT is the Hadamard gate. For multiple qudits, it decomposes into single-qudit Hadamards and controlled phase gates:

fn qft<const N: u32, const D: u32>(
    state: &mut Qstate<N, D>,
    omega: Complex,
) {
    for i in 0..N {
        // Hadamard on qudit i
        apply_gate(state, &hadamard_gate::<D>(omega), i);
        
        // Controlled phase gates
        for j in (i+1)..N {
            let phase_exp = D.pow(j - i);
            let phase_gate = phase_gate::<D>(complex_pow(omega, phase_exp));
            apply_controlled(state, &phase_gate, j, i);
        }
    }
    // Reverse qudit order
    reverse_qudits(state);
}

Classical cost: $O(N^2 \cdot D^N)$ field operations for $N$ qudits of dimension $D$. Enormous for simulation, but the STARK proof is polynomial in the circuit size.

Quantum cost: $O(N^2)$ gates. This is where quantum hardware provides exponential advantage — the same Trident code, compiled to Cirq, becomes a quantum circuit with polynomially many gates acting on exponentially large state spaces.

Part II: Quantum Algorithms as Trident Programs

Grover's Search

Grover's algorithm finds a marked element in an unstructured database of $N$ items in $O(\sqrt{N})$ queries. In Trident, it's a function that takes an oracle (a Trident function returning bool) and returns the marked element:

/// Grover's search over D^n states
fn grover_search<const N: u32, const D: u32>(
    oracle: fn(&[u32; N]) -> bool,
    iterations: u32,
) -> [u32; N] {
    // Initialize uniform superposition
    let mut state = uniform_superposition::<N, D>();
    
    for _ in 0..iterations {
        // Oracle: phase-flip marked states
        oracle_phase_flip(&mut state, oracle);
        
        // Diffusion: reflect about the mean amplitude
        grover_diffusion(&mut state);
    }
    
    // Measure all qudits
    measure_all(&state)
}

The oracle parameter is any Trident function. On classical Triton VM, Grover's algorithm simulates quantum evolution — exponentially expensive in the number of qudits, but the STARK proof is polynomial in the circuit size. The proof verifies the simulation was correct.

On quantum hardware, the oracle compiles to a quantum oracle circuit, and Grover's algorithm runs natively with $O(\sqrt{N})$ queries. The measurement result feeds back into the classical STARK prover for verification.

The key insight: the oracle is a regular Trident function. Any constraint system — a lock script, a neural network classifier, a graph search predicate — can be passed as the oracle. Grover's algorithm becomes a generic accelerator for any Trident computation that involves searching.

Quantum Phase Estimation

Phase estimation determines the eigenvalue of a unitary operator — the foundation of Shor's algorithm, quantum chemistry simulation, and quantum machine learning kernels.

fn phase_estimation<const N: u32, const D: u32>(
    unitary: &Gate<D>,
    eigenstate: &Qstate<1, D>,
    precision_qudits: u32,
) -> Field {
    // Prepare precision register in uniform superposition
    let mut precision = uniform_superposition_n(precision_qudits, D);
    
    // Controlled powers of unitary
    for k in 0..precision_qudits {
        let power = D.pow(k);
        let u_power = matrix_power(unitary, power);
        apply_controlled_on_register(
            &mut precision, eigenstate,
            &u_power, k
        );
    }
    
    // Inverse QFT on precision register
    inverse_qft(&mut precision);
    
    // Measure precision register → eigenvalue estimate
    let result = measure_all(&precision);
    field_from_digits(result, D)  // convert qudit digits to field element
}

Classical simulation: exponential cost, but STARK-provable.

Quantum execution: polynomial cost, with STARK proof of the classical post-processing.

For quantum chemistry: the unitary represents molecular Hamiltonian evolution. The eigenvalue is the molecular energy. The Trident proof verifies the entire computation — input molecule, Hamiltonian construction, phase estimation, energy output. Verifiable quantum chemistry.

Variational Quantum Eigensolver (VQE)

VQE is the hybrid quantum-classical algorithm that dominates near-term quantum computing. A parametrized quantum circuit (ansatz) is optimized by a classical loop:

fn vqe<const N: u32, const D: u32>(
    hamiltonian: &[PauliTerm],
    ansatz: fn(&[Field], &mut Qstate<N, D>),
    n_params: u32,
    n_iterations: u32,
) -> (Field, [Field]) {
    // Initialize parameters
    let mut params: [Field; n_params] = divine();  // prover provides initial guess
    
    for iter in 0..n_iterations {
        // Prepare state with current parameters
        let mut state = zero_state::<N, D>();
        ansatz(&params, &mut state);
        
        // Measure expectation value of Hamiltonian
        let energy = expectation_value(&state, hamiltonian);
        
        // Classical parameter update (gradient descent in F_p)
        let gradients = parameter_shift_gradient(
            &params, hamiltonian, ansatz
        );
        for i in 0..n_params {
            params[i] = params[i] - LEARNING_RATE * gradients[i];
        }
    }
    
    (energy, params)
}

This is quantum machine learning. The ansatz is a neural network analog — a parametrized transformation whose parameters are optimized to minimize a cost function. The parameter shift rule for gradient computation is a quantum-specific technique that Trident can express naturally.

On classical Triton VM: the entire VQE loop is simulated and STARK-proven. The proof verifies that the optimization was executed correctly — that the reported energy and parameters actually result from running the claimed ansatz on the claimed Hamiltonian for the claimed number of iterations.

On quantum hardware: the ansatz executes on qudits (quantum speedup for state preparation and measurement), while the classical optimizer runs on the classical controller. The hybrid loop crosses the classical-quantum boundary — and Trident's field-native architecture ensures zero overhead at this boundary.

Part III: std.nn.quantum — Where AI Meets Quantum

The deepest payoff comes from composing std.nn and std.quantum. Quantum machine learning algorithms are parametrized quantum circuits — they ARE neural networks whose "layers" are quantum gates.

Quantum Neural Network Layer

/// A quantum neural network layer: parametrized rotation gates
fn quantum_nn_layer<const N: u32, const D: u32>(
    state: &mut Qstate<N, D>,
    params: &[Field],
    omega: Complex,
) {
    // Single-qudit rotations (parametrized)
    for i in 0..N {
        let angle = params[i];
        let rotation = rotation_gate::<D>(omega, angle);
        apply_gate(state, &rotation, i);
    }
    
    // Entangling gates (fixed structure)
    for i in 0..N-1 {
        let entangle = controlled_shift::<D>();
        apply_controlled(state, &entangle, i, i + 1);
    }
}

This is equivalent to a std.nn.linear_layer — parametrized transformation followed by a fixed nonlinearity. The difference: classical layers transform $\mathbb{F}_p^n$ vectors, quantum layers transform $\mathbb{F}_{p^2}^{D^n}$ state vectors. Both are matrix operations over field elements.

Quantum Classifier

fn quantum_classifier<const N: u32, const D: u32>(
    input: [Field; N],
    model: &QuantumModel,
) -> [Field; D] {
    // Encode classical data into quantum state
    let mut state = encode_amplitude(input);
    
    // Apply quantum neural network layers
    for layer in 0..model.n_layers {
        quantum_nn_layer(&mut state, &model.params[layer], model.omega);
    }
    
    // Measure to get classification probabilities
    measurement_probabilities_single(&state, 0)
}

On Triton VM: classical simulation with STARK proof. Verifiable inference of a quantum ML model.

On quantum hardware: native quantum execution. The encoding, layers, and measurement all run on qudits. Quadratically fewer parameters than classical networks for certain tasks (proven for qutrits — 90× improvement on optimization).

On both targets: the proof is identical. Whether the quantum classifier ran on classical simulation or quantum hardware, the STARK proof verifies the same arithmetic constraints. This is the key property — quantum and classical execution produce the same proof format, verifiable by the same verifier, on the same blockchain.

Hybrid Classical-Quantum Model

The most practical architecture for near-term quantum advantage combines classical and quantum layers:

fn hybrid_model(
    input: [Field; 784],          // MNIST image, 28×28
    classical_layers: &NNModel,   // std.nn classical layers
    quantum_layers: &QuantumModel, // std.quantum parametrized circuit
) -> [Field; 10] {
    // Classical feature extraction (std.nn)
    let features = std_nn::linear(input, classical_layers.w1, classical_layers.b1);
    let features = std_nn::relu_lookup(features);
    let features = std_nn::linear(features, classical_layers.w2, classical_layers.b2);
    // features: [Field; 8] — compressed to 8 dimensions
    
    // Quantum processing (std.quantum)
    let mut qstate = std_quantum::encode_amplitude(features);
    for layer in 0..quantum_layers.n_layers {
        std_quantum::quantum_nn_layer(&mut qstate, &quantum_layers.params[layer]);
    }
    let quantum_features = std_quantum::measure_probabilities(&qstate);
    // quantum_features: [Field; 8] — quantum-processed
    
    // Classical output head (std.nn)
    let logits = std_nn::linear(quantum_features, classical_layers.w3, classical_layers.b3);
    std_nn::softmax(logits)
}

This is a complete hybrid classical-quantum neural network. The entire model — classical preprocessing, quantum circuit, classical output — is one Trident program. One compilation. One STARK proof. One verification.

The classical layers run on Triton VM. The quantum layers can run on Triton VM (simulation) or quantum hardware (native execution). The boundary between classical and quantum is the boundary between std.nn and std.quantum function calls — and it's algebraically seamless because both libraries operate over the same field.

Compare this to the current state of hybrid quantum ML: train a model in PyTorch, extract the quantum circuit, port to Qiskit, run on IBM hardware, get results back, hope they're correct, try to verify somehow. Every boundary is a translation layer. Every translation introduces potential error. Nothing is proven.

Part IV: The Compilation Duality

Every std.quantum function has two compilation targets. This duality is the core architectural property.

Classical Target: Triton VM

std.quantum function call
  → Matrix operations over F_{p^2}
    → Arithmetic circuit over F_p (extension field ops decompose)
      → TASM instructions
        → Triton VM execution
          → STARK proof

The quantum simulation is expensive — exponential in the number of qudits. But the STARK proof is polynomial in the circuit size. For small quantum computations (a few qudits), classical simulation with STARK proof is practical today.

This is not just a development tool. Classical simulation with STARK proof has inherent value:

  • Verifiable quantum algorithm development: Write and test quantum algorithms on classical hardware with mathematical proof of correctness. When quantum hardware is available, the same code runs natively.
  • Benchmark certification: Prove that a claimed quantum speedup is genuine by proving the classical simulation took a specific number of operations.
  • Quantum error analysis: Simulate noisy quantum circuits classically, prove the simulation correct, compare with hardware results to characterize error rates.

Quantum Target: Cirq / QuForge

std.quantum function call
  → Gate sequence over D-dimensional qudits
    → Cirq circuit (qutrit / ququint / native dimension)
      → Quantum hardware execution
        → Measurement results (field elements)
          → Classical STARK prover uses results as witness
            → STARK proof

The quantum execution is cheap — polynomial in circuit depth. The measurement results become inputs to the classical STARK prover, which proves that the post-measurement classical computation was correct.

The verification loop:

  1. Write hybrid model in Trident
  2. Classical portions execute on Triton VM (proven by STARK)
  3. Quantum portions execute on quantum hardware (results injected as witness)
  4. The witness injection uses divine() — the same mechanism as private inputs
  5. Constraints verify consistency: the quantum results must satisfy the expected relationships
  6. Complete STARK proof covers the entire computation

The verifier sees one proof. It doesn't know or care which parts ran classically and which ran quantumly. The proof is the same format either way.

Part V: Concrete Applications

1. Quantum Chemistry as a Smart Contract

/// Compute molecular ground state energy
/// Verifiable on-chain, executable on quantum hardware
fn molecular_energy(
    molecule: &MoleculeSpec,  // atomic positions, charges
    basis_set: &BasisSet,     // quantum chemistry basis
    vqe_depth: u32,           // circuit depth
    vqe_iterations: u32,      // optimization iterations
) -> Field {
    // Build Hamiltonian from molecular specification
    let hamiltonian = build_hamiltonian(molecule, basis_set);
    
    // Define ansatz circuit
    let ansatz = uccsd_ansatz(molecule.n_electrons, molecule.n_orbitals);
    
    // Run VQE
    let (energy, optimal_params) = vqe(
        &hamiltonian, ansatz, 
        ansatz.n_params(), vqe_iterations
    );
    
    energy  // ground state energy, STARK-proven
}

Deploy this as a Neptune smart contract. Anyone can call it with a molecule specification. The result is a STARK-proven ground state energy. Use cases:

  • Drug discovery: Prove binding affinity computations on-chain. Pharmaceutical IP as verifiable proofs.
  • Materials science: Prove material property predictions. Supply chain quality assurance via verified simulation.
  • Carbon credit verification: Prove molecular-level photosynthesis simulation. Carbon absorption certificates backed by quantum chemistry.

2. Quantum-Enhanced CyberRank for bostrom

/// Quantum walk on knowledge graph
fn quantum_cyberrank(
    graph: &AdjacencyMatrix,  // knowledge graph
    n_steps: u32,             // walk length
    query_bias: [Field; N],   // query-relevance weighting
) -> [Field; N] {
    // Encode graph structure into quantum walk operator
    let walk_op = quantum_walk_operator(graph);
    
    // Initialize state with query bias
    let mut state = encode_amplitude(query_bias);
    
    // Quantum walk: n_steps applications of walk operator
    for _ in 0..n_steps {
        apply_walk_step(&mut state, &walk_op);
    }
    
    // Measurement probabilities = relevance ranking
    measurement_probabilities_all(&state)
}

Classical simulation for small graphs (STARK-proven). Quantum execution for large graphs (exponential speedup on mixing time). The ranking is provably correct — not "trust the algorithm," but "here's a mathematical proof."

For bostrom: every cybergraph query could carry a STARK proof of ranking correctness. Each neuron submits cyberlinks connecting particles, and focus determines relevance. Users do not trust the search engine — they verify it.

3. Quantum Random Number Generation with Proof

/// Generate provably random numbers from quantum measurement
fn quantum_random(n_bits: u32) -> [Field; n_bits] {
    let mut results: [Field; n_bits] = [];
    
    for i in 0..n_bits {
        // Prepare qudit in uniform superposition
        let mut q = zero_state::<1, D>();
        apply_gate(&mut q, &hadamard_gate(), 0);
        
        // Measure — outcome is inherently random
        let (outcome, _) = measure(&q, 0);
        results[i] = outcome;
    }
    
    results
}

On quantum hardware: the randomness is guaranteed by quantum mechanics (Born rule). The STARK proof verifies the post-measurement computation but cannot (and need not) prove the randomness itself — that's a property of physics.

On classical Triton VM: the prover uses divine() to inject outcomes. The proof verifies consistency but the randomness comes from the prover's source. This is still useful for commit-reveal schemes and verifiable random functions.

Application: on-chain randomness for NFT mints, lottery contracts, fair ordering — backed by quantum physics rather than pseudorandom generators.

4. Quantum Key Distribution Integration

/// BB84-style key establishment with on-chain verification
fn quantum_key_exchange(
    alice_bases: [u32; N],     // Alice's measurement bases
    alice_bits: [Field; N],    // Alice's prepared states
) -> Digest {
    // Prepare qudits in Alice's chosen bases and states
    let mut qudits: [Qstate<1, D>; N] = [];
    for i in 0..N {
        qudits[i] = prepare_bb84_state(alice_bases[i], alice_bits[i]);
    }
    
    // Bob measures in his bases (injected via divine())
    let bob_bases: [u32; N] = divine();
    let bob_results: [Field; N] = [];
    for i in 0..N {
        let (result, _) = measure_in_basis(&qudits[i], bob_bases[i]);
        bob_results[i] = result;
    }
    
    // Sifting: keep only matching bases
    let shared_key = sift(alice_bases, bob_bases, bob_results);
    
    // Commit shared key on-chain
    hash(shared_key)
}

Quantum key distribution established, key commitment on-chain, STARK proof of protocol correctness. The key itself remains private (zero-knowledge). The proof demonstrates the protocol was followed honestly.

5. Verifiable Quantum Optimization for DeFi

/// QAOA for portfolio optimization
fn quantum_portfolio_optimization(
    returns: [Field; N],           // expected returns per asset
    covariance: [Field; N * N],    // covariance matrix
    risk_budget: Field,            // maximum acceptable risk
    qaoa_depth: u32,               // QAOA circuit depth
) -> [Field; N] {
    // Encode cost function as Hamiltonian
    let cost_hamiltonian = portfolio_hamiltonian(returns, covariance, risk_budget);
    let mixer_hamiltonian = standard_mixer(N);
    
    // QAOA parameters (optimized or provided via divine())
    let gamma: [Field; qaoa_depth] = divine();
    let beta: [Field; qaoa_depth] = divine();
    
    // Run QAOA circuit
    let mut state = uniform_superposition_n(N, D);
    for p in 0..qaoa_depth {
        // Cost layer: exp(-i·γ·C)
        apply_cost_unitary(&mut state, &cost_hamiltonian, gamma[p]);
        // Mixer layer: exp(-i·β·B)
        apply_mixer_unitary(&mut state, &mixer_hamiltonian, beta[p]);
    }
    
    // Measure → portfolio allocation
    let allocation = measure_all(&state);
    
    // Verify constraints
    assert(portfolio_risk(allocation, covariance) <= risk_budget);
    
    allocation
}

The divine() calls for gamma and beta parameters are where optimization happens. Classically: the prover runs a classical optimizer to find good parameters. Quantumly: Grover-enhanced search over parameter space.

The STARK proof verifies:

  • The QAOA circuit was applied correctly with the stated parameters
  • The resulting allocation satisfies risk constraints
  • The measurement was consistent with the quantum state

Deploy as a Neptune smart contract. A DeFi fund calls this function. Investors verify the optimization proof. The fund's strategy (parameters) remains private. The compliance (risk constraints satisfied) is public and proven.

Part VI: The Library Architecture

std/
├── quantum/
│   ├── state.tri          // Qstate type, initialization, normalization
│   ├── gates.tri          // Standard gate set (X, Z, H, CNOT, Toffoli, custom)
│   ├── circuit.tri        // Circuit builder, gate scheduling, optimization
│   ├── measure.tri        // Measurement, partial trace, Born rule
│   ├── qft.tri            // Quantum Fourier Transform and inverse
│   ├── grover.tri         // Grover's search algorithm
│   ├── phase_est.tri      // Quantum Phase Estimation
│   ├── walk.tri           // Quantum walks on graphs
│   ├── vqe.tri            // Variational Quantum Eigensolver
│   ├── qaoa.tri           // Quantum Approximate Optimization
│   ├── error.tri          // Error models, noise simulation
│   └── compile/
│       ├── cirq.tri       // Cirq backend compilation
│       ├── quforge.tri    // QuForge backend compilation
│       └── native.tri     // Native qudit hardware compilation
│
├── nn/
│   ├── linear.tri         // Linear layers (matmul over F_p)
│   ├── conv.tri           // Convolutional layers
│   ├── attention.tri      // Multi-head attention
│   ├── activation.tri     // Lookup-table activations (ReLU, GELU, SiLU)
│   ├── norm.tri           // LayerNorm, BatchNorm
│   ├── loss.tri           // Cross-entropy, MSE
│   ├── optim.tri          // SGD, Adam over F_p
│   └── onnx/
│       ├── import.tri     // ONNX → Trident transpiler
│       └── export.tri     // Trident → ONNX
│
├── nn_quantum/             // THE INTERSECTION
│   ├── encoding.tri       // Classical data → quantum state encoding
│   ├── variational.tri    // Parametrized quantum circuits (ansatz library)
│   ├── kernel.tri         // Quantum kernel methods
│   ├── hybrid.tri         // Hybrid classical-quantum models
│   ├── qnn_layer.tri      // Quantum neural network layers
│   └── train.tri          // Hybrid training loops (parameter shift rule)
│
└── crypto/
    ├── hash.tri           // Tip5, [[Poseidon2]]
    ├── merkle.tri         // Merkle trees
    ├── commit.tri         // Commitments
    └── verify.tri         // STARK verification (recursive)

The four directories represent the four capabilities of the tri-kernel architecture:

  • std.quantum — quantum computing primitives
  • std.nn — neural network primitives
  • std.nn_quantum — quantum machine learning (the intersection)
  • std.crypto — cryptographic primitives (already exists in Triton VM)

All four share the same foundation: arithmetic over $\mathbb{F}_p$. All four produce the same artifact: arithmetic circuits compiled to TASM, proven by STARK.

Part VII: Why std.quantum Changes the Game

For Quantum Computing

Current quantum programming (Q#, Qiskit, Cirq) produces unverifiable results. You send a circuit to a cloud quantum computer. You get a result. You trust the provider. There is no mathematical guarantee of correctness.

std.quantum makes quantum results provable. The STARK proof verifies that the classical post-processing (or the classical simulation) was correct. For quantum hardware results, the proof covers everything except the quantum measurement itself — which is inherently probabilistic and outside any proof system's scope.

This enables trustless quantum cloud computing. Send a Trident program to a quantum provider. Receive results plus STARK proof. Verify locally. No trust in the provider, the hardware, or the network.

For AI/ML

Current quantum ML (PennyLane, Qiskit ML, TensorFlow Quantum) is a research playground with no path to production deployment. Models are trained on simulators, tested on noisy hardware, and trusted on faith.

std.nn_quantum makes quantum ML deployable. The hybrid model runs on production quantum hardware (or classical simulation), produces a STARK proof, and deploys on-chain as a smart contract. The proof covers training correctness, inference correctness, and constraint satisfaction.

This enables verifiable quantum AI agents — the first AI systems that are simultaneously intelligent (learned models), quantum-enhanced (qudit circuits), and mathematically accountable (STARK proofs).

For Blockchain

Current blockchains have no quantum computing capability. Smart contracts are classical, deterministic, and slow. Quantum advantage is entirely off-chain, separated by trust boundaries.

std.quantum makes quantum computing a blockchain-native capability. A Neptune smart contract can include quantum subroutines that execute on quantum hardware, with results verified on-chain via STARK proofs. The blockchain becomes a settlement layer for quantum computation.

This enables quantum DeFi — financial instruments whose valuations, risk models, and optimizations leverage quantum advantage, with every computation proven and settled on-chain.

For the Trident Ecosystem

std.quantum completes the trinity. trident becomes not just a smart contract language, not just a provable computation language, not just a quantum-compatible language — but a universal verifiable computation language that spans classical, quantum, and AI workloads under one proof system.

The developer writes one program. It compiles to classical execution (today), quantum execution (tomorrow), and hybrid execution (the transition). The proof is the same. The verification is the same. The blockchain is the same.

Implementation Roadmap

Phase 1 — Core primitives (3-6 months): Implement Qstate, Gate, apply_gate, measure for small qudit counts (1-4 qudits, dimension 3 or 5). Demonstrate: Grover's search on 2-qutrit system, STARK-proven, verified on neptune. This is the "Hello World" of provable quantum computing.

Phase 2 — Algorithm library (6-12 months): Implement QFT, phase estimation, VQE, QAOA. Build Cirq compilation backend. Demonstrate: VQE for a small molecule (H₂) classically simulated with STARK proof, then the same code running on trapped-ion qutrit hardware via Cirq.

Phase 3 — std.nn_quantum (12-18 months): Build quantum neural network layers, hybrid models, training loops. Demonstrate: hybrid classical-quantum classifier on a real dataset, STARK-proven inference, deployed as neptune smart contract.

Phase 4 — Production applications (18-36 months): Quantum portfolio optimization, quantum CyberRank, quantum chemistry as smart contracts. Integrate with quantum cloud providers (IBM, Google, IonQ) via Cirq backend.

The Unification

std.nn gives Trident intelligence. std.quantum gives Trident quantum power. std.nn_quantum gives Trident quantum intelligence. std.crypto gives Trident provability. And all four are the same thing underneath: arithmetic over $\mathbb{F}_p$.

A single Trident program can:

  1. Run a neural network to make a prediction (std.nn)
  2. Use a quantum circuit to enhance that prediction (std.quantum)
  3. Combine them in a hybrid quantum-classical model (std.nn_quantum)
  4. Prove the entire computation correct (std.crypto / Triton VM STARK)
  5. Execute the proof on any blockchain (Level 1: Execute Anywhere)
  6. Settle economic consequences via smart contracts (Neptune)

This is the full stack of verifiable quantum AI. Not as a theoretical possibility — as a concrete library architecture with a concrete implementation roadmap, built on a concrete mathematical foundation.

The field is prime. The algebra is shared. The library is waiting to be written.

Cross-References

See trinity for how quantum fits into the three-pillar architecture. See trident-complete-stdlib for the full stdlib specification including std.quantum. See trident-trinity-zk-ai-quantum for the unified thesis.

Linked References

Local Graph