Code that writes code

Exponential-looking growth in computing emerges when systems can improve their own code. Once a loop exists that can generate, execute, measure, and refine programs, capability compounds: each improvement enhances the next round of search over the program phase space. The driver is not just device density, but a feedback process that compresses errors, increases information flow, and reduces effective entropy in the distribution of behaviors.

In mathematical terms, let the capability at iteration \(t\) be \(C(t)\). A simple model of compounded self-improvement is \[ C(t+1) = (1 + r)\,C(t), \qquad r > 0, \] whose solution is \[ C(t) = C(0)\,(1 + r)^t \approx C(0)\,e^{rt}, \] capturing the "exponential-looking" growth. A more general continuous-time feedback model writes \[ \frac{dC}{dt} = f(I,C,E), \] where \(I\) measures information flow through the system and \(E\) is the average error. A negative dependence \(\partial f/\partial E < 0\) and positive \(\partial f/\partial I > 0\) encode that reducing errors and increasing information flow accelerates capability.

Let the program hypothesis space be a high-dimensional manifold \(\mathcal{H}\) with coordinates \(\theta \in \mathbb{R}^d\) parameterizing programs, and let a loss (or energy-like) function be \(L : \mathcal{H} \to \mathbb{R}_+\). Guided search corresponds to iterates \[ \theta_{t+1} = \theta_t - \eta_t \, \nabla_\theta L(\theta_t) + \xi_t, \] where \(\eta_t\) is a learning rate and \(\xi_t\) represents stochastic exploration. The associated probability density over programs, \(p_t(\theta)\), can be modeled as a Gibbs-like distribution \[ p_t(\theta) = \frac{1}{Z_t} \exp\big(-\beta_t L(\theta)\big), \] with partition function \(Z_t\) and inverse temperature \(\beta_t\) reflecting how strongly the system prefers lower-loss code. As \(t\) increases and \(\beta_t\) grows, the distribution contracts around high-quality programs, reducing effective entropy \[ S[p_t] = - \int_{\mathcal{H}} p_t(\theta)\, \log p_t(\theta)\, d\theta. \]

Mechanism of self-improvement (physics and information terms)

Mechanism of self-improvement

• Program synthesis as guided search over a high-dimensional landscape; gradients, heuristics, and discrete exploration move candidates toward lower loss (energy-like objective).

Formally, represent each candidate program by parameters \(\theta \in \mathbb{R}^d\) and define an objective (loss/energy) \(L(\theta)\). Gradient-based guidance is \[ \theta_{t+1} = \theta_t - \eta_t \, \nabla_\theta L(\theta_t), \] while purely heuristic or discrete search can be modeled by a Markov chain with transition kernel \(K(\theta'\mid\theta)\) that satisfies \[ p_{t+1}(\theta') = \int K(\theta'\mid\theta)\,p_t(\theta)\,d\theta, \] where \(p_t(\theta)\) is the probability density over program parameters at iteration \(t\). The stationary distribution \(p_*(\theta)\) often approximates a Boltzmann form \(p_*(\theta) \propto e^{-\beta L(\theta)}\), concentrating mass near minima of \(L\).

• Error-correcting feedback reduces uncertainty: observations act as measurements that collapse hypotheses, concentrating probability mass on higher-fidelity code.

Let \(\Theta\) be a discrete hypothesis set of programs, with prior \(P(\theta)\) for \(\theta \in \Theta\). Given observed data or test outcomes \(D\), Bayes' rule updates beliefs via \[ P(\theta \mid D) = \frac{P(D \mid \theta)P(\theta)}{P(D)}, \qquad P(D) = \sum_{\theta' \in \Theta} P(D \mid \theta')P(\theta'). \] The Shannon entropy of the hypothesis distribution is \[ H[P] = -\sum_{\theta \in \Theta} P(\theta)\,\log P(\theta). \] Error-correcting feedback corresponds to sequences of measurements \(D_1, D_2, \dots\) such that \[ H\big[P(\cdot \mid D_1,\dots,D_t)\big] \searrow H_* \quad \text{as} \quad t \to \infty, \] where \(H_*\) is low and the posterior probability mass is concentrated on a small set of high-fidelity programs.

• Thermodynamics of computation: every irreversible operation dissipates energy; efficient self-editing favors representations and compilers that minimize erasures while maximizing useful work per joule.

Landauer's principle states that erasing one bit of information incurs a minimum heat dissipation \[ Q_{\min} = k_B T \ln 2, \] where \(k_B\) is Boltzmann's constant and \(T\) is the ambient temperature. If a self-editing system performs \(N_\text{erase}\) bit erasures per update, the minimal thermodynamic cost per update is \[ Q_\text{update} \ge N_\text{erase}\,k_B T \ln 2. \] The computational efficiency can be expressed as useful information gain per unit energy, \[ \eta_\text{info} = \frac{\Delta I}{Q_\text{update}}, \] where \(\Delta I\) is the increase in mutual information between internal model parameters \(\Theta\) and task outcomes \(Y\): \[ I(\Theta;Y) = \sum_{\theta,y} p(\theta,y)\, \log \frac{p(\theta,y)}{p(\theta)\,p(y)}. \] Efficient self-editing corresponds to strategies that maximize \(\eta_\text{info}\) by minimizing unnecessary erasures and maximizing \(\Delta I\).

• Compression and generalization: shorter effective descriptions (lower algorithmic complexity) tend to transfer across tasks, enabling accelerated reuse and faster convergence on new problems.

Let the Kolmogorov complexity (algorithmic information content) of a solution for task \(T\) be \(K(T)\), defined as the length (in bits) of the shortest program that solves \(T\) on a fixed universal Turing machine \(U\). For two tasks \(T_1, T_2\), the shared structure can be quantified by an information-theoretic overlap \[ I(T_1;T_2) \approx K(T_1) + K(T_2) - K(T_1,T_2), \] where \(K(T_1,T_2)\) is the complexity of jointly solving both tasks. Reusable compressed representations correspond to internal codes \(z\) of small description length \(L(z)\) that minimize expected description length over tasks: \[ \min_{z} \, \mathbb{E}_{T \sim \mathcal{D}}\big[ L(z) + L(T \mid z) \big], \] with \(\mathcal{D}\) a task distribution. Lower \(L(z)\) and \(L(T\mid z)\) imply faster adaptation and thus accelerated convergence on new problems.

Portfolio as a wave function: density matrices instead of variance–covariance matrices

Below is a compact, code-adjacent sketch of the idea: in a quantum-flavored actuarial / MPT model, the classical variance–covariance matrix \(\Sigma\) is upgraded to a density matrix \(\rho\), and a portfolio is treated as a wave function \(|\psi\rangle\). This mirrors what you might simulate with Stan, but in linear-algebra form.

Classical MPT / actuarial notation

// n assets, vector of random returns R
// mean vector μ, variance–covariance matrix Σ

E[R]   = μ               // n×1
Cov(R) = Σ               // n×n (symmetric, PSD)

// portfolio weights (deterministic)
vector[n] w;
real R_p = dot_product(w, R);     // portfolio return

real mean_p   = dot_product(w, μ);
real var_p    = quad_form(Σ, w);  // w^T Σ w

LaTeX form: \( \mathbb{E}[R] = \mu \), \( \operatorname{Cov}(R) = \Sigma \), \( R_p = w^\top R \), \( \mathbb{E}[R_p] = w^\top \mu \), \( \operatorname{Var}(R_p) = w^\top \Sigma w \).

\( \mathbb{E}[R] = \mu \), \( \operatorname{Cov}(R) = \Sigma \), \( R_p = w^\top R \), \( \mathbb{E}[R_p] = w^\top \mu \), \( \operatorname{Var}(R_p) = w^\top \Sigma w \).

Quantum-style upgrade

We now re-interpret these same objects in a Hilbert-space way:

• Basis states \(|e_i\rangle\): one-hot exposure to asset \(i\).
• Portfolio wave function \(|\psi\rangle = \sum_i \psi_i |e_i\rangle\) with \(\sum_i |\psi_i|^2 = 1\).
• Density matrix (pure state) \(\rho = |\psi\rangle\langle\psi|\).
• Return operator \(\hat R\) with components \(\hat R |e_i\rangle = R_i |e_i\rangle\) in the simplest diagonal case.

In code-flavored pseudomath:

// amplitudes ψ (complex allowed), normalized
complex psi[n];




// density matrix ρ_ij = ψ_i ψ*_j
complex rho[n, n] = outer_product(psi, conj(psi));

// return operator R̂ (for illustration, take it diagonal)

real R_vals[n];    // possible asset returns
complex R_op[n, n];
for (i in 1:n) {
  for (j in 1:n) R_op[i,j] = (i == j) ? R_vals[i] : 0;
}

// quantum expectation of portfolio return

real R_vals[n];                    // possible asset returns
complex R_op[n, n];

for (i in 1:n) {
  for (j in 1:n) {
    R_op[i,j] = (i == j) ? R_vals[i] : 0;
  }
}

// quantum expectation of portfolio return
complex mean_p_q = trace(rho * R_op);  // ≈ classical w^T μ

Mathematically,

ρ = |ψ><ψ|
<R̂>_ψ = <ψ| R̂ |ψ> = Tr(ρ R̂)

LaTeX form: \( \rho = |\psi\rangle\langle\psi| \), \( \langle \hat R \rangle_\psi = \langle \psi|\hat R|\psi\rangle = \operatorname{Tr}(\rho \hat R) \).

\( \rho = |\psi\rangle\langle\psi| \), \( \langle \hat R \rangle_\psi = \langle \psi|\hat R|\psi\rangle = \operatorname{Tr}(\rho \hat R) \).

If \(\hat R\) is diagonal in the \(|e_i\rangle\) basis and we ignore phases, \(|\psi_i|^2\) plays the same role as classical weights \(w_i\). But once off-diagonal elements are allowed, \(\rho\) carries richer cross-asset structure than \(\Sigma\) alone.

Stan-style simulation vs Schrödinger-style evolution

Stan would typically simulate posterior draws of parameters \(\theta\) (e.g., drifts, vols, correlations) and then simulate returns:

// very schematic Stan pseudo-flow
for (m in 1:M) {
  θ[m] ~ posterior(· | data);      // parameters: μ[m], Σ[m], etc.
  R[m] ~ mvnormal(μ[m], Σ[m]);     // draw returns
  R_p[m] = dot_product(w, R[m]);
}
// compute empirical mean/var/VaR of R_p from samples

In the wave-function view, instead of sampling many \(\theta\), we evolve the state itself under a Hamiltonian \(\hat H\) that encodes drift/volatility/market structure:

// time evolution of portfolio state ψ_t
// i ℏ d|ψ_t>/dt = Ĥ |ψ_t>

psi_t+Δt ≈ exp(-i Δt Ĥ / ℏ) * psi_t;

// at horizon T, density matrix ρ_T = |ψ_T><ψ_T|
// expected payoff under operator Π̂
price_0 ≈ discount * trace(ρ_T * Π̂);

This is the code-level version of “instead of running Stan simulations over parameter space, we let the whole portfolio turn into a wave function and flow under \(\hat H\).”

Black–Scholes as the one-asset limit

For a single risky asset in Black–Scholes, under the risk-neutral measure we have

dS_t = r S_t dt + σ S_t dW_t

The price \(V(S,t)\) of a European claim satisfies

∂V/∂t + (1/2) σ^2 S^2 ∂^2V/∂S^2 + r S ∂V/∂S - r V = 0

After changing variables \(x = \ln S\) and switching to forward time \(τ = T - t\), this maps to a heat equation which can be written in imaginary time as

∂φ/∂τ = (1/2) σ^2 ∂^2φ/∂x^2 - V_eff(x) φ

Under a Wick rotation \(τ → i t\), this resembles a Schrödinger equation

i ℏ ∂ψ/∂t = Ĥ ψ

with \(\hat H\) containing a kinetic term (diffusion from \(σ\)) plus an effective potential. In this limit:

• Classical risk-neutral density \(f_{S_T}(s)\) ↔ \(|\psi(s,T)|^2\).
• Option price \(V_0\) ↔ expectation \(\langle \psi_T| \hat \Pi |\psi_T\rangle\).

So Black–Scholes is already half-way to the quantum formalism: it evolves distributions through a linear PDE. The density-matrix formulation just generalizes this to multiple coupled assets and richer dependence than a single \(\Sigma\) can conveniently express.

Classical vs Quantum Computer: pseudo-code views

This section strips away hardware details and shows, in pseudo‑code, how a conventional laptop and a quantum processor “feel” different as machines.

Conventional computer (motherboard + CPU + RAM)

// physical picture
// --------------------------------------------------
// motherboard: connects CPU, RAM, storage, peripherals
// CPU: executes instructions sequentially (with some parallelism)
// RAM: holds bits (0 or 1) for active programs

machine ClassicalComputer {
  Motherboard board;
  CPU         cpu;
  RAM         ram;
  Storage     disk;
}

// run a program
function run_classical(program P, input bits_in[]):
  // 1. load code and data into RAM
  ram.load(P.code)
  ram.load(bits_in)

  // 2. CPU executes instructions one by one
  while cpu.instruction_pointer not at END(P):
    instr = ram.fetch(cpu.instruction_pointer)
    cpu.execute(instr, ram)
    cpu.instruction_pointer++

  // 3. read out output bits from RAM
  bits_out = ram.read(P.output_region)
  return bits_out

Quantum processor (QPU + classical control computer)

// physical picture
// --------------------------------------------------
// classical control computer: compiles code, sends pulses
// quantum processing unit (QPU): array of qubits on a chip
// cryostat: keeps QPU near absolute zero

machine QuantumComputer {
  ClassicalControl ctrl;     // compiler, schedulers
  QuantumProcessingUnit qpu; // qubits + control lines
}

// high-level quantum run
function run_quantum(circuit C, classical_input bits_in[]):
  // 1. classical pre-processing
  //    e.g., encode bits_in into initial qubit states
  compiled_pulses = ctrl.compile(C, bits_in)

  // 2. upload pulse schedule to QPU
  qpu.load(compiled_pulses)

  // 3. apply quantum operations (unitaries)
  qpu.apply_pulses()
  // internally, each gate is a unitary U on |ψ>:
  //    |ψ_new> = U |ψ_old>

  // 4. measure qubits
  measurement_record = qpu.measure_all() // collapses |ψ> → classical bits

  // 5. classical post-processing
  result = ctrl.post_process(measurement_record)
  return result

Conceptually:

• The classical CPU walks through a list of instructions, flipping bits in RAM.
• The QPU evolves a joint quantum state \(|\psi\rangle\) of many qubits by applying gates (unitaries), then collapses that state to classical bits via measurement.

Same abstract computation, two execution models

Imagine we want to compute the parity (even/odd) of \(N\) bits.

Classical laptop parity

function parity_classical(bits[]):
  acc = 0
  for b in bits:
    acc = acc XOR b
  return acc  // 0 = even, 1 = odd

Quantum parity sketch (conceptual)

function parity_quantum(bits[]):
  // 1. encode bits into computational basis states
  //    |b_1 b_2 ... b_N>
  |ψ> = |b_1 b_2 ... b_N>

  // 2. use a circuit of CNOTs to copy global parity into an ancilla qubit
  //    |ψ, 0>  →  |ψ, parity(bits)>
  for i in 1..N:
    CNOT(control = qubit_i, target = ancilla)

  // 3. measure only the ancilla qubit
  result = measure(ancilla)
  return result  // 0 = even, 1 = odd

On such a small task, the quantum route is not “better” than the classical one—it is just a different physical implementation of the same logical function. Real quantum advantage typically appears in problems that exploit superposition and interference over huge state spaces.

Hardware analogy table: consumer laptop vs quantum processor

The numbers below are intentionally rounded, order‑of‑magnitude style, to give intuition. Real devices vary wildly; treat these as cartoon benchmarks, not spec sheets.

"How many laptops equal one quantum processor?" (cartoon answers)

The trick is: you cannot fairly compare them just by counting operations per second, because a quantum gate on \(N\) qubits simultaneously transforms amplitudes across \(2^N\) basis states. Still, you can get a feeling by asking:

Example 1: simulating 30 qubits on laptops

• State vector size: \(2^{30} \approx 10^9\) complex amplitudes.
• One layer of single‑qubit gates might touch all \(10^9\) amplitudes.
• A single good laptop GPU at ~10¹² FLOP/s can simulate ~10³ such layers per second in theory, but memory bandwidth and overhead reduce this a lot.

A small physical QPU with 30–40 qubits applies the same logical layer directly in hardware; it is as if you had a cluster of hundreds to thousands of laptops working together to update all amplitudes every gate.

Example 2: 50‑qubit state

• State vector size: \(2^{50} \approx 10^{15}\) complex amplitudes.
• Storing this naively (16 bytes per complex) needs ~16 petabytes of RAM.
• Even a supercomputer cluster of millions of consumer‑class laptops would struggle to hold and update this in full generality.

So for generic 50‑qubit circuits, one real QPU is morally comparable to an enormous classical cluster. For structured problems, clever classical algorithms can do much better—that’s why the “how many laptops” question has no single honest number.

Very rough equivalence table (for mental models only)

This table only matches memory capacity, not speed. But it gives a rough intuition: once you pass ~40–50 entangled qubits in a generic circuit, brute‑force classical simulation starts looking like “you’d need an absurd number of consumer laptops.”

SHA-256 password hashing examples

The table below shows a few simple example passwords and their corresponding SHA-256 hash values. These are illustrative only; never hard‑code real passwords or reuse simple patterns like these in production.

Hashes above were computed with standard SHA-256; you can reproduce them using tools like sha256sum, openssl dgst -sha256, Python’s hashlib.sha256, or any reputable online hash calculator.

How long would it take to crack these passwords?

The table below uses deliberately rough, order‑of‑magnitude estimates to show how different passwords fare against brute‑force search on classical hardware and on a future, optimistic quantum attacker using Grover-style speedups. The point is not the exact numbers, but the contrast between weak and strong secrets—and why multi‑factor authentication (MFA) is so valuable.

Assumptions are deliberately aggressive in favor of the attacker and ignore I/O, memory, and protocol defenses; they’re meant as back‑of‑the‑envelope illustrations, not operational guarantees. Also, SHA‑256 is used here in isolation; real systems should wrap it in slow, memory‑hard KDFs such as bcrypt, scrypt, or Argon2.