Two kinds of twin. Lab 1 is what accelerator physicists mean by a digital twin: a model of a specific machine whose physics is settled, used to tune the real thing. Labs 2–4 are the twin that fits your three questions: a quantum-circuit simulator, which is exactly how researchers prototype quantum computers, teleportation protocols, and the information physics of time loops before (or instead of) touching hardware.
Lab 5 draws the geometry behind the questions: 4D shapes, the block universe, black-hole and wormhole surfaces, and the light cones that tip over into time loops.
Every lab states a hypothesis, the fastest test, and what would refute it. Log runs to the notebook at the bottom and copy them out as JSON.
Accelerator twin: FODO lattice
A beamline is a chain of focusing (QF) and defocusing (QD) quadrupole magnets separated by drifts. The twin propagates the beam envelope through the lattice with transfer matrices. Real twins (SLAC LCLS, Fermilab Twinac, STFC CLARA) do the same with thousands of elements, fed by the live magnet settings.
Lattice settings
Thin-lens model. Stability of a FODO cell needs |Tr M| < 2, which for thin lenses means f > L/2. Beam size σ = √(ε·β).
Readouts
Top: beam size along the ring (one cell = QF, drift, QD, drift). Bottom: one-cell trace vs focal length; inside the band the beam stays bounded cell after cell. A real twin adds the parts this one lacks: live settings read from the control system, diagnostics compared against the prediction, and a model that is re-fit when they disagree.
Experiments
| Hypothesis | Fastest test | Refuted if |
|---|---|---|
| A thin-lens FODO cell is stable iff f > L/2. | Sweep f on the bottom chart with L fixed. | Tr M stays inside ±2 for some f < L/2, or leaves the band for f > L/2. |
| Beam size scales as √ε, not ε. | Set ε = 5 then 20; size at QF should grow ×2. | Size grows ×4 or does not change. |
| Near the stability edge (f → L/2) the beam gets large even while stable. | Slide f down toward 2.1 with L = 4. | β max does not rise as the edge is approached. |
Qubit lab: state-vector simulator
A classical computer can hold the full quantum state of a few qubits (2ⁿ complex amplitudes) and apply gates exactly. This is the standard research twin for quantum computing: write a circuit, run it ideal, then add noise and measure how far the result drifts. Up to 8 qubits here; Qiskit Aer and Cirq do the same thing at 30+.
Circuit
Gates: H X Y Z S SDG T TDG · RX RY RZ q angle (angles accept pi, e.g. RY 0 pi/3) · CX c t · CZ a b · SWAP a b · M [qubits]. Bitstrings read q0 q1 q2… left to right.
Readouts
Experiments
| Hypothesis | Fastest test | Refuted if |
|---|---|---|
| Interference is physical, not statistical: H·Z·H sends |0⟩ to |1⟩ every time. | Preset “Interference”, noise 0. | Any count of “0”. |
| Fidelity decays roughly as (1−p)^gates under per-gate depolarizing noise. | GHZ preset at p = 0.02 and 0.05; compare fidelity to (1−p)^3. | Fidelity is flat or non-monotone in p. |
| Syndrome bits locate a single flip without measuring the data qubits. | Bit-flip code at p = 0.05; the syndrome histogram should be mostly 00 with 10 / 01 / 11 minorities. | Syndromes are uniform, or 00 is not the mode. |
| One Grover iteration on 2 qubits finds the target with certainty. | Grover preset, noise 0. | Any outcome other than 11. |
Teleportation lab: Bennett protocol with a noisy channel
Quantum teleportation moves a qubit’s state, not matter: Alice and Bob share an entangled pair, Alice measures her qubit together with the message qubit and sends Bob two classical bits, Bob applies a correction. The lab runs the full three-qubit circuit shot by shot, with depolarizing noise on Bob’s half of the pair, and compares Bob’s output to the message.
Message and channel
q0 message · q1 Alice’s half · q2 Bob’s half. The X and Z on q2 are applied only when the classical bits say so.
Readouts
Theory for depolarizing noise with probability p on Bob’s half: F = 1 − 2p/3 with the correction, F = 1/2 without it (Bob holds a random Pauli twist of ψ until the bits arrive, so no signal travels faster than the classical channel).
Experiments
| Hypothesis | Fastest test | Refuted if |
|---|---|---|
| Bob’s qubit carries no information until the two classical bits arrive. | Uncheck the correction; expect F ≈ 0.50 and mean Bloch length ≈ 0. | F stays above 0.5 by more than ~3 standard errors, or the mean vector points along ψ. |
| F = 1 − 2p/3 for a depolarizing channel. | p = 0.30, 4096 shots: expect 0.80 ± 0.01. | Measured F is more than ~3σ from 0.80. |
| Alice’s outcomes are uniform whatever ψ is. | Set θ = 0 (message |0⟩) and θ = 180; bits stay ≈ 25% each. | One outcome dominates beyond sampling noise. |
Time loops: closed timelike curves on a qubit
Nobody can build a closed timelike curve, so physicists simulate what one would do to information. Two models disagree. Deutsch (1991): the qubit inside the loop must come out in the same state it went in, ρ = Tr[U(ρ_in ⊗ ρ)U†], which can be solved and is nonlinear. Post-selected CTCs (Lloyd et al. 2011): the loop is a teleportation that only counts when it is self-consistent, so paradoxical histories get probability zero. Both were run on photons (Ringbauer et al. 2014; Arvidsson-Shukur et al. 2023).
Loop interaction
Readouts
Sweep: the traveler state is swept from |0⟩ (θ = 0) to |1⟩ (θ = 180°) at the chosen φ. Dashed: how distinguishable the input is from |0⟩ (trace distance). Solid: how distinguishable the Deutsch outputs are. Any linear quantum process keeps the solid line at or below the dashed one; where it rises above, the loop is doing something no ordinary channel can. Green: P-CTC success probability.
Your hypothesis, mapped to the physics
Forward-only without a closed loop; a loop is confined to a zone; it forks spacetime; the energy is planetary to cosmic.
| Claim | Status | Fastest test | Refuted if |
|---|---|---|---|
| You can only go forward in time unless there is a closed timelike curve. | established Time dilation (speed, gravity) moves you forward at different rates; measured with GPS, muons, flown clocks. Going back needs a CTC in the geometry (Gödel, Tipler, Morris–Thorne wormholes). | None needed; textbook relativity. | — |
| A CTC makes a loop inside a zone, not a free trip to the past. | matches GR CTCs live in a chronology-violating region bounded by a Cauchy (chronology) horizon. A wormhole time machine cannot reach any time before it was built. | Swap preset: the traveler emerges as itself; the loop closes on itself and nothing escapes it. | A model where the traveler’s output is unconstrained by the loop. |
| It forks spacetime from the rest. | model-dependent Forking is Deutsch’s picture (consistent with branching / many-worlds). The rival is Novikov self-consistency, modelled by P-CTCs: one history, paradoxes get probability zero. | Grandfather preset: Deutsch gives a 50/50 mixture; P-CTC gives probability 0. | A real CTC experiment that shows forbidden histories (never-seen outcomes) rather than mixed outputs would favour a single self-consistent timeline over forking. |
| Energy on the scale of planets or universes. | right order A metre-scale wormhole throat needs roughly a Jupiter’s worth of mass-energy, and it has to be negative (exotic matter). A Tipler cylinder needs infinite length; Gödel’s solution needs the whole universe to rotate. Refinement: the blocker is the sign and quantum back-reaction near the chronology horizon (Hawking’s chronology-protection conjecture; Ford–Roman limits on negative energy), not just the amount. | Exotic-matter slider below. | A classical-matter CTC solution with ordinary positive energy, which no known solution provides. |
Exotic-matter budget (order of magnitude)
Thin-shell estimate |M| ~ c²·r₀/G of negative mass-energy to hold the throat open (Morris–Thorne 1988; Visser).
Readouts
For scale: a 1 m throat ≈ 0.7 Jupiter masses; a 1 km throat ≈ 0.7 solar masses. Humanity’s annual energy use is ~6×10²⁰ J.
Experiments
| Hypothesis | Fastest test | Refuted if |
|---|---|---|
| Deutsch and P-CTC disagree on the grandfather loop. | Grandfather preset. | Both give the same output state and the same success probability. |
| A Deutsch loop can tell two overlapping states apart with certainty, which no linear channel can. | Tuned π-rotation preset at θ = 45°, φ = 0: Deutsch output fidelity vs input ≈ 0.15 (ψ is sent to |1⟩) while |0⟩ is sent to |0⟩; the solid sweep line reaches 1 at 45°. | The solid line stays at or below the dashed line at the chosen θ. |
| CNOT-then-SWAP squares the input’s z component: output = (0, 0, cos²θ). | CNOT-then-SWAP preset at θ = 60°: output z reads 0.25. | Output z ≠ cos²θ, or the output keeps a phase (x or y ≠ 0). |
| P-CTC “effective time travel” via a swap succeeds one time in four. | Swap preset; success probability reads 0.25 for every input. | The probability depends on the input or is not 0.25. |
| With no interaction the loop is harmless. | “No interaction” preset. | Output differs from input, or P-CTC success is below 1. |
Dimensions: spacetime surfaces and 4D shapes, drawn in 3D
The other labs compute; this one draws the geometry they assume. Every scene squeezes one extra axis into a 3D view you can orbit: a fourth spatial direction w shown as a projection or a moving slice, time shown as height, or curvature shown as an embedding surface. Drag the view to orbit it.
Five things “4D” means
| Meaning | The extra axis | Scene here |
|---|---|---|
| A fourth direction of space (Hinton’s tesseract, Abbott’s Flatland) | w, at right angles to x, y and z | 4D shape: projection, and hyperplane slice |
| Spacetime (Minkowski, Einstein) | time t; light cones fix what can reach what | Block universe; Misner and Gödel light cones |
| Curved space (Schwarzschild, Morris–Thorne) | none: the embedding height is a drawing aid, not a direction anyone can travel | Black hole and wormhole surfaces |
| Captured 4D: volumetric video, 4D Gaussian splats, a 3D globe with a replay clock | time attached to a 3D capture | Block universe: scrub “now” like a replay; events fade in and out around it |
| Interactive 4D (Roblox Cube: 3D plus behaviour) | function: parts, rigging, scripts | Not modelled. It is a product term for behaviour, not geometry |
Scene
View
Experiments
| Hypothesis | Fastest test | Refuted if |
|---|---|---|
| A tesseract met by a 3D hyperplane through its middle leaves a cube; tilt the tesseract in x–w and the slice becomes a different polyhedron. | Slice scene, tesseract, all rotations 0, w₀ = 0: 8 vertices, 12 edges. Then x–w = 40°. | The untilted count is not 8 / 12, or the tilted slice still reads 8 / 12 at every w₀. |
| Rotating through w turns the projection inside out: the inner cube and the outer cube trade places at x–w = 180°. | Projection scene, perspective, spin off; slide x–w from 0 to 180°. | The inner and outer cubes keep their colours (w sign) after 180°. |
| The black-hole funnel has the same shape at every mass; only its size in km changes. | Black-hole scene; slide the mass from 1 to 10¹⁰ Suns. | The drawn surface changes shape, or r_s does not scale linearly with mass. |
| A Misner loop is a time machine only above the chronology horizon, never before it. | Misner scene: T = −0.3, 0, 0.3. | The loop reads timelike at T ≤ 0, or spacelike at T > 0. |
| Gödel light cones tip past horizontal at r = ln(1 + √2) ≈ 0.881. | Gödel scene: r = 0.87 then 0.89. | The loop type does not flip between them. |
| Moving observers disagree about “now” but nothing leaves its light cone. | Block scene: β = 0.6, then tick the rest-frame box; the now slice tilts by 0.6, every worldline stays inside the cone. | A worldline is steeper than the cone (speed ≥ c), or the slope differs from β. |
Lab notebook
| When | Lab | Parameters | Results |
|---|
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