04. Detailed Explanation of Dynamical Parameters: Source Code of Physical Laws

Introduction: From Static Structure to Dynamic Evolution

In Article 03, we established universe’s “spatial skeleton”—lattice set $\Lambda$ and cell Hilbert space ${\mathcal{H}}_{\text{cell}}$. But this is only static “stage”.

Key Question: How does universe evolve? How does time advance? How do physical laws operate?

Answer lies in dynamical parameter ${\Theta }_{\text{dyn}}$.

Popular Analogy: From Architectural Blueprint to Construction Rules

Continuing our building analogy:

Article 03 (Structural Parameters):

• Architectural blueprint: How many floors? How large each floor?
• This is static information—describes “what house looks like”

Article 04 (Dynamical Parameters):

• Construction rules: How to lay bricks? How to pour concrete?
• This is dynamic information—describes “how house is built”

Universe’s Situation:

• ${\Theta }_{\text{str}}$: “What universe looks like” (spatial structure)
• ${\Theta }_{\text{dyn}}$: “How universe operates” (temporal evolution)

This article will explain in detail how to encode entire universe’s time evolution from finite bit string ${\Theta }_{\text{dyn}}$.

Part I: QCA Automorphism and Time Evolution

What Is Time Evolution?

Classical Mechanics:

• Initial state: $(x_0,p_0)$
• Hamiltonian: $H(x,p)$
• Time evolution: $\dot{x}=\partial H/\partial p$, $\dot{p}=-\partial H/\partial x$
• Result: Trajectory $(x(t),p(t))$

Quantum Mechanics:

• Initial state: $|{\psi }_0⟩$
• Hamiltonian: $\hat{H}$
• Time evolution: $|\psi (t)⟩=e^{-i\hat{H}t/\hslash }|{\psi }_0⟩$
• Result: Unitary evolution of quantum state over time

Quantum Cellular Automaton (QCA):

• Initial state: ${\omega }_0$ (state functional)
• Evolution operator: $U$ (unitary operator)
• Time evolution: ${\omega }_n(A)={\omega }_0(U^{†n}A{U}^n)$
• Result: Unitary automorphism at discrete time steps

Definition of QCA Automorphism

Definition 4.1 (QCA Time Evolution):

Given global Hilbert space ${\mathcal{H}}_{\Lambda }={⨂}_{x\in \Lambda }{\mathcal{H}}_x$ and quasi-local algebra $\mathcal{A}(\Lambda )$, one time step of QCA is realized by unitary operator $U:{\mathcal{H}}_{\Lambda }\to {\mathcal{H}}_{\Lambda }$:

$$\alpha (A)=U^{†}AU,A\in \mathcal{A}(\Lambda )$$

Properties:

1. Unitarity: $U^{†}U=U{U}^{†}=1$ (reversible)
2. Locality: $U$ can be represented as product of finite depth local gates
3. Causality: Information propagates at finite speed (Lieb-Robinson bound)

Physical Meaning:

• $\alpha$: Maps observables at time $t$ to observables at time $t+1$
• Analogy: Camera shutter each shot, world “jumps” to next frame

Popular Analogy:

• Continuous time (classical/quantum mechanics) = Movie film (infinite frame rate)
• Discrete time (QCA) = Stop-motion animation (finite frame rate)
• $U$: “Transformation rule” from one frame to next

Task of ${\Theta }_{\text{dyn}}$

Dynamical parameter ${\Theta }_{\text{dyn}}$ needs to completely specify unitary operator $U$.

Challenge:

• Dimension of ${\mathcal{H}}_{\Lambda }$: $d_{\text{cell}}^{N_{\text{cell}}}\sim 10^{6\times 10^{90}}$ (astronomical number)
• Degrees of freedom of unitary operator $U$: $\sim (d_{\text{cell}}^{N_{\text{cell}}}{)}^2$ (double exponential)
• Direct encoding: Need $\sim 10^{180}$ bits (far exceeds $I_{\max }$!)

Solution: Exploit locality!

• $U$ is not arbitrary unitary operator
• $U$ composed of finite depth local gate circuits
• Local gates act on few neighbors (e.g., 2-4 lattice points)
• Total degrees of freedom exponentially compressed

Part II: Finite Gate Set $\mathcal{G}$

Gate Set: QCA’s “Programming Language”

In classical computation, all logical operations can be composed from basic gates (e.g., NAND). In quantum computation, similar “universal gate sets” exist.

Definition 4.2 (Finite Gate Set):

Fix a finite set of local unitary operators:

$$\boxed{\mathcal{G}=\{G_1,G_2,\dots ,G_K\}}$$

where each $G_k$ satisfies:

1. Finite dimension: Acts on finite number of lattice points (within radius $r_0$ neighborhood)
2. Unitarity: $G_k^{†}{G}_k=1$
3. Parameterized: Matrix elements determined by finite precision angle parameters

Example 1 (Single Lattice Point Gate): Acts on single cell ${\mathcal{H}}_x\cong {\mathbb{C}}^2$:

• Pauli gates: $X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, $Y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, $Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$
• Hadamard gate: $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$
• Rotation gate: $R_y(\theta )=\exp (-i\theta {\sigma }_y/2)=\left(\begin{array}{cc}\cos (\theta /2)& -\sin (\theta /2)\\ \sin (\theta /2)& \cos (\theta /2)\end{array}\right)$

Example 2 (Two Lattice Point Gate): Acts on two adjacent cells ${\mathcal{H}}_x\otimes {\mathcal{H}}_{x+1}\cong {\mathbb{C}}^2\otimes {\mathbb{C}}^2$:

• CNOT gate: $\text{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$
• SWAP gate: $\text{SWAP}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$
• Controlled rotation: $C{R}_y(\theta )=|\uparrow ⟩⟨\uparrow |\otimes 1+|\downarrow ⟩⟨\downarrow |\otimes R_y(\theta )$

Example 3 (Dirac-QCA Gate Set):

• Coin gate: $C(\theta )=\exp (-i\theta {\sigma }_y)$ acts on spin degree of freedom
• Shift gate: $S={\sum }_{x,s}|x+s,s⟩⟨x,s|$ (spin-dependent translation)
• Parameter: Angle $\theta \in [0,2\pi )$ (needs discretization)

Gate Set Size and Universality

Theorem 4.3 (Universal Quantum Gate Set):

Exists finite gate set $\mathcal{G}$ such that any unitary operator can be arbitrarily approximated as finite depth combination of gates in $\mathcal{G}$.

Classical Result (Solovay-Kitaev Theorem):

• Gate set $\{H,T,\text{CNOT}\}$ is universal on ${\mathbb{C}}^2$
• $H=\text{Hadamard}$, $T=\left(\begin{array}{cc}1& 0\\ 0& e^{i\pi /4}\end{array}\right)$
• Approximation precision $ϵ$, need depth $D\sim {\log }^c(1/ϵ)$ (polylogarithmic growth)

Physical Meaning:

• Don’t need infinite variety of gates
• Finite gate set $\mathcal{G}$ (e.g., $K=10$) sufficient to express all physics
• Analogy: 10 basic notes can combine into all symphonies

Encoding Overhead:

• Gate set can be pre-agreed (e.g., choose “Standard Model QCA gate set”)
• Or encode in ${\Theta }_{\text{str}}$ (additional $\sim 50$ bits)
• Usually: Agree on fixed gate set, save encoding

Bit Encoding of Gate Set

If gate set has $K$ gates, selecting a gate needs:

$${\log }_{2}K\text{ bits}$$

Example:

• $K=16$: Need 4 bits
• $K=256$: Need 8 bits

Part III: Construction of Quantum Circuits

Finite Depth Circuits

Definition 4.4 (Quantum Circuit):

Global unitary operator $U$ composed of $D$ layers of local gates:

$$\boxed{U=U_D{U}_{D-1}\cdots U_2{U}_1}$$

where each layer $U_{\ell }$ is parallel application of several local gates $G_k$:

$$U_{\ell }=\prod _j{G}_{k_{\ell ,j}}^{(R_{\ell ,j})}$$

• $k_{\ell ,j}$: Type of $j$-th gate in layer $\ell$ (selected from $\mathcal{G}$)
• $R_{\ell ,j}\subset \Lambda$: Action region of this gate

Key Constraints:

• Finite depth: $D<\infty$ (usually $D\sim 10$-$10^3$)
• Locality: Gates in same layer act on disjoint regions (can parallelize)
• Finite radius: Each gate acts within radius $r_0$ neighborhood

Popular Analogy: Imagine factory assembly line:

• Each layer $U_{\ell }$: One workstation on assembly line
• Each gate $G_k$: One operation at workstation (e.g., “tighten screw”, “weld”)
• Depth $D$: How many workstations on assembly line
• Product passes through $D$ workstations to complete assembly = Universe passes one time step

Encoding Circuit Depth

Encoded Content:

1. Depth $D$:

   • Use $\lceil {\log }_2{D}_{\max }\rceil$ bits
   • If $D\le 1024$, need 10 bits

2. Gate Configuration for Each Layer (for $\ell =1,\dots ,D$):

   • Gate type $k_{\ell ,j}$: ${\log }_{2}K$ bits/gate
   • Action region $R_{\ell ,j}$: Depends on symmetry

Translation-Invariant Case (greatly simplified):

If gate configuration of each layer is translation-invariant (all odd/even lattice points apply same gate):

• Only need to specify: Gate type + odd/even
• Encoding per layer: ${\log }_{2}K+1$ bits

General Case (non-translation-invariant):

Need to specify position of each gate:

• Lattice point coordinates: $\sim d{\log }_{2}L$ bits/gate
• $n_{\text{gates}}$ gates: $\sim n_{\text{gates}}\times d{\log }_{2}L$ bits

Power of Symmetry Compression:

Conclusion: Translation symmetry is necessary, otherwise parameter explosion!

Part IV: Discrete Angle Parameters

Problem of Continuous Parameters

Many physical gates contain continuous parameters:

$$R_y(\theta )=\exp (-i\theta {\sigma }_y)$$

where $\theta \in [0,2\pi )$ is a real number.

Problem:

• Real number $\theta$ needs infinite bits for exact encoding (e.g., $\pi =3.14159\dots$)
• Contradicts finite information axiom!

Solution: Discretize angle parameters.

Discretization Scheme

Definition 4.5 (Discrete Angle):

Restrict angles to rational numbers:

$$\boxed{\theta =\frac{2\pi n}{2^m}}$$

where:

• $n\in \{0,1,\dots ,2^m-1\}$: Discrete label
• $m\in \mathbb{N}$: Precision bit number

Example ($m=3$):

• Available angles: $\frac{2\pi \times 0}{8},\frac{2\pi \times 1}{8},\dots ,\frac{2\pi \times 7}{8}$
• That is: $0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$
• Total 8 discrete values

Encoding:

• Only need to store integer $n$ (needs $m$ bits)
• Example: $n=5$ (binary: 101) represents $\theta =5\pi /4$

Precision Analysis:

Angle resolution: $$\Delta \theta =\frac{2\pi }{2^m}$$

Physical Distinguishability:

Current most precise atomic clock angular frequency measurement precision $\sim 10^{-18}$, therefore:

• $m=64$: Precision far exceeds current experiments (excessive)
• $m=32$: Sufficient for all foreseeable experiments
• $m=16$: Sufficient for many applications

Effect of Discrete Angles on Physical Constants

In Dirac-QCA, relation between electron mass and coin angle (from source theory Theorem 3.4):

$$m_{\text{eff}}{c}^2\approx \frac{\theta }{\Delta t}$$

Discretization Effect:

$$\Delta m_e\sim m_e\frac{\Delta \theta }{\theta }=m_e\frac{2\pi /2^m}{\theta }$$

Numerical Example:

• Assume $\theta \sim \pi /4$ (typical value)
• $m=16$: $\Delta m_e/m_e\sim 2\pi /(2^{16}\times \pi /4)=8/2^{16}\sim 10^{-4}$
• $m=32$: $\Delta m_e/m_e\sim 10^{-9}$ (comparable to current measurement precision)

Conclusion:

• $m=32$ sufficient to match all known physical constant measurement precisions
• $m=64$ excessive for foreseeable future
• Conservative choice: $m=50$ (middle value)

Part V: Lieb-Robinson Bound and Causality

Information Propagation Speed

Locality of QCA ensures information propagates at finite speed.

Theorem 4.6 (Lieb-Robinson Bound):

Given finite depth $D$ local unitary circuit $U$, define evolution $\alpha (A)=U^{†}AU$. If operator $A$ supported on region $X$ ($A\in {\mathcal{A}}_X$), then for any region $Y$ with distance $d(X,Y)>v_{\text{LR}}D$:

$$\boxed{\Vert [\alpha (A),B]\Vert \le C{e}^{-c(d(X,Y)-v_{\text{LR}}D)}}$$

where $B\in {\mathcal{A}}_Y$, $v_{\text{LR}}$ is Lieb-Robinson velocity, $C,c$ are constants.

Physical Meaning:

• $v_{\text{LR}}$: “Effective speed of light” for information propagation
• After $D$ time steps, information propagates at most distance $\sim v_{\text{LR}}D$
• Beyond this distance, operator commutator decays exponentially (almost no interaction)

Example (Nearest-Neighbor Gates):

• Each layer gate only acts on nearest neighbors → $v_{\text{LR}}=1$ lattice point/step
• Depth $D=10$ → Information propagates $\sim 10$ lattice points
• Regions at distance $>10$ almost unaffected

Popular Analogy:

• Imagine throwing stone into pond
• Ripples propagate outward (information transfer)
• Wave speed finite ($v_{\text{LR}}$)
• After time $t$, ripple radius $\sim v_{\text{LR}}t$
• Distant frogs haven’t felt yet (small commutator)

Light Cone Structure

Definition 4.7 (QCA Light Cone):

For lattice point $x\in \Lambda$ and time $n$, define:

$${\mathcal{C}}_n(x)=\{y\in \Lambda :d(x,y)\le v_{\text{LR}}n\}$$

as causal cone (or light cone) of $x$ after $n$ steps.

Causality Principle:

• Information at lattice point $x$ at time 0 can only affect lattice points in ${\mathcal{C}}_n(x)$ at time $n$
• External lattice points $y\notin {\mathcal{C}}_n(x)$ almost unaffected

Analogy with Relativity:

Numerical Example (Universe QCA):

• Lattice spacing $a={\ell }_p\sim 10^{-35}\text{m}$ (Planck length)
• Time step $\Delta t\sim t_p\sim 10^{-43}\text{s}$ (Planck time)
• Lieb-Robinson velocity: $v_{\text{LR}}=a/\Delta t\sim c$ (speed of light!)

This is no coincidence—QCA in continuous limit automatically recovers relativistic causality!

Part VI: Bit Count of Dynamical Parameters

Combining above parts:

$$\boxed{|{\Theta }_{\text{dyn}}|=I_{\text{depth}}+D\times (I_{\text{gate}}+I_{\text{region}}+I_{\text{angles}})}$$

Decomposition of Terms

(1) Circuit Depth: $$I_{\text{depth}}=\lceil {\log }_2{D}_{\max }\rceil$$

Example: $D_{\max }=1024$ → 10 bits

(2) Gate Type Per Layer (for each layer $\ell$): $$I_{\text{gate}}={\log }_{2}K$$

Example: $K=16$ → 4 bits/layer

(3) Action Region (per layer):

Translation-Invariant: $$I_{\text{region}}=1\text{ bit}$$ (Specify odd/even lattice points)

General Case: $$I_{\text{region}}=n_{\text{gates}}\times d{\log }_{2}L$$ (Coordinates of each gate)

(4) Angle Parameters (per layer):

If each layer has $n_{\text{angles}}$ gates needing angle parameters, each with precision $m$ bits: $$I_{\text{angles}}=n_{\text{angles}}\times m$$

Example: $n_{\text{angles}}=2$, $m=50$ → 100 bits/layer

Total (Translation-Invariant Dirac-QCA)

Parameter Settings:

• $D=10$ (depth)
• $K=16$ (gate set size)
• Translation-invariant ($I_{\text{region}}=1$)
• 2 angle parameters per layer, $m=50$

Calculation:

• $I_{\text{depth}}=10$ bits
• Per layer: $I_{\text{gate}}+I_{\text{region}}+I_{\text{angles}}=4+1+100=105$ bits
• Total: $10+10\times 105=1060$ bits

$$\boxed{|{\Theta }_{\text{dyn}}|\sim 1000\text{ bits}}$$

Key Observation: $$|{\Theta }_{\text{dyn}}|\sim 10^3\ll I_{\max }\sim 10^{123}$$

Dynamical parameter information negligible!

Part VII: From Discrete to Continuous—Continuous Limit Preview

How Does QCA Lead to Field Equations?

Core Idea: In limit of lattice spacing $a\to 0$, time step $\Delta t\to 0$, discrete QCA converges to continuous field theory.

Dirac-QCA Example (from source theory Theorem 3.4):

Discrete QCA:

• Update operator: $U=S\cdot C(\theta )$
• $C(\theta )=\exp (-i\theta {\sigma }_y)$ (coin gate)
• $S$: Spin-dependent translation

Scaling Limit:

• $a,\Delta t,\theta \to 0$
• $c_{\text{eff}}=a/\Delta t$ fixed (effective speed of light)
• $m_{\text{eff}}=\theta /(\Delta t{c}_{\text{eff}}^2)$ fixed (effective mass)

Continuous Limit:

$$\boxed{i{\partial }_t\psi =\left(-i{c}_{\text{eff}}{\sigma }_z{\partial }_x+m_{\text{eff}}{c}_{\text{eff}}^2{\sigma }_y\right)\psi }$$

This is exactly one-dimensional Dirac equation!

Key Relation (from source theory):

$$\boxed{m_{\text{eff}}\approx \frac{\theta }{\Delta t}}$$

Physical Meaning:

• Discrete angle parameter $\theta$ → Continuous field theory mass $m_{\text{eff}}$
• By adjusting angle parameters in ${\Theta }_{\text{dyn}}$, can analytically derive physical constants!

Example (Electron Mass):

• Experimental value: $m_e{c}^2\approx 0.511\text{MeV}$
• If $\Delta t=t_p\sim 10^{-43}\text{s}$
• Then $\theta =m_e\Delta t/\hslash \sim 10^{-22}$ (extremely small angle)

This will be detailed in Article 07.

Gauge Fields and Gravitational Constant

Similarly, gauge coupling constants and gravitational constant can also be derived from ${\Theta }_{\text{dyn}}$.

Theorem Preview (source theory Theorem 3.5):

In QCA with gauge registers, gauge coupling $g(\Theta )$ related to discrete angle combinations in ${\Theta }_{\text{dyn}}$:

$$g(\Theta )=f_g({\theta }_1,{\theta }_2,\dots )$$

Gravitational constant:

$$G(\Theta )={\ell }_{\text{cell}}^{d-2}/m_{\text{Planck}}^{d-2}(\Theta )$$

Philosophical Implication:

• Physical constants not “arbitrary numbers chosen by God”
• But mathematical consequences of finite parameter $\Theta$
• Analogy: $\pi$ not arbitrary number, but geometric consequence of circle

Part VIII: Connection with Universe Evolution

One Time Step = One QCA Update

Universe Evolution Picture:

graph LR
    A["t=0<br/>Initial State ω₀"] --> B["t=1<br/>α(ω₀)"]
    B --> C["t=2<br/>α²(ω₀)"]
    C --> D["t=3<br/>α³(ω₀)"]
    D --> E["...<br/>"]
    E --> F["t=n<br/>αⁿ(ω₀)"]

    style A fill:#ffe6e6
    style F fill:#e6f3ff

Each Step:

• Apply unitary operator $U_{{\Theta }_{\text{dyn}}}$
• State changes: ${\omega }_n\to {\omega }_{n+1}$
• Operator changes: $A\to \alpha (A)=U^{†}AU$

Nature of Time:

• Discrete time steps: $n=0,1,2,\dots$
• Physical time: $t=n\Delta t$
• Continuous limit: $\Delta t\to 0$ → Recover continuous time

Popular Analogy:

• Universe like giant “clock”
• Each “tick” (time step) applies $U$ once
• ${\Theta }_{\text{dyn}}$ is clock’s “gear design blueprint”
• 13.7 billion years = $\sim 10^{60}$ “ticks” (in Planck time units)

Universe’s “Program”

Computational Universe Perspective:

$$\text{Universe}=\text{Initial State}+\text{Evolution Rule}+\text{Time}$$

$$\mathfrak{U}(t)={\alpha }_{{\Theta }_{\text{dyn}}}^{t/\Delta t}({\omega }_0^{{\Theta }_{\text{ini}}})$$

Analogy:

• Initial state ${\omega }_0$: Program’s “input”
• Evolution $\alpha$: Program’s “algorithm” (encoded by ${\Theta }_{\text{dyn}}$)
• Time $t$: Program’s “number of iterations”
• Final state $\mathfrak{U}(t)$: Program’s “output”

Universe is a Quantum Program:

• Program length: $|\Theta |\sim 2000$ bits (extremely short!)
• Running time: $10^{60}$ steps
• State space: $\dim \mathcal{H}\sim 10^{10^{91}}$ (huge)
• Output: Physical universe we observe

Summary of Core Points of This Article

Five Components of Dynamical Parameters

$${\Theta }_{\text{dyn}}=(\mathcal{G},D,\{k_{\ell }\},\{R_{\ell }\},\{{\theta }_{\ell ,j}\})$$

QCA Automorphism

$$\alpha (A)=U^{†}AU,U=U_D\cdots U_1$$

Properties:

• Unitarity (reversible)
• Locality (finite depth)
• Causality (Lieb-Robinson bound)

Discrete Angle Parameters

$$\theta =\frac{2\pi n}{2^m},n\in \{0,\dots ,2^m-1\}$$

Precision:

• $m=16$: $\sim 10^{-4}$
• $m=32$: $\sim 10^{-9}$
• $m=50$: $\sim 10^{-15}$

Lieb-Robinson Bound

$$\Vert [\alpha (A),B]\Vert \le C{e}^{-c(d-v_{\text{LR}}D)}$$

Physical Meaning: Information propagation speed $v_{\text{LR}}$, similar to speed of light.

Continuous Limit Preview

$$\underset{a,\Delta t\to 0}{\lim }\text{QCA}\Rightarrow i{\partial }_t\psi =(-ic{\sigma }_z{\partial }_x+m{\sigma }_y)\psi$$

Mass-Angle Parameter Relation: $$m_{\text{eff}}\approx \theta /\Delta t$$

Core Insights

1. Locality compresses information: $|{\Theta }_{\text{dyn}}|\sim 10^3\ll I_{\max }$
2. Symmetry necessary: Translation-invariance → Information from $10^{90}$ reduced to $10^3$
3. Discretization necessary: Finite information → Angle parameters must be discrete
4. Causality natural: Lieb-Robinson bound → Relativistic causality
5. Physical constants derivable: $m,g,G$ are all functions of $\Theta$

Key Terminology

• QCA Automorphism: $\alpha (A)=U^{†}AU$
• Finite Gate Set: $\mathcal{G}=\{G_1,\dots ,G_K\}$
• Finite Depth Circuit: $U=U_D\cdots U_1$
• Discrete Angle: $\theta =2\pi n/2^m$
• Lieb-Robinson Bound: Upper bound on information propagation speed
• Lieb-Robinson Velocity: $v_{\text{LR}}$

Next Article Preview: 05. Detailed Explanation of Initial State Parameters: Universe’s Factory Settings

• Construction of initial state ${\omega }_0^{\Theta }$
• State preparation circuit $V_{{\Theta }_{\text{ini}}}$
• QCA version of Hartle-Hawking no-boundary state
• Initial entanglement structure and Lieb-Robinson bound
• Symmetry constraints on initial state
• Initial entropy and universe “age”