Insights from Quantum Information into Fundamental Physics

• The dissertation is identified as work completed during Ph.D. studies at Cambridge’s Department of Applied Mathematics and Theoretical Physics.Source: Declaration · p. 5
• The listed material covers lattice quantum field theory, local qubit representations, lattice QED, quantum cellular automata, causal models, and information flow.Source: Declaration · p. 8
• Later listed sections address causal discrete-time models, fermionic quantum cellular automata, local decomposition, quantum-computer simulation, and QCA evolution on a line.Source: Declaration · p. 8

• The thesis applies quantum-information ideas to relativistic quantum systems and quantum equilibration, divided into two corresponding parts.Source: Prologue · p. 13
• The first part incorporates results from [1] and, including the main theorems and corollaries identified for those papers.Source: Prologue · p. 13
• The second part incorporates material from [3] and [4], while noting that the slow-equilibration result from [4] is not the author’s own work.Source: Prologue · p. 13
• Some results were expanded or altered from the papers, and many sections are unpublished.Source: Prologue · pp. 13–14

Introduction

• Causality on a spatial lattice requires discrete time because continuous-time local Hamiltonians permit instantaneous, arbitrarily small-amplitude propagation.Source: Introduction · pp. 18–19
• Discrete spacetime offers finite memory requirements, momentum cutoffs, natural scales, and maximum propagation speeds, supporting simulation and regularization goals.Source: Introduction · pp. 17, 19
• The introduction also identifies broken spacetime symmetries, cubic-lattice naturalness, and fermion doubling as limitations or open concerns for discrete models.Source: Introduction · pp. 20–21
• Discrete-time quantum walks provide a general continuum-limit route to relativistic particles, while fermion doubling can be overcome in two dimensions and reduced in three.Source: Introduction · p. 21
• Discrete causal fermion fields require a nontrivial vacuum converging to the continuum vacuum, and the thesis constructs such limits for relativistic fermion fields.Source: Introduction · p. 22
• Adding ancillary particles makes causal fermionic dynamics constant-depth circuits, establishing equivalence between fermionic and regular quantum cellular automata.Source: Introduction · p. 22
• A decomposition of one-dimensional QCA dynamics into on-site unitaries and shifts is presented as an abstract closing result for the chapter.Source: Introduction · p. 22

2.1 Quantum Computing

• Quantum walks and quantum cellular automata are universal quantum-computation frameworks whose causality makes them useful as discretized models of relativistic systems.Causality bounds information propagation over finite distances and is closely aligned with relativistic causality.Source: §2.1 · p. 24

2.1.1 Computational Complexity and Efficiency

• Efficient quantum computation means solving size-n problems with polynomially many operations, while BQP permits bounded error with success probability at least 2/3.Gate complexity counts logical operations, whereas time complexity counts sequential time steps and can be smaller when gates run in parallel.Source: §2.1.1 · p. 25

2.1.2 Simulating Physics on a Quantum Computer

• Quantum computers efficiently simulate lattice systems with local Hamiltonians by decomposing short-time evolution into local operations and applying product-formula approximations.The total gate count is polynomial in the number of time steps and lattice sites; fermionic systems require specialized mappings, while fermionic and conventional quantum computers efficiently simulate each other with constant versus logarithmic slowdown.Source: §2.1.2 · p. 27

An Aside: Analogue Quantum Simulators

• Analogue quantum simulators implement related physical Hamiltonians without error correction and are closer to laboratory realization, including proposals using cold atoms or trapped ions to model relativistic systems.One proposal uses optical-lattice atoms to mimic interacting electrons and positrons through the Thirring model.Source: An Aside: Analogue Quantum Simulators · p. 28

Another Aside: The Computational Power of Nature

• Efficient simulation of φ4 theory suggests that quantum field theories may be simulable by quantum computers, despite their uncountably infinite degrees of freedom and unclear state spaces in interacting models.This also raises whether quantum field computers could exceed conventional quantum computers; efficient simulation by quantum computers would show they cannot.Source: Another Aside: The Computational Power… · pp. 28–29

2.1.3 Discrete-Time Quantum Walks

• Discrete-time quantum walks are causal, translationally invariant unitary dynamics on lattices, with a finite-dimensional internal degree of freedom that can represent spin or chirality in continuum limits.The internal coin mixes left- and right-moving components, producing an effective slowing analogous to mass mixing chiralities in the Dirac equation.Source: §2.1.3 · p. 30

2.1.4 Quantum Cellular Automata

• Quantum cellular automata are translationally invariant causal unitary evolutions of finite-dimensional quantum systems arranged on a lattice.A partitioned circuit of neighboring two-site unitaries provides a causal example, with an initially localized operator spreading only to a bounded set of nearby sites.Source: §2.1.4 · pp. 31–32
• Quantum cellular automata are causal, universal quantum-computing models that can serve as discretized relativistic systems and potentially practical quantum simulators.They can efficiently simulate the circuit model and vice versa, while free Dirac fermions on a line admit simple nearest-neighbour and onsite-unitary simulations [41].Source: §2.1.4 · p. 33

2.2 Physical Models

• Relativistic quantum systems become difficult with interactions because of antiparticles, vacuum entanglement, and infinities, motivating causal quantum simulation alongside lattice quantum field theory.The discussion uses Hamiltonians rather than Lagrangians or path integrals and adopts relativistic units with ℏ = c = 1.Source: §2.2 · pp. 33–34

2.2.1 Dirac and Weyl particles

• Free spin-1/2 particles evolve according to either the Dirac or Weyl equation, and their wavefunctions have two extra degrees of freedom in one or two dimensions and four in three.The Dirac Hamiltonian is linear in momentum, and massive spin-1/2 particles are called Dirac particles.Source: §2.2.1 · pp. 34–36

Chirality

• Chirality labels decoupled massless sectors in one and three dimensions, but no analogous chirality exists in two dimensions because the massless Hamiltonian has no nontrivial commuting rank-one projectors.Massless particles in these sectors are Weyl particles, and chiral symmetry occurs when m = 0.Source: Chirality · pp. 36–37

2.2.2 Fermions

• Fermionic lattice modes obey canonical anticommutation relations, whose consequences include the exclusion principle and physical observables restricted to even products of creation and annihilation operators.The Hubbard Hamiltonian illustrates nearest-neighbour hopping together with onsite Coulomb repulsion.Source: §2.2.2 · pp. 37–38

2.2.3 The Jordan-Wigner Transformation

• The Jordan-Wigner transformation represents fermionic modes with qubits by adding strings of Z operators that preserve anticommutation relations but make individual creation operators manifestly nonlocal.Local even fermionic operators can remain local on a line, whereas this generally fails in higher dimensions and can fail for periodic boundary conditions.Source: §2.2.3 · pp. 38–40

2.2.4 Second Quantization

• Second quantization maps single-particle states and operators to fermionic creation and annihilation operators, constructing multiparticle theories with the corresponding statistics.The construction preserves the canonical anticommutation relations and provides a corresponding map for single-particle Hamiltonians and momentum states.Source: §2.2.4 · pp. 40–41

2.2.5 Field Theory

• Free fermion field theory requires more than independently second-quantized Dirac particles because the physical vacuum is not annihilated by all annihilation operators.The chapter illustrates this construction by deriving the one-dimensional free Dirac fermion field theory from Dirac particles.Source: §2.2.5 · p. 41
• Second quantization promotes single-particle fields to multiparticle fermionic operators, with negative-energy modes reinterpreted through a filled vacuum and antiparticle holes.The vacuum fills all negative-energy modes, preventing further negative-energy creation and yielding the familiar particle–antiparticle field description.Source: §2.2.5 · pp. 41–43

An Aside: Vacuum Entanglement

• The relativistic vacuum is entangled, complicating simulation because preparing it is difficult, especially for interacting models.One preparation strategy turns on interactions adiabatically after preparing the free vacuum [12].Source: An Aside: Vacuum Entanglement · p. 43

2.2.6 Lattice Fermions and the Fermion Doubling Problem

• Lattice fermion doubling occurs because naive discretization produces low-energy modes near both zero and π momentum, despite recovering the continuum dispersion near zero.The dispersion relation therefore contains two low-energy mode sets: momenta near zero and momenta near π.Source: §2.2.6 · p. 44, Figure 2.4 · p. 44
• Interactions can couple the two low-momentum sectors, making the lattice theory appear to contain two fermion types; local chiral lattice Hamiltonians are subject to a general doubling theorem.Common remedies violate a theorem hypothesis, such as breaking chiral symmetry with Wilson fermions or reducing doublers with staggered fermions [50].Source: §2.2.6 · p. 45

3.1 Introduction

• Quantum walks can approach relativistic particles with Lorentz-symmetric continuum dynamics despite lacking continuous spacetime or lattice symmetries, while fermion doubling remains a significant limitation.The chapter examines whether doubling can be mitigated or removed in some cases.Source: §3.1 · pp. 47–48

3.2 Continuum Limits

• The one-dimensional walk with W = e−imσxa approximates Dirac evolution generated by h = Pσz + mσx, with short-time error O(a2).The construction requires the single-step evolution to approach the identity as a → 0.Source: §3.2 · p. 49

3.2.1 Approximating Continuum Models by Discrete Ones

• The continuum-limit framework separates approximation of initial states from approximation of dynamics, enabling discrete models to simulate continuous particles increasingly accurately as lattice spacing decreases.The discussion is framed primarily for simulation but can also describe discrete spacetime as a fundamental model.Source: §3.2.1 · pp. 50–51

The Block Mapping

• Block mapping preserves localization by embedding discrete position states into continuum wavefunctions, while momentum mapping identifies equal-momentum states but is physically less satisfying.Momentum mapping makes state convergence straightforward, whereas block mapping retains the localization of discrete particles.Source: §3.2.1 · p. 51, The Momentum Mapping · pp. 54–55
• Under block mapping, discrete and continuum momentum inner products converge to 2πδ(p −qd) below a cutoff Λ growing slower than 1/a, allowing arbitrary state-approximation accuracy.Accuracy at fixed lattice spacing depends on the smoothness of the continuum state.Source: The Block Mapping · pp. 52–53

The Embedded Evolution Operator

• The same embedded evolution operator works for both mappings because lattice shifts become exponentials of the corresponding continuum or discrete momentum operators with identical action after embedding.For block mapping, a shift is represented by e−i⃗q·⃗Pa; for momentum mapping, equal momentum identification makes the two shift operators equivalent.Source: The Embedded Evolution Operator · p. 55

3.2.2 The Lie-Trotter Product Formula

• With a momentum cutoff Λ growing sufficiently slowly as a → 0, the Lie-Trotter product formula proves convergence of repeated quantum-walk steps to continuum evolution.The unrestricted momentum operator norm grows too quickly with decreasing lattice spacing, motivating the cutoff.Source: §3.2.2 · pp. 55–56
• Quantum-walk convergence here concerns dynamics for fixed states rather than operator-norm convergence, with implications for studying external fields.The relevant convergence notion is explicitly distinguished from operator-norm convergence.Source: §3.2.2 · p. 56

3.2.3 A General Recipe

• A general product-of-shifts-and-coin recipe constructs walks whose continuum limits realize two-dimensional Dirac, three-dimensional Weyl, and Dirac equations.The three-dimensional Dirac construction combines left- and right-handed Weyl walks through a four-dimensional internal degree of freedom; the recipe also extends to Hamiltonians linear in momentum.Source: §3.2.3 · pp. 57–59

3.2.4 Faster Converging Quantum Walks

• The cited result states that the walk converges to evolution via H with error O(Λ3a2), while Strang splitting reduces the error from O(a2) to O(a3).Higher-order Suzuki-Trotter formulas may require terms like (eiPa)c to have integer c; otherwise, the evolution may not be causal.Source: §3.2.4 · pp. 59–60

3.2.5 Symmetries on the Lattice

• Standard ordered conditional shifts generally lack lattice rotational symmetry despite Lorentz symmetry in the continuum, but a four-dimensional coin can restore lattice rotational symmetry.The restoring construction is a direct sum of two continuum-Lorentz-symmetric walks and is considered somewhat unnatural because its symmetry-enabling degrees of freedom differ from those of the continuum.Source: §3.2.5 · pp. 61–62

3.3 Two Dimensional Coins are Special

• For massless quantum walks with two-dimensional coins, continuum limits generally yield massless relativistic equations after a coordinate change or by imposing continuum rotational symmetry, unlike higher-dimensional coins.The section motivates this as surprising because relativistic continuum behavior might otherwise be expected to require fine-tuning, and cites a three-dimensional-coin counterexample without relativistic symmetry.Source: §3.3 · p. 62

3.3.1 General Form of Quantum Walks

• Causal quantum walks have finitely supported transition operators A_q over a neighbourhood Q, while translational invariance removes dependence on the initial position.The general evolution is expressed through position shifts and internal operators, although a universal shift-and-coin decomposition is not established.Source: §3.3.1 · pp. 62–63

3.3.2 Mass for Quantum Walks

• Massless quantum walks are defined by a lattice unitary that remains fixed as lattice spacing and timestep vanish, unlike massive walks whose unitary approaches the identity.This distinction motivates defining mass so continuum limits can be taken consistently when the coin is nontrivial.Source: §3.3.2 · pp. 63–64

3.3.3 The Continuum Limit

• For massless walks, Theorem 3.4 establishes convergence to a continuum Hamiltonian as a →0 while the momentum cutoff grows with Λ^2a →0.This is identified as one of the main results of [1].Source: §3.3.3 · pp. 64, 66
• For massive walks with W = e^−iMa, Corollary 3.5 gives a continuum Hamiltonian obtained by adding the fixed self-adjoint mass operator M to the massless Hamiltonian.The convergence proof reduces the massive case to the massless result and shows convergence as lattice spacing vanishes.Source: §3.3.3 · pp. 66–67

3.3.4 The Continuum Hamiltonian

• After coordinate changes, every massless quantum walk with a two-dimensional coin has massless relativistic continuum dynamics, including lower-dimensional reductions and at most three active spatial dimensions.The proof removes constant velocity shifts, rescales axes, and uses singular-value decomposition; rotationally symmetric cases become Weyl Hamiltonians.Source: §3.3.4 · pp. 67–69
• Rotational symmetry in the continuum implies Lorentz symmetry for massless walks with two-dimensional coins, with the Hamiltonian proportional to a left- or right-handed Weyl Hamiltonian.The argument uses the induced SU(2) rotation representation to force orthogonal, equal-norm internal generators.Source: §3.3.4 · p. 69

Continuum Hamiltonians with Mass

• With mass, a one-dimensional choice M = mσx yields the Dirac Hamiltonian, but M = m1σz+m2σx need not produce relativistic evolution unless parity symmetry excludes it.Thus relativistic massive behavior requires constraints on the internal mass operator.Source: Continuum Hamiltonians with Mass · pp. 69–70

3.3.5 Higher Dimensional Coins are not so Special

• Higher-dimensional coins invalidate this implication: a three-dimensional coin can produce a rotationally symmetric continuum Hamiltonian that is not Lorentz symmetric.The counterexample uses the three-dimensional SO(3) representation generated by J_i and is realizable by conditional shifts.Source: §3.3.5 · p. 70
• Higher-dimensional coin constructions do not generally yield relativistic dynamics because H^2 − P⃗^2 is not Lorentz invariant.Adding a P̃^4 term or rescaling coordinate axes would break rotational symmetry, and rotations or unitary basis changes cannot remove the noninvariance.Source: §3.3.5 · p. 71

3.4 Fermion Doubling in Quantum Walks

• Discrete-time quantum simulators can evade the usual fermion-doubling obstruction because causal unitary evolution corresponds to a nonlocal Hamiltonian while still recovering relativistic fermions in the continuum limit.The models can also be implemented with simple quantum circuits.Source: §3.4 · p. 72
• Fermion doubling matters only when interactions are present, motivating the introduction of gauge interactions before analyzing the problem in one, two, and three spatial dimensions.The gauge construction uses U(1) fields and is related to the temporal-gauge approach in lattice gauge theory.Source: §3.4 · p. 72, §3.4.1 · pp. 72–73

3.4.1 Gauge Interactions for Quantum Walks

• Gauge interactions are introduced as link-dependent phases, producing gauge-invariant evolution whose continuum limit recovers a massless Dirac particle coupled to an electromagnetic field.For small lattice spacing, the link variable contributes as the continuum gauge field, although smoothness assumptions may be needed for a rigorous limit.Source: §3.4.1 · pp. 73–74
• In higher dimensions, each directional gauge field component is associated with lattice links, and particles acquire the corresponding phase when traversing those links.The construction extends the one-dimensional gauge-invariant shift to directions b ∈ {x, y, z}.Source: §3.4.1 · pp. 73–74

3.4.2 Fermion Doubling in Discrete Time

• Fermion doubling is defined by the survival of non-low-momentum modes as lattice spacing a approaches zero, and some discrete-time models can be modified to alleviate it.The analysis uses quantum walks as single-particle models whose results extend naturally to many-particle systems and discrete field theories.Source: §3.4 · p. 72, §3.4.2 · p. 74
• One-dimensional quantum walks avoid fermion doubling because their high-momentum modes acquire large phases and do not survive the continuum limit.The corresponding Hamiltonian is nonlocal, so the fermion-doubling theorem does not apply directly.Source: §3.4.2 · pp. 74–75

3.4.3 Fermion Doubling in Two Dimensions

• The two-dimensional walk has exactly two low-energy points, at zero momentum and at px = py = π/a, so the latter produces a massless Dirac doubler.The trace analysis shows no other points where the evolution operator is close to the identity.Source: §3.4.3 · pp. 75–76

Removing the Doubling Modes

• Restricting initial states to a body-centred cubic sublattice removes the two-dimensional doubler, and gauge or on-site interactions preserve this subspace.The two distinct b.c.c. sublattices can also be treated as different particle types.Source: Removing the Doubling Modes · pp. 76–78

3.4.4 Fermion Doubling in Three Dimensions

• The three-dimensional Weyl walk has eight modes obeying the Weyl equation in the continuum limit, comprising four π/a modes and four π/2a modes.A lemma establishes that these are all the modes for which the evolution operator becomes the identity.Source: §3.4.4 · pp. 79–80

Mitigating the Doubling problem

• In three dimensions, restricting states to a b.c.c. sublattice removes the π/a doublers but leaves one π/2a doubler, with four independent sublattices acting as particle types.Alternating the order of conditional shifts may make the remaining doubler high energy, but its continuum-limit implications remain unclear.Source: Mitigating the Doubling problem · pp. 81–82

3.5 Quantum Walks on a Line: Abstract Theory

• A translationally invariant quantum walk on a line can be represented using pairs of orthogonal projectors, conditional shifts, and a coin-space unitary.The proof derives orthogonal supports for the extreme-neighbourhood operators from causality and unitarity.Source: §3.5 · p. 83

3.5. QUANTUM WALKS ON A LINE: ABSTRACT THEORY

• On the even b.c.c. sublattice, the two-dimensional walk remains causal and unitary, and its evolution can be recast as a product of conditional shifts.The original factorization as T_xT_y is unavailable because odd-parity lattice sites are absent, but combinations of shifts along x and y act within the restricted lattice.Source: §3.5 · pp. 84–85
• The analogous three-dimensional walk also retains a conditional-shift decomposition when restricted to the b.c.c. sublattice containing (0, 0, 0).The relevant shift operators are translations by b.c.c. lattice vectors, so the restricted evolution remains decomposable into conditional shifts.Source: §3.5 · p. 85
• Because both restricted constructions preserve the decomposition, finding counterexamples likely requires more than simply restricting these walks to sublattices.The discussion explicitly identifies greater complexity as necessary if counterexamples exist.Source: §3.5 · p. 85

4.1 Introduction

• Causal fermionic systems in discrete spacetime are shown to connect discrete field theories with quantum cellular automata, including an equivalence between fermionic and regular quantum cellular automata.The chapter also presents a one-dimensional structure theorem for quantum cellular automata and motivates causal models as discretizations useful for regulation and simulation.Source: §4.1 · pp. 88–89

4.2 Continuum Limits of Discrete Free Fermion Fields

• Discrete fermion fields are constructed by upgrading quantum walks through second quantization, yielding evolutions decomposable into local one- and two-site unitaries or constant-depth circuits.The construction accounts for the nontrivial entangled vacuum required for a field-theoretic continuum limit.Source: §4.2 · p. 89, §4.2.1 · p. 90

Discrete Dirac Fermions in One Dimension

• In one dimension, second-quantized Dirac quantum walks recover the Dirac Hamiltonian in the continuum limit and implement their conditional shifts and mass evolution with local fermionic unitaries.The resulting evolution operator is a simple constant-depth circuit.Source: Discrete Dirac Fermions in One Dimension · pp. 91–92

Discrete Weyl Fermions in Three Dimensions

• In three dimensions, second-quantized Weyl quantum walks recover Weyl fermions in the continuum limit, with dynamics expressible entirely as products of local swaps.The same local decomposition remains available when restricting modes to the b.c.c. sublattice.Source: Discrete Weyl Fermions in Three… · p. 93

4.2.2 Discrete Fermion Fields and the Vacuum

• The discrete fermion vacuum is defined by filling all negative-energy modes, mirroring the continuum ground state rather than using a state annihilated by every annihilation operator.This construction addresses the fact that the continuum vacuum is a complicated entangled state.Source: §4.1 · p. 87, §4.2.2 · p. 94

4.2.3 Continuum Limits

• Continuum comparison is formulated through low-momentum observables and expectation values, because the discrete and continuum ground states differ in their momentum support and state spaces.A momentum cutoff Λ restricts the physically relevant states and measurements, with efficiently implementable unitaries relevant for simulation.Source: §4.2.3 · pp. 94–95
• In one dimension, Theorem 4.1 proves that the discrete vacua converge to the continuum vacuum as the lattice spacing tends to zero.The convergence is established in the sense specified by the theorem and is the key vacuum component of the continuum-limit argument.Source: §4.2.3 · p. 96
• With a sufficiently slowly growing cutoff, the discrete vacuum converges to the continuum vacuum, and the discrete fermion dynamics also converge.The proof uses a finite line with discrete momenta; for infinite lines, convergence should instead be formulated through expectation values of smeared observables.Source: §4.2.3 · pp. 99–100

4.3 Lattice QFT without Anticommutation

• Gauge symmetry allows some lattice fermion–gauge-field models to be mapped to completely local qubit or spin Hamiltonians without anticommuting fermion fields.Gauge degrees of freedom are partially represented by fermion pairs, and gauge symmetry prevents nonlocal observables in the qubit description.Source: §4.3 · p. 101

4.3.1 Representing Fermion Models by Local Qubit Models

• Introducing redundant fermion modes makes local fermion operators local in the qubit picture while preserving self-adjointness, Hamiltonian dynamics, and higher-order locality for even observables.The construction uses Majorana modes on links and maps the fermion Hilbert space into a joint +1 eigenspace of commuting link operators.Source: §4.3.1 · p. 103

Representation by Qubits

• Choosing a suitable Jordan–Wigner ordering maps the resulting local fermion Hamiltonian to a local qubit Hamiltonian, while auxiliary-mode requirements depend on spatial dimension rather than physical mode count.Auxiliary modes can be restricted to links associated with nonlocal hopping terms to reduce overhead.Source: Representation by Qubits · p. 104

4.3.2 Preparing the Entangled Initial State

• O(N^2) two-qubit operations prepare the +1 eigenstate of all Majorana link terms, because each parity computation uses O(N) operations and each site has constant local overhead.Parity flags reduce nonlocal Z strings to two-qubit unitaries, enabling efficient preparation of the entangled state.Source: §4.3.2 · pp. 105–106
• Entanglement replaces fermionic operator nonlocality in the qubit representation, while entanglement-based nonlocal correlations remain possible.The transformation removes nonlocality of fermion operators, not physical nonlocal correlations between entangled systems.Source: §4.3.2 · p. 106

4.3.3 Lattice Gauge Theories

• Lattice QED is constructed by coupling fermions to link gauge fields, enforcing gauge invariance and Gauss’ law on physical states.Link variables transform as U⃗n⃗m → V(⃗n)U⃗n⃗mV(⃗m)†, while the gauge generators must vanish on physical states.Source: §4.3.3 · pp. 107–108

4.3.4 Lattice QED as a Local Qubit Model

• Representing each gauge link by a parity qubit and fermionic modes makes the lattice-QED Hamiltonian a collection of local commuting operators.The link shift becomes a quantum-walk operator with parity as its coin, and the transformed hopping, electric-field, and plaquette terms commute across sites or links.Source: §4.3.4 · pp. 109–110

4.4. NATURE AS A QUANTUM CELLULAR AUTOMATON

• For local lattice Hamiltonians, Lie–Trotter decomposition yields a causal QCA whose evolution approximates lattice field dynamics and therefore the corresponding continuum theory.Local terms are partitioned into finitely many nonoverlapping subsets, producing finite-depth causal timesteps; the timestep may depend on lattice spacing to maintain accuracy.Source: §4.4 · p. 112

4.4 Nature as a Quantum Cellular Automaton

• The QCA approximation is conceptually valid but its simulation efficiency depends on coupling-constant scaling, state preparation, and Trotter errors as lattice spacing decreases.Renormalization and possible infinite bosonic degrees of freedom remain obstacles, and the construction assumes lattice QFT describes nature while ignoring gravity.Source: §4.4 · p. 113

4.5 Causal Discrete-Time Models on a Lattice

• Causal discrete-time models are developed as potentially more natural alternatives to Trotterized lattice QFT, with fermionic QCA localizability and equivalence to regular QCA as central results.Causality implies decomposition into constant-depth local-unitary circuits after adding a copy of the system, while fermionic QCA become regular QCA with constant ancillary systems per site.Source: §4.5 · pp. 114–115

4.5.1 Fermionic Quantum Cellular Automata

• Fermionic QCA place fermionic modes at lattice sites and define causality by bounding each timestep’s operator spread within a finite neighbourhood.For an annihilation operator a⃗n, the evolved operator is localized on sites satisfying |⃗n−⃗m| < L; nonoverlapping localized observables commute.Source: §4.5.1 · p. 115
• Causal fermionic unitaries have causal inverses, and fermionic cellular automata require translationally invariant, discrete-time causal evolution that leaves appended modes invariant.The inverse-causality result follows from the restriction that evolved annihilation operators are linear combinations of odd fermionic operator products.Source: §4.5.1 · pp. 116–117

4.5.2 Local Decomposition of Causal Models on a Lattice

• Causal evolution of two copies of a fermionic system decomposes into commuting local fermionic unitaries after appending a copy of every mode.The analogous decomposition extends to finite interacting fermion-boson systems, while a copy is generally necessary for fermionic shifts.Source: §4.5.2 · pp. 117–119

4.5.3 Representation by Quantum Cellular Automata

• Fermionic dynamics on qubits can be implemented as a constant-depth causal circuit, using auxiliary fermions to restore locality under Jordan–Wigner transformations.The number of added fermions per site is independent of the lattice size; bosonic representations remain unclear because bosonic modes are infinite dimensional.Source: §4.5.3 · pp. 119–121
• Any causal fermionic evolution is equivalent to a causal qubit evolution on a subsector, requiring at most a constant number of additional qubits per site.Moreover, quantum cellular automata and fermionic quantum cellular automata can simulate each other after adding a constant number of systems per site.Source: §4.5.3 · p. 121

4.5.4 Simulating Causal Systems on a Quantum Computer

• Causal fermion evolution can be simulated efficiently on a quantum computer using O(N) local unitaries, with many operations executed in parallel.The required simulation time per evolution step depends on the lattice dimension and causal neighborhood size, not on N.Source: §4.5.4 · pp. 122–123

4.6 Information Flow in QCAs

• For one-dimensional QCAs, the section develops local information-spreading tools toward a structure theorem expressing every QCA, after regrouping sites, as shifts followed by on-site unitaries.This framework generalizes the quantum-walk structure result and builds on the locally computable information-flow index.Source: §4.6 · p. 123, §4.6.1 · p. 123

4.6.1 Spreading of Information

• Support algebras track how QCA information spreads, allowing dynamics to be analyzed through one- and two-site algebras regrouped into nearest-neighbor blocks.Even and odd algebras capture information moving right and left; they generate the full QCA algebra and form subsystem algebras.Source: §4.6.1 · pp. 124–125

4.6.2 Structure of QCA Evolution on a Line

• After regrouping sites, every one-dimensional QCA evolution can be written as a product of on-site unitaries and partial shifts.The proof applies partial shifts to cancel net information transfer, leaving a product of local unitaries before inverting the construction.Source: §4.6.2 · p. 126
• For a finite periodic line with N sites, the shift decomposes into O(N) swaps whose order matters, so the resulting circuit is not constant depth.This contrasts with the infinite-system case, where constant-depth local decompositions can be obtained only by appending a copy of the original system.Source: §4.6.2 · p. 127

Conclusions and Open Problems

• Quantum walks with two-dimensional coins recover relativistic particles in the continuum limit, supporting their use as discretized relativistic models.The continuum limit naturally produces Hamiltonians linear in momentum, equivalent to relativistic Hamiltonians.Source: Conclusions and Open Problems · p. 129
• Discrete vacuum states converge to the continuum vacuum in one dimension, allowing causal discrete fermionic models to recover free Dirac fields.The interacting case remains harder because its vacuum is expected to be more complicated.Source: Conclusions and Open Problems · p. 130

5.2 Quantum Cellular Automata

• Fermionic systems interacting with abelian gauge fields can be represented as completely local qubit models, although the construction introduces redundancy and is somewhat artificial.Extending this approach to nonabelian gauge fields and reducing redundant degrees of freedom remain open problems.Source: Conclusions and Open Problems · p. 130, §5.2 · p. 133

Conclusions and Open Problems

• Fermionic quantum cellular automata are equivalent to regular quantum cellular automata, and causal discrete fermion models can be efficiently simulated by quantum computers.The thesis also proves a structure theorem for quantum cellular automata on a line.Source: Conclusions and Open Problems · p. 130

5.1. Quantum Walks

• Important open problems include interacting continuum limits, higher-dimensional quantum-walk structure theorems, and quantum-computational advantages beyond simulation.The thesis also notes that gravity is not accounted for and that fermion-doubling results remain preliminary.Source: Conclusions and Open Problems · p. 130, §5.2 · p. 132, §5.1 · p. 132

Introduction

• The thesis investigates whether quantum theory can justify or eliminate foundational statistical-physics principles such as equal a priori probability and equilibration.The motivation is to derive these principles from quantum dynamics rather than treating them as additional assumptions.Source: Introduction · pp. 137–138
• A practical definition of equilibration is introduced so that the concept reflects the capabilities of actual measurements.The thesis focuses on finite intervals and bounds on the time required for equilibration.Source: Introduction · pp. 138–139
• The equilibration-time bounds are broadly applicable but quite large, making generality both a strength and a weakness of the approach.The introduction explicitly identifies the size of the bounds as the main weakness.Source: Introduction · p. 139
• The equilibrium state's form is examined through initial-state independence and the question of when a subsystem reaches a Gibbs state.Standard Gibbs-state arguments invoke equal a priori probability, weak subsystem-environment interactions, and exponential environmental density of states.Source: Introduction · p. 139
• The main statistical-physics results address equilibration over finite times and bounds on equilibration time for fixed but general systems, measurements, and Hamiltonians.This contrasts with earlier results that often established equilibration only over infinite timescales or by averaging over states, Hamiltonians, or measurements.Source: Introduction · pp. 138–139

7.1 Quantum Information

• Quantum-information methods include Haar averaging, Lieb-Robinson bounds, and state-distance measures relevant to equilibration and thermalization.Haar averaging yields invariant average states, while Lieb-Robinson bounds limit information propagation and enable finite-region approximations.Source: §7.1 · pp. 141–142

7.1.1 Time Averages and Energy Filtering

• Time averages of unitary dynamics produce a dephased state determined by energy gaps, while weighted averages trade temporal duration against energy resolution and spatial locality.Lorentzian weighting can upper-bound top-hat time averages and may be interpreted as energy filtering.Source: §7.1.1 · p. 144, §7.1 · p. 145

7.1.2 Distinguishing States

• Distinguishability restricts state comparisons to experimentally available measurements, recovering trace distance when all projective measurements are allowed but potentially missing large physical differences.This operational restriction is useful for many-body systems where arbitrary global measurements are impractical.Source: §7.1.2 · pp. 146–147

7.2 Statistical Physics

• The statistical-physics analysis considers generally closed, time-independent Hamiltonian systems, especially macroscopic subsystem-environment settings, under weak energy-gap and high-effective-dimension conditions.The approach uses exact unitary evolution rather than a subsystem Lindblad approximation, whose accuracy can worsen over long times.Source: §7.2 · pp. 147–148

7.2.1 Equilibration

• Because discrete-spectrum unitary systems recur rather than relax permanently, equilibration is defined as remaining close to a fixed state for most times within a chosen interval.The equilibrium state can therefore depend on the timescale under consideration.Source: §7.2.1 · p. 149
• Equilibration is operationally specified through the time-averaged distinguishability relative to a measurement set, with the equilibration time chosen so this average falls below a fixed small threshold.The relevant concern is how this time scales with system size; the threshold’s precise constant value is immaterial when fixed.Source: §7.2.1 · pp. 149–150
• Expectation-value equilibration requires time-averaged fluctuations to remain small relative to ∥A∥, with the tolerance ϵA motivated by experimental resolution.Small values of ∥A∥ϵA correspond to fluctuations that are usually undetectable by the apparatus.Source: §7.2.1 · p. 150
• Equilibrated expectation values do not by themselves imply that ρ(t) is indistinguishable from its time average or that the whole system has equilibrated [95].For large Hilbert spaces, states can remain far from their time averages while observable expectation values equilibrate, motivating distinguishability-based equilibration.Source: §7.2.1 · pp. 150–151

7.2.2 Macroscopic Systems

• Macroscopic states are expected to have extremely large effective dimension because exponentially dense energy levels spread them over many energy eigenstates.The effective dimension measures this spread: it equals 1 for an energy eigenstate and dE for equal occupation of all energy eigenstates.Source: §7.2.2 · pp. 151–152

7.2.3 Distribution of Energy Gaps

• Equilibration-time bounds depend strongly on energy-gap density and degeneracy, because energy-basis coherences oscillate at frequencies set by energy differences.Very small gaps imply very long equilibration times, while many degenerate gaps can keep suitable expectation values large and obstruct equilibration.Source: §7.2.3 · pp. 152–153
• For macroscopic systems, the energy distribution is effectively exponential in a restricted window, with β = 1/kBT, while typical gaps are of order kBT and the smallest gap is expected to be exponentially small in particle number.The typical-gap estimate assumes β∆ ≫ 1; Chebyshev’s inequality shows most gaps are comparable to 1/β, despite exponentially small minima.Source: §7.2.3 · pp. 154–155

8.1 Introduction

• Equilibration is defined relative to physically restricted measurements because unrestricted observations would resolve ongoing microscopic dynamics rather than reveal equilibration.The discussion uses coarse-grained observables and highlights measurement limitations as essential to observing equilibration.Source: §8.1 · p. 157
• The framework applies to all finite-dimensional systems with minimal dynamical assumptions, but its equilibration-time upper bounds can be far too large for physical spectra and cannot generally be substantially improved without further assumptions.Some systems can equilibrate very slowly, motivating additional spectral or dynamical restrictions for realistic estimates.Source: §8.1 · pp. 157–158

8.2.3 Equilibration Times with Physical Spectra

• The subsequent analysis turns from proving finite-time equilibration to determining when equilibration has occurred and bounding the associated equilibration time.This follows the finite-interval results developed for expectation values, measurement sets, and subsystems.Source: §8.1 · p. 158, §8.2.3 · p. 168

8.2.2 Finite Time Equilibration for Systems and Subsystems

• The expectation-value bound yields finite-time equilibration for realistic measurement sets and for subsystems that are small compared with sufficiently large environments, even when some energy gaps are degenerate.For subsystems, the condition requires the subsystem dimension to be small compared with √deff; the finite-time extension permits degeneracy when the largest gap degeneracy DG is not too large.Source: §8.2 · p. 159, §8.2.2 · pp. 164–167

8.2.1 Finite Time Equilibration of Expectation Values

• Finite-interval equilibration of expectation values is generalized beyond nondegenerate energy gaps by requiring only that the most degenerate energy gap is not too highly degenerate.The resulting theorem improves the earlier bound in [3] by O(log2 dE) and slightly improves the bound in [4].Source: §8.2.1 · pp. 159, 163

8.2.2 Finite Time Equilibration for Systems and Subsystems

• The subsystem result recovers an infinite-time bound for arbitrary Hamiltonians and is more general than the nondegenerate-gap case, although it can be weaker by a constant factor when DG = 1.The maximal possible degeneracy is DG = dE −1, while nontrivial Hamiltonians are expected to have much smaller DG.Source: §8.2.2 · p. 167

8.2.3 Equilibration Times with Physical Spectra

• The equilibration bounds depend on balancing ε against the effective dimension and the number of gaps N(ε), while fixed measurement capabilities constrain how these quantities scale.For subsystem equilibration the relevant factor involves d_S^2, whereas finite-measurement equilibration involves S(M)^2; choosing ε requires balancing the bound against the condition ensuring equilibration.Source: §8.2.3 · p. 168
• For systems with physical spectra, the best general equilibration-time bound can scale as 1/d_E, which is exponentially long in system size.Although most energy gaps are not exponentially small, their exponentially large number forces ε to be exponentially small; the resulting bounds are otherwise close to optimal without extra assumptions.Source: §8.2.3 · p. 169

8.2.4 Slow Equilibration

• A constructed pure state and two-projector measurement can remain far from its infinite-time average for arbitrarily long intervals, despite eventually equilibrating.The construction uses a subspace spanned by K time snapshots and yields a timescale governed by the effective dimension.Source: §8.2.4 · pp. 169–171
• General equilibration bounds must accommodate exponentially slow examples, although whether local Hamiltonians and local measurements can realize such timescales remains unclear.These systems may fail to equilibrate to their infinite-time average over physically reasonable timescales while appearing equilibrated on shorter timescales.Source: §8.2.4 · p. 172

8.3 The Equilibrium state

• Thermalization asks when a subsystem equilibrates to a Gibbs state, with initial-state independence as a necessary requirement alongside independence from the environment’s detailed initial state.Existing results establishing Gibbs equilibration require interaction-strength and initial-state conditions that are most naturally compatible with one-dimensional systems.Source: §8.3 · pp. 172–173

8.3. THE EQUILIBRIUM STATE

• The qubit–oscillator example can be made arbitrarily close to an integrable noninteracting model while retaining initial-state independence, but it does not generally thermalize.The interaction strength affects equilibration times through energy gaps, and the model has degenerate gaps unless the coupling is made n-dependent.Source: §8.3 · pp. 173–174
• In a qubit–oscillator model, any nonzero interaction makes the subsystem’s long-time average maximally mixed, independent of the initial state and interaction strength.The interacting energy eigenstates are maximally entangled except for the ground state, so the conclusion holds for arbitrarily small nonzero interaction, subject to ground-state overlap.Source: §8.3 · pp. 173–174
• The toy model’s relevance to locally perturbed spin systems remains an open question, especially because integrability is commonly formulated for such models.The example is described as unnatural, motivating investigation of whether similar behavior occurs under local perturbations.Source: §8.3 · p. 174

Conclusions and Open Problems

• Quantum-information tools bound equilibration times generally, but exponentially small energy gaps can dominate estimates and produce long times for physical spectra.The proposed next step is to impose more structure on the measurement set and Hamiltonian.Source: Conclusions and Open Problems · p. 175

9.2 Thermalization

• Determining whether equilibrium is a Gibbs state requires understanding local observables and energy eigenstates, while trace-distance arguments remain sensitive to global projector changes.The eigenstate thermalization hypothesis shifts the difficulty toward proving that local reductions of energy eigenstates resemble Gibbs states.Source: Conclusions and Open Problems · p. 175, §9.2 · p. 179

9.1 Equilibration

• Finite-duration measurements may suppress rapidly oscillating terms, but slowly oscillating contributions from small gaps remain the central obstacle to non-exponential bounds.For gaps satisfying |G_nm|τ ≪ 1, the corresponding terms remain nearly unchanged during an experiment.Source: §9.1 · pp. 176–177

9.1. EQUILIBRATION

• Equilibration can occur to temporary states on distinct timescales, with substantially different temporary equilibria requiring progressively longer intervals than earlier ones.Thus, finite-time equilibration need not target the infinite-time average state.Source: §9.1 · pp. 176–178
• The analysis leaves open whether exponentially growing equilibration times can occur for local Hamiltonians with local measurements, and how locality affects equilibration more generally.A related lattice model has equilibration times diverging with system size, while the local-measurement case remains unresolved.Source: §9.1 · p. 178

Epilogue

• Discrete-time quantum models provided evidence that discretized dynamics may illuminate relativistic physics and support quantum simulation.The discussion also raises the possibility that nature itself could be described by a quantum cellular automaton.Source: Epilogue · p. 181
• The statistical-physics analysis sought general statements about equilibration by making as few assumptions as possible.Its broader conclusion is that equilibration and the form of equilibrium can depend crucially on the timescale.Source: Epilogue · p. 181
• The central theme is that quantum information remains a productive source of tools and questions for fundamental physics.The thesis places its contributions within earlier applications to high-energy physics and statistical-physics foundations [5] [6] [7] [8] [9] [90] [93] [94] [95] [96] [98] [99].Source: Epilogue · p. 181

Appendix: Dynamics on the b.c.c. Sublattice

• After second quantization, the same local construction applies to fermionic modes, and an analogous more complicated procedure works for the three-dimensional Weyl walk.The swap operation exchanges fermion modes at neighboring translated sites while leaving all other modes unchanged.Source: Appendix: Dynamics on the b.c.c.… · pp. 193–194
• Restricting the two-dimensional massless Dirac walk to a b.c.c. sublattice preserves the sublattice and defines a quantum walk with unitary V.The evolution is built from conditional shifts whose formulas are simplified by using the z-direction for the second shift.Source: Appendix: Dynamics on the b.c.c.… · p. 193
• The restricted walk is implemented through products of local unitaries, including swaps, coin operations, and local superpositions.The construction applies the conditional shift using local swaps and then combines neighboring states with local unitaries.Source: Appendix: Dynamics on the b.c.c.… · pp. 193–194