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Noise, Fault Tolerance, Concatenation, and Thresholds — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Classical and Quantum Error-Correcting Codes
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Averaging and Character-Theory Prerequisites
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability Spaces and Random Variables
- Finite-Dimensional Complex Inner-Product Spaces and Matrix Interfaces
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Independence Borel Cantelli and Zero One Laws
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Algebra Methods in Combinatorics
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measures and Their Basic Properties
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Noise, Fault Tolerance, Concatenation, and Thresholds
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Pauli and Clifford Groups, Binary Symplectic Tableaux
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- Qubits, Bloch Sphere, Pauli Algebra, and One-Qubit Gates
- Registers, Tensor Products, and Partial Trace
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Stabilizer and CSS Codes, Syndromes, and Procedures
- Stabilizer Circuits and Gottesman–Knill Simulation
- Strongly Continuous Semigroups and Hille Yosida
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The ZFC Axioms and the Basic Set Constructions
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
Objective
Work through one finite geometric-series instance of the iid-, perfect-syndrome planar memory bound, with all decoder and noise hypotheses stated. The example is a logical leaf and does not infer a faulty-gate threshold.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
Surface-Code Memory Bound Under IID Z Noise
Statement
Assume the distance-five relative square planar patch, which has data qubits, iid -error rate , one perfect syndrome round, and minimum-weight recovery. The finite series in the planar memory theorem gives the conditional upper bound . This calculation concerns an ideal-syndrome memory round only.
Example
Take the stated finite parameters. Since , each summand equals . The theorem's prefactor is .
Facts & Assumptions
Given: , , , iid -noise, perfect syndrome extraction, and minimum-weight recovery.
Planar Surface-Code Memory Bound Under IID Z Noise: Under those hypotheses, .
Verification
Compute , so .
The finite geometric sum is . Multiplying by gives .
Applying [F1] yields the stated conditional failure-probability bound. The hypotheses include perfect syndrome extraction; this example makes no faulty-gate threshold claim.