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Stabilizer Circuits and Gottesman–Knill Simulation — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Circuit Syntax, Denotational Semantics, and Measurement
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- 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
- Controlled Gates, Phase, and Entanglement
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- 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-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
- 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
- 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
- Measurements, POVMs, and Instruments
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Splitting Fields
- Stabilizer Circuits and Gottesman–Knill Simulation
- Strongly Continuous Semigroups and Hille Yosida
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- 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 Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
Objective
Work through repeated one-qubit measurements starting from . The example computes both fair branches and their signed stabilizer generators, then checks the probability-zero opposite branch on a repeated measurement.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
Stabilizer measurement on in the basis
Example
Start in the one-qubit basis state , whose stabilizer is . Measure , then measure again. The first measurement has two equiprobable outcomes and updates the signed stabilizer generator to the corresponding eigenvalue. The second measurement is deterministic; its opposite branch has probability zero and no normalized conditional state.
Facts & Assumptions
Given: The ordered one-qubit computational basis, , and the Pauli matrices .
The Pauli matrices: , , , and .
Pauli measurement branch semantics: for , the projectors are and .
An anticommuting stabilizer measurement is fair: if the measured Pauli anticommutes with a stabilizer generator, each bit occurs with probability .
Random Pauli measurement updates stabilizer generators: an allowed bit replaces the anticommuting pivot by .
Zero-probability Pauli measurement branch: a zero branch map is zero and does not define a normalized conditional state.
Verification
Given: The data in the Example.
The initial generator fixes , and it anticommutes with by [F1]. Hence [F3] gives for the first measurement.
Directly, and . Each vector has squared norm , so normalization gives and . By [F4], their stabilizers are and , respectively.
Measuring again on either state is deterministic: lies in and lies in . The opposite projector annihilates the state, so its branch has probability zero and no normalized conditional state by [F5].
Sources
The stabilizer and -eigenstate examples appear in de Wolf §20.1 and Watrous Lesson 14 §14.1, printed pp. 405–407 (PDF pp. 411–413). The branch probabilities and update follow the exact local calculations above: https://arxiv.org/pdf/1907.09415; https://jhwatrous.github.io/UQIC.pdf.