Quantum Circuit Theory and Implementations

Dependencies

Table of Contents

• 00_postulates
  • Dirac notation
  • Postulate 1 (State Space)
    • First case: discrete spectrum
    • Second case: continuous spectrum
  • Postulate 2 (Evolution)
  • Postulate 3 (Measurement)
    • Projective measurement (a.k.a von Neumann measurement) and Born's rule
    • Expectation value
  • Postulate 4 (Composite State)
  • System's Dynamics: Schrödinger equation
    • Wave Mechanics formalism
    • Matrix Mechanics formalism
      • Schrödinger Picture (S-P)
      • Heisenberg Picture (H-P)
  • Solution to the Schrödinger equation
• 01_formalism
  • Linear operators.
  • Outer product representation.
  • Basis states.
    • Z-basis.
    • X-basis.
    • Y-basis.
  • Pauli group.
    • Pauli gates.
    • Pauli algebra.
  • Clifford group.
    • Clifford gates.
  • Hermitian gates.
  • Spectral decomposition theorem.
  • Operator function.
• 02_single_qubit_gates
  • Pauli gates.
  • Hadamard gate a.k.a superposition gate.
  • $R_{\hat{n}}(\theta)$ standard rotation single-qubit gate.
  • $P(\lambda)$ single-qubit phase gate.
    • Phase gate S.
    • Phase gate T.
  • $U(\alpha, \beta, \gamma, \delta)$ arbitrary single-qubit gate.
  • $U(\theta, \phi, \lambda)$ three-parameter single-qubit gate.
  • Qiskit $U1(\lambda)\equiv U(0,0,\lambda)=P(\lambda)$.
  • Qiskit $U2(\phi, \lambda) \equiv U(\pi/2, \phi, \lambda)$.
• 03_change_of_basis.ipynb
  • Measurement.
  • From the $Z$-basis to the $X$-basis.
  • From the $Z$-basis to the $Y$-basis.
• 04_two_qubit_gates
• Deriving gates via spectral theorem, parity trick, and single-qubit rotations.
  • CNOT gate.
  • SWAP gate.
  • $ZZ(t)$ gate.
  • $XX(t)$ gate.
  • $YY(t)$ gate.
  • Verifying outer products.
• 05_multi_qubit_gates
  • Multi-qubit gate $C_n^{j}(U_{2^m})$ with $n$ control qubits and $m$ target qubits.
• 06_gate_circuit_identities
  • Qiskit little-endian convention.
  • Pauli gates.
  • Single-qubit gates.
  • Multi-qubit gates.
  • Rotations.
  • Conjugation by Unitary.
• 07_gate_decomposition
  • Pauli decomposition.
  • Single-qubit gate decomposition.
  • Two-qubit gate decomposition.
• 08_universal_gate_set
  • Obtaining the universal gate set.
• 09_trotterization
  • The Trotter-Susuki formula
  • The Heisenberg XXX Spin-1/2 Lattice Model for $N=3$ Three Particles
  • Decomposition of $U_{\text{Heis3}}(t)$ using Trotterization
  • About the Trotterized Evolution
    • Verifying identities with NumPy
    • Verifying identities with Qiskit Opflow
• 10_implementations
  • NumPy and Sympy implementations of:
    • Basis states.
    • Projector operators.
    • Single-qubit gates.
    • Two-qubit gates.
    • Eigenvalues and eigenvectors.
• algebraic_identities
  • Notation.
  • Representations.
  • Identities: algebraic proof of useful linear algebra identities for quantum circuits with SymPy and SciPy verification.
    • Retangular matrices.
    • Vectors.
    • Kronecker product between vectors.
    • Kronecker product between retangular matrices.
    • Kronecker product with vectors and matrices.
    • Commutation.
    • Matrix exponential.
    • Rotations.
• glossary
  • Jargon and Terminology.
• Q&A
  • Questions and Answers.

Conda Env.

1. Clone this repository and access the cloned directory:

git clone https://github.com/qucai-lab/quantum-circuit-theory.git && cd quantum-circuit-theory

2. Create env.:

conda create -yn qct python==3.11.2 && conda activate qct

3. Install core dependencies:

python -m pip install -r requirements.txt

References  

[1] Nielsen MA, Chuang IL. 2010. "Quantum Computation and Quantum Information." New York: Cambridge Univ. Press. 10th Anniv. Ed.

• Corollary 4.2, pg. 176: Gate decomposition.
• Theorem 4.3, pg. 207: Trotter formula.
• Chapter 4.7.2, pg. 206: The quantum simulation algorithm.

[2] Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A. and Weinfurter, H. (1995) Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467.

[3] Scott Aaronson and Daniel Gottesman. 2004. Improved simulation of stabilizer circuits. Phys. Rev. A 70, 5 (Nov. 2004), 052328.

[4] John Preskill. "Course Information for Physics 219/Computer Science 219 Quantum Computation." California Institute of Technology.

• Chapter 5: Classical and Quantum Circuits.

[5] Hans J. Weber and George B. Arfken. Essential Mathematical Methods for Physicists. Academic Press, NY.

License

This work is licensed under a Creative Commons Zero v1.0 Universal license.

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