Numerical Recipes in Quantum Information Theory and Quantum Computing: An Adventure in FORTRAN 90

Description

This first of a kind textbook provides computational tools in Fortran 90 that are fundamental to quantum information, quantum computing, linear algebra and one dimensional spin half condensed matter systems. Over 160 subroutines are included, and the numerical recipes are aided by detailed flowcharts. Suitable for beginner and advanced readers alike, students and researchers will find this textbook to be a helpful guide and a compendium. Key Features: Includes 160 subroutines all of which can be used either as a standalone program or integrated with any other main program without any issues. Every parameter in the input, output and execution has been provided while keeping both beginner and advanced users in mind. The output of every program is explained thoroughly with detailed examples. A detailed dependency chart is provided for every recipe. Cover Half Title Title Page Copyright Page Dedication Contents Preface List of Figures Acknowledgment 1. Introduction to FORTRAN 90 1.1. Printing on screen 1.2. Data types 1.2.1. Integer 1.2.2. Real 1.2.3. Complex 1.2.4. Character 1.2.5. Logical 1.3. Constants 1.4. Operators 1.4.1. Arithmetic operators 1.4.2. Relational operators 1.4.3. Logical operators 1.5. Decisions 1.5.1. If... then... end if 1.5.2. If...then...else 1.6. Loops 1.6.1. The do loop 1.6.2. The do while loop 1.7. EXIT, STOP and CYCLE 1.7.1. EXIT 1.7.2. STOP 1.7.3. CYCLE 1.8. Arrays 1.8.1. Declaration 1.8.2. Assignment 1.9. Dynamic arrays 1.10. Reading and writing 1.10.1. Writing in the file 1.10.2. Reading from file 1.11. Intrinsic functions 1.12. Subroutines 1.13. Some tit bits in FORTRAN for effective programming 1.14. LAPACK 1.14.1. DSYEV 1.14.2. ZHEEV 1.14.3. DGEEV 1.14.4. ZGEEV 1.14.5. DGEQRF 1.14.6. DORGQR 1.14.7. ZGEQRF 1.14.8. ZUNGQR 1.14.9. DSYTRD 1.14.10. DORGTR 1.14.11. ZHETRD 1.14.12. ZUNGTR 1.14.13. DGETRF 1.14.14. DGETRI 1.14.15. ZGETRF 1.14.16. ZGETRI 1.14.17. DGESVD 1.14.18. ZGESVD 1.15. Compilation and execution 1.16. Computation time 2. Nuts and Bolts of Quantum Mathematics 2.1. Inner product between two vectors 2.1.1. For real vectors 2.1.2. For complex vectors 2.2. Norm of a vector 2.2.1. For a real vector 2.2.2. For a complex vector 2.3. Normalization of a vector 2.3.1. For a real vector 2.3.2. For a complex vector 2.4. Outer product of a vector 2.4.1. For real vectors 2.4.2. For complex vectors 2.5. Matrix multiplication 2.5.1. For real matrices 2.5.2. For complex matrices 2.6. Tensor product between two matrices 2.6.1. For real matrices 2.6.2. For complex matrices 2.7. Trace of a matrix 2.7.1. For a real matrix 2.7.2. For a complex matrix 2.8. Commutator between two matrices 2.8.1. For real matrices 2.8.2. For complex matrices 2.9. Anticommutator between two matrices 2.9.1. For real matrices 2.9.2. For complex matrices 2.10. Binary to decimal conversion 2.11. Decimal to binary conversion 2.12. Bit test 2.13. Left shift 2.14. Right shift 2.15. Swapping or permutation operator 3. Numerical Linear Algebra and Matrix Operations 3.1. Inverse of a matrix 3.1.1. For a real matrix 3.1.2. For a complex matrix 3.2. Lanczos tridiagonalization 3.2.1. For a real symmetric matrix 3.2.2. For a complex Hermitian matrix 3.3. QR decomposition 3.3.1. For a real matrix 3.3.2. For a complex matrix 3.4. Function of a matrix 3.4.1. For a real symmetric matrix 3.4.2. For a complex Hermitian matrix 3.4.3. For a real non-symmetric matrix 3.4.4. For a complex non-symmetric matrix 3.5. Power of a matrix 3.5.1. For a real symmetric matrix 3.5.2. For a complex Hermitian matrix 3.5.3. For a real non-symmetric matrix 3.5.4. For a complex non-symmetric matrix 3.6. Trace of a power of the matrix 3.6.1. For a real symmetric matrix 3.6.2. For a complex Hermitian matrix 3.6.3. For a real non-symmetric matrix 3.6.4. For a complex non-symmetric matrix 3.7. Determinant of a Matrix 3.7.1. For a real symmetric matrix 3.7.2. For a complex Hermitian matrix 3.7.3. For a real non-symmetric matrix 3.7.4. For a complex non-symmetric matrix 3.8. Trace norm of a matrix 3.8.1. For a real matrix 3.8.2. For a complex matrix 3.9. Hilbert-Schmidt norm of a matrix 3.9.1. For a real matrix 3.9.2. For a complex matrix 3.10. Absolute value of a matrix 3.10.1. For a real matrix 3.10.2. For a complex matrix 3.11. The Hilbert-Schmidt inner product between two matrices 3.11.1. For real matrices 3.11.2. For complex matrices 3.12. Gram-Schmidt orthogonalization 3.12.1. For real vectors 3.12.2. For complex vectors 3.13. Singular value decomposition 3.13.1. For a real matrix 3.13.2. For a complex matrix 4. Tools of Quantum Information Theory 4.1. Frequently used quantum gates 4.1.1. The Pauli-X 4.1.2. The Pauli-Y 4.1.3. The Pauli-Z 4.1.4. The Hadamard gate 4.1.5. The phase gate 4.1.6. The rotation gate 4.1.7. The controlled NOT gate (CX gate) 4.1.8. The controlled Z gate (CZ gate) 4.1.9. The SWAP gate 4.1.10. The Toffoli gate (CCX, CCNOT, TOFF) 4.1.11. The Fredkin gate (CSWAP) 4.2. The N-qubit Hadamard gate 4.3. The N-qubit computational basis (identity matrix) 4.4. The Bell states 4.5. The N-qubit Greenberger-Horne-Zeilinger (GHZ) state 4.6. The N-qubit W state 4.7. The Generalized N-qubit Werner state 4.8. Shannon entropy 4.9. Linear entropy 4.9.1. For a real density matrix 4.9.2. For a complex density matrix 4.10. Relative entropy 4.10.1. Between two real density matrices 4.10.2. Between two complex density matrices 4.11. Trace distance 4.11.1. Between two real density matrices 4.11.2. Between two complex density matrices 4.12. Fidelity 4.12.1. Between two real states 4.12.2. Between two complex states 4.12.3. Between two real density matrices 4.12.4. Between two complex density matrices 4.12.5. Between a real state and a real density matrix 4.12.6. Between a complex state and a complex density matrix 4.13. Super fidelity 4.13.1. Between two real density matrices 4.13.2. Between two complex density matrices 4.14. Bures distance 4.14.1. Between two real states 4.14.2. Between two complex states 4.14.3. Between two real density matrices 4.14.4. Between two complex density matrices 4.15. Expectation value of an observable 4.15.1. For a real observable via a real state 4.15.2. For a complex observable via a complex state 4.15.3. For a real observable via a real density matrix 4.15.4. For a complex observable via a complex density matrix 4.16. The single-qubit measurement 4.16.1. For a real state 4.16.2. For a complex state 4.16.3. For a real density matrix 4.16.4. For a complex density matrix 5. Quantum Entanglement and Its Quantification 5.1. Partial trace 5.1.1. Prelude to partial tracing – rearrangement of indices 5.1.2. For a real state 5.1.3. For a complex state 5.1.4. For a real density matrix 5.1.5. For a complex density matrix 5.2. Partial transpose 5.2.1. For a real density matrix 5.2.2. For a complex density matrix 5.2.3. For a real state 5.2.4. For a complex state 5.3. Concurrence 5.3.1. For a real state 5.3.2. For a complex state 5.3.3. For a real density matrix 5.3.4. For a complex density matrix 5.4. Block entropy 5.4.1. For a real state 5.4.2. For a complex state 5.4.3. For a real density matrix 5.4.4. For a complex density matrix 5.5. Renyi entropy 5.5.1. For a real density matrix 5.5.2. For a complex density matrix 5.6. Negativity and logarithmic negativity 5.6.1. For a real state 5.6.2. For a complex state 5.6.3. For a real density matrix 5.6.4. For a complex density matrix 5.7. Q measure or the Meyer-Wallach-Brennen measure 5.7.1. For a real state 5.7.2. For a complex state 5.7.3. For a real density matrix 5.7.4. For a complex density matrix 5.8. Entanglement spectrum 5.8.1. For a real density matrix 5.8.2. For a complex density matrix 5.9. Residual entanglement for three qubits 5.9.1. For a real state 5.9.2. For a complex vector 5.9.3. For a real density matrix 5.9.4. For a complex density matrix 5.10. Detection of entanglement 5.10.1. PPT criteria or the Peres positive partial transpose criteria 5.10.2. Reduction criteria 6. One-Dimensional Spin Chain Models in Condensed Matter Theory 6.1. Hamiltonian of spins interacting with an external magnetic field 6.1.1. Hamiltonian of spins interacting with an external magnetic field in the X direction 6.1.2. Hamiltonian of spins interacting with an external magnetic field in the Y direction 6.1.3. Hamiltonian of spins interacting with an external magnetic field in the Z direction 6.2. Hamiltonian for the direct exchange spin-spin interaction 6.2.1. Spin-spin interaction in X direction 6.2.2. Spin-spin interaction in Y direction 6.2.3. Spin-spin interaction in Z direction 6.2.4. Variants of the Ising model 6.2.4.1. The isotropic Ising model in a non-zero uniform magnetic field 6.2.4.2. The isotropic transverse Ising model in a non-zero uniform magnetic field 6.3. The Heisenberg interaction 6.3.1. Variants of the Heisenberg model 6.3.1.1. The XXX model 6.3.1.2. The XYZ model 6.3.1.3. The Majumdar-Ghosh model 6.4. The Dzyaloshinskii-Moriya interaction 6.4.1. DM vector in X direction 6.4.2. DM vector in Y direction 6.4.3. DM vector in Z direction 6.4.4. Illustrative example of constructing a combination Hamiltonian 7. Random Matrices and Random Vectors 7.1. Random number generator for Gaussian distribution 7.1.1. For a real random matrix whose elements are chosen from the Gaussian distribution 7.1.2. For a complex random matrix whose real and imaginary parts are individually chosen from the Gaussian distribution 7.2. Random number generator for uniform distribution in the range (a,b) 7.2.1. For a real random matrix whose elements are chosen from uniform distribution 7.2.2. For a complex random matrix whose elements are chosen from uniform distribution 7.3. Random real symmetric matrices 7.4. Random complex Hermitian matrices 7.5. Random unitary matrices 7.5.1. Random real orthogonal matrix 7.5.2. Random complex unitary matrices 7.6. Real Ginibre matrices 7.7. Complex Ginibre matrices 7.8. Wishart matrices 7.8.1. Real Wishart matrix 7.8.2. Complex Wishart matrix 7.9. Random probability vector 7.10. Random pure quantum state vector 7.10.1. Random real state vector 7.10.2. Random pure complex state vector 7.11. Random density matrices 7.11.1. Random real density matrix 7.11.2. Random complex density matrix Dependency Chart Bibliography Subject Index