Perturbation Theory for Solid-State Quantum Computation with Many Quantum Bits
G P Berman, D I Kamenev & V I Tsifrinovich
224 pages 9x6 inches
March 2005 Hardcover
ISBN 1-58949-051-7
US$58

 

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A quantum computer is expected capable to solve a vast range of mathematical and physical problems much faster than a classical computer. In order to be useful, the register of a quantum computer must contain a large number of qubits. Numerical simulations of quantum dynamics of a many-qubit quantum computer (for optimization of parameters, benchmarking, and architecture design) requires diagonalization of exponentially large matrices or integration for a long time of an exponentially large system of coupled differential equations.
In this book, a theoretical approach is presented, which solves the above problem by using a quantum-mechanical perturbation theory. The perturbation theory is based on small parameters that naturally appear in the system. The results of numerical simulations are widely used to support the theoretical propositions.
It is demonstrated how to simulate simple quantum logic operations and quantum protocols involving a large number of qubits (up to 2000). It is especially useful for minimizing the effects of most types of errors encountered in solid-state quantum computers. Implementations of quantum algorithms are demonstrated in spin systems, quantum dots, and superconducting systems. Almost all results of the book are derived from the first principles.
This book is useful for scientists, engineers, and students who are interested in quantum computing.

 

1. Quantum and classical computation
2. Quantum dynamics of one qubit in an applied magnetic field

    2.1. Spin in a permanent magnetic field
    2.2. Spin in permanent and time-dependent fields
         2.2.1. The resonant condition
         2.2.2. The rotating frame
3. Spin chains
    3.1. Dynamics of two coupled spins
         3.1.1. Conditional quantum logic with two qubits
         3.1.2. The 2pK-method
    3.2. The spin chain
         3.2.1. Dynamics of the spin chain
         3.2.2. Dynamics in the rotating frame
         3.2.3. Comments on numerical simulations
4. Perturbation theory
    4.1. The two-level approximation
         4.1.1. The permutation procedure
         4.1.2. Dynamics generated by 2 2 blocks
         4.1.3. The generalized 2pK-method and conditional quantum logic for a spin chain
   4.2. Nonresonant transitions
         4.2.1. Small parameters
         4.2.2. Simulation of the dynamics including nonresonant transitions
         4.2.3. Minimization of the nonresonant transitions
5. Quantum algorithms
    5.1. A Control-Not gate involving remote qubits
         5.1.1. Errors generated by near-resonant transitions
         5.1.2. Total error for a remote Control-Not gate
    5.2. Full adder
         5.2.1. Implementation of F-gates using Control-Not and Not gates
         5.2.2. Addition of two quantum numbers
5.3. Universal quantum gates
         5.3.1. Phase corrections
         5.3.2. The Not gate for intermediate qubits
         5.3.3. The Control-Not gate for intermediate qubits
         5.3.4. The Not and Control-Not gate for end qubits
         5.3.5. Universal F-gates for a full adder
    5.4. A universal remote Control-Not gate
    5.5. Modeling the universal full adder with 1000 qubits using quantum maps
         5.5.1. Phase errors
         5.5.2. A test of the accuracy of the quantum map approach
         5.5.3. Modeling the full adder with 1000 addend qubits
         5.5.4. Using quantum maps for modeling quantum dynamics
    5.6. Simulation of the diffusion equation on a type-II quantum computer
         5.6.1. Modeling the diffusion equation using quantum bits
6. Other quantum computer models
    6.1. Quantum computation with quantum dots
         6.1.1. Logical qubits
         6.1.2. Single-qubit logic gates
         6.1.3. The Swap gate
         6.1.4. Modeling errors in the Swap gate
    6.2. A superconducting quantum computer
         6.2.1. An array of capacitively coupled SQUIDs as a quantum register
         6.2.2. Operator field
         6.2.3. Quantum dynamics
         6.2.4. Creation of entangled states
         6.2.5. Errors caused by long-range interaction
    6.3. Collective decoherence in the quantum Shor algorithm
         6.3.1. A simple scheme for quantum Shor algorithm
         6.3.2. Collective decoherence
 Bibliography
 Index