15 research outputs found
A Universal Quantum Circuit Scheme For Finding Complex Eigenvalues
We present a general quantum circuit design for finding eigenvalues of
non-unitary matrices on quantum computers using the iterative phase estimation
algorithm. In particular, we show how the method can be used for the simulation
of resonance states for quantum systems
Quantum Circuit Design for Solving Linear Systems of Equations
Recently, it is shown that quantum computers can be used for obtaining
certain information about the solution of a linear system Ax=b exponentially
faster than what is possible with classical computation. Here we first review
some key aspects of the algorithm from the standpoint of finding its efficient
quantum circuit implementation using only elementary quantum operations, which
is important for determining the potential usefulness of the algorithm in
practical settings. Then we present a small-scale quantum circuit that solves a
2x2 linear system. The quantum circuit uses only 4 qubits, implying a tempting
possibility for experimental realization. Furthermore, the circuit is
numerically simulated and its performance under different circuit parameter
settings is demonstrated.Comment: 7 pages, 3 figures. The errors are corrected. For the general case,
discussions are added to account for recent results. The 4x4 example is
replaced by a 2x2 one due to recent experimental efforts. The 2x2 example was
devised at the time of writing v1 but not included in v1 for brevit
Decomposition of Unitary Matrices for Finding Quantum Circuits: Application to Molecular Hamiltonians
Constructing appropriate unitary matrix operators for new quantum algorithms
and finding the minimum cost gate sequences for the implementation of these
unitary operators is of fundamental importance in the field of quantum
information and quantum computation. Evolution of quantum circuits faces two
major challenges: complex and huge search space and the high costs of
simulating quantum circuits on classical computers. Here, we use the group
leaders optimization algorithm to decompose a given unitary matrix into a
proper-minimum cost quantum gate sequence. We test the method on the known
decompositions of Toffoli gate, the amplification step of the Grover search
algorithm, the quantum Fourier transform, and the sender part of the quantum
teleportation. Using this procedure, we present the circuit designs for the
simulation of the unitary propagators of the Hamiltonians for the hydrogen and
the water molecules. The approach is general and can be applied to generate the
sequence of quantum gates for larger molecular systems
Universal Programmable Quantum Circuit Schemes to Emulate an Operator
Unlike fixed designs, programmable circuit designs support an infinite number
of operators. The functionality of a programmable circuit can be altered by
simply changing the angle values of the rotation gates in the circuit. Here, we
present a new quantum circuit design technique resulting in two general
programmable circuit schemes. The circuit schemes can be used to simulate any
given operator by setting the angle values in the circuit. This provides a
fixed circuit design whose angles are determined from the elements of the given
matrix-which can be non-unitary-in an efficient way. We also give both the
classical and quantum complexity analysis for these circuits and show that the
circuits require a few classical computations. They have almost the same
quantum complexities as non-general circuits. Since the presented circuit
designs are independent from the matrix decomposition techniques and the global
optimization processes used to find quantum circuits for a given operator, high
accuracy simulations can be done for the unitary propagators of molecular
Hamiltonians on quantum computers. As an example, we show how to build the
circuit design for the hydrogen molecule.Comment: combined with former arXiv:1207.174
Group Leaders Optimization Algorithm
We present a new global optimization algorithm in which the influence of the
leaders in social groups is used as an inspiration for the evolutionary
technique which is designed into a group architecture. To demonstrate the
efficiency of the method, a standard suite of single and multidimensional
optimization functions along with the energies and the geometric structures of
Lennard-Jones clusters are given as well as the application of the algorithm on
quantum circuit design problems. We show that as an improvement over previous
methods, the algorithm scales as N^2.5 for the Lennard-Jones clusters of
N-particles. In addition, an efficient circuit design is shown for two qubit
Grover search algorithm which is a quantum algorithm providing quadratic
speed-up over the classical counterpart