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. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator U = e(-iHt) for a given atomic and molecular Hamiltonian H in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. 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.
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. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator U = e(-iHt) for a given atomic and molecular Hamiltonian H in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. 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.
Constructing general programmable circuits to be able to run any given unitary operator efficiently on a quantum processor is of fundamental importance. We present a new quantum circuit design technique resulting 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 classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations, and the quantum complexities of them are almost the same as non-general circuits.
Simulating quantum mechanical evolutions in general is difficult on classical computers because the size of the Hilbert space grows exponentially with the number of particles involved in the simulations. However, a quantum computer, which obeys the laws of quantum mechanics, is believed to perform such simulations more efficiently. The time evolution operator, a unitary matrix, of a quantum system acts as a quantum gate changing the state of the system in time. Therefore, any computation on quantum computers can be interpreted as a unitary transformation on quantum state vectors. Generally, quantum gates are sufficiently simple operations so as to avoid difficulty in the physical implementation. The implementation of a general given computation represented by a unitary matrix on quantum computers requires finding an array of elementary quantum gates describing the desired computation. This describes a matrix decomposition problem known as the quantum circuit design problem. In this dissertation, we address the following questions concerning quantum circuit designs: How hard is it to find suitable quantum circuits for a given computation? What are the possible circuit design methods? How do we apply these to simulations of quantum problems, in particular quantum chemistry problems? To answer these questions, we categorize methods for designing quantum circuits as evolutionary algorithms and deterministic methods. We apply our developed group leaders optimization algorithm, an evolutionary algorithm, to find quantum circuits for known quantum algorithms and the simulation of molecular Hamiltonians. Then, we present a deterministic circuit design approach to produce universal circuits, which can be programmable. Moreover, we show how to use these programmable circuit designs within the quantum phase estimation algorithm to find eigenvalues of nonunitary matrices and study resonances in quantum systems. Finally, the quantum phase estimation algorithm is employed to efficiently solve ranking problems: in particular, multiple network alignment is considered to find the similarities between protein-protein interaction networks and molecular networks.
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
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.
In this paper, a new synthesis method for Boolean irreversible functions has been proposed which is based on rotation quantum operators. The primary motivation is that the implementation of these operators is commonly available in quantum technologies. We introduce an approach for synthesizing Boolean irreversible functions in quantum logic which is based on quantum rotation and CNOT operators and uses a prespecified fixed design where by changing the rotation angles a new functionality can be achieved. So this design can be viewed as a programmable array of rotation gates. If reduction in quantum cost is desired, after this step the optimization approach may be applied to reduce the number of gates by using some rules, which leads to a reduction of up to 48% in the quantum cost after optimization compared to the original nonoptimized design obtained from the synthesis method. The paper also introduces efficient designs with rotation operators for some important gates in the quantum circuit libraries. Our main contribution is introducing a novel and low complexity approach for synthesizing irreversible functions using elementary rotation operators with a regular fixed design. Also having as many functions as desired with the same inputs, the execution time just increases by one for each additional function and the circuit width is increased by one for each additional function. It is proved that our approach can always be applied to each Boolean function. We show the correctness of our approach and make an upper bound of the quantum cost of the circuit.