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.
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.
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.
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.
To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.
Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.
While advances in quantum hardware occur in modest steps, simulators running on classical computers provide a valuable test bed for the construction of quantum algorithms. Given a unitary matrix that performs certain operation, obtaining the equivalent quantum circuit, even if as an approximation of the input unitary, is a non-trivial task and can be modeled as a search problem. This work presents an evolutionary search algorithm based on the island model concept, for the decomposition of unitary matrices in their equivalent circuit. Three problems are explored: the coin for the quantum walker, the Toffoli gate and the Fredkin gate. The algorithm proposed proved to be efficient in decomposition of quantum circuits, and as a generic approach, it is limited only by the available computational power.
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.