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.
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.
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.
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.
Quantum computation has attracted much attention, among other things, due to its potentialities to solve classical NP problems in polynomial time. For this reason, there has been a growing interest to build a quantum computer. One of the basic steps is to implement the quantum circuit able to realize a given unitary operation. This task has been solved using decomposition of unitary matrices in simpler ones till reach quantum circuits having only single-qubits and CNOTs gates. Usually the goal is to find the minimal quantum circuit able to solve a given problem. In this paper we go in a different direction. We propose a general quantum circuit able to implement any specific quantum circuit by just setting correctly the parameters. In other words, we propose a programmable quantum circuit. This opens the possibility to construct a real quantum computer where several different quantum operations can be realized in the same hardware. The configuration is proposed and its optical implementation is discussed.
Quantum processors are now able to run quantum circuits that are infeasible to simulate classically, creating a need for benchmarks that assess a quantum processor's rate of errors when running these circuits. Here, we introduce a general technique for creating efficient benchmarks from any set of quantum computations, specified by unitary circuits. Our benchmarks assess the integrated performance of a quantum processor's classical compilation algorithms and its low-level quantum operations. Unlike existing “full-stack benchmarks”, our benchmarks do not require classical simulations of quantum circuits, and they use only efficient classical computations. We use our method to create randomized circuit benchmarks, including a computationally efficient version of the quantum volume benchmark, and an algorithm-based benchmark that uses Hamiltonian simulation circuits. We perform these benchmarks on IBM Q devices and in simulations, and we compare their results to the results of existing benchmarking methods.
Diagonal unitary operators are commonly found in many quantum algorithms. They find application as analytical potential operators for quantum simulation, as well as for complex oracles used in quantum searches. However, in order to implement a quantum algorithm on a given quantum device, each operator must be decomposed into a sequence of fault-tolerant, device-level instructions. In general, to implement an n-qubit diagonal unitary {\\em exactly} on a quantum computer generally requires 2n+1−3 one- and two-qubit gates. However, for most practical implementations of diagonal unitaries, some degree of approximation will be necessary if the circuit is to be efficient. In this thesis we develop two complementary methods for the approximate synthesis of quantum circuits for diagonal unitaries. We show how to apply these techniques to real-space quantum simulation and show how efficient high fidelity quantum simulations can be implemented with low-depth quantum circuits.
This paper explores a novel approach to quantum computation centered around self-referential quantum circuits. The core concept involves designing circuits that recursively incorporate themselves into their operational structure, effectively creating computational modules capable of self-replication. This is achieved through the strategic implementation of feedback loops and entanglement within the circuit architecture. The potential benefits of this paradigm include a reduction in circuit complexity and the possibility of constructing universal quantum computers from fundamentally simpler components. The theoretical framework presented herein outlines the key principles and mathematical underpinnings of this self-referential design, proposing a pathway toward more efficient and scalable quantum computation. The presented model uses the following notation: |ψ⟩ represents a quantum state, U(θ) represents a unitary operator with rotation angle θ, C represents a quantum circuit, and E represents the expectation value. Specifically, the expectation value of an observable A is denoted as E[A] = <ψ|A|ψ>, where |ψ> is the quantum state. The Hamiltonian of a quantum system is typically represented as H, and the time evolution of the state is governed by the Schrödinger equation: iℏ∂|ψ(t)⟩/∂t = H|ψ(t)⟩. Entanglement is represented by the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Quantum gates, such as the Hadamard gate (H) and the CNOT gate (CN), are represented as unitary transformations. The circuit representation involves sequential application of gates, and the overall unitary transformation is the product of these gates. The circuit depth, denoted as d, represents the number of consecutive gate operations applied to a quantum state. The circuit complexity, denoted as C, is a measure of the number of gates required to implement a specific quantum algorithm. The circuit fidelity, denoted as F, quantifies the accuracy of a quantum circuit's output, ranging from 0 to 1. We define the quantum state preparation as |ψ⟩ = ∑i ci|i⟩, where ci are complex coefficients and |i⟩ are the computational basis states. The unitary transformation is defined as U|ψ⟩ = ∑i (Uici)|i⟩, where Ui are the matrix elements of the unitary operator U. The propagation of quantum states through a circuit is represented by the Schrödinger equation, and the final state is obtained by applying the unitary transformation to the initial state. The concept of quantum coherence, denoted as χ, represents the ability of a quantum system to maintain its superposition state. The fidelity of a quantum state is defined as the probability that a measurement will yield the expected outcome.